Subgroups — C₂×C₇×#3324 #13238360

Order 1792.

2566 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×D₄²	C₂	1	1	yes	maximal, characteristic
3	C₇×#3324	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₇×#486	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇×#482	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#474	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#468	C₂	1	2	yes	maximal, complemented
8	C₂×C₇×#460	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₇×#450	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×C₇×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₇×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#418	C₂	1	2	yes	maximal, complemented
13	C₂³×C₇×D₄	—	2	1	no	—
14	C₂³×C₇×D₄	C₂²	1	2	yes	—
15	C₂²×C₇×SD₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₇×SD₁₆	C₂²	1	2	yes	complemented
17	C₂²×C₇×M₁₆	C₂²	1	2	yes	complemented
18	C₇×#556	—	2	4	no	—
19	C₇×#556	C₂²	1	8	yes	complemented
20	C₇×D₄²	—	2	4	no	—
21	C₇×D₄²	C₂²	1	8	yes	—
22	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₇×#127	C₂²	1	1	yes	characteristic
24	C₂×C₇×#126	—	2	2	no	—
25	C₂×C₇×#124	—	2	1	no	—
26	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	complemented
27	C₂×C₇×#121	—	2	1	no	—
28	C₂×C₇×#121	C₂²	1	2	yes	—
29	C₂×C₇×#120	—	2	2	no	—
30	C₂×C₄×C₇×Q₈	—	2	1	no	—
31	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
32	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
34	C₂×C₇×#117	C₂²	1	2	yes	complemented
35	C₇×#486	C₂²	1	8	yes	complemented
36	C₇×#482	C₂²	1	16	yes	complemented
37	C₇×#474	C₂²	1	8	yes	complemented
38	C₇×#468	C₂²	1	16	yes	complemented
39	C₇×#460	C₂²	1	8	yes	complemented
40	C₇×#450	C₂²	1	16	yes	complemented
41	C₄×C₇×SD₁₆	C₂²	1	8	yes	complemented
42	C₇×#436	C₂²	1	8	yes	complemented
43	C₂×C₇×#107	C₂²	1	2	yes	complemented
44	C₂×C₇×#106	C₂²	1	1	yes	characteristic, complemented
45	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
46	C₇×#418	C₂²	1	16	yes	complemented
47	C₂×C₇×#103	C₂²	1	1	yes	characteristic, complemented
48	C₂×C₇×#102	C₂²	1	1	yes	characteristic
49	C₂×C₇×#102	C₂²	1	2	yes	—
50	C₂×C₇×#102	C₂²	1	2	yes	complemented
51	C₂×C₇×#102	C₂²	1	2	yes	complemented
52	C₂×C₇×#102	C₂²	1	2	yes	complemented
53	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	complemented
54	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic, complemented
55	C₂×#3324	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂⁵×C₇	—	2	2	no	—
57	C₂²×C₇×Q₈	—	2	1	no	—
58	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
59	C₂²×C₇×D₄	—	2	8	no	—
60	C₂²×C₇×D₄	—	2	2	no	—
61	C₂²×C₇×D₄	—	2	8	no	—
62	C₂²×C₇×D₄	—	2	1	no	—
63	C₂²×C₇×D₄	—	2	2	no	—
64	C₂²×C₇×D₄	—	2	4	no	—
65	C₂²×C₇×D₄	—	2	1	no	—
66	C₂²×C₇×D₄	—	4	4	no	—
67	C₂²×C₇×D₄	—	4	8	no	—
68	C₂²×C₇×D₄	D₄	1	1	yes	characteristic
69	C₂²×C₇×D₄	C₂³	1	2	yes	—
70	C₂²×C₇×D₄	C₂³	1	8	yes	—
71	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₇	—	2	1	no	—
73	C₂³×C₄×C₇	C₂³	1	2	yes	—
74	C₂×C₇×SD₁₆	—	2	4	no	—
75	C₂×C₇×SD₁₆	—	4	4	no	—
76	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
77	C₂×C₇×SD₁₆	C₂³	1	8	yes	complemented
78	C₂×C₇×M₁₆	—	4	4	no	—
79	C₂×C₇×M₁₆	C₂³	1	8	yes	complemented
80	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
81	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
82	C₂²×C₇×C₈	C₂³	1	2	yes	—
83	C₇×C₄⋊Q₈	—	2	2	no	—
84	C₇×C₄⋊Q₈	C₂³	1	4	yes	complemented
85	C₇×#127	—	2	2	no	—
86	C₇×#127	C₂³	1	4	yes	—
87	C₇×#126	—	2	16	no	—
88	C₇×#124	—	2	8	no	—
89	C₇×C₂²⋊Q₈	—	2	4	no	—
90	C₇×C₂²⋊Q₈	C₂³	1	8	yes	complemented
91	C₇×#121	—	2	4	no	—
92	C₇×#121	—	2	8	no	—
93	C₇×#121	C₂³	1	8	yes	—
94	C₇×#120	—	2	16	no	—
95	C₄×C₇×Q₈	—	2	8	no	—
96	C₄×C₇×Q₈	—	2	2	no	—
97	C₄×C₇×Q₈	C₂³	1	4	yes	complemented
98	C₄×C₇×D₄	—	2	2	no	—
99	C₄×C₇×D₄	D₄	1	4	yes	—
100	C₄×C₇×D₄	C₂³	1	4	yes	—
101	C₄×C₇×D₄	C₂³	1	4	yes	—
102	C₇×#117	—	2	4	no	—
103	C₇×#117	C₂³	1	8	yes	complemented
104	C₂×C₇×C₄⋊C₄	—	2	2	no	—
105	C₂×C₇×C₄⋊C₄	—	2	1	no	—
106	C₂×C₇×C₄⋊C₄	—	2	1	no	—
107	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
108	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
109	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
110	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
111	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
112	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
113	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
114	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
115	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
116	C₂×C₄²×C₇	—	2	2	no	—
117	C₂×C₄²×C₇	—	2	1	no	—
118	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
119	C₇×#107	C₂³	1	8	yes	complemented
120	C₇×#106	C₂³	1	4	yes	complemented
121	C₇×C₄⋊C₈	C₂³	1	4	yes	complemented
122	C₇×#103	C₂³	1	4	yes	complemented
123	C₇×#102	C₂³	1	8	yes	—
124	C₇×#102	C₂³	1	4	yes	—
125	C₇×#102	C₂³	1	8	yes	complemented
126	C₇×#102	C₂³	1	8	yes	complemented
127	C₇×#102	C₂³	1	8	yes	complemented
128	C₇×C₂²⋊C₈	C₂³	1	8	yes	complemented
129	C₄×C₇×C₈	C₂³	1	4	yes	complemented
130	C₂×#556	C₂×C₇	1	1	yes	characteristic, complemented
131	C₂×D₄²	C₂×C₇	1	1	yes	characteristic
132	#3324	C₂×C₇	1	16	yes	direct product factor
133	C₂×#486	C₂×C₇	1	1	yes	characteristic, complemented
134	C₂×#482	C₂×C₇	1	2	yes	complemented
135	C₂×#474	C₂×C₇	1	1	yes	characteristic, complemented
136	C₂×#468	C₂×C₇	1	2	yes	complemented
137	C₂×#460	C₂×C₇	1	1	yes	characteristic, complemented
138	C₂×#450	C₂×C₇	1	2	yes	complemented
139	C₂×C₄×SD₁₆	C₂×C₇	1	1	yes	characteristic, complemented
140	C₂×#436	C₂×C₇	1	1	yes	characteristic, complemented
141	C₂×#418	C₂×C₇	1	2	yes	complemented
142	C₂⁴×C₇	—	2	1	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	2	yes	—
149	C₂×C₇×Q₈	—	2	1	no	—
150	C₂×C₇×Q₈	—	2	4	no	—
151	C₂×C₇×Q₈	—	2	2	no	—
152	C₂×C₇×Q₈	—	2	2	no	—
153	C₂×C₇×Q₈	—	4	4	no	—
154	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
155	C₂×C₇×D₄	—	2	16	no	—
156	C₂×C₇×D₄	—	2	8	no	—
157	C₂×C₇×D₄	—	2	4	no	—
158	C₂×C₇×D₄	—	2	1	no	—
159	C₂×C₇×D₄	—	2	8	no	—
160	C₂×C₇×D₄	—	2	2	no	—
161	C₂×C₇×D₄	—	2	4	no	—
162	C₂×C₇×D₄	—	2	4	no	—
163	C₂×C₇×D₄	—	2	4	no	—
164	C₂×C₇×D₄	—	2	2	no	—
165	C₂×C₇×D₄	—	2	4	no	—
166	C₂×C₇×D₄	—	4	8	no	—
167	C₂×C₇×D₄	—	4	16	no	—
168	C₂×C₇×D₄	—	4	8	no	—
169	C₂×C₇×D₄	—	4	32	no	—
170	C₂×C₇×D₄	—	4	8	no	—
171	C₂×C₇×D₄	—	4	4	no	—
172	C₂×C₇×D₄	—	8	2	no	—
173	C₂×C₇×D₄	C₂×D₄	1	4	yes	—
174	C₂×C₇×D₄	#41	1	4	yes	—
175	C₂×C₇×D₄	C₂⁴	1	4	yes	—
176	C₂×C₇×D₄	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₇	—	2	4	no	—
178	C₂²×C₄×C₇	—	2	4	no	—
179	C₂²×C₄×C₇	—	2	1	no	—
180	C₂²×C₄×C₇	—	2	1	no	—
181	C₂²×C₄×C₇	—	2	2	no	—
182	C₂²×C₄×C₇	—	2	1	no	—
183	C₂²×C₄×C₇	—	2	4	no	—
184	C₂²×C₄×C₇	—	4	4	no	—
185	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
186	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
187	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
189	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
190	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
191	C₇×SD₁₆	—	2	8	no	—
192	C₇×SD₁₆	—	4	8	no	—
193	C₇×M₁₆	—	4	8	no	—
194	C₂×C₇×C₈	—	2	2	no	—
195	C₂×C₇×C₈	—	2	2	no	—
196	C₂×C₇×C₈	—	2	4	no	—
197	C₂×C₇×C₈	C₂⁴	1	2	yes	—
198	C₂×C₇×C₈	C₂⁴	1	2	yes	—
199	C₂×C₇×C₈	C₂⁴	1	4	yes	—
200	C₇×C₄⋊C₄	—	2	8	no	—
201	C₇×C₄⋊C₄	—	2	4	no	—
202	C₇×C₄⋊C₄	—	2	4	no	—
203	C₇×C₄⋊C₄	—	2	1	no	—
204	C₇×C₄⋊C₄	—	2	2	no	—
205	C₇×C₄⋊C₄	—	2	1	no	—
206	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
207	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
208	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
209	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
210	C₇×C₂²⋊C₄	—	2	8	no	—
211	C₇×C₂²⋊C₄	—	2	8	no	—
212	C₇×C₂²⋊C₄	—	2	4	no	—
213	C₇×C₂²⋊C₄	—	2	8	no	—
214	C₇×C₂²⋊C₄	C₂×D₄	1	8	yes	—
215	C₄²×C₇	—	2	8	no	—
216	C₄²×C₇	—	2	4	no	—
217	C₄²×C₇	C₂×D₄	1	2	yes	—
218	C₄²×C₇	C₂⁴	1	2	yes	—
219	C₂³×D₄	—	2	1	no	—
220	C₂³×D₄	C₂²×C₇	1	2	yes	—
221	C₂²×SD₁₆	C₂²×C₇	1	1	yes	characteristic
222	C₂²×SD₁₆	C₂²×C₇	1	2	yes	complemented
223	C₂²×M₁₆	C₂²×C₇	1	2	yes	complemented
224	#556	—	2	4	no	—
225	#556	C₂²×C₇	1	8	yes	complemented
226	D₄²	—	2	4	no	—
227	D₄²	C₂²×C₇	1	8	yes	—
228	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
229	C₂×#127	C₂²×C₇	1	1	yes	characteristic
230	C₂×#126	—	2	2	no	—
231	C₂×#124	—	2	1	no	—
232	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	complemented
233	C₂×#121	—	2	1	no	—
234	C₂×#121	C₂²×C₇	1	2	yes	—
235	C₂×#120	—	2	2	no	—
236	C₂×C₄×Q₈	—	2	1	no	—
237	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
238	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
239	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
240	C₂×#117	C₂²×C₇	1	2	yes	complemented
241	#486	C₂²×C₇	1	8	yes	complemented
242	#482	C₂²×C₇	1	16	yes	complemented
243	#474	C₂²×C₇	1	8	yes	complemented
244	#468	C₂²×C₇	1	16	yes	complemented
245	#460	C₂²×C₇	1	8	yes	complemented
246	#450	C₂²×C₇	1	16	yes	complemented
247	C₄×SD₁₆	C₂²×C₇	1	8	yes	complemented
248	#436	C₂²×C₇	1	8	yes	complemented
249	C₂×#107	C₂²×C₇	1	2	yes	complemented
250	C₂×#106	C₂²×C₇	1	1	yes	characteristic, complemented
251	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic, complemented
252	#418	C₂²×C₇	1	16	yes	complemented
253	C₂×#103	C₂²×C₇	1	1	yes	characteristic, complemented
254	C₂×#102	C₂²×C₇	1	1	yes	characteristic
255	C₂×#102	C₂²×C₇	1	2	yes	—
256	C₂×#102	C₂²×C₇	1	2	yes	complemented
257	C₂×#102	C₂²×C₇	1	2	yes	complemented
258	C₂×#102	C₂²×C₇	1	2	yes	complemented
259	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	complemented
260	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic, complemented
261	C₂³×C₇	—	2	4	no	—
262	C₂³×C₇	—	2	4	no	—
263	C₂³×C₇	—	2	4	no	—
264	C₂³×C₇	—	2	4	no	—
265	C₂³×C₇	—	2	2	no	—
266	C₂³×C₇	—	2	4	no	—
267	C₂³×C₇	—	4	16	no	—
268	C₂³×C₇	—	4	16	no	—
269	C₂³×C₇	—	4	2	no	—
270	C₂³×C₇	—	4	2	no	—
271	C₂³×C₇	—	4	4	no	—
272	C₂³×C₇	—	4	4	no	—
273	C₂³×C₇	—	4	4	no	—
274	C₂³×C₇	—	8	8	no	—
275	C₂³×C₇	—	8	32	no	—
276	C₂³×C₇	C₂²×D₄	1	4	yes	—
277	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
278	C₇×Q₈	—	2	2	no	—
279	C₇×Q₈	—	2	2	no	—
280	C₇×Q₈	—	4	2	no	—
281	C₇×Q₈	—	4	8	no	—
282	C₇×D₄	—	2	4	no	—
283	C₇×D₄	—	2	8	no	—
284	C₇×D₄	—	2	4	no	—
285	C₇×D₄	—	2	2	no	—
286	C₇×D₄	—	2	4	no	—
287	C₇×D₄	—	4	16	no	—
288	C₇×D₄	—	4	16	no	—
289	C₇×D₄	—	4	4	no	—
290	C₇×D₄	—	4	8	no	—
291	C₇×D₄	—	8	4	no	—
292	C₇×D₄	C₂×#41	1	4	yes	—
293	C₂×C₄×C₇	—	2	4	no	—
294	C₂×C₄×C₇	—	2	2	no	—
295	C₂×C₄×C₇	—	2	8	no	—
296	C₂×C₄×C₇	—	2	4	no	—
297	C₂×C₄×C₇	—	2	4	no	—
298	C₂×C₄×C₇	—	2	2	no	—
299	C₂×C₄×C₇	—	2	2	no	—
300	C₂×C₄×C₇	—	2	4	no	—
301	C₂×C₄×C₇	—	2	2	no	—
302	C₂×C₄×C₇	—	4	2	no	—
303	C₂×C₄×C₇	—	4	2	no	—
304	C₂×C₄×C₇	—	4	4	no	—
305	C₂×C₄×C₇	—	4	2	no	—
306	C₂×C₄×C₇	—	4	8	no	—
307	C₂×C₄×C₇	AGL(1,ℤ/8)	1	4	yes	—
308	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
309	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
310	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
311	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
312	C₂×C₄×C₇	C₂×#41	1	2	yes	—
313	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
314	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
315	C₇×C₈	—	2	2	no	—
316	C₇×C₈	—	2	2	no	—
317	C₇×C₈	—	2	4	no	—
318	C₂⁵	—	2	2	no	—
319	C₂²×Q₈	—	2	1	no	—
320	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
321	C₂²×D₄	—	2	8	no	—
322	C₂²×D₄	—	2	2	no	—
323	C₂²×D₄	—	2	8	no	—
324	C₂²×D₄	—	2	2	no	—
325	C₂²×D₄	—	2	1	no	—
326	C₂²×D₄	—	2	4	no	—
327	C₂²×D₄	—	2	1	no	—
328	C₂²×D₄	—	4	4	no	—
329	C₂²×D₄	—	4	8	no	—
330	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
331	C₂²×D₄	C₂³×C₇	1	2	yes	—
332	C₂²×D₄	C₂³×C₇	1	8	yes	—
333	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
334	C₂³×C₄	—	2	1	no	—
335	C₂³×C₄	C₂³×C₇	1	2	yes	—
336	C₂×SD₁₆	—	2	4	no	—
337	C₂×SD₁₆	—	4	4	no	—
338	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
339	C₂×SD₁₆	C₂³×C₇	1	8	yes	complemented
340	C₂×M₁₆	—	4	4	no	—
341	C₂×M₁₆	C₂³×C₇	1	8	yes	complemented
342	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
343	C₂²×C₈	C₂³×C₇	1	2	yes	—
344	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
345	C₄⋊Q₈	—	2	2	no	—
346	C₄⋊Q₈	C₂³×C₇	1	4	yes	complemented
347	#127	—	2	2	no	—
348	#127	C₂³×C₇	1	4	yes	—
349	#126	—	2	16	no	—
350	#124	—	2	8	no	—
351	C₂²⋊Q₈	—	2	4	no	—
352	C₂²⋊Q₈	C₂³×C₇	1	8	yes	complemented
353	#121	—	2	4	no	—
354	#121	—	2	8	no	—
355	#121	C₂³×C₇	1	8	yes	—
356	#120	—	2	16	no	—
357	C₄×Q₈	—	2	8	no	—
358	C₄×Q₈	—	2	2	no	—
359	C₄×Q₈	C₂³×C₇	1	4	yes	complemented
360	C₄×D₄	—	2	2	no	—
361	C₄×D₄	C₇×D₄	1	4	yes	—
362	C₄×D₄	C₂³×C₇	1	4	yes	—
363	C₄×D₄	C₂³×C₇	1	4	yes	—
364	#117	—	2	4	no	—
365	#117	C₂³×C₇	1	8	yes	complemented
366	C₂×C₄⋊C₄	—	2	2	no	—
367	C₂×C₄⋊C₄	—	2	1	no	—
368	C₂×C₄⋊C₄	—	2	1	no	—
369	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
370	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
371	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
372	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
373	C₂×C₂²⋊C₄	—	2	2	no	—
374	C₂×C₂²⋊C₄	—	2	2	no	—
375	C₂×C₂²⋊C₄	—	2	1	no	—
376	C₂×C₂²⋊C₄	—	2	2	no	—
377	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
378	C₂×C₄²	—	2	2	no	—
379	C₂×C₄²	—	2	1	no	—
380	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
381	#107	C₂³×C₇	1	8	yes	complemented
382	#106	C₂³×C₇	1	4	yes	complemented
383	C₄⋊C₈	C₂³×C₇	1	4	yes	complemented
384	#103	C₂³×C₇	1	4	yes	complemented
385	#102	C₂³×C₇	1	4	yes	—
386	#102	C₂³×C₇	1	8	yes	—
387	#102	C₂³×C₇	1	8	yes	complemented
388	#102	C₂³×C₇	1	8	yes	complemented
389	#102	C₂³×C₇	1	8	yes	complemented
390	C₂²⋊C₈	C₂³×C₇	1	8	yes	complemented
391	C₄×C₈	C₂³×C₇	1	4	yes	complemented
392	C₂²×C₇	—	2	4	no	—
393	C₂²×C₇	—	2	4	no	—
394	C₂²×C₇	—	2	4	no	—
395	C₂²×C₇	—	2	4	no	—
396	C₂²×C₇	—	4	2	no	—
397	C₂²×C₇	—	4	8	no	—
398	C₂²×C₇	—	4	8	no	—
399	C₂²×C₇	—	4	2	no	—
400	C₂²×C₇	—	4	4	no	—
401	C₂²×C₇	—	4	4	no	—
402	C₂²×C₇	—	4	4	no	—
403	C₂²×C₇	—	8	32	no	—
404	C₂²×C₇	—	8	8	no	—
405	C₂²×C₇	#546	1	2	yes	—
406	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
407	C₂²×C₇	#577	1	2	yes	—
408	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
409	C₄×C₇	—	2	4	no	—
410	C₄×C₇	—	2	2	no	—
411	C₄×C₇	—	4	2	no	—
412	C₄×C₇	—	4	2	no	—
413	C₄×C₇	—	4	4	no	—
414	C₄×C₇	—	4	2	no	—
415	C₄×C₇	#546	1	2	yes	—
416	C₄×C₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
417	C₄×C₇	C₂²×#41	1	1	yes	characteristic
418	C₄×C₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
419	C₂⁴	—	2	1	no	—
420	C₂⁴	—	2	2	no	—
421	C₂⁴	—	2	8	no	—
422	C₂⁴	—	4	8	no	—
423	C₂⁴	—	4	8	no	—
424	C₂⁴	—	8	4	no	—
425	C₂⁴	C₂×C₇×D₄	1	2	yes	—
426	C₂×Q₈	—	2	1	no	—
427	C₂×Q₈	—	2	4	no	—
428	C₂×Q₈	—	2	2	no	—
429	C₂×Q₈	—	2	2	no	—
430	C₂×Q₈	—	4	4	no	—
431	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
432	C₂×D₄	—	2	16	no	—
433	C₂×D₄	—	2	8	no	—
434	C₂×D₄	—	2	4	no	—
435	C₂×D₄	—	2	1	no	—
436	C₂×D₄	—	2	8	no	—
437	C₂×D₄	—	2	2	no	—
438	C₂×D₄	—	2	4	no	—
439	C₂×D₄	—	2	4	no	—
440	C₂×D₄	—	2	4	no	—
441	C₂×D₄	—	2	2	no	—
442	C₂×D₄	—	2	4	no	—
443	C₂×D₄	—	4	8	no	—
444	C₂×D₄	—	4	16	no	—
445	C₂×D₄	—	4	8	no	—
446	C₂×D₄	—	4	32	no	—
447	C₂×D₄	—	4	8	no	—
448	C₂×D₄	—	4	4	no	—
449	C₂×D₄	—	8	2	no	—
450	C₂×D₄	C₂×C₇×D₄	1	4	yes	—
451	C₂×D₄	C₇×#41	1	4	yes	—
452	C₂×D₄	C₂⁴×C₇	1	4	yes	—
453	C₂×D₄	C₂⁴×C₇	1	2	yes	—
454	C₂²×C₄	—	2	4	no	—
455	C₂²×C₄	—	2	4	no	—
456	C₂²×C₄	—	2	1	no	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	2	2	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	4	no	—
461	C₂²×C₄	—	4	4	no	—
462	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
463	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
464	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
465	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
466	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
467	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
468	SD₁₆	—	2	8	no	—
469	SD₁₆	—	4	8	no	—
470	M₁₆	—	4	8	no	—
471	C₂×C₈	—	2	2	no	—
472	C₂×C₈	—	2	2	no	—
473	C₂×C₈	—	2	4	no	—
474	C₂×C₈	C₂⁴×C₇	1	2	yes	—
475	C₂×C₈	C₂⁴×C₇	1	2	yes	—
476	C₂×C₈	C₂⁴×C₇	1	4	yes	—
477	C₄⋊C₄	—	2	8	no	—
478	C₄⋊C₄	—	2	4	no	—
479	C₄⋊C₄	—	2	4	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
484	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
485	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
486	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
487	C₂²⋊C₄	—	2	8	no	—
488	C₂²⋊C₄	—	2	8	no	—
489	C₂²⋊C₄	—	2	4	no	—
490	C₂²⋊C₄	—	2	8	no	—
491	C₂²⋊C₄	C₂×C₇×D₄	1	8	yes	—
492	C₄²	—	2	8	no	—
493	C₄²	—	2	4	no	—
494	C₄²	C₂×C₇×D₄	1	2	yes	—
495	C₄²	C₂⁴×C₇	1	2	yes	—
496	C₂×C₇	—	4	4	no	—
497	C₂×C₇	—	4	4	no	—
498	C₂×C₇	—	8	4	no	—
499	C₂×C₇	#3324	1	4	yes	direct product factor
500	C₂×C₇	C₂×#546	1	1	yes	characteristic
501	C₂×C₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
502	C₂×C₇	C₂×#577	1	1	yes	characteristic
503	C₂³	—	2	2	no	—
504	C₂³	—	2	4	no	—
505	C₂³	—	2	4	no	—
506	C₂³	—	2	4	no	—
507	C₂³	—	2	4	no	—
508	C₂³	—	2	4	no	—
509	C₂³	—	4	16	no	—
510	C₂³	—	4	16	no	—
511	C₂³	—	4	2	no	—
512	C₂³	—	4	2	no	—
513	C₂³	—	4	4	no	—
514	C₂³	—	4	4	no	—
515	C₂³	—	4	4	no	—
516	C₂³	—	8	8	no	—
517	C₂³	—	8	32	no	—
518	C₂³	C₂²×C₇×D₄	1	4	yes	—
519	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
520	Q₈	—	2	2	no	—
521	Q₈	—	2	2	no	—
522	Q₈	—	4	2	no	—
523	Q₈	—	4	8	no	—
524	D₄	—	2	4	no	—
525	D₄	—	2	2	no	—
526	D₄	—	2	4	no	—
527	D₄	—	2	4	no	—
528	D₄	—	2	8	no	—
529	D₄	—	4	16	no	—
530	D₄	—	4	16	no	—
531	D₄	—	4	4	no	—
532	D₄	—	4	8	no	—
533	D₄	—	8	4	no	—
534	D₄	C₂×C₇×#41	1	4	yes	—
535	C₂×C₄	—	2	4	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	8	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	4	2	no	—
545	C₂×C₄	—	4	2	no	—
546	C₂×C₄	—	4	4	no	—
547	C₂×C₄	—	4	2	no	—
548	C₂×C₄	—	4	8	no	—
549	C₂×C₄	C₇×AGL(1,ℤ/8)	1	4	yes	—
550	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
551	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
552	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
553	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
554	C₂×C₄	C₂×C₇×#41	1	2	yes	—
555	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
556	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
557	C₈	—	2	2	no	—
558	C₈	—	2	2	no	—
559	C₈	—	2	4	no	—
560	C₇	C₂×#3324	1	1	yes	characteristic, direct product factor, Sylow
561	C₂²	—	2	4	no	—
562	C₂²	—	2	4	no	—
563	C₂²	—	2	4	no	—
564	C₂²	—	2	4	no	—
565	C₂²	—	4	2	no	—
566	C₂²	—	4	2	no	—
567	C₂²	—	4	8	no	—
568	C₂²	—	4	8	no	—
569	C₂²	—	4	4	no	—
570	C₂²	—	4	4	no	—
571	C₂²	—	4	4	no	—
572	C₂²	—	8	32	no	—
573	C₂²	—	8	8	no	—
574	C₂²	C₇×#546	1	2	yes	—
575	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	2	yes	—
576	C₂²	C₇×#577	1	2	yes	—
577	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
578	C₄	—	2	2	no	—
579	C₄	—	2	4	no	—
580	C₄	—	4	2	no	—
581	C₄	—	4	4	no	—
582	C₄	—	4	2	no	—
583	C₄	—	4	2	no	—
584	C₄	C₇×#546	1	2	yes	—
585	C₄	C₂×C₇×AGL(1,ℤ/8)	1	4	yes	—
586	C₄	C₂²×C₇×#41	1	1	yes	characteristic
587	C₄	C₂×C₇×H(5,𝔽₂)	1	1	yes	characteristic
588	C₂	—	4	4	no	—
589	C₂	—	4	4	no	—
590	C₂	—	8	4	no	—
591	C₂	C₇×#3324	1	4	yes	direct product factor
592	C₂	C₂×C₇×#546	1	1	yes	characteristic
593	C₂	C₂²×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
594	C₂	C₂×C₇×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₂×C₇×D₄²	C₂
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. C₂×C₇×#556	C₂	semidirect
3. C₇×#3324	C₂	direct (N × H)
4. C₂×C₇×#486	C₂	semidirect
5. C₂×C₇×#482	C₂	semidirect
6. C₂×C₇×#474	C₂	semidirect
7. C₂×C₇×#468	C₂	semidirect
8. C₂×C₇×#460	C₂	semidirect
9. C₂×C₇×#450	C₂	semidirect
10. C₂×C₄×C₇×SD₁₆	C₂	semidirect
11. C₂×C₇×#436	C₂	semidirect
12. C₂×C₇×#418	C₂	semidirect
16. C₂²×C₇×SD₁₆	C₂²	semidirect
17. C₂²×C₇×M₁₆	C₂²	semidirect
19. C₇×#556	C₂²	semidirect
22. C₂×C₇×C₄⋊Q₈	C₂²	semidirect
26. C₂×C₇×C₂²⋊Q₈	C₂²	semidirect
31. C₂×C₄×C₇×Q₈	C₂²	semidirect
34. C₂×C₇×#117	C₂²	semidirect
35. C₇×#486	C₂²	semidirect
36. C₇×#482	C₂²	semidirect
37. C₇×#474	C₂²	semidirect
38. C₇×#468	C₂²	semidirect
39. C₇×#460	C₂²	semidirect
40. C₇×#450	C₂²	semidirect
41. C₄×C₇×SD₁₆	C₂²	semidirect
42. C₇×#436	C₂²	semidirect
43. C₂×C₇×#107	C₂²	semidirect
44. C₂×C₇×#106	C₂²	semidirect
45. C₂×C₇×C₄⋊C₈	C₂²	semidirect
46. C₇×#418	C₂²	semidirect
47. C₂×C₇×#103	C₂²	semidirect
50. C₂×C₇×#102	C₂²	semidirect
51. C₂×C₇×#102	C₂²	semidirect
52. C₂×C₇×#102	C₂²	semidirect
53. C₂×C₇×C₂²⋊C₈	C₂²	semidirect
54. C₂×C₄×C₇×C₈	C₂²	semidirect
55. C₂×#3324	C₇	direct (N × H)
77. C₂×C₇×SD₁₆	C₂³	semidirect
79. C₂×C₇×M₁₆	C₂³	semidirect
84. C₇×C₄⋊Q₈	C₂³	semidirect
90. C₇×C₂²⋊Q₈	C₂³	semidirect
97. C₄×C₇×Q₈	C₂³	semidirect
103. C₇×#117	C₂³	semidirect
119. C₇×#107	C₂³	semidirect
120. C₇×#106	C₂³	semidirect
121. C₇×C₄⋊C₈	C₂³	semidirect
122. C₇×#103	C₂³	semidirect
125. C₇×#102	C₂³	semidirect
126. C₇×#102	C₂³	semidirect
127. C₇×#102	C₂³	semidirect
128. C₇×C₂²⋊C₈	C₂³	semidirect
129. C₄×C₇×C₈	C₂³	semidirect
130. C₂×#556	C₂×C₇	semidirect
132. #3324	C₂×C₇	direct (N × H)
133. C₂×#486	C₂×C₇	semidirect
134. C₂×#482	C₂×C₇	semidirect
135. C₂×#474	C₂×C₇	semidirect
136. C₂×#468	C₂×C₇	semidirect
137. C₂×#460	C₂×C₇	semidirect
138. C₂×#450	C₂×C₇	semidirect
139. C₂×C₄×SD₁₆	C₂×C₇	semidirect
140. C₂×#436	C₂×C₇	semidirect
141. C₂×#418	C₂×C₇	semidirect
222. C₂²×SD₁₆	C₂²×C₇	semidirect
223. C₂²×M₁₆	C₂²×C₇	semidirect
225. #556	C₂²×C₇	semidirect
228. C₂×C₄⋊Q₈	C₂²×C₇	semidirect
232. C₂×C₂²⋊Q₈	C₂²×C₇	semidirect
237. C₂×C₄×Q₈	C₂²×C₇	semidirect
240. C₂×#117	C₂²×C₇	semidirect
241. #486	C₂²×C₇	semidirect
242. #482	C₂²×C₇	semidirect
243. #474	C₂²×C₇	semidirect
244. #468	C₂²×C₇	semidirect
245. #460	C₂²×C₇	semidirect
246. #450	C₂²×C₇	semidirect
247. C₄×SD₁₆	C₂²×C₇	semidirect
248. #436	C₂²×C₇	semidirect
249. C₂×#107	C₂²×C₇	semidirect
250. C₂×#106	C₂²×C₇	semidirect
251. C₂×C₄⋊C₈	C₂²×C₇	semidirect
252. #418	C₂²×C₇	semidirect
253. C₂×#103	C₂²×C₇	semidirect
256. C₂×#102	C₂²×C₇	semidirect
257. C₂×#102	C₂²×C₇	semidirect
258. C₂×#102	C₂²×C₇	semidirect
259. C₂×C₂²⋊C₈	C₂²×C₇	semidirect
260. C₂×C₄×C₈	C₂²×C₇	semidirect
339. C₂×SD₁₆	C₂³×C₇	semidirect
341. C₂×M₁₆	C₂³×C₇	semidirect
346. C₄⋊Q₈	C₂³×C₇	semidirect
352. C₂²⋊Q₈	C₂³×C₇	semidirect
359. C₄×Q₈	C₂³×C₇	semidirect
365. #117	C₂³×C₇	semidirect
381. #107	C₂³×C₇	semidirect
382. #106	C₂³×C₇	semidirect
383. C₄⋊C₈	C₂³×C₇	semidirect
384. #103	C₂³×C₇	semidirect
387. #102	C₂³×C₇	semidirect
388. #102	C₂³×C₇	semidirect
389. #102	C₂³×C₇	semidirect
390. C₂²⋊C₈	C₂³×C₇	semidirect
391. C₄×C₈	C₂³×C₇	semidirect
499. C₂×C₇	#3324	direct (N × H)
560. C₇	C₂×#3324	direct (N × H)
591. C₂	C₇×#3324	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#556	2
C₂×C₇×D₄²	2
C₇×#3324	2
C₂×C₇×#486	2
C₂×C₇×#482	2
C₂×C₇×#474	2
C₂×C₇×#468	2
C₂×C₇×#460	2
C₂×C₇×#450	2
C₂×C₄×C₇×SD₁₆	2
C₂×C₇×#436	2
C₂×C₇×#418	2
C₂×#3324	7

Subgroups — C2×C7×#3324 #13238360

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#3324 #13238360