Subgroups — C₇×#61869 #13239085

Order 1792.

1644 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3503	C₂	1	1	yes	maximal, characteristic
3	C₇×#3387	C₂	1	2	yes	maximal, complemented
4	C₇×#3386	C₂	1	2	yes	maximal, complemented
5	C₇×#3383	C₂	1	2	yes	maximal, complemented
6	C₇×#3381	C₂	1	2	yes	maximal, complemented
7	C₇×#3374	C₂	1	2	yes	maximal, complemented
8	C₇×#3372	C₂	1	2	yes	maximal, complemented
9	C₇×#3363	C₂	1	2	yes	maximal, complemented
10	C₇×#3359	C₂	1	2	yes	maximal, complemented
11	C₇×#3242	C₂	1	4	yes	maximal, complemented
12	C₇×#3230	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#3229	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#3120	C₂	1	4	yes	maximal
15	C₇×#3107	C₂	1	1	yes	maximal, characteristic
16	C₇×#3107	C₂	1	1	yes	maximal, characteristic
17	C₇×#2986	C₂	1	1	yes	maximal, characteristic, complemented
18	C₇×#564	—	2	2	no	—
19	C₇×Q₈²	C₂²	1	2	yes	—
20	C₇×#557	—	2	1	no	—
21	C₇×#557	—	2	1	no	—
22	C₇×#557	C₂²	1	2	yes	—
23	C₇×#557	C₂²	1	2	yes	—
24	C₇×#557	C₂²	1	2	yes	—
25	C₇×#557	C₂²	1	2	yes	—
26	C₇×#557	C₂²	1	2	yes	—
27	C₇×#554	—	2	2	no	—
28	C₇×#554	C₂²	1	2	yes	—
29	C₇×D₄×Q₈	—	2	2	no	—
30	C₇×D₄×Q₈	C₂²	1	2	yes	—
31	C₇×#544	—	2	2	no	—
32	C₇×#544	C₂²	1	2	yes	—
33	C₇×#541	—	2	2	no	—
34	C₇×#541	—	2	2	no	—
35	C₇×#541	C₂²	1	4	yes	—
36	C₇×#533	—	2	1	no	—
37	C₇×#533	C₂²	1	1	yes	characteristic
38	C₇×#533	C₂²	1	1	yes	characteristic
39	C₇×#533	C₂²	1	1	yes	characteristic
40	C₂×C₇×C₄⋊Q₈	—	2	1	no	—
41	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
42	C₇×#520	C₂²	1	2	yes	—
43	C₄×C₇×#41	C₂²	1	1	yes	characteristic
44	C₇×#501	C₂²	1	4	yes	—
45	C₇×#500	C₂²	1	2	yes	—
46	C₇×#499	C₂²	1	4	yes	—
47	C₇×#498	C₂²	1	2	yes	—
48	C₇×#491	C₂²	1	4	yes	—
49	C₇×#487	C₂²	1	2	yes	—
50	C₇×#487	C₂²	1	2	yes	—
51	C₇×#487	C₂²	1	4	yes	—
52	C₇×#485	C₂²	1	2	yes	—
53	C₇×#485	C₂²	1	2	yes	—
54	C₇×#484	C₂²	1	4	yes	—
55	C₇×#483	C₂²	1	4	yes	—
56	C₇×#479	C₂²	1	4	yes	—
57	C₇×#479	C₂²	1	4	yes	—
58	C₇×#479	C₂²	1	4	yes	—
59	C₇×#478	C₂²	1	4	yes	—
60	C₇×#477	C₂²	1	2	yes	—
61	C₇×#477	C₂²	1	2	yes	—
62	C₇×#477	C₂²	1	4	yes	—
63	C₇×#475	C₂²	1	2	yes	—
64	C₇×#475	C₂²	1	2	yes	—
65	C₇×#475	C₂²	1	4	yes	—
66	C₇×#475	C₂²	1	1	yes	characteristic
67	C₇×#475	C₂²	1	1	yes	characteristic
68	C₇×#474	C₂²	1	1	yes	characteristic
69	C₇×#474	C₂²	1	1	yes	characteristic
70	C₇×#462	C₂²	1	1	yes	characteristic
71	C₇×#462	C₂²	1	1	yes	characteristic
72	C₇×#462	C₂²	1	4	yes	—
73	C₇×#461	C₂²	1	1	yes	characteristic
74	C₇×#461	C₂²	1	4	yes	—
75	C₇×#461	C₂²	1	1	yes	characteristic
76	C₇×#452	C₂²	1	2	yes	—
77	C₇×#452	C₂²	1	2	yes	—
78	C₇×#446	C₂²	1	2	yes	—
79	C₇×#446	C₂²	1	2	yes	—
80	C₇×C₈×Q₈	C₂²	1	2	yes	—
81	C₇×C₈×Q₈	C₂²	1	2	yes	—
82	C₇×#441	C₂²	1	4	yes	—
83	C₇×#440	C₂²	1	4	yes	—
84	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
85	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
86	C₄×C₇×Q₁₆	C₂²	1	4	yes	—
87	C₄×C₇×SD₁₆	C₂²	1	2	yes	—
88	C₄×C₇×SD₁₆	C₂²	1	4	yes	—
89	C₄×C₇×SD₁₆	C₂²	1	2	yes	—
90	C₇×#433	C₂²	1	4	yes	—
91	C₇×#431	C₂²	1	1	yes	characteristic
92	C₇×#431	C₂²	1	1	yes	characteristic
93	#61869	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂×C₇×#41	C₂³	1	1	yes	characteristic
95	C₂²×C₇×Q₈	—	2	2	no	—
96	C₂²×C₇×Q₈	—	2	1	no	—
97	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
98	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
99	C₂×C₇×SD₁₆	C₂³	1	2	yes	—
100	C₂×C₇×SD₁₆	C₂³	1	2	yes	—
101	C₇×C₄⋊Q₈	—	2	2	no	—
102	C₇×C₄⋊Q₈	—	2	2	no	—
103	C₇×C₄⋊Q₈	—	2	2	no	—
104	C₇×C₄⋊Q₈	—	2	2	no	—
105	C₇×C₄⋊Q₈	—	2	2	no	—
106	C₇×C₄⋊Q₈	—	2	4	no	—
107	C₇×C₄⋊Q₈	—	2	1	no	—
108	C₇×C₄⋊Q₈	—	2	1	no	—
109	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
110	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
111	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
112	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
113	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
114	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
115	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
116	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
117	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
118	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
119	C₇×#126	—	2	4	no	—
120	C₇×#126	—	2	4	no	—
121	C₇×#125	—	2	2	no	—
122	C₇×#125	—	2	2	no	—
123	C₇×#125	—	2	2	no	—
124	C₇×#125	—	2	4	no	—
125	C₇×#125	—	2	4	no	—
126	C₇×#125	—	2	4	no	—
127	C₇×#125	—	2	4	no	—
128	C₇×#125	—	2	4	no	—
129	C₇×#125	—	2	4	no	—
130	C₇×#125	—	2	4	no	—
131	C₇×#125	—	2	1	no	—
132	C₇×#125	—	2	1	no	—
133	C₇×#125	D₄	1	2	yes	—
134	C₇×#125	C₂³	1	4	yes	—
135	C₇×#125	C₂³	1	4	yes	—
136	C₇×C₂²⋊Q₈	—	2	2	no	—
137	C₇×C₂²⋊Q₈	—	2	1	no	—
138	C₇×C₂²⋊Q₈	—	2	1	no	—
139	C₇×C₂²⋊Q₈	—	2	2	no	—
140	C₇×C₂²⋊Q₈	—	2	4	no	—
141	C₇×C₂²⋊Q₈	—	2	4	no	—
142	C₇×C₂²⋊Q₈	—	2	4	no	—
143	C₇×C₂²⋊Q₈	D₄	1	2	yes	—
144	C₇×C₂²⋊Q₈	D₄	1	2	yes	—
145	C₇×C₂²⋊Q₈	C₂³	1	2	yes	—
146	C₇×C₂²⋊Q₈	C₂³	1	2	yes	—
147	C₄×C₇×Q₈	—	2	2	no	—
148	C₄×C₇×Q₈	—	2	2	no	—
149	C₄×C₇×Q₈	—	2	2	no	—
150	C₄×C₇×Q₈	—	2	2	no	—
151	C₄×C₇×Q₈	—	2	2	no	—
152	C₄×C₇×Q₈	—	2	2	no	—
153	C₄×C₇×Q₈	—	2	1	no	—
154	C₄×C₇×Q₈	—	2	1	no	—
155	C₄×C₇×Q₈	C₂³	1	2	yes	—
156	C₄×C₇×Q₈	C₂³	1	4	yes	—
157	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
158	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
159	C₄×C₇×Q₈	C₂³	1	2	yes	—
160	C₄×C₇×Q₈	C₂³	1	4	yes	—
161	C₄×C₇×Q₈	C₂³	1	4	yes	—
162	C₄×C₇×Q₈	C₂³	1	2	yes	—
163	C₄×C₇×Q₈	C₂³	1	2	yes	—
164	C₄×C₇×D₄	—	2	1	no	—
165	C₄×C₇×D₄	—	2	1	no	—
166	C₄×C₇×D₄	—	2	4	no	—
167	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
168	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
169	C₄×C₇×D₄	C₂³	1	4	yes	—
170	C₇×#117	—	2	1	no	—
171	C₇×#117	—	2	1	no	—
172	C₇×#117	—	2	2	no	—
173	C₇×#117	—	2	1	no	—
174	C₇×#117	C₂³	1	1	yes	characteristic
175	C₇×#117	C₂³	1	2	yes	—
176	C₂×C₇×C₄⋊C₄	—	2	1	no	—
177	C₂×C₇×C₄⋊C₄	—	2	2	no	—
178	C₂×C₇×C₄⋊C₄	—	2	2	no	—
179	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
180	C₂×C₄²×C₇	—	2	1	no	—
181	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
182	C₇×#107	C₂³	1	2	yes	—
183	C₇×#107	C₂³	1	2	yes	—
184	C₇×#107	C₂³	1	4	yes	—
185	C₇×#107	C₂³	1	4	yes	—
186	C₇×#106	C₂³	1	2	yes	—
187	C₇×#106	C₂³	1	4	yes	—
188	C₇×#106	C₂³	1	4	yes	—
189	C₇×#106	C₂³	1	2	yes	—
190	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
191	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
192	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
193	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
194	C₇×C₄⋊C₈	C₂³	1	4	yes	—
195	C₇×C₄⋊C₈	C₂³	1	4	yes	—
196	C₇×#103	C₂³	1	2	yes	—
197	C₇×#103	C₂³	1	4	yes	—
198	C₇×#103	C₂³	1	4	yes	—
199	C₇×#103	C₂³	1	2	yes	—
200	C₇×#103	C₂³	1	2	yes	—
201	C₇×#103	C₂³	1	4	yes	—
202	C₇×#103	C₂³	1	2	yes	—
203	C₇×#103	C₂³	1	4	yes	—
204	C₇×#103	C₂³	1	2	yes	—
205	C₇×#103	C₂³	1	2	yes	—
206	C₇×#102	C₂³	1	2	yes	—
207	C₇×#102	C₂³	1	2	yes	—
208	C₇×C₂²⋊C₈	C₂³	1	2	yes	—
209	C₇×C₂²⋊C₈	C₂³	1	2	yes	—
210	C₇×#97	C₂³	1	4	yes	—
211	C₄×C₇×C₈	C₂³	1	2	yes	—
212	C₄×C₇×C₈	C₂³	1	4	yes	—
213	C₄×C₇×C₈	C₂³	1	2	yes	—
214	#3556	C₂×C₇	1	1	yes	characteristic, complemented
215	#3503	C₂×C₇	1	1	yes	characteristic
216	#3387	C₂×C₇	1	2	yes	complemented
217	#3386	C₂×C₇	1	2	yes	complemented
218	#3383	C₂×C₇	1	2	yes	complemented
219	#3381	C₂×C₇	1	2	yes	complemented
220	#3374	C₂×C₇	1	2	yes	complemented
221	#3372	C₂×C₇	1	2	yes	complemented
222	#3363	C₂×C₇	1	2	yes	complemented
223	#3359	C₂×C₇	1	2	yes	complemented
224	#3242	C₂×C₇	1	4	yes	complemented
225	#3230	C₂×C₇	1	1	yes	characteristic, complemented
226	#3229	C₂×C₇	1	1	yes	characteristic, complemented
227	#3120	C₂×C₇	1	4	yes	—
228	#3107	C₂×C₇	1	1	yes	characteristic
229	#3107	C₂×C₇	1	1	yes	characteristic
230	#2986	C₂×C₇	1	1	yes	characteristic, complemented
231	C₇×#41	—	2	2	no	—
232	C₇×#41	—	2	2	no	—
233	C₂×C₇×Q₈	—	2	1	no	—
234	C₂×C₇×Q₈	—	2	1	no	—
235	C₂×C₇×Q₈	—	2	2	no	—
236	C₂×C₇×Q₈	—	2	4	no	—
237	C₂×C₇×Q₈	—	2	2	no	—
238	C₂×C₇×Q₈	—	8	2	no	—
239	C₂×C₇×Q₈	—	8	4	no	—
240	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
241	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
242	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
243	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
244	C₂×C₇×Q₈	C₂⁴	1	1	yes	characteristic
245	C₂×C₇×D₄	—	2	1	no	—
246	C₂×C₇×D₄	C₂⁴	1	1	yes	characteristic
247	C₂²×C₄×C₇	—	2	1	no	—
248	C₂²×C₄×C₇	—	2	2	no	—
249	C₂²×C₄×C₇	—	2	1	no	—
250	C₂²×C₄×C₇	—	2	1	no	—
251	C₂²×C₄×C₇	—	2	4	no	—
252	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
253	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
254	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
255	C₇×Q₁₆	—	2	4	no	—
256	C₇×Q₁₆	—	2	4	no	—
257	C₇×SD₁₆	—	2	4	no	—
258	C₇×SD₁₆	—	2	4	no	—
259	C₂×C₇×C₈	C₂⁴	1	2	yes	—
260	C₂×C₇×C₈	C₂⁴	1	2	yes	—
261	C₂×C₇×C₈	C₂⁴	1	2	yes	—
262	C₂×C₇×C₈	C₂⁴	1	2	yes	—
263	C₇×C₄⋊C₄	—	2	2	no	—
264	C₇×C₄⋊C₄	—	2	2	no	—
265	C₇×C₄⋊C₄	—	2	4	no	—
266	C₇×C₄⋊C₄	—	2	2	no	—
267	C₇×C₄⋊C₄	—	2	4	no	—
268	C₇×C₄⋊C₄	—	2	1	no	—
269	C₇×C₄⋊C₄	—	2	1	no	—
270	C₇×C₄⋊C₄	—	2	1	no	—
271	C₇×C₄⋊C₄	—	2	1	no	—
272	C₇×C₄⋊C₄	—	2	1	no	—
273	C₇×C₄⋊C₄	—	2	1	no	—
274	C₇×C₄⋊C₄	—	2	1	no	—
275	C₇×C₄⋊C₄	—	2	1	no	—
276	C₇×C₄⋊C₄	—	2	4	no	—
277	C₇×C₄⋊C₄	—	2	4	no	—
278	C₇×C₄⋊C₄	—	2	2	no	—
279	C₇×C₄⋊C₄	—	2	2	no	—
280	C₇×C₄⋊C₄	—	2	4	no	—
281	C₇×C₄⋊C₄	—	2	4	no	—
282	C₇×C₄⋊C₄	—	2	4	no	—
283	C₇×C₄⋊C₄	—	2	4	no	—
284	C₇×C₄⋊C₄	—	2	2	no	—
285	C₇×C₄⋊C₄	—	2	2	no	—
286	C₇×C₄⋊C₄	—	2	4	no	—
287	C₇×C₄⋊C₄	—	2	4	no	—
288	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
289	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
290	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
291	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
292	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
293	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
294	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
295	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
296	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
297	C₇×C₄⋊C₄	C₂⁴	1	1	yes	characteristic
298	C₇×C₄⋊C₄	C₂⁴	1	1	yes	characteristic
299	C₇×C₂²⋊C₄	—	2	2	no	—
300	C₇×C₂²⋊C₄	—	2	2	no	—
301	C₇×C₂²⋊C₄	—	2	1	no	—
302	C₇×C₂²⋊C₄	—	2	1	no	—
303	C₇×C₂²⋊C₄	—	2	1	no	—
304	C₇×C₂²⋊C₄	—	2	4	no	—
305	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
306	C₄²×C₇	—	2	1	no	—
307	C₄²×C₇	—	2	1	no	—
308	C₄²×C₇	—	2	1	no	—
309	C₄²×C₇	—	2	4	no	—
310	C₄²×C₇	—	2	2	no	—
311	C₄²×C₇	—	2	2	no	—
312	C₄²×C₇	—	2	4	no	—
313	C₄²×C₇	—	2	4	no	—
314	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
315	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
316	C₄²×C₇	C₂⁴	1	1	yes	characteristic
317	C₄²×C₇	C₂⁴	1	1	yes	characteristic
318	C₄²×C₇	C₂⁴	1	4	yes	—
319	#564	—	2	2	no	—
320	Q₈²	C₂²×C₇	1	2	yes	—
321	#557	—	2	1	no	—
322	#557	—	2	1	no	—
323	#557	C₂²×C₇	1	2	yes	—
324	#557	C₂²×C₇	1	2	yes	—
325	#557	C₂²×C₇	1	2	yes	—
326	#557	C₂²×C₇	1	2	yes	—
327	#557	C₂²×C₇	1	2	yes	—
328	#554	—	2	2	no	—
329	#554	C₂²×C₇	1	2	yes	—
330	D₄×Q₈	—	2	2	no	—
331	D₄×Q₈	C₂²×C₇	1	2	yes	—
332	#544	—	2	2	no	—
333	#544	C₂²×C₇	1	2	yes	—
334	#541	—	2	2	no	—
335	#541	—	2	2	no	—
336	#541	C₂²×C₇	1	4	yes	—
337	#533	—	2	1	no	—
338	#533	C₂²×C₇	1	1	yes	characteristic
339	#533	C₂²×C₇	1	1	yes	characteristic
340	#533	C₂²×C₇	1	1	yes	characteristic
341	C₂×C₄⋊Q₈	—	2	1	no	—
342	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
343	#520	C₂²×C₇	1	2	yes	—
344	C₄×#41	C₂²×C₇	1	1	yes	characteristic
345	#501	C₂²×C₇	1	4	yes	—
346	#500	C₂²×C₇	1	2	yes	—
347	#499	C₂²×C₇	1	4	yes	—
348	#498	C₂²×C₇	1	2	yes	—
349	#491	C₂²×C₇	1	4	yes	—
350	#487	C₂²×C₇	1	4	yes	—
351	#487	C₂²×C₇	1	2	yes	—
352	#487	C₂²×C₇	1	2	yes	—
353	#485	C₂²×C₇	1	2	yes	—
354	#485	C₂²×C₇	1	2	yes	—
355	#484	C₂²×C₇	1	4	yes	—
356	#483	C₂²×C₇	1	4	yes	—
357	#479	C₂²×C₇	1	4	yes	—
358	#479	C₂²×C₇	1	4	yes	—
359	#479	C₂²×C₇	1	4	yes	—
360	#478	C₂²×C₇	1	4	yes	—
361	#477	C₂²×C₇	1	2	yes	—
362	#477	C₂²×C₇	1	4	yes	—
363	#477	C₂²×C₇	1	2	yes	—
364	#475	C₂²×C₇	1	2	yes	—
365	#475	C₂²×C₇	1	4	yes	—
366	#475	C₂²×C₇	1	1	yes	characteristic
367	#475	C₂²×C₇	1	1	yes	characteristic
368	#475	C₂²×C₇	1	2	yes	—
369	#474	C₂²×C₇	1	1	yes	characteristic
370	#474	C₂²×C₇	1	1	yes	characteristic
371	#462	C₂²×C₇	1	1	yes	characteristic
372	#462	C₂²×C₇	1	1	yes	characteristic
373	#462	C₂²×C₇	1	4	yes	—
374	#461	C₂²×C₇	1	1	yes	characteristic
375	#461	C₂²×C₇	1	4	yes	—
376	#461	C₂²×C₇	1	1	yes	characteristic
377	#452	C₂²×C₇	1	2	yes	—
378	#452	C₂²×C₇	1	2	yes	—
379	#446	C₂²×C₇	1	2	yes	—
380	#446	C₂²×C₇	1	2	yes	—
381	C₈×Q₈	C₂²×C₇	1	2	yes	—
382	C₈×Q₈	C₂²×C₇	1	2	yes	—
383	#441	C₂²×C₇	1	4	yes	—
384	#440	C₂²×C₇	1	4	yes	—
385	C₄×Q₁₆	C₂²×C₇	1	2	yes	—
386	C₄×Q₁₆	C₂²×C₇	1	2	yes	—
387	C₄×Q₁₆	C₂²×C₇	1	4	yes	—
388	C₄×SD₁₆	C₂²×C₇	1	2	yes	—
389	C₄×SD₁₆	C₂²×C₇	1	4	yes	—
390	C₄×SD₁₆	C₂²×C₇	1	2	yes	—
391	#433	C₂²×C₇	1	4	yes	—
392	#431	C₂²×C₇	1	1	yes	characteristic
393	#431	C₂²×C₇	1	1	yes	characteristic
394	C₂³×C₇	—	2	1	no	—
395	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
396	C₇×Q₈	—	2	4	no	—
397	C₇×Q₈	—	2	4	no	—
398	C₇×Q₈	—	2	1	no	—
399	C₇×Q₈	—	2	2	no	—
400	C₇×Q₈	—	2	2	no	—
401	C₇×Q₈	—	2	2	no	—
402	C₇×Q₈	—	2	2	no	—
403	C₇×Q₈	—	4	2	no	—
404	C₇×Q₈	—	4	4	no	—
405	C₇×Q₈	—	4	2	no	—
406	C₇×Q₈	—	8	2	no	—
407	C₇×Q₈	—	8	4	no	—
408	C₇×Q₈	2₋¹⁺⁴	1	2	yes	—
409	C₇×D₄	—	2	4	no	—
410	C₇×D₄	—	2	1	no	—
411	C₇×D₄	2₋¹⁺⁴	1	2	yes	—
412	C₂×C₄×C₇	—	2	4	no	—
413	C₂×C₄×C₇	—	2	1	no	—
414	C₂×C₄×C₇	—	2	2	no	—
415	C₂×C₄×C₇	—	2	2	no	—
416	C₂×C₄×C₇	—	2	1	no	—
417	C₂×C₄×C₇	—	2	1	no	—
418	C₂×C₄×C₇	—	2	2	no	—
419	C₂×C₄×C₇	—	2	2	no	—
420	C₂×C₄×C₇	—	2	2	no	—
421	C₂×C₄×C₇	—	2	2	no	—
422	C₂×C₄×C₇	—	2	1	no	—
423	C₂×C₄×C₇	—	2	1	no	—
424	C₂×C₄×C₇	—	2	4	no	—
425	C₂×C₄×C₇	—	4	2	no	—
426	C₂×C₄×C₇	—	4	2	no	—
427	C₂×C₄×C₇	—	8	4	no	—
428	C₂×C₄×C₇	—	8	2	no	—
429	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
430	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
431	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
432	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
433	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
434	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
435	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
436	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
437	C₇×C₈	—	2	2	no	—
438	C₇×C₈	—	2	2	no	—
439	C₇×C₈	—	2	2	no	—
440	C₇×C₈	—	2	2	no	—
441	C₂×#41	C₂³×C₇	1	1	yes	characteristic
442	C₂²×Q₈	—	2	2	no	—
443	C₂²×Q₈	—	2	1	no	—
444	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
445	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
446	C₂×SD₁₆	C₂³×C₇	1	2	yes	—
447	C₂×SD₁₆	C₂³×C₇	1	2	yes	—
448	C₄⋊Q₈	—	2	2	no	—
449	C₄⋊Q₈	—	2	2	no	—
450	C₄⋊Q₈	—	2	2	no	—
451	C₄⋊Q₈	—	2	2	no	—
452	C₄⋊Q₈	—	2	2	no	—
453	C₄⋊Q₈	—	2	4	no	—
454	C₄⋊Q₈	—	2	1	no	—
455	C₄⋊Q₈	—	2	1	no	—
456	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
457	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
458	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
459	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
460	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
461	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
462	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
463	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
464	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
465	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
466	#126	—	2	4	no	—
467	#126	—	2	4	no	—
468	#125	—	2	2	no	—
469	#125	—	2	2	no	—
470	#125	—	2	2	no	—
471	#125	—	2	4	no	—
472	#125	—	2	4	no	—
473	#125	—	2	4	no	—
474	#125	—	2	4	no	—
475	#125	—	2	4	no	—
476	#125	—	2	4	no	—
477	#125	—	2	4	no	—
478	#125	—	2	1	no	—
479	#125	—	2	1	no	—
480	#125	C₇×D₄	1	2	yes	—
481	#125	C₂³×C₇	1	4	yes	—
482	#125	C₂³×C₇	1	4	yes	—
483	C₂²⋊Q₈	—	2	2	no	—
484	C₂²⋊Q₈	—	2	1	no	—
485	C₂²⋊Q₈	—	2	1	no	—
486	C₂²⋊Q₈	—	2	2	no	—
487	C₂²⋊Q₈	—	2	4	no	—
488	C₂²⋊Q₈	—	2	4	no	—
489	C₂²⋊Q₈	—	2	4	no	—
490	C₂²⋊Q₈	C₇×D₄	1	2	yes	—
491	C₂²⋊Q₈	C₇×D₄	1	2	yes	—
492	C₂²⋊Q₈	C₂³×C₇	1	2	yes	—
493	C₂²⋊Q₈	C₂³×C₇	1	2	yes	—
494	C₄×Q₈	—	2	2	no	—
495	C₄×Q₈	—	2	2	no	—
496	C₄×Q₈	—	2	2	no	—
497	C₄×Q₈	—	2	2	no	—
498	C₄×Q₈	—	2	2	no	—
499	C₄×Q₈	—	2	2	no	—
500	C₄×Q₈	—	2	1	no	—
501	C₄×Q₈	—	2	1	no	—
502	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
503	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
504	C₄×Q₈	C₂³×C₇	1	4	yes	—
505	C₄×Q₈	C₂³×C₇	1	2	yes	—
506	C₄×Q₈	C₂³×C₇	1	2	yes	—
507	C₄×Q₈	C₂³×C₇	1	4	yes	—
508	C₄×Q₈	C₂³×C₇	1	4	yes	—
509	C₄×Q₈	C₂³×C₇	1	2	yes	—
510	C₄×Q₈	C₂³×C₇	1	2	yes	—
511	C₄×D₄	—	2	1	no	—
512	C₄×D₄	—	2	1	no	—
513	C₄×D₄	—	2	4	no	—
514	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
515	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
516	C₄×D₄	C₂³×C₇	1	4	yes	—
517	#117	—	2	1	no	—
518	#117	—	2	1	no	—
519	#117	—	2	2	no	—
520	#117	—	2	1	no	—
521	#117	C₂³×C₇	1	1	yes	characteristic
522	#117	C₂³×C₇	1	2	yes	—
523	C₂×C₄⋊C₄	—	2	1	no	—
524	C₂×C₄⋊C₄	—	2	2	no	—
525	C₂×C₄⋊C₄	—	2	2	no	—
526	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
527	C₂×C₄²	—	2	1	no	—
528	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
529	#107	C₂³×C₇	1	2	yes	—
530	#107	C₂³×C₇	1	2	yes	—
531	#107	C₂³×C₇	1	4	yes	—
532	#107	C₂³×C₇	1	4	yes	—
533	#106	C₂³×C₇	1	2	yes	—
534	#106	C₂³×C₇	1	4	yes	—
535	#106	C₂³×C₇	1	2	yes	—
536	#106	C₂³×C₇	1	4	yes	—
537	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
538	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
539	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
540	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
541	C₄⋊C₈	C₂³×C₇	1	4	yes	—
542	C₄⋊C₈	C₂³×C₇	1	4	yes	—
543	#103	C₂³×C₇	1	2	yes	—
544	#103	C₂³×C₇	1	4	yes	—
545	#103	C₂³×C₇	1	2	yes	—
546	#103	C₂³×C₇	1	2	yes	—
547	#103	C₂³×C₇	1	2	yes	—
548	#103	C₂³×C₇	1	2	yes	—
549	#103	C₂³×C₇	1	4	yes	—
550	#103	C₂³×C₇	1	2	yes	—
551	#103	C₂³×C₇	1	4	yes	—
552	#103	C₂³×C₇	1	4	yes	—
553	#102	C₂³×C₇	1	2	yes	—
554	#102	C₂³×C₇	1	2	yes	—
555	C₂²⋊C₈	C₂³×C₇	1	2	yes	—
556	C₂²⋊C₈	C₂³×C₇	1	2	yes	—
557	#97	C₂³×C₇	1	4	yes	—
558	C₄×C₈	C₂³×C₇	1	2	yes	—
559	C₄×C₈	C₂³×C₇	1	2	yes	—
560	C₄×C₈	C₂³×C₇	1	4	yes	—
561	C₂²×C₇	—	2	1	no	—
562	C₂²×C₇	—	2	2	no	—
563	C₂²×C₇	—	2	1	no	—
564	C₂²×C₇	—	2	1	no	—
565	C₂²×C₇	—	4	1	no	—
566	C₂²×C₇	—	4	1	no	—
567	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
568	C₄×C₇	—	2	1	no	—
569	C₄×C₇	—	2	2	no	—
570	C₄×C₇	—	2	1	no	—
571	C₄×C₇	—	2	4	no	—
572	C₄×C₇	—	2	1	no	—
573	C₄×C₇	—	2	4	no	—
574	C₄×C₇	—	4	2	no	—
575	C₄×C₇	—	4	2	no	—
576	C₄×C₇	—	4	1	no	—
577	C₄×C₇	—	4	4	no	—
578	C₄×C₇	—	4	2	no	—
579	C₄×C₇	—	4	1	no	—
580	C₄×C₇	—	4	2	no	—
581	C₄×C₇	#577	1	2	yes	—
582	C₄×C₇	#578	1	2	yes	—
583	C₄×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
584	C₄×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
585	#41	—	2	2	no	—
586	#41	—	2	2	no	—
587	C₂×Q₈	—	2	1	no	—
588	C₂×Q₈	—	2	1	no	—
589	C₂×Q₈	—	2	2	no	—
590	C₂×Q₈	—	2	4	no	—
591	C₂×Q₈	—	2	2	no	—
592	C₂×Q₈	—	8	2	no	—
593	C₂×Q₈	—	8	4	no	—
594	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
595	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
596	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
597	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
598	C₂×Q₈	C₂⁴×C₇	1	1	yes	characteristic
599	C₂×D₄	—	2	1	no	—
600	C₂×D₄	C₂⁴×C₇	1	1	yes	characteristic
601	C₂²×C₄	—	2	1	no	—
602	C₂²×C₄	—	2	2	no	—
603	C₂²×C₄	—	2	1	no	—
604	C₂²×C₄	—	2	1	no	—
605	C₂²×C₄	—	2	4	no	—
606	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
607	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
608	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
609	Q₁₆	—	2	4	no	—
610	Q₁₆	—	2	4	no	—
611	SD₁₆	—	2	4	no	—
612	SD₁₆	—	2	4	no	—
613	C₂×C₈	C₂⁴×C₇	1	2	yes	—
614	C₂×C₈	C₂⁴×C₇	1	2	yes	—
615	C₂×C₈	C₂⁴×C₇	1	2	yes	—
616	C₂×C₈	C₂⁴×C₇	1	2	yes	—
617	C₄⋊C₄	—	2	2	no	—
618	C₄⋊C₄	—	2	2	no	—
619	C₄⋊C₄	—	2	4	no	—
620	C₄⋊C₄	—	2	2	no	—
621	C₄⋊C₄	—	2	4	no	—
622	C₄⋊C₄	—	2	1	no	—
623	C₄⋊C₄	—	2	1	no	—
624	C₄⋊C₄	—	2	1	no	—
625	C₄⋊C₄	—	2	1	no	—
626	C₄⋊C₄	—	2	1	no	—
627	C₄⋊C₄	—	2	1	no	—
628	C₄⋊C₄	—	2	1	no	—
629	C₄⋊C₄	—	2	1	no	—
630	C₄⋊C₄	—	2	4	no	—
631	C₄⋊C₄	—	2	4	no	—
632	C₄⋊C₄	—	2	2	no	—
633	C₄⋊C₄	—	2	2	no	—
634	C₄⋊C₄	—	2	4	no	—
635	C₄⋊C₄	—	2	4	no	—
636	C₄⋊C₄	—	2	4	no	—
637	C₄⋊C₄	—	2	4	no	—
638	C₄⋊C₄	—	2	2	no	—
639	C₄⋊C₄	—	2	2	no	—
640	C₄⋊C₄	—	2	4	no	—
641	C₄⋊C₄	—	2	4	no	—
642	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
643	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
644	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
645	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
646	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
647	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
648	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
649	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
650	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
651	C₄⋊C₄	C₂⁴×C₇	1	1	yes	characteristic
652	C₄⋊C₄	C₂⁴×C₇	1	1	yes	characteristic
653	C₂²⋊C₄	—	2	2	no	—
654	C₂²⋊C₄	—	2	2	no	—
655	C₂²⋊C₄	—	2	1	no	—
656	C₂²⋊C₄	—	2	1	no	—
657	C₂²⋊C₄	—	2	1	no	—
658	C₂²⋊C₄	—	2	4	no	—
659	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
660	C₄²	—	2	1	no	—
661	C₄²	—	2	1	no	—
662	C₄²	—	2	1	no	—
663	C₄²	—	2	4	no	—
664	C₄²	—	2	2	no	—
665	C₄²	—	2	2	no	—
666	C₄²	—	2	4	no	—
667	C₄²	—	2	4	no	—
668	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
669	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
670	C₄²	C₂⁴×C₇	1	1	yes	characteristic
671	C₄²	C₂⁴×C₇	1	1	yes	characteristic
672	C₄²	C₂⁴×C₇	1	4	yes	—
673	C₂×C₇	—	4	1	no	—
674	C₂×C₇	—	4	2	no	—
675	C₂×C₇	#3486	1	1	yes	characteristic
676	C₂×C₇	C₂×#577	1	1	yes	characteristic
677	C₂×C₇	C₂×#578	1	1	yes	characteristic
678	C₂³	—	2	1	no	—
679	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
680	Q₈	—	2	1	no	—
681	Q₈	—	2	4	no	—
682	Q₈	—	2	4	no	—
683	Q₈	—	2	2	no	—
684	Q₈	—	2	2	no	—
685	Q₈	—	2	2	no	—
686	Q₈	—	2	2	no	—
687	Q₈	—	4	2	no	—
688	Q₈	—	4	4	no	—
689	Q₈	—	4	2	no	—
690	Q₈	—	8	2	no	—
691	Q₈	—	8	4	no	—
692	Q₈	C₇×2₋¹⁺⁴	1	2	yes	—
693	D₄	—	2	4	no	—
694	D₄	—	2	1	no	—
695	D₄	C₇×2₋¹⁺⁴	1	2	yes	—
696	C₂×C₄	—	2	4	no	—
697	C₂×C₄	—	2	1	no	—
698	C₂×C₄	—	2	1	no	—
699	C₂×C₄	—	2	1	no	—
700	C₂×C₄	—	2	2	no	—
701	C₂×C₄	—	2	2	no	—
702	C₂×C₄	—	2	2	no	—
703	C₂×C₄	—	2	2	no	—
704	C₂×C₄	—	2	2	no	—
705	C₂×C₄	—	2	2	no	—
706	C₂×C₄	—	2	1	no	—
707	C₂×C₄	—	2	1	no	—
708	C₂×C₄	—	2	4	no	—
709	C₂×C₄	—	4	2	no	—
710	C₂×C₄	—	4	2	no	—
711	C₂×C₄	—	8	4	no	—
712	C₂×C₄	—	8	2	no	—
713	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
714	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
715	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
716	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
717	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
718	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
719	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
720	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
721	C₈	—	2	2	no	—
722	C₈	—	2	2	no	—
723	C₈	—	2	2	no	—
724	C₈	—	2	2	no	—
725	C₇	#61869	1	1	yes	characteristic, direct product factor, Sylow
726	C₂²	—	2	1	no	—
727	C₂²	—	2	1	no	—
728	C₂²	—	2	1	no	—
729	C₂²	—	2	2	no	—
730	C₂²	—	4	1	no	—
731	C₂²	—	4	1	no	—
732	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
733	C₄	—	2	1	no	—
734	C₄	—	2	4	no	—
735	C₄	—	2	1	no	—
736	C₄	—	2	4	no	—
737	C₄	—	2	1	no	—
738	C₄	—	2	2	no	—
739	C₄	—	4	1	no	—
740	C₄	—	4	4	no	—
741	C₄	—	4	2	no	—
742	C₄	—	4	1	no	—
743	C₄	—	4	2	no	—
744	C₄	—	4	2	no	—
745	C₄	—	4	2	no	—
746	C₄	C₇×#577	1	2	yes	—
747	C₄	C₇×#578	1	2	yes	—
748	C₄	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
749	C₄	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
750	C₂	—	4	1	no	—
751	C₂	—	4	2	no	—
752	C₂	C₇×#3486	1	1	yes	characteristic
753	C₂	C₂×C₇×#577	1	1	yes	characteristic
754	C₂	C₂×C₇×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₂×C₇×#41	C₂³
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×#3556	C₂	semidirect
3. C₇×#3387	C₂	semidirect
4. C₇×#3386	C₂	semidirect
5. C₇×#3383	C₂	semidirect
6. C₇×#3381	C₂	semidirect
7. C₇×#3374	C₂	semidirect
8. C₇×#3372	C₂	semidirect
9. C₇×#3363	C₂	semidirect
10. C₇×#3359	C₂	semidirect
11. C₇×#3242	C₂	semidirect
12. C₇×#3230	C₂	semidirect
13. C₇×#3229	C₂	semidirect
17. C₇×#2986	C₂	semidirect
93. #61869	C₇	direct (N × H)
214. #3556	C₂×C₇	semidirect
216. #3387	C₂×C₇	semidirect
217. #3386	C₂×C₇	semidirect
218. #3383	C₂×C₇	semidirect
219. #3381	C₂×C₇	semidirect
220. #3374	C₂×C₇	semidirect
221. #3372	C₂×C₇	semidirect
222. #3363	C₂×C₇	semidirect
223. #3359	C₂×C₇	semidirect
224. #3242	C₂×C₇	semidirect
225. #3230	C₂×C₇	semidirect
226. #3229	C₂×C₇	semidirect
230. #2986	C₂×C₇	semidirect
725. C₇	#61869	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×#3556	2
C₇×#3503	2
C₇×#3387	2
C₇×#3386	2
C₇×#3383	2
C₇×#3381	2
C₇×#3374	2
C₇×#3372	2
C₇×#3363	2
C₇×#3359	2
C₇×#3242	2
C₇×#3230	2
C₇×#3229	2
C₇×#3120	2
C₇×#3107	2
C₇×#2986	2
#61869	7

Subgroups — C7×#61869 #13239085

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#61869 #13239085