Subgroups — C₇×#61365 #13238581

Order 1792.

1766 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×Q₈×#41	C₂	1	2	yes	maximal, complemented
2	C₇×#3390	C₂	1	2	yes	maximal, complemented
3	C₇×#3389	C₂	1	2	yes	maximal, complemented
4	C₇×#3388	C₂	1	4	yes	maximal, complemented
5	C₇×#3387	C₂	1	2	yes	maximal, complemented
6	C₇×#3386	C₂	1	2	yes	maximal, complemented
7	C₇×#3385	C₂	1	2	yes	maximal, complemented
8	C₇×#3384	C₂	1	2	yes	maximal, complemented
9	C₇×#3162	C₂	1	2	yes	maximal, complemented
10	C₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#3076	C₂	1	2	yes	maximal, complemented
12	C₇×#3076	C₂	1	4	yes	maximal, complemented
13	C₂×C₇×#446	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#2943	C₂	1	2	yes	maximal
15	C₄×C₇×#135	C₂	1	1	yes	maximal, characteristic
16	C₂×C₇×#135	C₂²	1	1	yes	characteristic
17	C₇×Q₈²	C₂²	1	2	yes	—
18	C₇×#557	C₂²	1	2	yes	—
19	C₇×#557	C₂²	1	4	yes	—
20	C₇×#554	—	2	4	no	—
21	C₇×#554	—	2	2	no	—
22	C₇×#554	C₂²	1	2	yes	—
23	C₇×#554	C₂²	1	4	yes	—
24	C₇×D₄×Q₈	—	2	2	no	—
25	C₇×D₄×Q₈	C₂²	1	2	yes	—
26	C₇×#533	—	2	2	no	—
27	C₇×#533	—	2	4	no	—
28	C₇×#533	C₂²	1	2	yes	—
29	C₇×#533	C₂²	1	4	yes	—
30	C₄×C₇×#41	C₂²	1	2	yes	—
31	C₄×C₇×#41	C₂²	1	4	yes	—
32	C₂×C₄×C₇×Q₈	—	2	2	no	—
33	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
34	C₇×#501	C₂²	1	4	yes	—
35	C₇×#501	C₂²	1	4	yes	—
36	C₇×#501	C₂²	1	4	yes	—
37	C₇×#501	C₂²	1	4	yes	—
38	C₇×#500	C₂²	1	1	yes	characteristic
39	C₇×#500	C₂²	1	1	yes	characteristic
40	C₇×#499	C₂²	1	2	yes	—
41	C₇×#499	C₂²	1	2	yes	—
42	C₇×#498	C₂²	1	2	yes	—
43	C₇×#479	C₂²	1	4	yes	—
44	C₇×#479	C₂²	1	4	yes	—
45	C₇×#479	C₂²	1	4	yes	—
46	C₇×#478	C₂²	1	4	yes	—
47	C₇×#478	C₂²	1	4	yes	—
48	C₇×#478	C₂²	1	4	yes	—
49	C₇×#477	C₂²	1	2	yes	—
50	C₇×#477	C₂²	1	4	yes	—
51	C₇×#476	C₂²	1	2	yes	—
52	C₇×#476	C₂²	1	4	yes	—
53	C₇×#475	C₂²	1	2	yes	—
54	C₇×#475	C₂²	1	4	yes	—
55	C₇×#474	C₂²	1	2	yes	—
56	C₇×#474	C₂²	1	4	yes	—
57	C₇×#446	C₂²	1	2	yes	—
58	C₇×#446	C₂²	1	2	yes	—
59	C₇×#446	C₂²	1	2	yes	—
60	C₇×#446	C₂²	1	2	yes	—
61	C₇×#442	C₂²	1	4	yes	—
62	C₇×#441	C₂²	1	4	yes	—
63	C₇×#440	C₂²	1	4	yes	—
64	C₇×#440	C₂²	1	4	yes	—
65	C₄×C₇×Q₁₆	C₂²	1	1	yes	characteristic
66	C₄×C₇×Q₁₆	C₂²	1	1	yes	characteristic
67	C₄×C₇×SD₁₆	C₂²	1	2	yes	—
68	C₄×C₇×SD₁₆	C₂²	1	2	yes	—
69	C₄×C₇×D₈	C₂²	1	1	yes	characteristic
70	C₄×C₇×D₈	C₂²	1	1	yes	characteristic
71	C₇×#427	C₂²	1	2	yes	—
72	C₇×#427	C₂²	1	2	yes	—
73	C₇×#427	C₂²	1	1	yes	characteristic
74	C₇×#427	C₂²	1	4	yes	—
75	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
76	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
77	C₇×#416	C₂²	1	2	yes	—
78	C₇×#416	C₂²	1	4	yes	—
79	C₂×C₇×#97	C₂²	1	2	yes	—
80	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
81	#61365	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₇×#41	C₂³	1	2	yes	—
83	C₂²×C₇×Q₈	—	2	2	no	—
84	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
85	C₇×#135	Q₈	1	8	yes	complemented
86	C₂×C₇×Q₁₆	C₂³	1	1	yes	characteristic
87	C₂×C₇×SD₁₆	C₂³	1	2	yes	—
88	C₂×C₇×D₈	C₂³	1	1	yes	characteristic
89	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
90	C₂²×C₇×C₈	C₂³	1	2	yes	—
91	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
92	C₇×C₄⋊Q₈	—	2	2	no	—
93	C₇×C₄⋊Q₈	—	2	4	no	—
94	C₇×C₄⋊Q₈	—	2	2	no	—
95	C₇×C₄⋊Q₈	—	2	4	no	—
96	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
97	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
98	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
99	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
100	C₇×#125	—	2	4	no	—
101	C₇×#125	—	2	4	no	—
102	C₇×#125	—	2	4	no	—
103	C₇×#125	C₂³	1	4	yes	—
104	C₇×#125	C₂³	1	4	yes	—
105	C₇×#125	C₂³	1	4	yes	—
106	C₇×C₂²⋊Q₈	—	2	2	no	—
107	C₇×C₂²⋊Q₈	—	2	4	no	—
108	C₇×C₂²⋊Q₈	—	2	2	no	—
109	C₇×C₂²⋊Q₈	—	2	4	no	—
110	C₇×C₂²⋊Q₈	—	2	2	no	—
111	C₇×C₂²⋊Q₈	—	2	4	no	—
112	C₇×C₂²⋊Q₈	—	2	2	no	—
113	C₇×C₂²⋊Q₈	—	2	4	no	—
114	C₇×C₂²⋊Q₈	—	2	4	no	—
115	C₇×C₂²⋊Q₈	—	2	4	no	—
116	C₇×C₂²⋊Q₈	—	2	4	no	—
117	C₄×C₇×Q₈	—	2	2	no	—
118	C₄×C₇×Q₈	—	2	4	no	—
119	C₄×C₇×Q₈	—	2	2	no	—
120	C₄×C₇×Q₈	—	2	2	no	—
121	C₄×C₇×Q₈	—	2	4	no	—
122	C₄×C₇×Q₈	—	2	4	no	—
123	C₄×C₇×Q₈	—	2	2	no	—
124	C₄×C₇×Q₈	D₄	1	2	yes	—
125	C₄×C₇×Q₈	D₄	1	2	yes	—
126	C₄×C₇×Q₈	C₂³	1	4	yes	—
127	C₄×C₇×Q₈	C₂³	1	2	yes	—
128	C₄×C₇×Q₈	C₂³	1	2	yes	—
129	C₄×C₇×Q₈	C₂³	1	4	yes	—
130	C₄×C₇×Q₈	C₂³	1	2	yes	—
131	C₄×C₇×Q₈	C₂³	1	2	yes	—
132	C₄×C₇×D₄	—	2	2	no	—
133	C₄×C₇×D₄	—	2	4	no	—
134	C₄×C₇×D₄	—	2	2	no	—
135	C₄×C₇×D₄	—	2	4	no	—
136	C₄×C₇×D₄	C₂³	1	2	yes	—
137	C₄×C₇×D₄	C₂³	1	4	yes	—
138	C₄×C₇×D₄	C₂³	1	4	yes	—
139	C₄×C₇×D₄	C₂³	1	2	yes	—
140	C₇×#117	—	2	4	no	—
141	C₇×#117	—	2	2	no	—
142	C₇×#117	C₂³	1	2	yes	—
143	C₇×#117	C₂³	1	4	yes	—
144	C₂×C₇×C₄⋊C₄	—	2	4	no	—
145	C₂×C₇×C₄⋊C₄	—	2	2	no	—
146	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
147	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
148	C₂×C₄²×C₇	—	2	2	no	—
149	C₂×C₄²×C₇	—	2	4	no	—
150	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
151	C₂×C₄²×C₇	C₂³	1	2	yes	—
152	C₇×#107	C₂³	1	2	yes	—
153	C₇×#107	C₂³	1	4	yes	—
154	C₇×#107	C₂³	1	4	yes	—
155	C₇×#107	C₂³	1	2	yes	—
156	C₇×#107	C₂³	1	2	yes	—
157	C₇×#107	C₂³	1	2	yes	—
158	C₇×#107	C₂³	1	1	yes	characteristic
159	C₇×#107	C₂³	1	1	yes	characteristic
160	C₇×#106	C₂³	1	4	yes	—
161	C₇×#106	C₂³	1	4	yes	—
162	C₇×#106	C₂³	1	4	yes	—
163	C₇×#106	C₂³	1	4	yes	—
164	C₇×#106	C₂³	1	2	yes	—
165	C₇×C₄⋊C₈	C₂³	1	2	yes	—
166	C₇×C₄⋊C₈	C₂³	1	4	yes	—
167	C₇×C₄⋊C₈	C₂³	1	4	yes	—
168	C₇×C₄⋊C₈	C₂³	1	2	yes	—
169	C₇×#103	C₂³	1	2	yes	—
170	C₇×#103	C₂³	1	4	yes	—
171	C₇×#103	C₂³	1	4	yes	—
172	C₇×#103	C₂³	1	2	yes	—
173	C₇×#102	C₂³	1	2	yes	—
174	C₇×#102	C₂³	1	4	yes	—
175	C₇×#102	C₂³	1	4	yes	—
176	C₇×#102	C₂³	1	2	yes	—
177	C₇×#97	C₂³	1	4	yes	—
178	C₇×#97	C₂³	1	4	yes	—
179	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
180	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
181	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
182	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
183	Q₈×#41	C₂×C₇	1	2	yes	complemented
184	#3390	C₂×C₇	1	2	yes	complemented
185	#3389	C₂×C₇	1	2	yes	complemented
186	#3388	C₂×C₇	1	4	yes	complemented
187	#3387	C₂×C₇	1	2	yes	complemented
188	#3386	C₂×C₇	1	2	yes	complemented
189	#3385	C₂×C₇	1	2	yes	complemented
190	#3384	C₂×C₇	1	2	yes	complemented
191	#3162	C₂×C₇	1	2	yes	complemented
192	#3160	C₂×C₇	1	1	yes	characteristic, complemented
193	#3076	C₂×C₇	1	2	yes	complemented
194	#3076	C₂×C₇	1	4	yes	complemented
195	C₂×#446	C₂×C₇	1	1	yes	characteristic, complemented
196	#2943	C₂×C₇	1	2	yes	—
197	C₄×#135	C₂×C₇	1	1	yes	characteristic
198	C₇×#41	—	2	4	no	—
199	C₇×#41	C₂×Q₈	1	8	yes	complemented
200	C₂×C₇×Q₈	—	2	4	no	—
201	C₂×C₇×Q₈	—	2	2	no	—
202	C₂×C₇×Q₈	—	2	4	no	—
203	C₂×C₇×Q₈	—	2	2	no	—
204	C₂×C₇×Q₈	—	4	2	no	—
205	C₂×C₇×Q₈	—	4	8	no	—
206	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
207	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
208	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
209	C₂×C₇×D₄	—	2	2	no	—
210	C₂×C₇×D₄	C₂⁴	1	2	yes	—
211	C₂²×C₄×C₇	—	2	2	no	—
212	C₂²×C₄×C₇	—	2	4	no	—
213	C₂²×C₄×C₇	—	2	2	no	—
214	C₂²×C₄×C₇	—	2	2	no	—
215	C₂²×C₄×C₇	—	2	4	no	—
216	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
217	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
218	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
219	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
220	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
221	C₇×Q₁₆	C₂×Q₈	1	4	yes	complemented
222	C₇×SD₁₆	C₂×Q₈	1	8	yes	complemented
223	C₇×D₈	C₂×Q₈	1	4	yes	complemented
224	C₂×C₇×C₈	—	2	2	no	—
225	C₂×C₇×C₈	—	2	4	no	—
226	C₂×C₇×C₈	C₂×Q₈	1	4	yes	complemented
227	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
228	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
229	C₂×C₇×C₈	C₂⁴	1	4	yes	—
230	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
231	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
232	C₇×C₄⋊C₄	—	2	4	no	—
233	C₇×C₄⋊C₄	—	2	4	no	—
234	C₇×C₄⋊C₄	—	2	4	no	—
235	C₇×C₄⋊C₄	—	2	4	no	—
236	C₇×C₄⋊C₄	—	2	2	no	—
237	C₇×C₄⋊C₄	—	2	4	no	—
238	C₇×C₄⋊C₄	—	2	4	no	—
239	C₇×C₄⋊C₄	—	2	2	no	—
240	C₇×C₄⋊C₄	—	2	2	no	—
241	C₇×C₄⋊C₄	—	2	4	no	—
242	C₇×C₄⋊C₄	—	2	2	no	—
243	C₇×C₄⋊C₄	—	2	4	no	—
244	C₇×C₄⋊C₄	—	2	4	no	—
245	C₇×C₄⋊C₄	—	2	4	no	—
246	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
247	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
248	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
249	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
250	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
251	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
252	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
253	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
254	C₇×C₂²⋊C₄	—	2	2	no	—
255	C₇×C₂²⋊C₄	—	2	4	no	—
256	C₇×C₂²⋊C₄	—	2	4	no	—
257	C₇×C₂²⋊C₄	—	2	4	no	—
258	C₇×C₂²⋊C₄	—	2	2	no	—
259	C₇×C₂²⋊C₄	—	2	2	no	—
260	C₄²×C₇	—	2	2	no	—
261	C₄²×C₇	—	2	4	no	—
262	C₄²×C₇	—	2	4	no	—
263	C₄²×C₇	—	2	2	no	—
264	C₄²×C₇	—	2	2	no	—
265	C₄²×C₇	—	2	4	no	—
266	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
267	C₄²×C₇	C₂×D₄	1	4	yes	—
268	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
269	C₄²×C₇	C₂⁴	1	1	yes	characteristic
270	C₄²×C₇	C₂⁴	1	1	yes	characteristic
271	C₄²×C₇	C₂⁴	1	4	yes	—
272	C₂×#135	C₂²×C₇	1	1	yes	characteristic
273	Q₈²	C₂²×C₇	1	2	yes	—
274	#557	C₂²×C₇	1	2	yes	—
275	#557	C₂²×C₇	1	4	yes	—
276	#554	—	2	4	no	—
277	#554	—	2	2	no	—
278	#554	C₂²×C₇	1	2	yes	—
279	#554	C₂²×C₇	1	4	yes	—
280	D₄×Q₈	—	2	2	no	—
281	D₄×Q₈	C₂²×C₇	1	2	yes	—
282	#533	—	2	2	no	—
283	#533	—	2	4	no	—
284	#533	C₂²×C₇	1	2	yes	—
285	#533	C₂²×C₇	1	4	yes	—
286	C₄×#41	C₂²×C₇	1	2	yes	—
287	C₄×#41	C₂²×C₇	1	4	yes	—
288	C₂×C₄×Q₈	—	2	2	no	—
289	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
290	#501	C₂²×C₇	1	4	yes	—
291	#501	C₂²×C₇	1	4	yes	—
292	#501	C₂²×C₇	1	4	yes	—
293	#501	C₂²×C₇	1	4	yes	—
294	#500	C₂²×C₇	1	1	yes	characteristic
295	#500	C₂²×C₇	1	1	yes	characteristic
296	#499	C₂²×C₇	1	2	yes	—
297	#499	C₂²×C₇	1	2	yes	—
298	#498	C₂²×C₇	1	2	yes	—
299	#479	C₂²×C₇	1	4	yes	—
300	#479	C₂²×C₇	1	4	yes	—
301	#479	C₂²×C₇	1	4	yes	—
302	#478	C₂²×C₇	1	4	yes	—
303	#478	C₂²×C₇	1	4	yes	—
304	#478	C₂²×C₇	1	4	yes	—
305	#477	C₂²×C₇	1	2	yes	—
306	#477	C₂²×C₇	1	4	yes	—
307	#476	C₂²×C₇	1	2	yes	—
308	#476	C₂²×C₇	1	4	yes	—
309	#475	C₂²×C₇	1	2	yes	—
310	#475	C₂²×C₇	1	4	yes	—
311	#474	C₂²×C₇	1	2	yes	—
312	#474	C₂²×C₇	1	4	yes	—
313	#446	C₂²×C₇	1	2	yes	—
314	#446	C₂²×C₇	1	2	yes	—
315	#446	C₂²×C₇	1	2	yes	—
316	#446	C₂²×C₇	1	2	yes	—
317	#442	C₂²×C₇	1	4	yes	—
318	#441	C₂²×C₇	1	4	yes	—
319	#440	C₂²×C₇	1	4	yes	—
320	#440	C₂²×C₇	1	4	yes	—
321	C₄×Q₁₆	C₂²×C₇	1	1	yes	characteristic
322	C₄×Q₁₆	C₂²×C₇	1	1	yes	characteristic
323	C₄×SD₁₆	C₂²×C₇	1	2	yes	—
324	C₄×SD₁₆	C₂²×C₇	1	2	yes	—
325	C₄×D₈	C₂²×C₇	1	1	yes	characteristic
326	C₄×D₈	C₂²×C₇	1	1	yes	characteristic
327	#427	C₂²×C₇	1	2	yes	—
328	#427	C₂²×C₇	1	4	yes	—
329	#427	C₂²×C₇	1	2	yes	—
330	#427	C₂²×C₇	1	1	yes	characteristic
331	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
332	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
333	#416	C₂²×C₇	1	2	yes	—
334	#416	C₂²×C₇	1	4	yes	—
335	C₂×#97	C₂²×C₇	1	2	yes	—
336	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
337	C₂³×C₇	—	2	2	no	—
338	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
339	C₇×Q₈	—	2	4	no	—
340	C₇×Q₈	—	2	2	no	—
341	C₇×Q₈	—	4	2	no	—
342	C₇×Q₈	—	4	8	no	—
343	C₇×Q₈	—	4	4	no	—
344	C₇×Q₈	—	8	4	no	—
345	C₇×Q₈	—	8	2	no	—
346	C₇×Q₈	C₂²×Q₈	1	4	yes	—
347	C₇×D₄	—	2	8	no	—
348	C₇×D₄	—	2	2	no	—
349	C₇×D₄	C₂²×Q₈	1	4	yes	—
350	C₂×C₄×C₇	—	2	2	no	—
351	C₂×C₄×C₇	—	2	8	no	—
352	C₂×C₄×C₇	—	2	2	no	—
353	C₂×C₄×C₇	—	2	4	no	—
354	C₂×C₄×C₇	—	2	2	no	—
355	C₂×C₄×C₇	—	2	4	no	—
356	C₂×C₄×C₇	—	2	2	no	—
357	C₂×C₄×C₇	—	4	1	no	—
358	C₂×C₄×C₇	—	4	2	no	—
359	C₂×C₄×C₇	—	4	2	no	—
360	C₂×C₄×C₇	—	4	8	no	—
361	C₂×C₄×C₇	—	4	4	no	—
362	C₂×C₄×C₇	—	8	2	no	—
363	C₂×C₄×C₇	—	8	4	no	—
364	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
365	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
366	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
367	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
368	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
369	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
370	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
371	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
372	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
373	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
374	C₇×C₈	—	2	1	no	—
375	C₇×C₈	—	2	1	no	—
376	C₇×C₈	—	2	4	no	—
377	C₇×C₈	C₂²×Q₈	1	2	yes	—
378	C₇×C₈	C₂²×Q₈	1	2	yes	—
379	C₂×#41	C₂³×C₇	1	2	yes	—
380	C₂²×Q₈	—	2	2	no	—
381	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
382	#135	C₇×Q₈	1	8	yes	complemented
383	C₂×Q₁₆	C₂³×C₇	1	1	yes	characteristic
384	C₂×SD₁₆	C₂³×C₇	1	2	yes	—
385	C₂×D₈	C₂³×C₇	1	1	yes	characteristic
386	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
387	C₂²×C₈	C₂³×C₇	1	2	yes	—
388	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
389	C₄⋊Q₈	—	2	2	no	—
390	C₄⋊Q₈	—	2	4	no	—
391	C₄⋊Q₈	—	2	2	no	—
392	C₄⋊Q₈	—	2	4	no	—
393	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
394	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
395	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
396	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
397	#125	—	2	4	no	—
398	#125	—	2	4	no	—
399	#125	—	2	4	no	—
400	#125	C₂³×C₇	1	4	yes	—
401	#125	C₂³×C₇	1	4	yes	—
402	#125	C₂³×C₇	1	4	yes	—
403	C₂²⋊Q₈	—	2	2	no	—
404	C₂²⋊Q₈	—	2	4	no	—
405	C₂²⋊Q₈	—	2	2	no	—
406	C₂²⋊Q₈	—	2	4	no	—
407	C₂²⋊Q₈	—	2	2	no	—
408	C₂²⋊Q₈	—	2	4	no	—
409	C₂²⋊Q₈	—	2	2	no	—
410	C₂²⋊Q₈	—	2	4	no	—
411	C₂²⋊Q₈	—	2	4	no	—
412	C₂²⋊Q₈	—	2	4	no	—
413	C₂²⋊Q₈	—	2	4	no	—
414	C₄×Q₈	—	2	2	no	—
415	C₄×Q₈	—	2	4	no	—
416	C₄×Q₈	—	2	2	no	—
417	C₄×Q₈	—	2	2	no	—
418	C₄×Q₈	—	2	4	no	—
419	C₄×Q₈	—	2	4	no	—
420	C₄×Q₈	—	2	2	no	—
421	C₄×Q₈	C₇×D₄	1	2	yes	—
422	C₄×Q₈	C₇×D₄	1	2	yes	—
423	C₄×Q₈	C₂³×C₇	1	2	yes	—
424	C₄×Q₈	C₂³×C₇	1	4	yes	—
425	C₄×Q₈	C₂³×C₇	1	2	yes	—
426	C₄×Q₈	C₂³×C₇	1	4	yes	—
427	C₄×Q₈	C₂³×C₇	1	2	yes	—
428	C₄×Q₈	C₂³×C₇	1	2	yes	—
429	C₄×D₄	—	2	2	no	—
430	C₄×D₄	—	2	4	no	—
431	C₄×D₄	—	2	2	no	—
432	C₄×D₄	—	2	4	no	—
433	C₄×D₄	C₂³×C₇	1	2	yes	—
434	C₄×D₄	C₂³×C₇	1	4	yes	—
435	C₄×D₄	C₂³×C₇	1	2	yes	—
436	C₄×D₄	C₂³×C₇	1	4	yes	—
437	#117	—	2	4	no	—
438	#117	—	2	2	no	—
439	#117	C₂³×C₇	1	2	yes	—
440	#117	C₂³×C₇	1	4	yes	—
441	C₂×C₄⋊C₄	—	2	4	no	—
442	C₂×C₄⋊C₄	—	2	2	no	—
443	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
444	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
445	C₂×C₄²	—	2	2	no	—
446	C₂×C₄²	—	2	4	no	—
447	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
448	C₂×C₄²	C₂³×C₇	1	2	yes	—
449	#107	C₂³×C₇	1	1	yes	characteristic
450	#107	C₂³×C₇	1	1	yes	characteristic
451	#107	C₂³×C₇	1	2	yes	—
452	#107	C₂³×C₇	1	2	yes	—
453	#107	C₂³×C₇	1	2	yes	—
454	#107	C₂³×C₇	1	4	yes	—
455	#107	C₂³×C₇	1	4	yes	—
456	#107	C₂³×C₇	1	2	yes	—
457	#106	C₂³×C₇	1	2	yes	—
458	#106	C₂³×C₇	1	4	yes	—
459	#106	C₂³×C₇	1	4	yes	—
460	#106	C₂³×C₇	1	4	yes	—
461	#106	C₂³×C₇	1	4	yes	—
462	C₄⋊C₈	C₂³×C₇	1	2	yes	—
463	C₄⋊C₈	C₂³×C₇	1	4	yes	—
464	C₄⋊C₈	C₂³×C₇	1	4	yes	—
465	C₄⋊C₈	C₂³×C₇	1	2	yes	—
466	#103	C₂³×C₇	1	2	yes	—
467	#103	C₂³×C₇	1	2	yes	—
468	#103	C₂³×C₇	1	4	yes	—
469	#103	C₂³×C₇	1	4	yes	—
470	#102	C₂³×C₇	1	2	yes	—
471	#102	C₂³×C₇	1	2	yes	—
472	#102	C₂³×C₇	1	4	yes	—
473	#102	C₂³×C₇	1	4	yes	—
474	#97	C₂³×C₇	1	4	yes	—
475	#97	C₂³×C₇	1	4	yes	—
476	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
477	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
478	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
479	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
480	C₂²×C₇	—	2	1	no	—
481	C₂²×C₇	—	2	1	no	—
482	C₂²×C₇	—	2	4	no	—
483	C₂²×C₇	—	4	2	no	—
484	C₂²×C₇	—	4	2	no	—
485	C₂²×C₇	D₄×Q₈	1	2	yes	—
486	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
487	C₄×C₇	—	2	1	no	—
488	C₄×C₇	—	2	4	no	—
489	C₄×C₇	—	2	1	no	—
490	C₄×C₇	—	2	4	no	—
491	C₄×C₇	—	2	1	no	—
492	C₄×C₇	—	2	4	no	—
493	C₄×C₇	—	4	2	no	—
494	C₄×C₇	—	4	4	no	—
495	C₄×C₇	—	4	4	no	—
496	C₄×C₇	—	4	2	no	—
497	C₄×C₇	D₄×Q₈	1	2	yes	—
498	C₄×C₇	#575	1	2	yes	—
499	C₄×C₇	C₂³×Q₈	1	1	yes	characteristic
500	C₄×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
501	#41	—	2	4	no	—
502	#41	C₂×C₇×Q₈	1	8	yes	complemented
503	C₂×Q₈	—	2	4	no	—
504	C₂×Q₈	—	2	2	no	—
505	C₂×Q₈	—	2	4	no	—
506	C₂×Q₈	—	2	2	no	—
507	C₂×Q₈	—	4	2	no	—
508	C₂×Q₈	—	4	8	no	—
509	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
510	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
511	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
512	C₂×D₄	—	2	2	no	—
513	C₂×D₄	C₂⁴×C₇	1	2	yes	—
514	C₂²×C₄	—	2	2	no	—
515	C₂²×C₄	—	2	4	no	—
516	C₂²×C₄	—	2	2	no	—
517	C₂²×C₄	—	2	2	no	—
518	C₂²×C₄	—	2	4	no	—
519	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
520	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
521	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
522	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
523	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
524	Q₁₆	C₂×C₇×Q₈	1	4	yes	complemented
525	SD₁₆	C₂×C₇×Q₈	1	8	yes	complemented
526	D₈	C₂×C₇×Q₈	1	4	yes	complemented
527	C₂×C₈	—	2	2	no	—
528	C₂×C₈	—	2	4	no	—
529	C₂×C₈	C₂×C₇×Q₈	1	4	yes	complemented
530	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
531	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
532	C₂×C₈	C₂⁴×C₇	1	4	yes	—
533	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
534	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
535	C₄⋊C₄	—	2	4	no	—
536	C₄⋊C₄	—	2	4	no	—
537	C₄⋊C₄	—	2	4	no	—
538	C₄⋊C₄	—	2	4	no	—
539	C₄⋊C₄	—	2	2	no	—
540	C₄⋊C₄	—	2	4	no	—
541	C₄⋊C₄	—	2	4	no	—
542	C₄⋊C₄	—	2	2	no	—
543	C₄⋊C₄	—	2	2	no	—
544	C₄⋊C₄	—	2	4	no	—
545	C₄⋊C₄	—	2	2	no	—
546	C₄⋊C₄	—	2	4	no	—
547	C₄⋊C₄	—	2	4	no	—
548	C₄⋊C₄	—	2	4	no	—
549	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
550	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
551	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
552	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
553	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
554	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
555	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
556	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
557	C₂²⋊C₄	—	2	2	no	—
558	C₂²⋊C₄	—	2	4	no	—
559	C₂²⋊C₄	—	2	4	no	—
560	C₂²⋊C₄	—	2	4	no	—
561	C₂²⋊C₄	—	2	2	no	—
562	C₂²⋊C₄	—	2	2	no	—
563	C₄²	—	2	2	no	—
564	C₄²	—	2	4	no	—
565	C₄²	—	2	2	no	—
566	C₄²	—	2	4	no	—
567	C₄²	—	2	4	no	—
568	C₄²	—	2	2	no	—
569	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
570	C₄²	C₂×C₇×D₄	1	4	yes	—
571	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
572	C₄²	C₂⁴×C₇	1	1	yes	characteristic
573	C₄²	C₂⁴×C₇	1	1	yes	characteristic
574	C₄²	C₂⁴×C₇	1	4	yes	—
575	C₂×C₇	—	2	2	no	—
576	C₂×C₇	—	4	4	no	—
577	C₂×C₇	C₂×D₄×Q₈	1	1	yes	characteristic
578	C₂×C₇	C₂×#575	1	1	yes	characteristic
579	C₂×C₇	#3584	1	1	yes	characteristic
580	C₂³	—	2	2	no	—
581	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
582	Q₈	—	2	2	no	—
583	Q₈	—	2	4	no	—
584	Q₈	—	4	2	no	—
585	Q₈	—	4	8	no	—
586	Q₈	—	4	4	no	—
587	Q₈	—	8	4	no	—
588	Q₈	—	8	2	no	—
589	Q₈	C₂²×C₇×Q₈	1	4	yes	—
590	D₄	—	2	8	no	—
591	D₄	—	2	2	no	—
592	D₄	C₂²×C₇×Q₈	1	4	yes	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	8	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	4	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	4	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	4	1	no	—
601	C₂×C₄	—	4	2	no	—
602	C₂×C₄	—	4	2	no	—
603	C₂×C₄	—	4	8	no	—
604	C₂×C₄	—	4	4	no	—
605	C₂×C₄	—	8	2	no	—
606	C₂×C₄	—	8	4	no	—
607	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
608	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
609	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
610	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
611	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
612	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
613	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
614	C₂×C₄	C₂²×C₇×Q₈	1	2	yes	—
615	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
616	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
617	C₈	—	2	1	no	—
618	C₈	—	2	1	no	—
619	C₈	—	2	4	no	—
620	C₈	C₂²×C₇×Q₈	1	2	yes	—
621	C₈	C₂²×C₇×Q₈	1	2	yes	—
622	C₇	#61365	1	1	yes	characteristic, direct product factor, Sylow
623	C₂²	—	2	1	no	—
624	C₂²	—	2	1	no	—
625	C₂²	—	2	4	no	—
626	C₂²	—	4	2	no	—
627	C₂²	—	4	2	no	—
628	C₂²	C₇×D₄×Q₈	1	2	yes	—
629	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
630	C₄	—	2	1	no	—
631	C₄	—	2	4	no	—
632	C₄	—	2	1	no	—
633	C₄	—	2	4	no	—
634	C₄	—	2	1	no	—
635	C₄	—	2	4	no	—
636	C₄	—	4	4	no	—
637	C₄	—	4	2	no	—
638	C₄	—	4	2	no	—
639	C₄	—	4	4	no	—
640	C₄	C₇×D₄×Q₈	1	2	yes	—
641	C₄	C₇×#575	1	2	yes	—
642	C₄	C₂³×C₇×Q₈	1	1	yes	characteristic
643	C₄	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
644	C₂	—	2	2	no	—
645	C₂	—	4	4	no	—
646	C₂	C₂×C₇×D₄×Q₈	1	1	yes	characteristic
647	C₂	C₂×C₇×#575	1	1	yes	characteristic
648	C₂	C₇×#3584	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
socle	C₂²×C₇	C₂³×D₄
prime radical	C₂×C₇×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×Q₈×#41	C₂	semidirect
2. C₇×#3390	C₂	semidirect
3. C₇×#3389	C₂	semidirect
4. C₇×#3388	C₂	semidirect
5. C₇×#3387	C₂	semidirect
6. C₇×#3386	C₂	semidirect
7. C₇×#3385	C₂	semidirect
8. C₇×#3384	C₂	semidirect
9. C₇×#3162	C₂	semidirect
10. C₇×#3160	C₂	semidirect
11. C₇×#3076	C₂	semidirect
12. C₇×#3076	C₂	semidirect
13. C₂×C₇×#446	C₂	semidirect
81. #61365	C₇	direct (N × H)
85. C₇×#135	Q₈	semidirect
183. Q₈×#41	C₂×C₇	semidirect
184. #3390	C₂×C₇	semidirect
185. #3389	C₂×C₇	semidirect
186. #3388	C₂×C₇	semidirect
187. #3387	C₂×C₇	semidirect
188. #3386	C₂×C₇	semidirect
189. #3385	C₂×C₇	semidirect
190. #3384	C₂×C₇	semidirect
191. #3162	C₂×C₇	semidirect
192. #3160	C₂×C₇	semidirect
193. #3076	C₂×C₇	semidirect
194. #3076	C₂×C₇	semidirect
195. C₂×#446	C₂×C₇	semidirect
199. C₇×#41	C₂×Q₈	semidirect
221. C₇×Q₁₆	C₂×Q₈	semidirect
222. C₇×SD₁₆	C₂×Q₈	semidirect
223. C₇×D₈	C₂×Q₈	semidirect
226. C₂×C₇×C₈	C₂×Q₈	semidirect
382. #135	C₇×Q₈	semidirect
502. #41	C₂×C₇×Q₈	semidirect
524. Q₁₆	C₂×C₇×Q₈	semidirect
525. SD₁₆	C₂×C₇×Q₈	semidirect
526. D₈	C₂×C₇×Q₈	semidirect
529. C₂×C₈	C₂×C₇×Q₈	semidirect
622. C₇	#61365	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×Q₈×#41	2
C₇×#3390	2
C₇×#3389	2
C₇×#3388	2
C₇×#3387	2
C₇×#3386	2
C₇×#3385	2
C₇×#3384	2
C₇×#3162	2
C₇×#3160	2
C₇×#3076	2
C₂×C₇×#446	2
C₇×#2943	2
C₄×C₇×#135	2
#61365	7

Subgroups — C7×#61365 #13238581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#61365 #13238581