Subgroups — C₂×C₇×#3280 #13238256

Order 1792.

3074 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₇×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#546	C₂	1	2	yes	maximal, complemented
4	C₇×#3280	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₇×#495	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×#467	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#466	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×#463	C₂	1	2	yes	maximal, complemented
9	C₂×C₇×#449	C₂	1	2	yes	maximal, complemented
10	C₂×C₇×#450	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×C₇×D₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×C₇×#41	C₂²	1	2	yes	complemented
14	C₂³×C₇×D₄	C₂²	1	2	yes	—
15	C₂×C₇×#135	—	2	8	no	—
16	C₂×C₇×#135	C₂²	1	8	yes	complemented
17	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic, complemented
18	C₂²×C₇×SD₁₆	C₂²	1	2	yes	complemented
19	C₂²×C₇×SD₁₆	C₂²	1	2	yes	complemented
20	C₂²×C₇×D₈	—	2	8	no	—
21	C₂²×C₇×D₈	C₂²	1	2	yes	—
22	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
23	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
24	C₂²×C₇×D₈	C₂²	1	8	yes	complemented
25	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
26	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic, complemented
27	C₇×#546	—	2	8	no	—
28	C₇×#546	C₂²	1	16	yes	complemented
29	C₂×C₇×#124	C₂²	1	2	yes	complemented
30	C₂×C₇×#123	—	2	2	no	—
31	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
32	C₂×C₇×#121	—	2	2	no	—
33	C₂×C₇×#121	C₂²	1	2	yes	complemented
34	C₂×C₇×#120	—	2	2	no	—
35	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₇×D₄	C₂²	1	2	yes	complemented
37	C₂²×C₇×C₂²⋊C₄	—	2	2	no	—
38	C₇×#495	C₂²	1	8	yes	complemented
39	C₇×#467	C₂²	1	8	yes	complemented
40	C₇×#466	C₂²	1	8	yes	complemented
41	C₇×#463	C₂²	1	16	yes	complemented
42	C₇×#450	C₂²	1	16	yes	complemented
43	C₇×#449	C₂²	1	16	yes	complemented
44	C₄×C₇×D₈	C₂²	1	8	yes	complemented
45	C₇×C₈×D₄	C₂²	1	8	yes	complemented
46	C₂×C₇×#107	C₂²	1	1	yes	characteristic, complemented
47	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
48	C₂×C₇×#103	C₂²	1	2	yes	complemented
49	C₂×C₇×#102	C₂²	1	2	yes	complemented
50	C₂×C₇×#102	C₂²	1	2	yes	complemented
51	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
52	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic, complemented
53	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic, complemented
54	C₂×#3280	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂⁵×C₇	—	2	2	no	—
56	C₂×C₇×#41	—	2	4	no	—
57	C₂×C₇×#41	—	2	8	no	—
58	C₂×C₇×#41	—	4	4	no	—
59	C₂×C₇×#41	C₂³	1	8	yes	complemented
60	C₂²×C₇×Q₈	C₂³	1	2	yes	—
61	C₂²×C₇×D₄	—	2	8	no	—
62	C₂²×C₇×D₄	—	2	2	no	—
63	C₂²×C₇×D₄	—	2	2	no	—
64	C₂²×C₇×D₄	—	2	2	no	—
65	C₂²×C₇×D₄	—	2	4	no	—
66	C₂²×C₇×D₄	—	4	4	no	—
67	C₂²×C₇×D₄	D₄	1	1	yes	characteristic
68	C₂²×C₇×D₄	C₂³	1	2	yes	—
69	C₂²×C₇×D₄	C₂³	1	8	yes	—
70	C₂²×C₇×D₄	C₂³	1	2	yes	—
71	C₂³×C₄×C₇	—	2	2	no	—
72	C₂³×C₄×C₇	—	2	2	no	—
73	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
75	C₇×#135	—	2	32	no	—
76	C₂×C₇×Q₁₆	—	2	4	no	—
77	C₂×C₇×Q₁₆	C₂³	1	4	yes	complemented
78	C₂×C₇×SD₁₆	—	2	8	no	—
79	C₂×C₇×SD₁₆	—	4	4	no	—
80	C₂×C₇×SD₁₆	C₂³	1	8	yes	complemented
81	C₂×C₇×SD₁₆	C₂³	1	8	yes	complemented
82	C₂×C₇×D₈	—	2	8	no	—
83	C₂×C₇×D₈	—	2	32	no	—
84	C₂×C₇×D₈	—	2	4	no	—
85	C₂×C₇×D₈	D₄	1	8	yes	complemented
86	C₂×C₇×D₈	C₂³	1	4	yes	—
87	C₂×C₇×D₈	C₂³	1	8	yes	—
88	C₂×C₇×D₈	C₂³	1	4	yes	—
89	C₂²×C₇×C₈	—	2	4	no	—
90	C₂²×C₇×C₈	—	2	4	no	—
91	C₂²×C₇×C₈	C₂³	1	4	yes	—
92	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
93	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
94	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
95	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
96	C₂²×C₇×C₈	C₂³	1	4	yes	complemented
97	C₇×#124	—	2	4	no	—
98	C₇×#124	C₂³	1	8	yes	complemented
99	C₇×#123	—	2	16	no	—
100	C₇×C₂²⋊Q₈	—	2	4	no	—
101	C₇×C₂²⋊Q₈	C₂³	1	8	yes	—
102	C₇×#121	—	2	4	no	—
103	C₇×#121	—	2	16	no	—
104	C₇×#121	C₂³	1	8	yes	complemented
105	C₇×#120	—	2	16	no	—
106	C₄×C₇×D₄	—	2	4	no	—
107	C₄×C₇×D₄	D₄	1	4	yes	—
108	C₄×C₇×D₄	C₂³	1	4	yes	—
109	C₄×C₇×D₄	C₂³	1	8	yes	complemented
110	C₂×C₇×C₄⋊C₄	—	2	2	no	—
111	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
112	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
113	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
114	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
115	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
116	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
117	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
118	C₂×C₇×C₂²⋊C₄	—	2	16	no	—
119	C₂×C₇×C₂²⋊C₄	D₄	1	1	yes	characteristic
120	C₂×C₇×C₂²⋊C₄	D₄	1	1	yes	characteristic
121	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
122	C₇×#107	C₂³	1	4	yes	complemented
123	C₇×C₄⋊C₈	C₂³	1	4	yes	complemented
124	C₇×#103	C₂³	1	8	yes	complemented
125	C₇×#102	C₂³	1	8	yes	complemented
126	C₇×#102	C₂³	1	8	yes	complemented
127	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
128	C₇×C₂²⋊C₈	C₂³	1	4	yes	complemented
129	C₄×C₇×C₈	C₂³	1	4	yes	complemented
130	C₂²×#135	C₂×C₇	1	1	yes	characteristic, complemented
131	C₂³×D₈	C₂×C₇	1	1	yes	characteristic, complemented
132	C₂×#546	C₂×C₇	1	2	yes	complemented
133	#3280	C₂×C₇	1	16	yes	direct product factor
134	C₂×#495	C₂×C₇	1	1	yes	characteristic, complemented
135	C₂×#467	C₂×C₇	1	1	yes	characteristic, complemented
136	C₂×#466	C₂×C₇	1	1	yes	characteristic, complemented
137	C₂×#463	C₂×C₇	1	2	yes	complemented
138	C₂×#449	C₂×C₇	1	2	yes	complemented
139	C₂×#450	C₂×C₇	1	2	yes	complemented
140	C₂×C₄×D₈	C₂×C₇	1	1	yes	characteristic, complemented
141	C₂×C₈×D₄	C₂×C₇	1	1	yes	characteristic, complemented
142	C₂⁴×C₇	—	2	2	no	—
143	C₂⁴×C₇	—	2	2	no	—
144	C₂⁴×C₇	—	2	8	no	—
145	C₂⁴×C₇	—	4	8	no	—
146	C₂⁴×C₇	—	4	8	no	—
147	C₂⁴×C₇	—	8	4	no	—
148	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
149	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
150	C₇×#41	—	2	16	no	—
151	C₇×#41	—	4	8	no	—
152	C₇×#41	—	4	16	no	—
153	C₂×C₇×Q₈	—	2	2	no	—
154	C₂×C₇×Q₈	—	2	4	no	—
155	C₂×C₇×Q₈	—	2	4	no	—
156	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
157	C₂×C₇×D₄	—	2	4	no	—
158	C₂×C₇×D₄	—	2	16	no	—
159	C₂×C₇×D₄	—	2	8	no	—
160	C₂×C₇×D₄	—	2	8	no	—
161	C₂×C₇×D₄	—	2	2	no	—
162	C₂×C₇×D₄	—	2	4	no	—
163	C₂×C₇×D₄	—	2	8	no	—
164	C₂×C₇×D₄	—	2	2	no	—
165	C₂×C₇×D₄	—	2	4	no	—
166	C₂×C₇×D₄	—	2	4	no	—
167	C₂×C₇×D₄	—	4	8	no	—
168	C₂×C₇×D₄	—	4	16	no	—
169	C₂×C₇×D₄	—	4	8	no	—
170	C₂×C₇×D₄	—	4	8	no	—
171	C₂×C₇×D₄	—	8	4	no	—
172	C₂×C₇×D₄	C₂×D₄	1	4	yes	—
173	C₂×C₇×D₄	C₂×D₄	1	8	yes	—
174	C₂×C₇×D₄	C₂⁴	1	4	yes	—
175	C₂×C₇×D₄	C₂⁴	1	4	yes	—
176	C₂²×C₄×C₇	—	2	2	no	—
177	C₂²×C₄×C₇	—	2	2	no	—
178	C₂²×C₄×C₇	—	2	2	no	—
179	C₂²×C₄×C₇	—	2	2	no	—
180	C₂²×C₄×C₇	—	2	2	no	—
181	C₂²×C₄×C₇	—	2	2	no	—
182	C₂²×C₄×C₇	—	2	8	no	—
183	C₂²×C₄×C₇	—	2	8	no	—
184	C₂²×C₄×C₇	—	4	8	no	—
185	C₂²×C₄×C₇	—	8	4	no	—
186	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
187	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
188	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
189	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
190	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
191	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
192	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
193	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
194	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
195	C₇×Q₁₆	—	2	8	no	—
196	C₇×SD₁₆	—	2	16	no	—
197	C₇×SD₁₆	—	4	8	no	—
198	C₇×D₈	—	2	16	no	—
199	C₇×D₈	—	2	8	no	—
200	C₇×D₈	C₂×D₄	1	16	yes	complemented
201	C₂×C₇×C₈	—	2	2	no	—
202	C₂×C₇×C₈	—	2	2	no	—
203	C₂×C₇×C₈	—	2	2	no	—
204	C₂×C₇×C₈	—	2	8	no	—
205	C₂×C₇×C₈	—	2	8	no	—
206	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
207	C₂×C₇×C₈	C₂⁴	1	2	yes	—
208	C₂×C₇×C₈	C₂⁴	1	2	yes	—
209	C₂×C₇×C₈	C₂⁴	1	2	yes	—
210	C₂×C₇×C₈	C₂⁴	1	2	yes	—
211	C₇×C₄⋊C₄	—	2	8	no	—
212	C₇×C₄⋊C₄	—	2	2	no	—
213	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
214	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
215	C₇×C₂²⋊C₄	—	2	8	no	—
216	C₇×C₂²⋊C₄	—	2	8	no	—
217	C₇×C₂²⋊C₄	—	2	8	no	—
218	C₇×C₂²⋊C₄	—	2	8	no	—
219	C₇×C₂²⋊C₄	—	2	8	no	—
220	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
221	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
222	C₄²×C₇	C₂×D₄	1	2	yes	—
223	C₄²×C₇	C₂⁴	1	2	yes	—
224	C₂²×#41	C₂²×C₇	1	2	yes	complemented
225	C₂³×D₄	C₂²×C₇	1	2	yes	—
226	C₂×#135	—	2	8	no	—
227	C₂×#135	C₂²×C₇	1	8	yes	complemented
228	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic, complemented
229	C₂²×SD₁₆	C₂²×C₇	1	2	yes	complemented
230	C₂²×SD₁₆	C₂²×C₇	1	2	yes	complemented
231	C₂²×D₈	—	2	8	no	—
232	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
233	C₂²×D₈	C₂²×C₇	1	2	yes	—
234	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
235	C₂²×D₈	C₂²×C₇	1	8	yes	complemented
236	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
237	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic, complemented
238	#546	—	2	8	no	—
239	#546	C₂²×C₇	1	16	yes	complemented
240	C₂×#124	C₂²×C₇	1	2	yes	complemented
241	C₂×#123	—	2	2	no	—
242	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
243	C₂×#121	—	2	2	no	—
244	C₂×#121	C₂²×C₇	1	2	yes	complemented
245	C₂×#120	—	2	2	no	—
246	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
247	C₂×C₄×D₄	C₂²×C₇	1	2	yes	complemented
248	C₂²×C₂²⋊C₄	—	2	2	no	—
249	#495	C₂²×C₇	1	8	yes	complemented
250	#467	C₂²×C₇	1	8	yes	complemented
251	#466	C₂²×C₇	1	8	yes	complemented
252	#463	C₂²×C₇	1	16	yes	complemented
253	#450	C₂²×C₇	1	16	yes	complemented
254	#449	C₂²×C₇	1	16	yes	complemented
255	C₄×D₈	C₂²×C₇	1	8	yes	complemented
256	C₈×D₄	C₂²×C₇	1	8	yes	complemented
257	C₂×#107	C₂²×C₇	1	1	yes	characteristic, complemented
258	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic, complemented
259	C₂×#103	C₂²×C₇	1	2	yes	complemented
260	C₂×#102	C₂²×C₇	1	2	yes	complemented
261	C₂×#102	C₂²×C₇	1	2	yes	complemented
262	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
263	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic, complemented
264	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic, complemented
265	C₂³×C₇	—	2	2	no	—
266	C₂³×C₇	—	2	2	no	—
267	C₂³×C₇	—	2	2	no	—
268	C₂³×C₇	—	2	8	no	—
269	C₂³×C₇	—	2	2	no	—
270	C₂³×C₇	—	2	2	no	—
271	C₂³×C₇	—	2	4	no	—
272	C₂³×C₇	—	2	4	no	—
273	C₂³×C₇	—	4	16	no	—
274	C₂³×C₇	—	4	4	no	—
275	C₂³×C₇	—	4	4	no	—
276	C₂³×C₇	—	4	4	no	—
277	C₂³×C₇	—	4	4	no	—
278	C₂³×C₇	—	4	16	no	—
279	C₂³×C₇	—	4	4	no	—
280	C₂³×C₇	—	8	8	no	—
281	C₂³×C₇	—	8	32	no	—
282	C₂³×C₇	#135	1	4	yes	complemented
283	C₂³×C₇	C₂²×D₄	1	2	yes	—
284	C₂³×C₇	C₂²×D₄	1	2	yes	—
285	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
286	C₇×Q₈	—	2	4	no	—
287	C₇×Q₈	—	2	4	no	—
288	C₇×Q₈	—	4	4	no	—
289	C₇×D₄	—	2	8	no	—
290	C₇×D₄	—	2	8	no	—
291	C₇×D₄	—	2	4	no	—
292	C₇×D₄	—	2	4	no	—
293	C₇×D₄	—	2	4	no	—
294	C₇×D₄	—	4	16	no	—
295	C₇×D₄	—	4	16	no	—
296	C₇×D₄	—	4	4	no	—
297	C₇×D₄	—	8	8	no	—
298	C₇×D₄	C₂²×D₄	1	8	yes	—
299	C₂×C₄×C₇	—	2	2	no	—
300	C₂×C₄×C₇	—	2	2	no	—
301	C₂×C₄×C₇	—	2	2	no	—
302	C₂×C₄×C₇	—	2	2	no	—
303	C₂×C₄×C₇	—	2	4	no	—
304	C₂×C₄×C₇	—	2	4	no	—
305	C₂×C₄×C₇	—	2	2	no	—
306	C₂×C₄×C₇	—	2	2	no	—
307	C₂×C₄×C₇	—	2	2	no	—
308	C₂×C₄×C₇	—	2	2	no	—
309	C₂×C₄×C₇	—	2	4	no	—
310	C₂×C₄×C₇	—	2	4	no	—
311	C₂×C₄×C₇	—	4	16	no	—
312	C₂×C₄×C₇	—	4	4	no	—
313	C₂×C₄×C₇	—	4	4	no	—
314	C₂×C₄×C₇	—	8	8	no	—
315	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
316	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
317	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
318	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
319	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
320	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
321	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
322	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
323	C₇×C₈	—	2	2	no	—
324	C₇×C₈	—	2	2	no	—
325	C₇×C₈	—	2	2	no	—
326	C₇×C₈	C₂²×D₄	1	4	yes	—
327	C₂⁵	—	2	2	no	—
328	C₂×#41	—	2	4	no	—
329	C₂×#41	—	2	8	no	—
330	C₂×#41	—	4	4	no	—
331	C₂×#41	C₂³×C₇	1	8	yes	complemented
332	C₂²×Q₈	C₂³×C₇	1	2	yes	—
333	C₂²×D₄	—	2	8	no	—
334	C₂²×D₄	—	2	2	no	—
335	C₂²×D₄	—	2	2	no	—
336	C₂²×D₄	—	2	2	no	—
337	C₂²×D₄	—	2	4	no	—
338	C₂²×D₄	—	4	4	no	—
339	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
340	C₂²×D₄	C₂³×C₇	1	2	yes	—
341	C₂²×D₄	C₂³×C₇	1	8	yes	—
342	C₂²×D₄	C₂³×C₇	1	2	yes	—
343	C₂³×C₄	—	2	2	no	—
344	C₂³×C₄	—	2	2	no	—
345	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
346	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
347	#135	—	2	32	no	—
348	C₂×Q₁₆	—	2	4	no	—
349	C₂×Q₁₆	C₂³×C₇	1	4	yes	complemented
350	C₂×SD₁₆	—	2	8	no	—
351	C₂×SD₁₆	—	4	4	no	—
352	C₂×SD₁₆	C₂³×C₇	1	8	yes	complemented
353	C₂×SD₁₆	C₂³×C₇	1	8	yes	complemented
354	C₂×D₈	—	2	8	no	—
355	C₂×D₈	—	2	32	no	—
356	C₂×D₈	—	2	4	no	—
357	C₂×D₈	C₇×D₄	1	8	yes	complemented
358	C₂×D₈	C₂³×C₇	1	8	yes	—
359	C₂×D₈	C₂³×C₇	1	4	yes	—
360	C₂×D₈	C₂³×C₇	1	4	yes	—
361	C₂²×C₈	—	2	4	no	—
362	C₂²×C₈	—	2	4	no	—
363	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
364	C₂²×C₈	C₂³×C₇	1	4	yes	—
365	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
366	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
367	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
368	C₂²×C₈	C₂³×C₇	1	4	yes	complemented
369	#124	—	2	4	no	—
370	#124	C₂³×C₇	1	8	yes	complemented
371	#123	—	2	16	no	—
372	C₂²⋊Q₈	—	2	4	no	—
373	C₂²⋊Q₈	C₂³×C₇	1	8	yes	—
374	#121	—	2	4	no	—
375	#121	—	2	16	no	—
376	#121	C₂³×C₇	1	8	yes	complemented
377	#120	—	2	16	no	—
378	C₄×D₄	—	2	4	no	—
379	C₄×D₄	C₇×D₄	1	4	yes	—
380	C₄×D₄	C₂³×C₇	1	4	yes	—
381	C₄×D₄	C₂³×C₇	1	8	yes	complemented
382	C₂×C₄⋊C₄	—	2	2	no	—
383	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
384	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
385	C₂×C₂²⋊C₄	—	2	2	no	—
386	C₂×C₂²⋊C₄	—	2	2	no	—
387	C₂×C₂²⋊C₄	—	2	2	no	—
388	C₂×C₂²⋊C₄	—	2	2	no	—
389	C₂×C₂²⋊C₄	—	2	2	no	—
390	C₂×C₂²⋊C₄	—	2	16	no	—
391	C₂×C₂²⋊C₄	C₇×D₄	1	1	yes	characteristic
392	C₂×C₂²⋊C₄	C₇×D₄	1	1	yes	characteristic
393	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
394	#107	C₂³×C₇	1	4	yes	complemented
395	C₄⋊C₈	C₂³×C₇	1	4	yes	complemented
396	#103	C₂³×C₇	1	8	yes	complemented
397	#102	C₂³×C₇	1	8	yes	complemented
398	#102	C₂³×C₇	1	8	yes	complemented
399	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
400	C₂²⋊C₈	C₂³×C₇	1	4	yes	complemented
401	C₄×C₈	C₂³×C₇	1	4	yes	complemented
402	C₂²×C₇	—	2	2	no	—
403	C₂²×C₇	—	2	8	no	—
404	C₂²×C₇	—	2	2	no	—
405	C₂²×C₇	—	2	2	no	—
406	C₂²×C₇	—	2	8	no	—
407	C₂²×C₇	—	2	2	no	—
408	C₂²×C₇	—	2	2	no	—
409	C₂²×C₇	—	4	4	no	—
410	C₂²×C₇	—	4	16	no	—
411	C₂²×C₇	—	4	4	no	—
412	C₂²×C₇	—	4	4	no	—
413	C₂²×C₇	—	4	4	no	—
414	C₂²×C₇	—	4	4	no	—
415	C₂²×C₇	—	4	4	no	—
416	C₂²×C₇	—	8	32	no	—
417	C₂²×C₇	—	8	8	no	—
418	C₂²×C₇	D₄²	1	2	yes	—
419	C₂²×C₇	C₂×#135	1	2	yes	—
420	C₂²×C₇	C₂×#135	1	4	yes	complemented
421	C₂²×C₇	#576	1	2	yes	—
422	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
423	C₄×C₇	—	2	2	no	—
424	C₄×C₇	—	2	2	no	—
425	C₄×C₇	—	2	2	no	—
426	C₄×C₇	—	2	2	no	—
427	C₄×C₇	—	4	4	no	—
428	C₄×C₇	—	4	4	no	—
429	C₄×C₇	D₄²	1	2	yes	—
430	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
431	C₄×C₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
432	C₂⁴	—	2	2	no	—
433	C₂⁴	—	2	2	no	—
434	C₂⁴	—	2	8	no	—
435	C₂⁴	—	4	8	no	—
436	C₂⁴	—	4	8	no	—
437	C₂⁴	—	8	4	no	—
438	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
439	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
440	#41	—	2	16	no	—
441	#41	—	4	8	no	—
442	#41	—	4	16	no	—
443	C₂×Q₈	—	2	2	no	—
444	C₂×Q₈	—	2	4	no	—
445	C₂×Q₈	—	2	4	no	—
446	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
447	C₂×D₄	—	2	4	no	—
448	C₂×D₄	—	2	16	no	—
449	C₂×D₄	—	2	8	no	—
450	C₂×D₄	—	2	8	no	—
451	C₂×D₄	—	2	2	no	—
452	C₂×D₄	—	2	4	no	—
453	C₂×D₄	—	2	8	no	—
454	C₂×D₄	—	2	2	no	—
455	C₂×D₄	—	2	4	no	—
456	C₂×D₄	—	2	4	no	—
457	C₂×D₄	—	4	8	no	—
458	C₂×D₄	—	4	16	no	—
459	C₂×D₄	—	4	8	no	—
460	C₂×D₄	—	4	8	no	—
461	C₂×D₄	—	8	4	no	—
462	C₂×D₄	C₂×C₇×D₄	1	4	yes	—
463	C₂×D₄	C₂×C₇×D₄	1	8	yes	—
464	C₂×D₄	C₂⁴×C₇	1	4	yes	—
465	C₂×D₄	C₂⁴×C₇	1	4	yes	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	8	no	—
473	C₂²×C₄	—	2	8	no	—
474	C₂²×C₄	—	4	8	no	—
475	C₂²×C₄	—	8	4	no	—
476	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
477	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
478	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
479	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
480	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
481	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
482	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
483	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
484	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
485	Q₁₆	—	2	8	no	—
486	SD₁₆	—	2	16	no	—
487	SD₁₆	—	4	8	no	—
488	D₈	—	2	16	no	—
489	D₈	—	2	8	no	—
490	D₈	C₂×C₇×D₄	1	16	yes	complemented
491	C₂×C₈	—	2	2	no	—
492	C₂×C₈	—	2	2	no	—
493	C₂×C₈	—	2	2	no	—
494	C₂×C₈	—	2	8	no	—
495	C₂×C₈	—	2	8	no	—
496	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
497	C₂×C₈	C₂⁴×C₇	1	2	yes	—
498	C₂×C₈	C₂⁴×C₇	1	2	yes	—
499	C₂×C₈	C₂⁴×C₇	1	2	yes	—
500	C₂×C₈	C₂⁴×C₇	1	2	yes	—
501	C₄⋊C₄	—	2	8	no	—
502	C₄⋊C₄	—	2	2	no	—
503	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
504	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
505	C₂²⋊C₄	—	2	8	no	—
506	C₂²⋊C₄	—	2	8	no	—
507	C₂²⋊C₄	—	2	8	no	—
508	C₂²⋊C₄	—	2	8	no	—
509	C₂²⋊C₄	—	2	8	no	—
510	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
511	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
512	C₄²	C₂×C₇×D₄	1	2	yes	—
513	C₄²	C₂⁴×C₇	1	2	yes	—
514	C₂×C₇	—	2	4	no	—
515	C₂×C₇	—	4	2	no	—
516	C₂×C₇	—	4	8	no	—
517	C₂×C₇	—	8	4	no	—
518	C₂×C₇	#3280	1	4	yes	direct product factor
519	C₂×C₇	C₂×D₄²	1	1	yes	characteristic
520	C₂×C₇	C₂²×#135	1	1	yes	characteristic
521	C₂×C₇	C₂×#576	1	1	yes	characteristic
522	C₂³	—	2	2	no	—
523	C₂³	—	2	2	no	—
524	C₂³	—	2	4	no	—
525	C₂³	—	2	2	no	—
526	C₂³	—	2	8	no	—
527	C₂³	—	2	2	no	—
528	C₂³	—	2	2	no	—
529	C₂³	—	2	4	no	—
530	C₂³	—	4	16	no	—
531	C₂³	—	4	4	no	—
532	C₂³	—	4	4	no	—
533	C₂³	—	4	4	no	—
534	C₂³	—	4	4	no	—
535	C₂³	—	4	16	no	—
536	C₂³	—	4	4	no	—
537	C₂³	—	8	8	no	—
538	C₂³	—	8	32	no	—
539	C₂³	C₇×#135	1	4	yes	complemented
540	C₂³	C₂²×C₇×D₄	1	2	yes	—
541	C₂³	C₂²×C₇×D₄	1	2	yes	—
542	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
543	Q₈	—	2	4	no	—
544	Q₈	—	2	4	no	—
545	Q₈	—	4	4	no	—
546	D₄	—	2	8	no	—
547	D₄	—	2	4	no	—
548	D₄	—	2	4	no	—
549	D₄	—	2	4	no	—
550	D₄	—	2	8	no	—
551	D₄	—	4	16	no	—
552	D₄	—	4	16	no	—
553	D₄	—	4	4	no	—
554	D₄	—	8	8	no	—
555	D₄	C₂²×C₇×D₄	1	8	yes	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	4	no	—
565	C₂×C₄	—	2	4	no	—
566	C₂×C₄	—	2	4	no	—
567	C₂×C₄	—	2	4	no	—
568	C₂×C₄	—	4	16	no	—
569	C₂×C₄	—	4	4	no	—
570	C₂×C₄	—	4	4	no	—
571	C₂×C₄	—	8	8	no	—
572	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
573	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
574	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
575	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
576	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
577	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
578	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
579	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
580	C₈	—	2	2	no	—
581	C₈	—	2	2	no	—
582	C₈	—	2	2	no	—
583	C₈	C₂²×C₇×D₄	1	4	yes	—
584	C₇	C₂×#3280	1	1	yes	characteristic, direct product factor, Sylow
585	C₂²	—	2	2	no	—
586	C₂²	—	2	8	no	—
587	C₂²	—	2	2	no	—
588	C₂²	—	2	2	no	—
589	C₂²	—	2	2	no	—
590	C₂²	—	2	2	no	—
591	C₂²	—	2	8	no	—
592	C₂²	—	4	4	no	—
593	C₂²	—	4	4	no	—
594	C₂²	—	4	4	no	—
595	C₂²	—	4	16	no	—
596	C₂²	—	4	4	no	—
597	C₂²	—	4	4	no	—
598	C₂²	—	4	4	no	—
599	C₂²	—	8	32	no	—
600	C₂²	—	8	8	no	—
601	C₂²	C₇×D₄²	1	2	yes	—
602	C₂²	C₂×C₇×#135	1	2	yes	—
603	C₂²	C₂×C₇×#135	1	4	yes	complemented
604	C₂²	C₇×#576	1	2	yes	—
605	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
606	C₄	—	2	2	no	—
607	C₄	—	2	2	no	—
608	C₄	—	2	2	no	—
609	C₄	—	2	2	no	—
610	C₄	—	4	4	no	—
611	C₄	—	4	4	no	—
612	C₄	C₇×D₄²	1	2	yes	—
613	C₄	C₂³×C₇×D₄	1	1	yes	characteristic
614	C₄	C₂×C₇×H(5,𝔽₂)	1	1	yes	characteristic
615	C₂	—	2	4	no	—
616	C₂	—	4	2	no	—
617	C₂	—	4	8	no	—
618	C₂	—	8	4	no	—
619	C₂	C₇×#3280	1	4	yes	direct product factor
620	C₂	C₂×C₇×D₄²	1	1	yes	characteristic
621	C₂	C₂²×C₇×#135	1	1	yes	characteristic
622	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×#135	C₂	semidirect
2. C₂³×C₇×D₈	C₂	semidirect
3. C₂×C₇×#546	C₂	semidirect
4. C₇×#3280	C₂	direct (N × H)
5. C₂×C₇×#495	C₂	semidirect
6. C₂×C₇×#467	C₂	semidirect
7. C₂×C₇×#466	C₂	semidirect
8. C₂×C₇×#463	C₂	semidirect
9. C₂×C₇×#449	C₂	semidirect
10. C₂×C₇×#450	C₂	semidirect
11. C₂×C₄×C₇×D₈	C₂	semidirect
12. C₂×C₇×C₈×D₄	C₂	semidirect
13. C₂²×C₇×#41	C₂²	semidirect
16. C₂×C₇×#135	C₂²	semidirect
17. C₂²×C₇×Q₁₆	C₂²	semidirect
18. C₂²×C₇×SD₁₆	C₂²	semidirect
19. C₂²×C₇×SD₁₆	C₂²	semidirect
24. C₂²×C₇×D₈	C₂²	semidirect
26. C₂³×C₇×C₈	C₂²	semidirect
28. C₇×#546	C₂²	semidirect
29. C₂×C₇×#124	C₂²	semidirect
33. C₂×C₇×#121	C₂²	semidirect
36. C₂×C₄×C₇×D₄	C₂²	semidirect
38. C₇×#495	C₂²	semidirect
39. C₇×#467	C₂²	semidirect
40. C₇×#466	C₂²	semidirect
41. C₇×#463	C₂²	semidirect
42. C₇×#450	C₂²	semidirect
43. C₇×#449	C₂²	semidirect
44. C₄×C₇×D₈	C₂²	semidirect
45. C₇×C₈×D₄	C₂²	semidirect
46. C₂×C₇×#107	C₂²	semidirect
47. C₂×C₇×C₄⋊C₈	C₂²	semidirect
48. C₂×C₇×#103	C₂²	semidirect
49. C₂×C₇×#102	C₂²	semidirect
50. C₂×C₇×#102	C₂²	semidirect
52. C₂×C₇×C₂²⋊C₈	C₂²	semidirect
53. C₂×C₄×C₇×C₈	C₂²	semidirect
54. C₂×#3280	C₇	direct (N × H)
59. C₂×C₇×#41	C₂³	semidirect
77. C₂×C₇×Q₁₆	C₂³	semidirect
80. C₂×C₇×SD₁₆	C₂³	semidirect
81. C₂×C₇×SD₁₆	C₂³	semidirect
85. C₂×C₇×D₈	D₄	semidirect
96. C₂²×C₇×C₈	C₂³	semidirect
98. C₇×#124	C₂³	semidirect
104. C₇×#121	C₂³	semidirect
109. C₄×C₇×D₄	C₂³	semidirect
122. C₇×#107	C₂³	semidirect
123. C₇×C₄⋊C₈	C₂³	semidirect
124. C₇×#103	C₂³	semidirect
125. C₇×#102	C₂³	semidirect
126. C₇×#102	C₂³	semidirect
128. C₇×C₂²⋊C₈	C₂³	semidirect
129. C₄×C₇×C₈	C₂³	semidirect
130. C₂²×#135	C₂×C₇	semidirect
131. C₂³×D₈	C₂×C₇	semidirect
132. C₂×#546	C₂×C₇	semidirect
133. #3280	C₂×C₇	direct (N × H)
134. C₂×#495	C₂×C₇	semidirect
135. C₂×#467	C₂×C₇	semidirect
136. C₂×#466	C₂×C₇	semidirect
137. C₂×#463	C₂×C₇	semidirect
138. C₂×#449	C₂×C₇	semidirect
139. C₂×#450	C₂×C₇	semidirect
140. C₂×C₄×D₈	C₂×C₇	semidirect
141. C₂×C₈×D₄	C₂×C₇	semidirect
200. C₇×D₈	C₂×D₄	semidirect
224. C₂²×#41	C₂²×C₇	semidirect
227. C₂×#135	C₂²×C₇	semidirect
228. C₂²×Q₁₆	C₂²×C₇	semidirect
229. C₂²×SD₁₆	C₂²×C₇	semidirect
230. C₂²×SD₁₆	C₂²×C₇	semidirect
235. C₂²×D₈	C₂²×C₇	semidirect
237. C₂³×C₈	C₂²×C₇	semidirect
239. #546	C₂²×C₇	semidirect
240. C₂×#124	C₂²×C₇	semidirect
244. C₂×#121	C₂²×C₇	semidirect
247. C₂×C₄×D₄	C₂²×C₇	semidirect
249. #495	C₂²×C₇	semidirect
250. #467	C₂²×C₇	semidirect
251. #466	C₂²×C₇	semidirect
252. #463	C₂²×C₇	semidirect
253. #450	C₂²×C₇	semidirect
254. #449	C₂²×C₇	semidirect
255. C₄×D₈	C₂²×C₇	semidirect
256. C₈×D₄	C₂²×C₇	semidirect
257. C₂×#107	C₂²×C₇	semidirect
258. C₂×C₄⋊C₈	C₂²×C₇	semidirect
259. C₂×#103	C₂²×C₇	semidirect
260. C₂×#102	C₂²×C₇	semidirect
261. C₂×#102	C₂²×C₇	semidirect
263. C₂×C₂²⋊C₈	C₂²×C₇	semidirect
264. C₂×C₄×C₈	C₂²×C₇	semidirect
282. C₂³×C₇	#135	semidirect
331. C₂×#41	C₂³×C₇	semidirect
349. C₂×Q₁₆	C₂³×C₇	semidirect
352. C₂×SD₁₆	C₂³×C₇	semidirect
353. C₂×SD₁₆	C₂³×C₇	semidirect
357. C₂×D₈	C₇×D₄	semidirect
368. C₂²×C₈	C₂³×C₇	semidirect
370. #124	C₂³×C₇	semidirect
376. #121	C₂³×C₇	semidirect
381. C₄×D₄	C₂³×C₇	semidirect
394. #107	C₂³×C₇	semidirect
395. C₄⋊C₈	C₂³×C₇	semidirect
396. #103	C₂³×C₇	semidirect
397. #102	C₂³×C₇	semidirect
398. #102	C₂³×C₇	semidirect
400. C₂²⋊C₈	C₂³×C₇	semidirect
401. C₄×C₈	C₂³×C₇	semidirect
420. C₂²×C₇	C₂×#135	semidirect
490. D₈	C₂×C₇×D₄	semidirect
518. C₂×C₇	#3280	direct (N × H)
539. C₂³	C₇×#135	semidirect
584. C₇	C₂×#3280	direct (N × H)
603. C₂²	C₂×C₇×#135	semidirect
619. C₂	C₇×#3280	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×#135	2
C₂³×C₇×D₈	2
C₂×C₇×#546	2
C₇×#3280	2
C₂×C₇×#495	2
C₂×C₇×#467	2
C₂×C₇×#466	2
C₂×C₇×#463	2
C₂×C₇×#449	2
C₂×C₇×#450	2
C₂×C₄×C₇×D₈	2
C₂×C₇×C₈×D₄	2
C₂×#3280	7

Subgroups — C2×C7×#3280 #13238256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#3280 #13238256