Subgroups — C₇×#22708 #13199924

Order 1792.

910 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₇×#2354	C₂	1	2	yes	maximal
3	C₇×#1948	C₂	1	2	yes	maximal, complemented
4	C₇×#1854	C₂	1	4	yes	maximal, complemented
5	C₇×#1834	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1807	C₂	1	2	yes	maximal, complemented
7	C₄×C₇×#107	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1735	C₂	1	2	yes	maximal, complemented
9	C₂³×C₇×C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
11	C₂×C₇×#117	C₂²	1	4	yes	—
12	C₇×C₈×D₄	C₄	1	8	yes	complemented
13	C₂×C₇×#107	C₂²	1	2	yes	—
14	C₂×C₇×#106	C₂²	1	4	yes	—
15	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
18	C₇×#390	—	2	2	no	—
19	C₇×#388	—	2	4	no	—
20	C₇×#384	C₂²	1	2	yes	—
21	C₇×#380	C₂²	1	4	yes	—
22	C₄×C₇×C₄⋊C₄	—	2	2	no	—
23	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
24	C₇×#340	C₂²	1	1	yes	characteristic
25	C₇×#340	C₂²	1	2	yes	—
26	C₇×#340	C₂²	1	4	yes	—
27	C₇×#340	C₂²	1	1	yes	characteristic
28	C₇×#340	C₂²	1	2	yes	—
29	#22708	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₇×D₄	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₇	C₂³	1	2	yes	—
32	C₂²×C₇×C₈	—	4	4	no	—
33	C₂²×C₇×C₈	C₂×C₄	1	8	yes	—
34	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
36	C₂²×C₇×C₈	C₂³	1	2	yes	—
37	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
38	C₄×C₇×D₄	Q₈	1	4	yes	—
39	C₇×#117	—	2	8	no	—
40	C₇×#117	—	4	4	no	—
41	C₂×C₇×C₄⋊C₄	—	2	2	no	—
42	C₂×C₇×C₄⋊C₄	—	2	2	no	—
43	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
44	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
45	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
46	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
48	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
49	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
50	C₂×C₄²×C₇	—	2	4	no	—
51	C₂×C₄²×C₇	—	2	2	no	—
52	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
53	C₇×#107	—	2	4	no	—
54	C₇×#106	—	4	4	no	—
55	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
56	C₇×C₂²⋊C₈	C₂×C₄	1	8	yes	—
57	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
58	C₇×#95	—	2	4	no	—
59	C₇×#95	—	2	2	no	—
60	C₂×C₈×D₄	C₂×C₇	1	1	yes	characteristic
61	#2354	C₂×C₇	1	2	yes	—
62	#1948	C₂×C₇	1	2	yes	complemented
63	#1854	C₂×C₇	1	4	yes	complemented
64	#1834	C₂×C₇	1	1	yes	characteristic, complemented
65	#1807	C₂×C₇	1	2	yes	complemented
66	C₄×#107	C₂×C₇	1	1	yes	characteristic, complemented
67	#1735	C₂×C₇	1	2	yes	complemented
68	C₂⁴×C₇	C₂×D₄	1	2	yes	—
69	C₂×C₇×D₄	—	2	2	no	—
70	C₂×C₇×D₄	—	2	2	no	—
71	C₂×C₇×D₄	C₄⋊C₄	1	4	yes	—
72	C₂²×C₄×C₇	—	2	4	no	—
73	C₂²×C₄×C₇	—	2	4	no	—
74	C₂²×C₄×C₇	—	2	2	no	—
75	C₂²×C₄×C₇	—	2	2	no	—
76	C₂²×C₄×C₇	—	2	4	no	—
77	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
82	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
83	C₂×C₇×C₈	—	2	2	no	—
84	C₂×C₇×C₈	—	2	4	no	—
85	C₂×C₇×C₈	—	2	2	no	—
86	C₂×C₇×C₈	—	4	8	no	—
87	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
88	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
90	C₇×C₄⋊C₄	—	2	2	no	—
91	C₇×C₄⋊C₄	—	2	2	no	—
92	C₇×C₄⋊C₄	—	2	4	no	—
93	C₇×C₄⋊C₄	—	2	2	no	—
94	C₇×C₄⋊C₄	—	2	4	no	—
95	C₇×C₄⋊C₄	—	4	4	no	—
96	C₇×C₄⋊C₄	—	8	2	no	—
97	C₇×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
98	C₇×C₄⋊C₄	#41	1	4	yes	—
99	C₇×C₂²⋊C₄	—	4	8	no	—
100	C₇×C₂²⋊C₄	—	8	4	no	—
101	C₇×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
102	C₄²×C₇	—	4	8	no	—
103	C₄²×C₇	—	4	4	no	—
104	C₄²×C₇	C₂²×C₄	1	2	yes	—
105	C₄²×C₇	C₂×Q₈	1	2	yes	—
106	C₂³×C₈	C₂²×C₇	1	2	yes	—
107	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
108	C₂×#117	C₂²×C₇	1	4	yes	—
109	C₈×D₄	C₄×C₇	1	8	yes	complemented
110	C₂×#107	C₂²×C₇	1	2	yes	—
111	C₂×#106	C₂²×C₇	1	4	yes	—
112	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
113	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
114	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
115	#390	—	2	2	no	—
116	#388	—	2	4	no	—
117	#384	C₂²×C₇	1	2	yes	—
118	#380	C₂²×C₇	1	4	yes	—
119	C₄×C₄⋊C₄	—	2	2	no	—
120	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
121	#340	C₂²×C₇	1	1	yes	characteristic
122	#340	C₂²×C₇	1	2	yes	—
123	#340	C₂²×C₇	1	2	yes	—
124	#340	C₂²×C₇	1	4	yes	—
125	#340	C₂²×C₇	1	1	yes	characteristic
126	C₂³×C₇	—	2	4	no	—
127	C₂³×C₇	—	2	2	no	—
128	C₂³×C₇	—	2	2	no	—
129	C₂³×C₇	—	2	4	no	—
130	C₂³×C₇	C₂×C₄⋊C₄	1	4	yes	—
131	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
132	C₇×D₄	—	2	8	no	—
133	C₂×C₄×C₇	—	2	4	no	—
134	C₂×C₄×C₇	—	2	2	no	—
135	C₂×C₄×C₇	—	2	4	no	—
136	C₂×C₄×C₇	—	2	4	no	—
137	C₂×C₄×C₇	—	2	8	no	—
138	C₂×C₄×C₇	—	2	4	no	—
139	C₂×C₄×C₇	—	4	2	no	—
140	C₂×C₄×C₇	—	4	2	no	—
141	C₂×C₄×C₇	—	4	4	no	—
142	C₂×C₄×C₇	—	4	2	no	—
143	C₂×C₄×C₇	—	4	2	no	—
144	C₂×C₄×C₇	—	4	4	no	—
145	C₂×C₄×C₇	—	4	2	no	—
146	C₂×C₄×C₇	—	4	4	no	—
147	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
148	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
149	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
150	C₂×C₄×C₇	#135	1	2	yes	—
151	C₂×C₄×C₇	#135	1	2	yes	—
152	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
154	C₂×C₄×C₇	C₂×#41	1	2	yes	—
155	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
156	C₇×C₈	—	2	2	no	—
157	C₇×C₈	—	2	2	no	—
158	C₇×C₈	—	2	4	no	—
159	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
160	C₂³×C₄	C₂³×C₇	1	2	yes	—
161	C₂²×C₈	—	4	4	no	—
162	C₂²×C₈	C₂×C₄×C₇	1	8	yes	—
163	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
164	C₂²×C₈	C₂³×C₇	1	2	yes	—
165	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
166	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
167	C₄×D₄	C₇×Q₈	1	4	yes	—
168	#117	—	2	8	no	—
169	#117	—	4	4	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
173	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
174	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
175	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
176	C₂×C₂²⋊C₄	—	2	4	no	—
177	C₂×C₂²⋊C₄	—	2	4	no	—
178	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
179	C₂×C₄²	—	2	4	no	—
180	C₂×C₄²	—	2	2	no	—
181	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
182	#107	—	2	4	no	—
183	#106	—	4	4	no	—
184	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	—
185	C₂²⋊C₈	C₂×C₄×C₇	1	8	yes	—
186	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
187	#95	—	2	4	no	—
188	#95	—	2	2	no	—
189	C₂²×C₇	—	2	4	no	—
190	C₂²×C₇	—	2	4	no	—
191	C₂²×C₇	—	2	4	no	—
192	C₂²×C₇	—	4	4	no	—
193	C₂²×C₇	—	4	4	no	—
194	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
195	C₂²×C₇	#546	1	2	yes	—
196	C₂²×C₇	C₂×#135	1	1	yes	characteristic
197	C₂²×C₇	C₂×#135	1	1	yes	characteristic
198	C₂²×C₇	#577	1	2	yes	—
199	C₄×C₇	—	2	4	no	—
200	C₄×C₇	—	2	2	no	—
201	C₄×C₇	—	4	4	no	—
202	C₄×C₇	—	8	2	no	—
203	C₄×C₇	—	8	4	no	—
204	C₄×C₇	#427	1	4	yes	—
205	C₄×C₇	C₄×#41	1	1	yes	characteristic
206	C₄×C₇	#518	1	1	yes	characteristic
207	C₄×C₇	#554	1	2	yes	—
208	C₂⁴	C₂×C₇×D₄	1	2	yes	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	C₇×C₄⋊C₄	1	4	yes	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
218	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
219	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
222	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	4	8	no	—
227	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
228	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
229	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₄⋊C₄	—	8	2	no	—
237	C₄⋊C₄	C₇×C₄⋊C₄	1	4	yes	—
238	C₄⋊C₄	C₇×#41	1	4	yes	—
239	C₂²⋊C₄	—	4	8	no	—
240	C₂²⋊C₄	—	8	4	no	—
241	C₂²⋊C₄	C₇×C₄⋊C₄	1	8	yes	—
242	C₄²	—	4	8	no	—
243	C₄²	—	4	4	no	—
244	C₄²	C₂²×C₄×C₇	1	2	yes	—
245	C₄²	C₂×C₇×Q₈	1	2	yes	—
246	C₂×C₇	—	4	4	no	—
247	C₂×C₇	#2349	1	1	yes	characteristic
248	C₂×C₇	C₂×#427	1	1	yes	characteristic
249	C₂×C₇	#2912	1	1	yes	characteristic
250	C₂×C₇	#3303	1	2	yes	—
251	C₂×C₇	#3304	1	2	yes	—
252	C₂³	—	2	4	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	4	no	—
256	C₂³	C₂×C₇×C₄⋊C₄	1	4	yes	—
257	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
258	D₄	—	2	8	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
274	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
275	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
276	C₂×C₄	C₇×#135	1	2	yes	—
277	C₂×C₄	C₇×#135	1	2	yes	—
278	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
279	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
280	C₂×C₄	C₂×C₇×#41	1	2	yes	—
281	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
282	C₈	—	2	2	no	—
283	C₈	—	2	2	no	—
284	C₈	—	2	4	no	—
285	C₇	#22708	1	1	yes	characteristic, direct product factor, Sylow
286	C₂²	—	2	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	4	4	no	—
290	C₂²	—	4	4	no	—
291	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
292	C₂²	C₇×#546	1	2	yes	—
293	C₂²	C₂×C₇×#135	1	1	yes	characteristic
294	C₂²	C₂×C₇×#135	1	1	yes	characteristic
295	C₂²	C₇×#577	1	2	yes	—
296	C₄	—	2	2	no	—
297	C₄	—	2	4	no	—
298	C₄	—	4	4	no	—
299	C₄	—	8	2	no	—
300	C₄	—	8	4	no	—
301	C₄	C₇×#427	1	4	yes	—
302	C₄	C₄×C₇×#41	1	1	yes	characteristic
303	C₄	C₇×#518	1	1	yes	characteristic
304	C₄	C₇×#554	1	2	yes	—
305	C₂	—	4	4	no	—
306	C₂	C₇×#2349	1	1	yes	characteristic
307	C₂	C₂×C₇×#427	1	1	yes	characteristic
308	C₂	C₇×#2912	1	1	yes	characteristic
309	C₂	C₇×#3303	1	2	yes	—
310	C₂	C₇×#3304	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂²×C₇×D₄	D₄
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
3. C₇×#1948	C₂	semidirect
4. C₇×#1854	C₂	semidirect
5. C₇×#1834	C₂	semidirect
6. C₇×#1807	C₂	semidirect
7. C₄×C₇×#107	C₂	semidirect
8. C₇×#1735	C₂	semidirect
12. C₇×C₈×D₄	C₄	semidirect
29. #22708	C₇	direct (N × H)
62. #1948	C₂×C₇	semidirect
63. #1854	C₂×C₇	semidirect
64. #1834	C₂×C₇	semidirect
65. #1807	C₂×C₇	semidirect
66. C₄×#107	C₂×C₇	semidirect
67. #1735	C₂×C₇	semidirect
109. C₈×D₄	C₄×C₇	semidirect
285. C₇	#22708	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×C₈×D₄	2
C₇×#2354	2
C₇×#1948	2
C₇×#1854	2
C₇×#1834	2
C₇×#1807	2
C₄×C₇×#107	2
C₇×#1735	2
#22708	7

Subgroups — C7×#22708 #13199924

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22708 #13199924