Subgroups — C₇×#61370 #13238586

Order 1792.

1818 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3501	C₂	1	2	yes	maximal, complemented
2	C₇×#3389	C₂	1	4	yes	maximal, complemented
3	C₇×#3384	C₂	1	4	yes	maximal, complemented
4	C₇×#3380	C₂	1	2	yes	maximal, complemented
5	C₇×#3380	C₂	1	4	yes	maximal, complemented
6	C₇×Q₈×D₈	C₂	1	2	yes	maximal, complemented
7	C₇×#3164	C₂	1	2	yes	maximal, complemented
8	C₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#3081	C₂	1	4	yes	maximal, complemented
10	C₇×#3077	C₂	1	2	yes	maximal, complemented
11	C₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#2948	C₂	1	2	yes	maximal
13	C₂×C₄×C₇×D₈	C₂	1	1	yes	maximal, characteristic
14	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
15	C₇×#554	—	2	2	no	—
16	C₇×#554	—	2	4	no	—
17	C₇×#554	C₂²	1	4	yes	complemented
18	C₇×#554	C₂²	1	8	yes	complemented
19	C₇×D₄×Q₈	—	2	2	no	—
20	C₇×D₄×Q₈	C₂²	1	4	yes	complemented
21	C₇×#544	C₂²	1	4	yes	—
22	C₇×#543	—	2	4	no	—
23	C₇×#533	C₂²	1	2	yes	—
24	C₇×#533	C₂²	1	1	yes	characteristic
25	C₂×C₇×C₂²⋊Q₈	—	2	4	no	—
26	C₇×#518	C₂²	1	4	yes	—
27	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
28	C₇×#501	C₂²	1	8	yes	complemented
29	C₇×#500	C₂²	1	1	yes	characteristic, complemented
30	C₇×#500	C₂²	1	1	yes	characteristic, complemented
31	C₇×#500	C₂²	1	4	yes	complemented
32	C₇×#499	C₂²	1	2	yes	complemented
33	C₇×#499	C₂²	1	2	yes	complemented
34	C₇×#499	C₂²	1	4	yes	complemented
35	C₇×#498	C₂²	1	2	yes	complemented
36	C₇×#478	C₂²	1	8	yes	complemented
37	C₇×#478	C₂²	1	8	yes	complemented
38	C₇×#478	C₂²	1	8	yes	complemented
39	C₇×#476	C₂²	1	4	yes	complemented
40	C₇×#476	C₂²	1	8	yes	complemented
41	C₇×#474	C₂²	1	4	yes	complemented
42	C₇×#474	C₂²	1	8	yes	complemented
43	C₇×#446	C₂²	1	2	yes	complemented
44	C₇×#446	C₂²	1	2	yes	complemented
45	C₇×C₈×Q₈	C₂²	1	2	yes	complemented
46	C₇×C₈×Q₈	C₂²	1	2	yes	complemented
47	C₇×#442	C₂²	1	8	yes	—
48	C₄×C₇×D₈	C₂²	1	2	yes	—
49	C₄×C₇×D₈	C₂²	1	2	yes	—
50	C₄×C₇×D₈	C₂²	1	2	yes	—
51	C₄×C₇×D₈	C₂²	1	2	yes	—
52	C₄×C₇×D₈	C₂²	1	4	yes	—
53	C₄×C₇×D₈	C₂²	1	4	yes	—
54	C₇×#428	C₂²	1	4	yes	—
55	C₇×#427	C₂²	1	2	yes	—
56	C₇×#427	C₂²	1	2	yes	—
57	C₂×C₇×#107	C₂²	1	1	yes	characteristic
58	C₇×#424	C₂²	1	2	yes	—
59	C₇×#423	C₂²	1	1	yes	characteristic
60	C₇×#418	C₂²	1	4	yes	—
61	C₂×C₇×#102	C₂²	1	2	yes	—
62	C₇×#405	C₂²	1	2	yes	—
63	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
64	#61370	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂²×C₇×Q₈	—	2	4	no	—
66	C₂²×C₇×D₄	C₂³	1	2	yes	—
67	C₂³×C₄×C₇	—	2	2	no	—
68	C₂×C₇×D₈	Q₈	1	8	yes	complemented
69	C₂×C₇×D₈	C₂³	1	2	yes	—
70	C₂×C₇×D₈	C₂³	1	2	yes	—
71	C₂×C₇×M₁₆	C₂³	1	2	yes	—
72	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
73	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
74	C₇×C₄⋊Q₈	—	2	4	no	—
75	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
76	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
77	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
78	C₇×C₄⋊Q₈	C₂³	1	8	yes	—
79	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
80	C₇×#125	—	2	4	no	—
81	C₇×#125	D₄	1	2	yes	—
82	C₇×#125	C₂³	1	4	yes	—
83	C₇×#125	C₂³	1	8	yes	—
84	C₇×C₂²⋊Q₈	—	2	4	no	—
85	C₇×C₂²⋊Q₈	—	2	8	no	—
86	C₇×C₂²⋊Q₈	—	2	2	no	—
87	C₇×C₂²⋊Q₈	—	2	2	no	—
88	C₇×C₂²⋊Q₈	—	2	4	no	—
89	C₇×C₂²⋊Q₈	—	2	8	no	—
90	C₇×C₂²⋊Q₈	—	2	8	no	—
91	C₇×C₂²⋊Q₈	—	2	8	no	—
92	C₇×C₂²⋊Q₈	—	2	8	no	—
93	C₇×C₂²⋊Q₈	—	2	8	no	—
94	C₇×C₂²⋊Q₈	D₄	1	2	yes	—
95	C₇×C₂²⋊Q₈	D₄	1	2	yes	—
96	C₄×C₇×Q₈	—	2	2	no	—
97	C₄×C₇×Q₈	—	2	2	no	—
98	C₄×C₇×Q₈	C₂³	1	2	yes	—
99	C₄×C₇×Q₈	C₂³	1	2	yes	—
100	C₄×C₇×D₄	—	2	4	no	—
101	C₄×C₇×D₄	—	2	4	no	—
102	C₄×C₇×D₄	—	2	2	no	—
103	C₄×C₇×D₄	—	2	2	no	—
104	C₄×C₇×D₄	C₂³	1	4	yes	—
105	C₄×C₇×D₄	C₂³	1	8	yes	—
106	C₄×C₇×D₄	C₂³	1	8	yes	—
107	C₄×C₇×D₄	C₂³	1	4	yes	—
108	C₇×#117	C₂³	1	4	yes	—
109	C₇×#117	C₂³	1	2	yes	—
110	C₂×C₇×C₄⋊C₄	—	2	8	no	—
111	C₂×C₇×C₄⋊C₄	—	2	4	no	—
112	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
113	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
114	C₂×C₇×C₂²⋊C₄	—	2	8	no	—
115	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
116	C₇×#107	C₂³	1	2	yes	—
117	C₇×#107	C₂³	1	8	yes	—
118	C₇×#107	C₂³	1	2	yes	—
119	C₇×#107	C₂³	1	2	yes	—
120	C₇×#107	C₂³	1	2	yes	—
121	C₇×#107	C₂³	1	2	yes	—
122	C₇×#107	C₂³	1	2	yes	—
123	C₇×#106	C₂³	1	4	yes	—
124	C₇×#106	C₂³	1	8	yes	—
125	C₇×#106	C₂³	1	4	yes	—
126	C₇×C₄⋊C₈	C₂³	1	2	yes	—
127	C₇×C₄⋊C₈	C₂³	1	4	yes	—
128	C₇×C₄⋊C₈	C₂³	1	4	yes	—
129	C₇×C₄⋊C₈	C₂³	1	2	yes	—
130	C₇×#102	C₂³	1	8	yes	—
131	C₇×#102	C₂³	1	4	yes	—
132	C₇×#102	C₂³	1	4	yes	—
133	C₇×#102	C₂³	1	8	yes	—
134	C₇×#97	C₂³	1	4	yes	—
135	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
136	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
137	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
138	C₄×C₇×C₈	C₂³	1	4	yes	—
139	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
140	#3501	C₂×C₇	1	2	yes	complemented
141	#3389	C₂×C₇	1	4	yes	complemented
142	#3384	C₂×C₇	1	4	yes	complemented
143	#3380	C₂×C₇	1	2	yes	complemented
144	#3380	C₂×C₇	1	4	yes	complemented
145	Q₈×D₈	C₂×C₇	1	2	yes	complemented
146	#3164	C₂×C₇	1	2	yes	complemented
147	#3160	C₂×C₇	1	1	yes	characteristic, complemented
148	#3081	C₂×C₇	1	4	yes	complemented
149	#3077	C₂×C₇	1	2	yes	complemented
150	#2960	C₂×C₇	1	1	yes	characteristic, complemented
151	#2948	C₂×C₇	1	2	yes	—
152	C₂×C₄×D₈	C₂×C₇	1	1	yes	characteristic
153	C₂⁴×C₇	—	2	2	no	—
154	C₂×C₇×Q₈	—	2	2	no	—
155	C₂×C₇×Q₈	—	2	2	no	—
156	C₂×C₇×Q₈	—	2	8	no	—
157	C₂×C₇×Q₈	—	8	8	no	—
158	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
159	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
160	C₂×C₇×D₄	—	2	2	no	—
161	C₂×C₇×D₄	—	2	4	no	—
162	C₂×C₇×D₄	C₂×Q₈	1	8	yes	complemented
163	C₂×C₇×D₄	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₇	—	2	2	no	—
165	C₂²×C₄×C₇	—	2	4	no	—
166	C₂²×C₄×C₇	—	2	8	no	—
167	C₂²×C₄×C₇	—	2	8	no	—
168	C₂²×C₄×C₇	—	2	4	no	—
169	C₂²×C₄×C₇	—	8	4	no	—
170	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
173	C₇×D₈	C₂×Q₈	1	8	yes	complemented
174	C₇×D₈	C₂×Q₈	1	8	yes	complemented
175	C₇×M₁₆	—	2	4	no	—
176	C₂×C₇×C₈	—	2	2	no	—
177	C₂×C₇×C₈	C₂×Q₈	1	4	yes	complemented
178	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
179	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
180	C₂×C₇×C₈	C₂⁴	1	4	yes	—
181	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
182	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
183	C₇×C₄⋊C₄	—	2	4	no	—
184	C₇×C₄⋊C₄	—	2	2	no	—
185	C₇×C₄⋊C₄	—	2	8	no	—
186	C₇×C₄⋊C₄	—	2	8	no	—
187	C₇×C₄⋊C₄	—	2	8	no	—
188	C₇×C₄⋊C₄	—	2	2	no	—
189	C₇×C₄⋊C₄	—	2	4	no	—
190	C₇×C₄⋊C₄	—	2	2	no	—
191	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
192	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
193	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
194	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
195	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
196	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
197	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
198	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
199	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
200	C₇×C₂²⋊C₄	—	2	4	no	—
201	C₇×C₂²⋊C₄	—	2	8	no	—
202	C₇×C₂²⋊C₄	—	2	8	no	—
203	C₇×C₂²⋊C₄	—	2	4	no	—
204	C₇×C₂²⋊C₄	—	2	4	no	—
205	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
206	C₄²×C₇	—	2	4	no	—
207	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
208	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
209	C₄²×C₇	C₂⁴	1	1	yes	characteristic
210	C₄²×C₇	C₂⁴	1	1	yes	characteristic
211	C₄²×C₇	C₂⁴	1	4	yes	—
212	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
213	#554	—	2	2	no	—
214	#554	—	2	4	no	—
215	#554	C₂²×C₇	1	4	yes	complemented
216	#554	C₂²×C₇	1	8	yes	complemented
217	D₄×Q₈	—	2	2	no	—
218	D₄×Q₈	C₂²×C₇	1	4	yes	complemented
219	#544	C₂²×C₇	1	4	yes	—
220	#543	—	2	4	no	—
221	#533	C₂²×C₇	1	2	yes	—
222	#533	C₂²×C₇	1	1	yes	characteristic
223	C₂×C₂²⋊Q₈	—	2	4	no	—
224	#518	C₂²×C₇	1	4	yes	—
225	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
226	#501	C₂²×C₇	1	8	yes	complemented
227	#500	C₂²×C₇	1	1	yes	characteristic, complemented
228	#500	C₂²×C₇	1	4	yes	complemented
229	#500	C₂²×C₇	1	1	yes	characteristic, complemented
230	#499	C₂²×C₇	1	2	yes	complemented
231	#499	C₂²×C₇	1	4	yes	complemented
232	#499	C₂²×C₇	1	2	yes	complemented
233	#498	C₂²×C₇	1	2	yes	complemented
234	#478	C₂²×C₇	1	8	yes	complemented
235	#478	C₂²×C₇	1	8	yes	complemented
236	#478	C₂²×C₇	1	8	yes	complemented
237	#476	C₂²×C₇	1	4	yes	complemented
238	#476	C₂²×C₇	1	8	yes	complemented
239	#474	C₂²×C₇	1	4	yes	complemented
240	#474	C₂²×C₇	1	8	yes	complemented
241	#446	C₂²×C₇	1	2	yes	complemented
242	#446	C₂²×C₇	1	2	yes	complemented
243	C₈×Q₈	C₂²×C₇	1	2	yes	complemented
244	C₈×Q₈	C₂²×C₇	1	2	yes	complemented
245	#442	C₂²×C₇	1	8	yes	—
246	C₄×D₈	C₂²×C₇	1	2	yes	—
247	C₄×D₈	C₂²×C₇	1	4	yes	—
248	C₄×D₈	C₂²×C₇	1	2	yes	—
249	C₄×D₈	C₂²×C₇	1	4	yes	—
250	C₄×D₈	C₂²×C₇	1	2	yes	—
251	C₄×D₈	C₂²×C₇	1	2	yes	—
252	#428	C₂²×C₇	1	4	yes	—
253	#427	C₂²×C₇	1	2	yes	—
254	#427	C₂²×C₇	1	2	yes	—
255	C₂×#107	C₂²×C₇	1	1	yes	characteristic
256	#424	C₂²×C₇	1	2	yes	—
257	#423	C₂²×C₇	1	1	yes	characteristic
258	#418	C₂²×C₇	1	4	yes	—
259	C₂×#102	C₂²×C₇	1	2	yes	—
260	#405	C₂²×C₇	1	2	yes	—
261	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
262	C₂³×C₇	—	2	4	no	—
263	C₂³×C₇	—	2	8	no	—
264	C₂³×C₇	—	8	2	no	—
265	C₂³×C₇	—	8	2	no	—
266	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
267	C₇×Q₈	—	2	4	no	—
268	C₇×Q₈	—	4	2	no	—
269	C₇×Q₈	—	4	4	no	—
270	C₇×Q₈	—	8	2	no	—
271	C₇×Q₈	—	8	8	no	—
272	C₇×D₄	—	2	8	no	—
273	C₇×D₄	—	2	4	no	—
274	C₇×D₄	C₂²×Q₈	1	8	yes	—
275	C₂×C₄×C₇	—	2	2	no	—
276	C₂×C₄×C₇	—	2	4	no	—
277	C₂×C₄×C₇	—	2	4	no	—
278	C₂×C₄×C₇	—	2	4	no	—
279	C₂×C₄×C₇	—	4	1	no	—
280	C₂×C₄×C₇	—	4	8	no	—
281	C₂×C₄×C₇	—	8	8	no	—
282	C₂×C₄×C₇	—	8	4	no	—
283	C₂×C₄×C₇	—	8	8	no	—
284	C₂×C₄×C₇	—	8	2	no	—
285	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
286	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
287	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
288	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
289	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
290	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
291	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
292	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
293	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
294	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
295	C₇×C₈	—	2	1	no	—
296	C₇×C₈	—	2	1	no	—
297	C₇×C₈	—	2	4	no	—
298	C₇×C₈	C₂²×Q₈	1	2	yes	—
299	C₇×C₈	C₂²×Q₈	1	2	yes	—
300	C₂²×Q₈	—	2	4	no	—
301	C₂²×D₄	C₂³×C₇	1	2	yes	—
302	C₂³×C₄	—	2	2	no	—
303	C₂×D₈	C₇×Q₈	1	8	yes	complemented
304	C₂×D₈	C₂³×C₇	1	2	yes	—
305	C₂×D₈	C₂³×C₇	1	2	yes	—
306	C₂×M₁₆	C₂³×C₇	1	2	yes	—
307	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
308	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
309	C₄⋊Q₈	—	2	4	no	—
310	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
311	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
312	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
313	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
314	C₄⋊Q₈	C₂³×C₇	1	8	yes	—
315	#125	—	2	4	no	—
316	#125	C₇×D₄	1	2	yes	—
317	#125	C₂³×C₇	1	4	yes	—
318	#125	C₂³×C₇	1	8	yes	—
319	C₂²⋊Q₈	—	2	4	no	—
320	C₂²⋊Q₈	—	2	8	no	—
321	C₂²⋊Q₈	—	2	2	no	—
322	C₂²⋊Q₈	—	2	2	no	—
323	C₂²⋊Q₈	—	2	4	no	—
324	C₂²⋊Q₈	—	2	8	no	—
325	C₂²⋊Q₈	—	2	8	no	—
326	C₂²⋊Q₈	—	2	8	no	—
327	C₂²⋊Q₈	—	2	8	no	—
328	C₂²⋊Q₈	—	2	8	no	—
329	C₂²⋊Q₈	C₇×D₄	1	2	yes	—
330	C₂²⋊Q₈	C₇×D₄	1	2	yes	—
331	C₄×Q₈	—	2	2	no	—
332	C₄×Q₈	—	2	2	no	—
333	C₄×Q₈	C₂³×C₇	1	2	yes	—
334	C₄×Q₈	C₂³×C₇	1	2	yes	—
335	C₄×D₄	—	2	4	no	—
336	C₄×D₄	—	2	4	no	—
337	C₄×D₄	—	2	2	no	—
338	C₄×D₄	—	2	2	no	—
339	C₄×D₄	C₂³×C₇	1	4	yes	—
340	C₄×D₄	C₂³×C₇	1	4	yes	—
341	C₄×D₄	C₂³×C₇	1	8	yes	—
342	C₄×D₄	C₂³×C₇	1	8	yes	—
343	#117	C₂³×C₇	1	4	yes	—
344	#117	C₂³×C₇	1	2	yes	—
345	C₂×C₄⋊C₄	—	2	8	no	—
346	C₂×C₄⋊C₄	—	2	4	no	—
347	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
348	C₂×C₂²⋊C₄	—	2	2	no	—
349	C₂×C₂²⋊C₄	—	2	8	no	—
350	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
351	#107	C₂³×C₇	1	2	yes	—
352	#107	C₂³×C₇	1	2	yes	—
353	#107	C₂³×C₇	1	2	yes	—
354	#107	C₂³×C₇	1	2	yes	—
355	#107	C₂³×C₇	1	2	yes	—
356	#107	C₂³×C₇	1	2	yes	—
357	#107	C₂³×C₇	1	8	yes	—
358	#106	C₂³×C₇	1	4	yes	—
359	#106	C₂³×C₇	1	4	yes	—
360	#106	C₂³×C₇	1	8	yes	—
361	C₄⋊C₈	C₂³×C₇	1	2	yes	—
362	C₄⋊C₈	C₂³×C₇	1	2	yes	—
363	C₄⋊C₈	C₂³×C₇	1	4	yes	—
364	C₄⋊C₈	C₂³×C₇	1	4	yes	—
365	#102	C₂³×C₇	1	4	yes	—
366	#102	C₂³×C₇	1	4	yes	—
367	#102	C₂³×C₇	1	8	yes	—
368	#102	C₂³×C₇	1	8	yes	—
369	#97	C₂³×C₇	1	4	yes	—
370	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
371	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
372	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
373	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
374	C₄×C₈	C₂³×C₇	1	4	yes	—
375	C₂²×C₇	—	2	1	no	—
376	C₂²×C₇	—	2	1	no	—
377	C₂²×C₇	—	2	8	no	—
378	C₂²×C₇	—	4	4	no	—
379	C₂²×C₇	—	4	4	no	—
380	C₂²×C₇	—	8	8	no	—
381	C₂²×C₇	D₄×Q₈	1	2	yes	—
382	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
383	C₄×C₇	—	2	1	no	—
384	C₄×C₇	—	2	1	no	—
385	C₄×C₇	—	2	4	no	—
386	C₄×C₇	—	2	1	no	—
387	C₄×C₇	—	2	4	no	—
388	C₄×C₇	—	4	4	no	—
389	C₄×C₇	—	4	4	no	—
390	C₄×C₇	—	4	4	no	—
391	C₄×C₇	D₄×Q₈	1	2	yes	—
392	C₄×C₇	#576	1	2	yes	—
393	C₄×C₇	C₂³×Q₈	1	1	yes	characteristic
394	C₄×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
395	C₂⁴	—	2	2	no	—
396	C₂×Q₈	—	2	2	no	—
397	C₂×Q₈	—	2	2	no	—
398	C₂×Q₈	—	2	8	no	—
399	C₂×Q₈	—	8	8	no	—
400	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
401	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	2	4	no	—
404	C₂×D₄	C₂×C₇×Q₈	1	8	yes	complemented
405	C₂×D₄	C₂⁴×C₇	1	4	yes	—
406	C₂²×C₄	—	2	2	no	—
407	C₂²×C₄	—	2	4	no	—
408	C₂²×C₄	—	2	8	no	—
409	C₂²×C₄	—	2	8	no	—
410	C₂²×C₄	—	2	4	no	—
411	C₂²×C₄	—	8	4	no	—
412	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
413	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
414	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
415	D₈	C₂×C₇×Q₈	1	8	yes	complemented
416	D₈	C₂×C₇×Q₈	1	8	yes	complemented
417	M₁₆	—	2	4	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	C₂×C₇×Q₈	1	4	yes	complemented
420	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
421	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
422	C₂×C₈	C₂⁴×C₇	1	4	yes	—
423	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
424	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
425	C₄⋊C₄	—	2	4	no	—
426	C₄⋊C₄	—	2	2	no	—
427	C₄⋊C₄	—	2	8	no	—
428	C₄⋊C₄	—	2	8	no	—
429	C₄⋊C₄	—	2	8	no	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	2	4	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
434	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
435	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
436	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
437	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
438	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
439	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
440	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
441	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
442	C₂²⋊C₄	—	2	4	no	—
443	C₂²⋊C₄	—	2	8	no	—
444	C₂²⋊C₄	—	2	8	no	—
445	C₂²⋊C₄	—	2	4	no	—
446	C₂²⋊C₄	—	2	4	no	—
447	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
448	C₄²	—	2	4	no	—
449	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
450	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
451	C₄²	C₂⁴×C₇	1	1	yes	characteristic
452	C₄²	C₂⁴×C₇	1	1	yes	characteristic
453	C₄²	C₂⁴×C₇	1	4	yes	—
454	C₂×C₇	—	2	2	no	—
455	C₂×C₇	—	4	8	no	—
456	C₂×C₇	C₂×D₄×Q₈	1	1	yes	characteristic
457	C₂×C₇	C₂×#576	1	1	yes	characteristic
458	C₂×C₇	#3584	1	1	yes	characteristic
459	C₂³	—	2	4	no	—
460	C₂³	—	2	8	no	—
461	C₂³	—	8	2	no	—
462	C₂³	—	8	2	no	—
463	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
464	Q₈	—	2	4	no	—
465	Q₈	—	4	2	no	—
466	Q₈	—	4	4	no	—
467	Q₈	—	8	2	no	—
468	Q₈	—	8	8	no	—
469	D₄	—	2	4	no	—
470	D₄	—	2	8	no	—
471	D₄	C₂²×C₇×Q₈	1	8	yes	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	8	no	—
478	C₂×C₄	—	8	8	no	—
479	C₂×C₄	—	8	4	no	—
480	C₂×C₄	—	8	8	no	—
481	C₂×C₄	—	8	2	no	—
482	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
483	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
484	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
485	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
486	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
487	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
488	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
489	C₂×C₄	C₂²×C₇×Q₈	1	2	yes	—
490	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
491	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
492	C₈	—	2	1	no	—
493	C₈	—	2	1	no	—
494	C₈	—	2	4	no	—
495	C₈	C₂²×C₇×Q₈	1	2	yes	—
496	C₈	C₂²×C₇×Q₈	1	2	yes	—
497	C₇	#61370	1	1	yes	characteristic, direct product factor, Sylow
498	C₂²	—	2	1	no	—
499	C₂²	—	2	1	no	—
500	C₂²	—	2	8	no	—
501	C₂²	—	4	4	no	—
502	C₂²	—	4	4	no	—
503	C₂²	—	8	8	no	—
504	C₂²	C₇×D₄×Q₈	1	2	yes	—
505	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
506	C₄	—	2	1	no	—
507	C₄	—	2	4	no	—
508	C₄	—	2	1	no	—
509	C₄	—	2	1	no	—
510	C₄	—	2	4	no	—
511	C₄	—	4	4	no	—
512	C₄	—	4	4	no	—
513	C₄	—	4	4	no	—
514	C₄	C₇×D₄×Q₈	1	2	yes	—
515	C₄	C₇×#576	1	2	yes	—
516	C₄	C₂³×C₇×Q₈	1	1	yes	characteristic
517	C₄	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
518	C₂	—	2	2	no	—
519	C₂	—	4	8	no	—
520	C₂	C₂×C₇×D₄×Q₈	1	1	yes	characteristic
521	C₂	C₂×C₇×#576	1	1	yes	characteristic
522	C₂	C₇×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₂²×C₇×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₇×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3501	C₂	semidirect
2. C₇×#3389	C₂	semidirect
3. C₇×#3384	C₂	semidirect
4. C₇×#3380	C₂	semidirect
5. C₇×#3380	C₂	semidirect
6. C₇×Q₈×D₈	C₂	semidirect
7. C₇×#3164	C₂	semidirect
8. C₇×#3160	C₂	semidirect
9. C₇×#3081	C₂	semidirect
10. C₇×#3077	C₂	semidirect
11. C₇×#2960	C₂	semidirect
17. C₇×#554	C₂²	semidirect
18. C₇×#554	C₂²	semidirect
20. C₇×D₄×Q₈	C₂²	semidirect
28. C₇×#501	C₂²	semidirect
29. C₇×#500	C₂²	semidirect
30. C₇×#500	C₂²	semidirect
31. C₇×#500	C₂²	semidirect
32. C₇×#499	C₂²	semidirect
33. C₇×#499	C₂²	semidirect
34. C₇×#499	C₂²	semidirect
35. C₇×#498	C₂²	semidirect
36. C₇×#478	C₂²	semidirect
37. C₇×#478	C₂²	semidirect
38. C₇×#478	C₂²	semidirect
39. C₇×#476	C₂²	semidirect
40. C₇×#476	C₂²	semidirect
41. C₇×#474	C₂²	semidirect
42. C₇×#474	C₂²	semidirect
43. C₇×#446	C₂²	semidirect
44. C₇×#446	C₂²	semidirect
45. C₇×C₈×Q₈	C₂²	semidirect
46. C₇×C₈×Q₈	C₂²	semidirect
64. #61370	C₇	direct (N × H)
68. C₂×C₇×D₈	Q₈	semidirect
140. #3501	C₂×C₇	semidirect
141. #3389	C₂×C₇	semidirect
142. #3384	C₂×C₇	semidirect
143. #3380	C₂×C₇	semidirect
144. #3380	C₂×C₇	semidirect
145. Q₈×D₈	C₂×C₇	semidirect
146. #3164	C₂×C₇	semidirect
147. #3160	C₂×C₇	semidirect
148. #3081	C₂×C₇	semidirect
149. #3077	C₂×C₇	semidirect
150. #2960	C₂×C₇	semidirect
162. C₂×C₇×D₄	C₂×Q₈	semidirect
173. C₇×D₈	C₂×Q₈	semidirect
174. C₇×D₈	C₂×Q₈	semidirect
177. C₂×C₇×C₈	C₂×Q₈	semidirect
215. #554	C₂²×C₇	semidirect
216. #554	C₂²×C₇	semidirect
218. D₄×Q₈	C₂²×C₇	semidirect
226. #501	C₂²×C₇	semidirect
227. #500	C₂²×C₇	semidirect
228. #500	C₂²×C₇	semidirect
229. #500	C₂²×C₇	semidirect
230. #499	C₂²×C₇	semidirect
231. #499	C₂²×C₇	semidirect
232. #499	C₂²×C₇	semidirect
233. #498	C₂²×C₇	semidirect
234. #478	C₂²×C₇	semidirect
235. #478	C₂²×C₇	semidirect
236. #478	C₂²×C₇	semidirect
237. #476	C₂²×C₇	semidirect
238. #476	C₂²×C₇	semidirect
239. #474	C₂²×C₇	semidirect
240. #474	C₂²×C₇	semidirect
241. #446	C₂²×C₇	semidirect
242. #446	C₂²×C₇	semidirect
243. C₈×Q₈	C₂²×C₇	semidirect
244. C₈×Q₈	C₂²×C₇	semidirect
303. C₂×D₈	C₇×Q₈	semidirect
404. C₂×D₄	C₂×C₇×Q₈	semidirect
415. D₈	C₂×C₇×Q₈	semidirect
416. D₈	C₂×C₇×Q₈	semidirect
419. C₂×C₈	C₂×C₇×Q₈	semidirect
497. C₇	#61370	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#3501	2
C₇×#3389	2
C₇×#3384	2
C₇×#3380	2
C₇×Q₈×D₈	2
C₇×#3164	2
C₇×#3160	2
C₇×#3081	2
C₇×#3077	2
C₇×#2960	2
C₇×#2948	2
C₂×C₄×C₇×D₈	2
#61370	7

Subgroups — C7×#61370 #13238586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#61370 #13238586