Subgroups — C₂×C₇×#3274 #13238249

Order 1792.

3278 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
2	C₂×C₇×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×D₄²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3274	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×#469	C₂	1	2	yes	maximal, complemented
7	C₂×C₇×#460	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×#459	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₇×#449	C₂	1	2	yes	maximal, complemented
10	C₂×C₇×#447	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×C₇×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#436	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×C₇×#41	C₂²	1	2	yes	complemented
14	C₂³×C₇×D₄	—	2	1	no	—
15	C₂³×C₇×D₄	C₂²	1	2	yes	—
16	C₂×C₇×AGL(1,ℤ/8)	—	2	16	no	—
17	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	16	yes	complemented
18	C₂²×C₇×SD₁₆	C₂²	1	1	yes	characteristic
19	C₂²×C₇×SD₁₆	C₂²	1	2	yes	complemented
20	C₂²×C₇×D₈	C₂²	1	2	yes	—
21	C₂²×C₇×D₈	C₂²	1	2	yes	complemented
22	C₂²×C₇×D₈	C₂²	1	2	yes	complemented
23	C₂²×C₇×M₁₆	C₂²	1	2	yes	complemented
24	C₇×#550	—	2	4	no	—
25	C₇×#550	C₂²	1	8	yes	complemented
26	C₇×D₄²	—	2	4	no	—
27	C₇×D₄²	C₂²	1	8	yes	complemented
28	C₂×C₇×#127	—	2	1	no	—
29	C₂×C₇×#127	C₂²	1	1	yes	characteristic
30	C₂×C₇×#127	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₇×#121	—	2	1	no	—
32	C₂×C₇×#121	—	2	2	no	—
33	C₂×C₇×#121	C₂²	1	2	yes	—
34	C₂×C₇×#121	C₂²	1	2	yes	complemented
35	C₂×C₇×#120	—	2	2	no	—
36	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
37	C₂×C₄×C₇×D₄	—	2	1	no	—
38	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
40	C₇×#493	C₂²	1	8	yes	complemented
41	C₇×#469	C₂²	1	16	yes	complemented
42	C₇×#460	C₂²	1	8	yes	complemented
43	C₇×#459	C₂²	1	8	yes	complemented
44	C₇×#449	C₂²	1	16	yes	complemented
45	C₇×#447	C₂²	1	16	yes	complemented
46	C₄×C₇×SD₁₆	C₂²	1	8	yes	complemented
47	C₇×#436	C₂²	1	8	yes	complemented
48	C₂×C₇×#106	C₂²	1	1	yes	characteristic, complemented
49	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
50	C₂×C₇×#103	C₂²	1	1	yes	characteristic, complemented
51	C₂×C₇×#102	C₂²	1	1	yes	characteristic
52	C₂×C₇×#102	C₂²	1	2	yes	complemented
53	C₂×C₇×#102	C₂²	1	2	yes	complemented
54	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	complemented
55	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic, complemented
56	C₂×#3274	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂⁵×C₇	—	2	2	no	—
58	C₂×C₇×#41	—	2	4	no	—
59	C₂×C₇×#41	—	2	8	no	—
60	C₂×C₇×#41	—	4	4	no	—
61	C₂×C₇×#41	C₂³	1	8	yes	complemented
62	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
63	C₂²×C₇×D₄	—	2	8	no	—
64	C₂²×C₇×D₄	—	2	2	no	—
65	C₂²×C₇×D₄	—	2	2	no	—
66	C₂²×C₇×D₄	—	2	2	no	—
67	C₂²×C₇×D₄	—	2	8	no	—
68	C₂²×C₇×D₄	—	2	1	no	—
69	C₂²×C₇×D₄	—	2	2	no	—
70	C₂²×C₇×D₄	—	2	4	no	—
71	C₂²×C₇×D₄	—	2	1	no	—
72	C₂²×C₇×D₄	—	2	1	no	—
73	C₂²×C₇×D₄	—	2	1	no	—
74	C₂²×C₇×D₄	—	4	4	no	—
75	C₂²×C₇×D₄	—	4	8	no	—
76	C₂²×C₇×D₄	D₄	1	1	yes	characteristic
77	C₂²×C₇×D₄	C₂³	1	2	yes	—
78	C₂²×C₇×D₄	C₂³	1	2	yes	—
79	C₂²×C₇×D₄	C₂³	1	8	yes	—
80	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₇	—	2	1	no	—
82	C₂³×C₄×C₇	—	2	2	no	—
83	C₂³×C₄×C₇	C₂³	1	2	yes	—
84	C₇×AGL(1,ℤ/8)	—	2	64	no	—
85	C₂×C₇×SD₁₆	—	2	8	no	—
86	C₂×C₇×SD₁₆	D₄	1	8	yes	complemented
87	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
88	C₂×C₇×SD₁₆	C₂³	1	8	yes	complemented
89	C₂×C₇×D₈	—	2	8	no	—
90	C₂×C₇×D₈	—	2	8	no	—
91	C₂×C₇×D₈	—	4	4	no	—
92	C₂×C₇×D₈	C₂³	1	8	yes	—
93	C₂×C₇×D₈	C₂³	1	8	yes	complemented
94	C₂×C₇×D₈	C₂³	1	8	yes	complemented
95	C₂×C₇×M₁₆	—	2	8	no	—
96	C₂×C₇×M₁₆	C₂³	1	8	yes	complemented
97	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
98	C₂²×C₇×C₈	C₂³	1	2	yes	—
99	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
100	C₇×#127	—	2	8	no	—
101	C₇×#127	—	2	2	no	—
102	C₇×#127	—	2	2	no	—
103	C₇×#127	C₂³	1	4	yes	—
104	C₇×#127	C₂³	1	4	yes	complemented
105	C₇×#121	—	2	4	no	—
106	C₇×#121	—	2	4	no	—
107	C₇×#121	—	2	8	no	—
108	C₇×#121	—	2	16	no	—
109	C₇×#121	C₂³	1	8	yes	—
110	C₇×#121	C₂³	1	8	yes	complemented
111	C₇×#120	—	2	16	no	—
112	C₄×C₇×Q₈	—	2	2	no	—
113	C₄×C₇×Q₈	C₂³	1	4	yes	complemented
114	C₄×C₇×D₄	—	2	8	no	—
115	C₄×C₇×D₄	—	2	2	no	—
116	C₄×C₇×D₄	D₄	1	4	yes	—
117	C₄×C₇×D₄	C₂³	1	4	yes	—
118	C₄×C₇×D₄	C₂³	1	4	yes	—
119	C₂×C₇×C₄⋊C₄	—	2	1	no	—
120	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
121	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
122	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
123	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
124	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
125	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
126	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
127	C₂×C₄²×C₇	—	2	1	no	—
128	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
129	C₇×#106	C₂³	1	4	yes	complemented
130	C₇×C₄⋊C₈	C₂³	1	4	yes	complemented
131	C₇×#103	C₂³	1	4	yes	complemented
132	C₇×#102	C₂³	1	4	yes	—
133	C₇×#102	C₂³	1	8	yes	complemented
134	C₇×#102	C₂³	1	8	yes	complemented
135	C₇×C₂²⋊C₈	C₂³	1	8	yes	complemented
136	C₄×C₇×C₈	C₂³	1	4	yes	complemented
137	C₂²×AGL(1,ℤ/8)	C₂×C₇	1	2	yes	complemented
138	C₂×#550	C₂×C₇	1	1	yes	characteristic, complemented
139	C₂×D₄²	C₂×C₇	1	1	yes	characteristic, complemented
140	#3274	C₂×C₇	1	16	yes	direct product factor
141	C₂×#493	C₂×C₇	1	1	yes	characteristic, complemented
142	C₂×#469	C₂×C₇	1	2	yes	complemented
143	C₂×#460	C₂×C₇	1	1	yes	characteristic, complemented
144	C₂×#459	C₂×C₇	1	1	yes	characteristic, complemented
145	C₂×#449	C₂×C₇	1	2	yes	complemented
146	C₂×#447	C₂×C₇	1	2	yes	complemented
147	C₂×C₄×SD₁₆	C₂×C₇	1	1	yes	characteristic, complemented
148	C₂×#436	C₂×C₇	1	1	yes	characteristic, complemented
149	C₂⁴×C₇	—	2	1	no	—
150	C₂⁴×C₇	—	2	2	no	—
151	C₂⁴×C₇	—	2	2	no	—
152	C₂⁴×C₇	—	2	8	no	—
153	C₂⁴×C₇	—	4	8	no	—
154	C₂⁴×C₇	—	4	8	no	—
155	C₂⁴×C₇	—	8	4	no	—
156	C₂⁴×C₇	C₂×D₄	1	2	yes	—
157	C₇×#41	—	2	16	no	—
158	C₇×#41	—	4	8	no	—
159	C₇×#41	—	4	16	no	—
160	C₂×C₇×Q₈	—	2	1	no	—
161	C₂×C₇×Q₈	—	2	2	no	—
162	C₂×C₇×Q₈	C₂×D₄	1	4	yes	complemented
163	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
164	C₂×C₇×D₄	—	2	16	no	—
165	C₂×C₇×D₄	—	2	8	no	—
166	C₂×C₇×D₄	—	2	8	no	—
167	C₂×C₇×D₄	—	2	8	no	—
168	C₂×C₇×D₄	—	2	4	no	—
169	C₂×C₇×D₄	—	2	1	no	—
170	C₂×C₇×D₄	—	2	4	no	—
171	C₂×C₇×D₄	—	2	4	no	—
172	C₂×C₇×D₄	—	2	4	no	—
173	C₂×C₇×D₄	—	2	8	no	—
174	C₂×C₇×D₄	—	2	2	no	—
175	C₂×C₇×D₄	—	2	2	no	—
176	C₂×C₇×D₄	—	2	4	no	—
177	C₂×C₇×D₄	—	2	4	no	—
178	C₂×C₇×D₄	—	2	4	no	—
179	C₂×C₇×D₄	—	2	2	no	—
180	C₂×C₇×D₄	—	2	4	no	—
181	C₂×C₇×D₄	—	2	4	no	—
182	C₂×C₇×D₄	—	4	8	no	—
183	C₂×C₇×D₄	—	4	16	no	—
184	C₂×C₇×D₄	—	4	8	no	—
185	C₂×C₇×D₄	—	4	8	no	—
186	C₂×C₇×D₄	—	4	32	no	—
187	C₂×C₇×D₄	—	4	8	no	—
188	C₂×C₇×D₄	—	4	4	no	—
189	C₂×C₇×D₄	—	4	4	no	—
190	C₂×C₇×D₄	—	8	2	no	—
191	C₂×C₇×D₄	—	8	2	no	—
192	C₂×C₇×D₄	—	8	4	no	—
193	C₂×C₇×D₄	C₂×D₄	1	4	yes	—
194	C₂×C₇×D₄	C₂×D₄	1	4	yes	—
195	C₂×C₇×D₄	C₂⁴	1	4	yes	—
196	C₂×C₇×D₄	C₂⁴	1	4	yes	—
197	C₂×C₇×D₄	C₂⁴	1	2	yes	—
198	C₂²×C₄×C₇	—	2	1	no	—
199	C₂²×C₄×C₇	—	2	4	no	—
200	C₂²×C₄×C₇	—	2	4	no	—
201	C₂²×C₄×C₇	—	2	1	no	—
202	C₂²×C₄×C₇	—	2	1	no	—
203	C₂²×C₄×C₇	—	2	8	no	—
204	C₂²×C₄×C₇	—	2	4	no	—
205	C₂²×C₄×C₇	—	4	8	no	—
206	C₂²×C₄×C₇	—	4	4	no	—
207	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
208	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
209	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
210	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
211	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
212	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
213	C₇×SD₁₆	—	2	16	no	—
214	C₇×SD₁₆	C₂×D₄	1	16	yes	complemented
215	C₇×D₈	—	2	16	no	—
216	C₇×D₈	—	2	16	no	—
217	C₇×D₈	—	4	8	no	—
218	C₇×M₁₆	—	2	16	no	—
219	C₂×C₇×C₈	—	2	4	no	—
220	C₂×C₇×C₈	—	2	2	no	—
221	C₂×C₇×C₈	C₂×D₄	1	4	yes	complemented
222	C₂×C₇×C₈	C₂⁴	1	2	yes	—
223	C₂×C₇×C₈	C₂⁴	1	4	yes	—
224	C₂×C₇×C₈	C₂⁴	1	2	yes	—
225	C₇×C₄⋊C₄	—	2	4	no	—
226	C₇×C₄⋊C₄	—	2	1	no	—
227	C₇×C₄⋊C₄	—	2	1	no	—
228	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
229	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
230	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
231	C₇×C₂²⋊C₄	—	2	8	no	—
232	C₇×C₂²⋊C₄	—	2	4	no	—
233	C₇×C₂²⋊C₄	—	2	8	no	—
234	C₇×C₂²⋊C₄	C₂×D₄	1	8	yes	—
235	C₄²×C₇	—	2	4	no	—
236	C₄²×C₇	C₂×D₄	1	2	yes	—
237	C₄²×C₇	C₂⁴	1	2	yes	—
238	C₂²×#41	C₂²×C₇	1	2	yes	complemented
239	C₂³×D₄	—	2	1	no	—
240	C₂³×D₄	C₂²×C₇	1	2	yes	—
241	C₂×AGL(1,ℤ/8)	—	2	16	no	—
242	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	16	yes	complemented
243	C₂²×SD₁₆	C₂²×C₇	1	1	yes	characteristic
244	C₂²×SD₁₆	C₂²×C₇	1	2	yes	complemented
245	C₂²×D₈	C₂²×C₇	1	2	yes	—
246	C₂²×D₈	C₂²×C₇	1	2	yes	complemented
247	C₂²×D₈	C₂²×C₇	1	2	yes	complemented
248	C₂²×M₁₆	C₂²×C₇	1	2	yes	complemented
249	#550	—	2	4	no	—
250	#550	C₂²×C₇	1	8	yes	complemented
251	D₄²	—	2	4	no	—
252	D₄²	C₂²×C₇	1	8	yes	complemented
253	C₂×#127	—	2	1	no	—
254	C₂×#127	C₂²×C₇	1	1	yes	characteristic
255	C₂×#127	C₂²×C₇	1	1	yes	characteristic, complemented
256	C₂×#121	—	2	1	no	—
257	C₂×#121	—	2	2	no	—
258	C₂×#121	C₂²×C₇	1	2	yes	—
259	C₂×#121	C₂²×C₇	1	2	yes	complemented
260	C₂×#120	—	2	2	no	—
261	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
262	C₂×C₄×D₄	—	2	1	no	—
263	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
264	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
265	#493	C₂²×C₇	1	8	yes	complemented
266	#469	C₂²×C₇	1	16	yes	complemented
267	#460	C₂²×C₇	1	8	yes	complemented
268	#459	C₂²×C₇	1	8	yes	complemented
269	#449	C₂²×C₇	1	16	yes	complemented
270	#447	C₂²×C₇	1	16	yes	complemented
271	C₄×SD₁₆	C₂²×C₇	1	8	yes	complemented
272	#436	C₂²×C₇	1	8	yes	complemented
273	C₂×#106	C₂²×C₇	1	1	yes	characteristic, complemented
274	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic, complemented
275	C₂×#103	C₂²×C₇	1	1	yes	characteristic, complemented
276	C₂×#102	C₂²×C₇	1	1	yes	characteristic
277	C₂×#102	C₂²×C₇	1	2	yes	complemented
278	C₂×#102	C₂²×C₇	1	2	yes	complemented
279	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	complemented
280	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic, complemented
281	C₂³×C₇	—	2	4	no	—
282	C₂³×C₇	—	2	4	no	—
283	C₂³×C₇	—	2	4	no	—
284	C₂³×C₇	—	2	4	no	—
285	C₂³×C₇	—	2	2	no	—
286	C₂³×C₇	—	2	4	no	—
287	C₂³×C₇	—	2	4	no	—
288	C₂³×C₇	—	4	16	no	—
289	C₂³×C₇	—	4	16	no	—
290	C₂³×C₇	—	4	2	no	—
291	C₂³×C₇	—	4	2	no	—
292	C₂³×C₇	—	4	4	no	—
293	C₂³×C₇	—	4	4	no	—
294	C₂³×C₇	—	4	4	no	—
295	C₂³×C₇	—	4	4	no	—
296	C₂³×C₇	—	4	4	no	—
297	C₂³×C₇	—	4	4	no	—
298	C₂³×C₇	—	8	8	no	—
299	C₂³×C₇	—	8	32	no	—
300	C₂³×C₇	C₂²×D₄	1	4	yes	—
301	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
302	C₇×Q₈	—	2	4	no	—
303	C₇×Q₈	—	2	2	no	—
304	C₇×Q₈	C₂²×D₄	1	4	yes	complemented
305	C₇×D₄	—	2	4	no	—
306	C₇×D₄	—	2	8	no	—
307	C₇×D₄	—	2	4	no	—
308	C₇×D₄	—	2	4	no	—
309	C₇×D₄	—	2	2	no	—
310	C₇×D₄	—	2	4	no	—
311	C₇×D₄	—	2	4	no	—
312	C₇×D₄	—	4	16	no	—
313	C₇×D₄	—	4	16	no	—
314	C₇×D₄	—	4	16	no	—
315	C₇×D₄	—	4	4	no	—
316	C₇×D₄	—	4	4	no	—
317	C₇×D₄	—	4	8	no	—
318	C₇×D₄	—	4	8	no	—
319	C₇×D₄	—	8	4	no	—
320	C₇×D₄	—	8	4	no	—
321	C₇×D₄	—	8	8	no	—
322	C₇×D₄	C₂²×D₄	1	4	yes	—
323	C₂×C₄×C₇	—	2	4	no	—
324	C₂×C₄×C₇	—	2	4	no	—
325	C₂×C₄×C₇	—	2	2	no	—
326	C₂×C₄×C₇	—	2	2	no	—
327	C₂×C₄×C₇	—	2	8	no	—
328	C₂×C₄×C₇	—	2	4	no	—
329	C₂×C₄×C₇	—	2	4	no	—
330	C₂×C₄×C₇	—	2	2	no	—
331	C₂×C₄×C₇	—	2	2	no	—
332	C₂×C₄×C₇	—	4	16	no	—
333	C₂×C₄×C₇	—	4	2	no	—
334	C₂×C₄×C₇	—	4	2	no	—
335	C₂×C₄×C₇	—	4	8	no	—
336	C₂×C₄×C₇	AGL(1,ℤ/8)	1	4	yes	complemented
337	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
338	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
339	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
340	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
341	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
342	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
343	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
344	C₇×C₈	—	2	4	no	—
345	C₇×C₈	—	2	2	no	—
346	C₇×C₈	C₂²×D₄	1	4	yes	complemented
347	C₂⁵	—	2	2	no	—
348	C₂×#41	—	2	4	no	—
349	C₂×#41	—	2	8	no	—
350	C₂×#41	—	4	4	no	—
351	C₂×#41	C₂³×C₇	1	8	yes	complemented
352	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
353	C₂²×D₄	—	2	8	no	—
354	C₂²×D₄	—	2	2	no	—
355	C₂²×D₄	—	2	2	no	—
356	C₂²×D₄	—	2	2	no	—
357	C₂²×D₄	—	2	8	no	—
358	C₂²×D₄	—	2	1	no	—
359	C₂²×D₄	—	2	2	no	—
360	C₂²×D₄	—	2	1	no	—
361	C₂²×D₄	—	2	4	no	—
362	C₂²×D₄	—	2	1	no	—
363	C₂²×D₄	—	2	1	no	—
364	C₂²×D₄	—	4	4	no	—
365	C₂²×D₄	—	4	8	no	—
366	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
367	C₂²×D₄	C₂³×C₇	1	2	yes	—
368	C₂²×D₄	C₂³×C₇	1	2	yes	—
369	C₂²×D₄	C₂³×C₇	1	8	yes	—
370	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
371	C₂³×C₄	—	2	2	no	—
372	C₂³×C₄	—	2	1	no	—
373	C₂³×C₄	C₂³×C₇	1	2	yes	—
374	AGL(1,ℤ/8)	—	2	64	no	—
375	C₂×SD₁₆	—	2	8	no	—
376	C₂×SD₁₆	C₇×D₄	1	8	yes	complemented
377	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
378	C₂×SD₁₆	C₂³×C₇	1	8	yes	complemented
379	C₂×D₈	—	2	8	no	—
380	C₂×D₈	—	2	8	no	—
381	C₂×D₈	—	4	4	no	—
382	C₂×D₈	C₂³×C₇	1	8	yes	—
383	C₂×D₈	C₂³×C₇	1	8	yes	complemented
384	C₂×D₈	C₂³×C₇	1	8	yes	complemented
385	C₂×M₁₆	—	2	8	no	—
386	C₂×M₁₆	C₂³×C₇	1	8	yes	complemented
387	C₂²×C₈	C₂³×C₇	1	2	yes	—
388	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
389	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
390	#127	—	2	8	no	—
391	#127	—	2	2	no	—
392	#127	—	2	2	no	—
393	#127	C₂³×C₇	1	4	yes	—
394	#127	C₂³×C₇	1	4	yes	complemented
395	#121	—	2	4	no	—
396	#121	—	2	4	no	—
397	#121	—	2	8	no	—
398	#121	—	2	16	no	—
399	#121	C₂³×C₇	1	8	yes	—
400	#121	C₂³×C₇	1	8	yes	complemented
401	#120	—	2	16	no	—
402	C₄×Q₈	—	2	2	no	—
403	C₄×Q₈	C₂³×C₇	1	4	yes	complemented
404	C₄×D₄	—	2	8	no	—
405	C₄×D₄	—	2	2	no	—
406	C₄×D₄	C₇×D₄	1	4	yes	—
407	C₄×D₄	C₂³×C₇	1	4	yes	—
408	C₄×D₄	C₂³×C₇	1	4	yes	—
409	C₂×C₄⋊C₄	—	2	1	no	—
410	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
411	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
412	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
413	C₂×C₂²⋊C₄	—	2	2	no	—
414	C₂×C₂²⋊C₄	—	2	1	no	—
415	C₂×C₂²⋊C₄	—	2	2	no	—
416	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
417	C₂×C₄²	—	2	1	no	—
418	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
419	#106	C₂³×C₇	1	4	yes	complemented
420	C₄⋊C₈	C₂³×C₇	1	4	yes	complemented
421	#103	C₂³×C₇	1	4	yes	complemented
422	#102	C₂³×C₇	1	4	yes	—
423	#102	C₂³×C₇	1	8	yes	complemented
424	#102	C₂³×C₇	1	8	yes	complemented
425	C₂²⋊C₈	C₂³×C₇	1	8	yes	complemented
426	C₄×C₈	C₂³×C₇	1	4	yes	complemented
427	C₂²×C₇	—	2	4	no	—
428	C₂²×C₇	—	2	4	no	—
429	C₂²×C₇	—	2	4	no	—
430	C₂²×C₇	—	2	4	no	—
431	C₂²×C₇	—	4	8	no	—
432	C₂²×C₇	—	4	8	no	—
433	C₂²×C₇	—	4	2	no	—
434	C₂²×C₇	—	4	2	no	—
435	C₂²×C₇	—	4	4	no	—
436	C₂²×C₇	—	4	4	no	—
437	C₂²×C₇	—	4	4	no	—
438	C₂²×C₇	—	4	4	no	—
439	C₂²×C₇	—	4	4	no	—
440	C₂²×C₇	—	4	4	no	—
441	C₂²×C₇	—	8	32	no	—
442	C₂²×C₇	—	8	8	no	—
443	C₂²×C₇	—	8	8	no	—
444	C₂²×C₇	D₄²	1	2	yes	—
445	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
446	C₂²×C₇	#576	1	2	yes	—
447	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
448	C₄×C₇	—	2	4	no	—
449	C₄×C₇	—	2	4	no	—
450	C₄×C₇	—	2	2	no	—
451	C₄×C₇	—	4	2	no	—
452	C₄×C₇	—	4	2	no	—
453	C₄×C₇	D₄²	1	2	yes	—
454	C₄×C₇	C₂×AGL(1,ℤ/8)	1	4	yes	complemented
455	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
456	C₄×C₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
457	C₂⁴	—	2	1	no	—
458	C₂⁴	—	2	2	no	—
459	C₂⁴	—	2	2	no	—
460	C₂⁴	—	2	8	no	—
461	C₂⁴	—	4	8	no	—
462	C₂⁴	—	4	8	no	—
463	C₂⁴	—	8	4	no	—
464	C₂⁴	C₂×C₇×D₄	1	2	yes	—
465	#41	—	2	16	no	—
466	#41	—	4	8	no	—
467	#41	—	4	16	no	—
468	C₂×Q₈	—	2	1	no	—
469	C₂×Q₈	—	2	2	no	—
470	C₂×Q₈	C₂×C₇×D₄	1	4	yes	complemented
471	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
472	C₂×D₄	—	2	16	no	—
473	C₂×D₄	—	2	8	no	—
474	C₂×D₄	—	2	8	no	—
475	C₂×D₄	—	2	8	no	—
476	C₂×D₄	—	2	4	no	—
477	C₂×D₄	—	2	1	no	—
478	C₂×D₄	—	2	4	no	—
479	C₂×D₄	—	2	4	no	—
480	C₂×D₄	—	2	4	no	—
481	C₂×D₄	—	2	8	no	—
482	C₂×D₄	—	2	2	no	—
483	C₂×D₄	—	2	2	no	—
484	C₂×D₄	—	2	4	no	—
485	C₂×D₄	—	2	4	no	—
486	C₂×D₄	—	2	4	no	—
487	C₂×D₄	—	2	2	no	—
488	C₂×D₄	—	2	4	no	—
489	C₂×D₄	—	2	4	no	—
490	C₂×D₄	—	4	8	no	—
491	C₂×D₄	—	4	16	no	—
492	C₂×D₄	—	4	8	no	—
493	C₂×D₄	—	4	8	no	—
494	C₂×D₄	—	4	32	no	—
495	C₂×D₄	—	4	8	no	—
496	C₂×D₄	—	4	4	no	—
497	C₂×D₄	—	4	4	no	—
498	C₂×D₄	—	8	2	no	—
499	C₂×D₄	—	8	2	no	—
500	C₂×D₄	—	8	4	no	—
501	C₂×D₄	C₂×C₇×D₄	1	4	yes	—
502	C₂×D₄	C₂×C₇×D₄	1	4	yes	—
503	C₂×D₄	C₂⁴×C₇	1	4	yes	—
504	C₂×D₄	C₂⁴×C₇	1	4	yes	—
505	C₂×D₄	C₂⁴×C₇	1	2	yes	—
506	C₂²×C₄	—	2	1	no	—
507	C₂²×C₄	—	2	4	no	—
508	C₂²×C₄	—	2	4	no	—
509	C₂²×C₄	—	2	1	no	—
510	C₂²×C₄	—	2	1	no	—
511	C₂²×C₄	—	2	8	no	—
512	C₂²×C₄	—	2	4	no	—
513	C₂²×C₄	—	4	8	no	—
514	C₂²×C₄	—	4	4	no	—
515	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
516	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
517	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
518	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
519	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
520	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
521	SD₁₆	—	2	16	no	—
522	SD₁₆	C₂×C₇×D₄	1	16	yes	complemented
523	D₈	—	2	16	no	—
524	D₈	—	2	16	no	—
525	D₈	—	4	8	no	—
526	M₁₆	—	2	16	no	—
527	C₂×C₈	—	2	4	no	—
528	C₂×C₈	—	2	2	no	—
529	C₂×C₈	C₂×C₇×D₄	1	4	yes	complemented
530	C₂×C₈	C₂⁴×C₇	1	2	yes	—
531	C₂×C₈	C₂⁴×C₇	1	4	yes	—
532	C₂×C₈	C₂⁴×C₇	1	2	yes	—
533	C₄⋊C₄	—	2	4	no	—
534	C₄⋊C₄	—	2	1	no	—
535	C₄⋊C₄	—	2	1	no	—
536	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
537	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
538	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
539	C₂²⋊C₄	—	2	8	no	—
540	C₂²⋊C₄	—	2	4	no	—
541	C₂²⋊C₄	—	2	8	no	—
542	C₂²⋊C₄	C₂×C₇×D₄	1	8	yes	—
543	C₄²	—	2	4	no	—
544	C₄²	C₂×C₇×D₄	1	2	yes	—
545	C₄²	C₂⁴×C₇	1	2	yes	—
546	C₂×C₇	—	4	4	no	—
547	C₂×C₇	—	4	4	no	—
548	C₂×C₇	—	8	4	no	—
549	C₂×C₇	—	8	4	no	—
550	C₂×C₇	#3274	1	4	yes	direct product factor
551	C₂×C₇	C₂×D₄²	1	1	yes	characteristic
552	C₂×C₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
553	C₂×C₇	C₂×#576	1	1	yes	characteristic
554	C₂³	—	2	2	no	—
555	C₂³	—	2	4	no	—
556	C₂³	—	2	4	no	—
557	C₂³	—	2	4	no	—
558	C₂³	—	2	4	no	—
559	C₂³	—	2	4	no	—
560	C₂³	—	2	4	no	—
561	C₂³	—	4	16	no	—
562	C₂³	—	4	16	no	—
563	C₂³	—	4	2	no	—
564	C₂³	—	4	2	no	—
565	C₂³	—	4	4	no	—
566	C₂³	—	4	4	no	—
567	C₂³	—	4	4	no	—
568	C₂³	—	4	4	no	—
569	C₂³	—	4	4	no	—
570	C₂³	—	4	4	no	—
571	C₂³	—	8	8	no	—
572	C₂³	—	8	32	no	—
573	C₂³	C₂²×C₇×D₄	1	4	yes	—
574	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
575	Q₈	—	2	2	no	—
576	Q₈	—	2	4	no	—
577	Q₈	C₂²×C₇×D₄	1	4	yes	complemented
578	D₄	—	2	4	no	—
579	D₄	—	2	4	no	—
580	D₄	—	2	2	no	—
581	D₄	—	2	4	no	—
582	D₄	—	2	4	no	—
583	D₄	—	2	4	no	—
584	D₄	—	2	8	no	—
585	D₄	—	4	16	no	—
586	D₄	—	4	16	no	—
587	D₄	—	4	16	no	—
588	D₄	—	4	4	no	—
589	D₄	—	4	4	no	—
590	D₄	—	4	8	no	—
591	D₄	—	4	8	no	—
592	D₄	—	8	4	no	—
593	D₄	—	8	4	no	—
594	D₄	—	8	8	no	—
595	D₄	C₂²×C₇×D₄	1	4	yes	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	4	no	—
598	C₂×C₄	—	2	4	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	4	no	—
601	C₂×C₄	—	2	4	no	—
602	C₂×C₄	—	2	8	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	4	16	no	—
606	C₂×C₄	—	4	2	no	—
607	C₂×C₄	—	4	2	no	—
608	C₂×C₄	—	4	8	no	—
609	C₂×C₄	C₇×AGL(1,ℤ/8)	1	4	yes	complemented
610	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
611	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
612	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
613	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
614	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
615	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
616	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
617	C₈	—	2	4	no	—
618	C₈	—	2	2	no	—
619	C₈	C₂²×C₇×D₄	1	4	yes	complemented
620	C₇	C₂×#3274	1	1	yes	characteristic, direct product factor, Sylow
621	C₂²	—	2	4	no	—
622	C₂²	—	2	4	no	—
623	C₂²	—	2	4	no	—
624	C₂²	—	2	4	no	—
625	C₂²	—	4	2	no	—
626	C₂²	—	4	2	no	—
627	C₂²	—	4	8	no	—
628	C₂²	—	4	8	no	—
629	C₂²	—	4	4	no	—
630	C₂²	—	4	4	no	—
631	C₂²	—	4	4	no	—
632	C₂²	—	4	4	no	—
633	C₂²	—	4	4	no	—
634	C₂²	—	4	4	no	—
635	C₂²	—	8	32	no	—
636	C₂²	—	8	8	no	—
637	C₂²	—	8	8	no	—
638	C₂²	C₇×D₄²	1	2	yes	—
639	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	2	yes	—
640	C₂²	C₇×#576	1	2	yes	—
641	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
642	C₄	—	2	2	no	—
643	C₄	—	2	4	no	—
644	C₄	—	2	4	no	—
645	C₄	—	4	2	no	—
646	C₄	—	4	2	no	—
647	C₄	C₇×D₄²	1	2	yes	—
648	C₄	C₂×C₇×AGL(1,ℤ/8)	1	4	yes	complemented
649	C₄	C₂³×C₇×D₄	1	1	yes	characteristic
650	C₄	C₂×C₇×H(5,𝔽₂)	1	1	yes	characteristic
651	C₂	—	4	4	no	—
652	C₂	—	4	4	no	—
653	C₂	—	8	4	no	—
654	C₂	—	8	4	no	—
655	C₂	C₇×#3274	1	4	yes	direct product factor
656	C₂	C₂×C₇×D₄²	1	1	yes	characteristic
657	C₂	C₂²×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
658	C₂	C₂×C₇×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄×C₇×D₄ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×C₇×AGL(1,ℤ/8)	C₂	semidirect
2. C₂×C₇×#550	C₂	semidirect
3. C₂×C₇×D₄²	C₂	semidirect
4. C₇×#3274	C₂	direct (N × H)
5. C₂×C₇×#493	C₂	semidirect
6. C₂×C₇×#469	C₂	semidirect
7. C₂×C₇×#460	C₂	semidirect
8. C₂×C₇×#459	C₂	semidirect
9. C₂×C₇×#449	C₂	semidirect
10. C₂×C₇×#447	C₂	semidirect
11. C₂×C₄×C₇×SD₁₆	C₂	semidirect
12. C₂×C₇×#436	C₂	semidirect
13. C₂²×C₇×#41	C₂²	semidirect
17. C₂×C₇×AGL(1,ℤ/8)	C₂²	semidirect
19. C₂²×C₇×SD₁₆	C₂²	semidirect
21. C₂²×C₇×D₈	C₂²	semidirect
22. C₂²×C₇×D₈	C₂²	semidirect
23. C₂²×C₇×M₁₆	C₂²	semidirect
25. C₇×#550	C₂²	semidirect
27. C₇×D₄²	C₂²	semidirect
30. C₂×C₇×#127	C₂²	semidirect
34. C₂×C₇×#121	C₂²	semidirect
36. C₂×C₄×C₇×Q₈	C₂²	semidirect
40. C₇×#493	C₂²	semidirect
41. C₇×#469	C₂²	semidirect
42. C₇×#460	C₂²	semidirect
43. C₇×#459	C₂²	semidirect
44. C₇×#449	C₂²	semidirect
45. C₇×#447	C₂²	semidirect
46. C₄×C₇×SD₁₆	C₂²	semidirect
47. C₇×#436	C₂²	semidirect
48. C₂×C₇×#106	C₂²	semidirect
49. C₂×C₇×C₄⋊C₈	C₂²	semidirect
50. C₂×C₇×#103	C₂²	semidirect
52. C₂×C₇×#102	C₂²	semidirect
53. C₂×C₇×#102	C₂²	semidirect
54. C₂×C₇×C₂²⋊C₈	C₂²	semidirect
55. C₂×C₄×C₇×C₈	C₂²	semidirect
56. C₂×#3274	C₇	direct (N × H)
61. C₂×C₇×#41	C₂³	semidirect
86. C₂×C₇×SD₁₆	D₄	semidirect
88. C₂×C₇×SD₁₆	C₂³	semidirect
93. C₂×C₇×D₈	C₂³	semidirect
94. C₂×C₇×D₈	C₂³	semidirect
96. C₂×C₇×M₁₆	C₂³	semidirect
104. C₇×#127	C₂³	semidirect
110. C₇×#121	C₂³	semidirect
113. C₄×C₇×Q₈	C₂³	semidirect
129. C₇×#106	C₂³	semidirect
130. C₇×C₄⋊C₈	C₂³	semidirect
131. C₇×#103	C₂³	semidirect
133. C₇×#102	C₂³	semidirect
134. C₇×#102	C₂³	semidirect
135. C₇×C₂²⋊C₈	C₂³	semidirect
136. C₄×C₇×C₈	C₂³	semidirect
137. C₂²×AGL(1,ℤ/8)	C₂×C₇	semidirect
138. C₂×#550	C₂×C₇	semidirect
139. C₂×D₄²	C₂×C₇	semidirect
140. #3274	C₂×C₇	direct (N × H)
141. C₂×#493	C₂×C₇	semidirect
142. C₂×#469	C₂×C₇	semidirect
143. C₂×#460	C₂×C₇	semidirect
144. C₂×#459	C₂×C₇	semidirect
145. C₂×#449	C₂×C₇	semidirect
146. C₂×#447	C₂×C₇	semidirect
147. C₂×C₄×SD₁₆	C₂×C₇	semidirect
148. C₂×#436	C₂×C₇	semidirect
162. C₂×C₇×Q₈	C₂×D₄	semidirect
214. C₇×SD₁₆	C₂×D₄	semidirect
221. C₂×C₇×C₈	C₂×D₄	semidirect
238. C₂²×#41	C₂²×C₇	semidirect
242. C₂×AGL(1,ℤ/8)	C₂²×C₇	semidirect
244. C₂²×SD₁₆	C₂²×C₇	semidirect
246. C₂²×D₈	C₂²×C₇	semidirect
247. C₂²×D₈	C₂²×C₇	semidirect
248. C₂²×M₁₆	C₂²×C₇	semidirect
250. #550	C₂²×C₇	semidirect
252. D₄²	C₂²×C₇	semidirect
255. C₂×#127	C₂²×C₇	semidirect
259. C₂×#121	C₂²×C₇	semidirect
261. C₂×C₄×Q₈	C₂²×C₇	semidirect
265. #493	C₂²×C₇	semidirect
266. #469	C₂²×C₇	semidirect
267. #460	C₂²×C₇	semidirect
268. #459	C₂²×C₇	semidirect
269. #449	C₂²×C₇	semidirect
270. #447	C₂²×C₇	semidirect
271. C₄×SD₁₆	C₂²×C₇	semidirect
272. #436	C₂²×C₇	semidirect
273. C₂×#106	C₂²×C₇	semidirect
274. C₂×C₄⋊C₈	C₂²×C₇	semidirect
275. C₂×#103	C₂²×C₇	semidirect
277. C₂×#102	C₂²×C₇	semidirect
278. C₂×#102	C₂²×C₇	semidirect
279. C₂×C₂²⋊C₈	C₂²×C₇	semidirect
280. C₂×C₄×C₈	C₂²×C₇	semidirect
304. C₇×Q₈	C₂²×D₄	semidirect
336. C₂×C₄×C₇	AGL(1,ℤ/8)	semidirect
346. C₇×C₈	C₂²×D₄	semidirect
351. C₂×#41	C₂³×C₇	semidirect
376. C₂×SD₁₆	C₇×D₄	semidirect
378. C₂×SD₁₆	C₂³×C₇	semidirect
383. C₂×D₈	C₂³×C₇	semidirect
384. C₂×D₈	C₂³×C₇	semidirect
386. C₂×M₁₆	C₂³×C₇	semidirect
394. #127	C₂³×C₇	semidirect
400. #121	C₂³×C₇	semidirect
403. C₄×Q₈	C₂³×C₇	semidirect
419. #106	C₂³×C₇	semidirect
420. C₄⋊C₈	C₂³×C₇	semidirect
421. #103	C₂³×C₇	semidirect
423. #102	C₂³×C₇	semidirect
424. #102	C₂³×C₇	semidirect
425. C₂²⋊C₈	C₂³×C₇	semidirect
426. C₄×C₈	C₂³×C₇	semidirect
454. C₄×C₇	C₂×AGL(1,ℤ/8)	semidirect
470. C₂×Q₈	C₂×C₇×D₄	semidirect
522. SD₁₆	C₂×C₇×D₄	semidirect
529. C₂×C₈	C₂×C₇×D₄	semidirect
550. C₂×C₇	#3274	direct (N × H)
577. Q₈	C₂²×C₇×D₄	semidirect
609. C₂×C₄	C₇×AGL(1,ℤ/8)	semidirect
619. C₈	C₂²×C₇×D₄	semidirect
620. C₇	C₂×#3274	direct (N × H)
648. C₄	C₂×C₇×AGL(1,ℤ/8)	semidirect
655. C₂	C₇×#3274	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₇×AGL(1,ℤ/8)	2
C₂×C₇×#550	2
C₂×C₇×D₄²	2
C₇×#3274	2
C₂×C₇×#493	2
C₂×C₇×#469	2
C₂×C₇×#460	2
C₂×C₇×#459	2
C₂×C₇×#449	2
C₂×C₇×#447	2
C₂×C₄×C₇×SD₁₆	2
C₂×C₇×#436	2
C₂×#3274	7

Subgroups — C2×C7×#3274 #13238249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#3274 #13238249