Subgroups — C₇×#3194 #11742503

Order 896.

466 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#525	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#121	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#483	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#482	C₂	1	2	yes	maximal, complemented
5	C₇×#480	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#469	C₂	1	2	yes	maximal, complemented
7	C₇×#468	C₂	1	2	yes	maximal, complemented
8	C₇×#450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#449	C₂	1	2	yes	maximal, complemented
10	C₇×#447	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#407	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#41	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₇×D₄	—	2	1	no	—
14	C₂²×C₇×D₄	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
16	C₂×C₇×SD₁₆	C₂²	1	2	yes	complemented
17	C₂×C₇×D₈	C₂²	1	2	yes	complemented
18	C₂×C₇×M₁₆	C₂²	1	2	yes	complemented
19	C₇×#123	—	2	1	no	—
20	C₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
21	C₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
22	C₇×#121	—	2	1	no	—
23	C₇×#121	—	2	1	no	—
24	C₇×#121	C₂²	1	2	yes	—
25	C₇×#121	C₂²	1	1	yes	characteristic
26	C₇×#121	C₂²	1	2	yes	—
27	C₇×#121	C₂²	1	1	yes	characteristic, complemented
28	C₇×#120	—	2	1	no	—
29	C₄×C₇×D₄	—	2	1	no	—
30	C₄×C₇×D₄	—	2	1	no	—
31	C₇×#117	C₂²	1	1	yes	characteristic, complemented
32	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
33	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
34	C₇×#107	C₂²	1	2	yes	complemented
35	C₇×#106	C₂²	1	2	yes	complemented
36	C₇×#103	C₂²	1	2	yes	complemented
37	C₇×#102	C₂²	1	2	yes	complemented
38	C₇×#102	C₂²	1	2	yes	complemented
39	C₇×#102	C₂²	1	2	yes	complemented
40	C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
41	C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	C₇×C₂²⋊C₈	C₂²	1	2	yes	complemented
43	#3194	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂⁴×C₇	—	2	1	no	—
45	C₂⁴×C₇	D₄	1	1	yes	characteristic
46	C₇×#41	—	4	2	no	—
47	C₂×C₇×Q₈	C₂³	1	1	yes	characteristic
48	C₂×C₇×D₄	—	2	1	no	—
49	C₂×C₇×D₄	—	2	1	no	—
50	C₂×C₇×D₄	—	2	1	no	—
51	C₂×C₇×D₄	—	2	2	no	—
52	C₂×C₇×D₄	—	2	2	no	—
53	C₂×C₇×D₄	—	2	1	no	—
54	C₂×C₇×D₄	—	4	2	no	—
55	C₂×C₇×D₄	—	4	4	no	—
56	C₂×C₇×D₄	C₂³	1	2	yes	—
57	C₂×C₇×D₄	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₇	—	2	1	no	—
59	C₂²×C₄×C₇	—	2	1	no	—
60	C₂²×C₄×C₇	—	4	2	no	—
61	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
62	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
63	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
67	C₇×SD₁₆	—	4	2	no	—
68	C₇×D₈	—	4	2	no	—
69	C₇×M₁₆	—	4	2	no	—
70	C₂×C₇×C₈	C₂³	1	2	yes	—
71	C₂×C₇×C₈	C₂³	1	2	yes	—
72	C₇×C₄⋊C₄	—	2	1	no	—
73	C₇×C₄⋊C₄	—	2	1	no	—
74	C₇×C₄⋊C₄	—	2	1	no	—
75	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
76	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
77	C₇×C₄⋊C₄	C₂³	1	2	yes	—
78	C₇×C₂²⋊C₄	—	2	1	no	—
79	C₇×C₂²⋊C₄	—	2	1	no	—
80	C₇×C₂²⋊C₄	—	2	1	no	—
81	C₇×C₂²⋊C₄	—	2	2	no	—
82	C₇×C₂²⋊C₄	—	2	2	no	—
83	C₇×C₂²⋊C₄	—	2	1	no	—
84	C₇×C₂²⋊C₄	—	2	1	no	—
85	C₄²×C₇	—	2	1	no	—
86	#525	C₂×C₇	1	1	yes	characteristic, complemented
87	C₂×#121	C₂×C₇	1	1	yes	characteristic, complemented
88	#483	C₂×C₇	1	1	yes	characteristic, complemented
89	#482	C₂×C₇	1	2	yes	complemented
90	#480	C₂×C₇	1	1	yes	characteristic, complemented
91	#469	C₂×C₇	1	2	yes	complemented
92	#468	C₂×C₇	1	2	yes	complemented
93	#450	C₂×C₇	1	1	yes	characteristic, complemented
94	#449	C₂×C₇	1	2	yes	complemented
95	#447	C₂×C₇	1	1	yes	characteristic, complemented
96	#407	C₂×C₇	1	1	yes	characteristic, complemented
97	C₂³×C₇	—	2	2	no	—
98	C₂³×C₇	—	2	1	no	—
99	C₂³×C₇	—	2	2	no	—
100	C₂³×C₇	—	4	1	no	—
101	C₂³×C₇	—	4	1	no	—
102	C₂³×C₇	—	4	2	no	—
103	C₂³×C₇	—	4	2	no	—
104	C₂³×C₇	—	4	2	no	—
105	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
106	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
107	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
108	C₇×Q₈	—	2	2	no	—
109	C₇×D₄	—	2	2	no	—
110	C₇×D₄	—	2	2	no	—
111	C₇×D₄	—	2	2	no	—
112	C₇×D₄	—	4	2	no	—
113	C₇×D₄	—	4	2	no	—
114	C₇×D₄	—	4	4	no	—
115	C₇×D₄	—	4	4	no	—
116	C₇×D₄	—	4	2	no	—
117	C₇×D₄	—	4	2	no	—
118	C₂×C₄×C₇	—	2	2	no	—
119	C₂×C₄×C₇	—	2	2	no	—
120	C₂×C₄×C₇	—	2	2	no	—
121	C₂×C₄×C₇	—	2	1	no	—
122	C₂×C₄×C₇	—	2	1	no	—
123	C₂×C₄×C₇	—	2	1	no	—
124	C₂×C₄×C₇	—	2	2	no	—
125	C₂×C₄×C₇	—	4	1	no	—
126	C₂×C₄×C₇	—	4	1	no	—
127	C₂×C₄×C₇	—	4	1	no	—
128	C₂×C₄×C₇	—	4	2	no	—
129	C₂×C₄×C₇	—	4	2	no	—
130	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
134	C₇×C₈	—	2	2	no	—
135	C₇×C₈	—	2	2	no	—
136	C₂×#41	C₂²×C₇	1	1	yes	characteristic, complemented
137	C₂²×D₄	—	2	1	no	—
138	C₂²×D₄	C₂²×C₇	1	1	yes	characteristic
139	C₂³×C₄	C₂²×C₇	1	1	yes	characteristic
140	C₂×SD₁₆	C₂²×C₇	1	2	yes	complemented
141	C₂×D₈	C₂²×C₇	1	2	yes	complemented
142	C₂×M₁₆	C₂²×C₇	1	2	yes	complemented
143	#123	—	2	1	no	—
144	C₂²⋊Q₈	C₂²×C₇	1	1	yes	characteristic
145	C₂²⋊Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
146	#121	—	2	1	no	—
147	#121	—	2	1	no	—
148	#121	C₂²×C₇	1	2	yes	—
149	#121	C₂²×C₇	1	2	yes	—
150	#121	C₂²×C₇	1	1	yes	characteristic
151	#121	C₂²×C₇	1	1	yes	characteristic, complemented
152	#120	—	2	1	no	—
153	C₄×D₄	—	2	1	no	—
154	C₄×D₄	—	2	1	no	—
155	#117	C₂²×C₇	1	1	yes	characteristic, complemented
156	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
157	C₂×C₂²⋊C₄	—	2	1	no	—
158	#107	C₂²×C₇	1	2	yes	complemented
159	#106	C₂²×C₇	1	2	yes	complemented
160	#103	C₂²×C₇	1	2	yes	complemented
161	#102	C₂²×C₇	1	2	yes	complemented
162	#102	C₂²×C₇	1	2	yes	complemented
163	#102	C₂²×C₇	1	2	yes	complemented
164	C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
165	C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
166	C₂²⋊C₈	C₂²×C₇	1	2	yes	complemented
167	C₂²×C₇	—	2	1	no	—
168	C₂²×C₇	—	2	1	no	—
169	C₂²×C₇	—	2	1	no	—
170	C₂²×C₇	—	2	1	no	—
171	C₂²×C₇	—	2	1	no	—
172	C₂²×C₇	—	2	1	no	—
173	C₂²×C₇	—	2	1	no	—
174	C₂²×C₇	—	2	1	no	—
175	C₂²×C₇	—	4	4	no	—
176	C₂²×C₇	—	4	1	no	—
177	C₂²×C₇	—	4	2	no	—
178	C₂²×C₇	—	4	2	no	—
179	C₂²×C₇	—	4	1	no	—
180	C₂²×C₇	—	4	1	no	—
181	C₂²×C₇	—	4	2	no	—
182	C₂²×C₇	—	8	4	no	—
183	C₂²×C₇	AGL(1,ℤ/8)	1	2	yes	—
184	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
185	C₄×C₇	—	2	1	no	—
186	C₄×C₇	—	2	1	no	—
187	C₄×C₇	—	2	1	no	—
188	C₄×C₇	—	4	1	no	—
189	C₄×C₇	—	4	2	no	—
190	C₄×C₇	—	4	1	no	—
191	C₄×C₇	—	4	1	no	—
192	C₄×C₇	H(5,𝔽₂)	1	2	yes	—
193	C₂⁴	—	2	1	no	—
194	C₂⁴	C₇×D₄	1	1	yes	characteristic
195	#41	—	4	2	no	—
196	C₂×Q₈	C₂³×C₇	1	1	yes	characteristic
197	C₂×D₄	—	2	1	no	—
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	4	4	no	—
205	C₂×D₄	C₂³×C₇	1	2	yes	—
206	C₂×D₄	C₂³×C₇	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	C₇×D₄	1	1	yes	characteristic
211	C₂²×C₄	C₇×D₄	1	1	yes	characteristic
212	C₂²×C₄	C₇×D₄	1	1	yes	characteristic
213	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
214	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
215	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
216	SD₁₆	—	4	2	no	—
217	D₈	—	4	2	no	—
218	M₁₆	—	4	2	no	—
219	C₂×C₈	C₂³×C₇	1	2	yes	—
220	C₂×C₈	C₂³×C₇	1	2	yes	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
225	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
226	C₄⋊C₄	C₂³×C₇	1	2	yes	—
227	C₂²⋊C₄	—	2	1	no	—
228	C₂²⋊C₄	—	2	1	no	—
229	C₂²⋊C₄	—	2	1	no	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₂²⋊C₄	—	2	2	no	—
232	C₂²⋊C₄	—	2	1	no	—
233	C₂²⋊C₄	—	2	1	no	—
234	C₄²	—	2	1	no	—
235	C₂×C₇	—	2	2	no	—
236	C₂×C₇	—	4	1	no	—
237	C₂×C₇	—	4	1	no	—
238	C₂×C₇	—	8	1	no	—
239	C₂×C₇	—	8	2	no	—
240	C₂×C₇	#534	1	1	yes	characteristic
241	C₂×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
242	C₂×C₇	#575	1	1	yes	characteristic
243	C₂³	—	2	2	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	2	no	—
246	C₂³	—	4	1	no	—
247	C₂³	—	4	1	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	4	2	no	—
251	C₂³	C₂×C₇×D₄	1	1	yes	characteristic
252	C₂³	C₂×C₇×D₄	1	1	yes	characteristic
253	C₂³	C₂×C₇×D₄	1	1	yes	characteristic
254	Q₈	—	2	2	no	—
255	D₄	—	2	2	no	—
256	D₄	—	2	2	no	—
257	D₄	—	2	2	no	—
258	D₄	—	4	2	no	—
259	D₄	—	4	2	no	—
260	D₄	—	4	4	no	—
261	D₄	—	4	4	no	—
262	D₄	—	4	2	no	—
263	D₄	—	4	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
277	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
278	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
279	C₂×C₄	C₂⁴×C₇	1	1	yes	characteristic
280	C₈	—	2	2	no	—
281	C₈	—	2	2	no	—
282	C₇	#3194	1	1	yes	characteristic, direct product factor, Sylow
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	4	4	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	2	no	—
298	C₂²	—	8	4	no	—
299	C₂²	C₇×AGL(1,ℤ/8)	1	2	yes	—
300	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	2	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	C₇×H(5,𝔽₂)	1	2	yes	—
309	C₂	—	2	2	no	—
310	C₂	—	4	1	no	—
311	C₂	—	4	1	no	—
312	C₂	—	8	2	no	—
313	C₂	—	8	1	no	—
314	C₂	C₇×#534	1	1	yes	characteristic
315	C₂	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
316	C₂	C₇×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#525	C₂	semidirect
2. C₂×C₇×#121	C₂	semidirect
3. C₇×#483	C₂	semidirect
4. C₇×#482	C₂	semidirect
5. C₇×#480	C₂	semidirect
6. C₇×#469	C₂	semidirect
7. C₇×#468	C₂	semidirect
8. C₇×#450	C₂	semidirect
9. C₇×#449	C₂	semidirect
10. C₇×#447	C₂	semidirect
11. C₇×#407	C₂	semidirect
12. C₂×C₇×#41	C₂²	semidirect
16. C₂×C₇×SD₁₆	C₂²	semidirect
17. C₂×C₇×D₈	C₂²	semidirect
18. C₂×C₇×M₁₆	C₂²	semidirect
21. C₇×C₂²⋊Q₈	C₂²	semidirect
27. C₇×#121	C₂²	semidirect
31. C₇×#117	C₂²	semidirect
34. C₇×#107	C₂²	semidirect
35. C₇×#106	C₂²	semidirect
36. C₇×#103	C₂²	semidirect
37. C₇×#102	C₂²	semidirect
38. C₇×#102	C₂²	semidirect
39. C₇×#102	C₂²	semidirect
42. C₇×C₂²⋊C₈	C₂²	semidirect
43. #3194	C₇	direct (N × H)
86. #525	C₂×C₇	semidirect
87. C₂×#121	C₂×C₇	semidirect
88. #483	C₂×C₇	semidirect
89. #482	C₂×C₇	semidirect
90. #480	C₂×C₇	semidirect
91. #469	C₂×C₇	semidirect
92. #468	C₂×C₇	semidirect
93. #450	C₂×C₇	semidirect
94. #449	C₂×C₇	semidirect
95. #447	C₂×C₇	semidirect
96. #407	C₂×C₇	semidirect
136. C₂×#41	C₂²×C₇	semidirect
140. C₂×SD₁₆	C₂²×C₇	semidirect
141. C₂×D₈	C₂²×C₇	semidirect
142. C₂×M₁₆	C₂²×C₇	semidirect
145. C₂²⋊Q₈	C₂²×C₇	semidirect
151. #121	C₂²×C₇	semidirect
155. #117	C₂²×C₇	semidirect
158. #107	C₂²×C₇	semidirect
159. #106	C₂²×C₇	semidirect
160. #103	C₂²×C₇	semidirect
161. #102	C₂²×C₇	semidirect
162. #102	C₂²×C₇	semidirect
163. #102	C₂²×C₇	semidirect
166. C₂²⋊C₈	C₂²×C₇	semidirect
282. C₇	#3194	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×#525	2
C₂×C₇×#121	2
C₇×#483	2
C₇×#482	2
C₇×#480	2
C₇×#469	2
C₇×#468	2
C₇×#450	2
C₇×#449	2
C₇×#447	2
C₇×#407	2
#3194	7

Subgroups — C7×#3194 #11742503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#3194 #11742503