Subgroups — C₇×#3171 #11742480

Order 896.

408 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#529	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#501	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#496	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#490	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#489	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#469	C₂	1	2	yes	maximal, complemented
8	C₇×#468	C₂	1	2	yes	maximal, complemented
9	C₄×C₇×SD₁₆	C₂	1	2	yes	maximal, complemented
10	C₄×C₇×D₈	C₂	1	2	yes	maximal, complemented
11	C₄×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#41	—	2	1	no	—
13	C₂×C₇×SD₁₆	C₂²	1	2	yes	—
14	C₂×C₇×D₈	C₂²	1	2	yes	—
15	C₂×C₇×M₁₆	C₂²	1	2	yes	—
16	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₇×#127	C₂²	1	1	yes	characteristic
18	C₇×#126	—	2	1	no	—
19	C₇×#125	C₂²	1	1	yes	characteristic
20	C₇×#124	—	2	1	no	—
21	C₇×#124	C₂²	1	1	yes	characteristic
22	C₇×#123	—	2	1	no	—
23	C₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
24	C₇×#121	—	2	1	no	—
25	C₇×#121	C₂²	1	2	yes	—
26	C₇×#121	C₂²	1	1	yes	characteristic
27	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
28	C₄×C₇×D₄	—	2	1	no	—
29	C₄×C₇×D₄	—	2	1	no	—
30	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
31	C₄×C₇×D₄	C₂²	1	2	yes	—
32	C₇×#117	—	2	1	no	—
33	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
34	C₇×#107	C₂²	1	2	yes	—
35	C₇×#106	C₂²	1	2	yes	—
36	C₇×#103	C₂²	1	2	yes	—
37	C₇×#102	C₂²	1	2	yes	—
38	C₇×#102	C₂²	1	2	yes	—
39	C₇×#102	C₂²	1	2	yes	—
40	C₇×#97	C₂²	1	1	yes	characteristic
41	C₇×#97	C₂²	1	1	yes	characteristic
42	C₄×C₇×C₈	C₂²	1	2	yes	—
43	#3171	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₇×#41	—	4	4	no	—
45	C₂×C₇×Q₈	—	2	1	no	—
46	C₂×C₇×Q₈	C₂³	1	1	yes	characteristic
47	C₂×C₇×D₄	—	2	1	no	—
48	C₂×C₇×D₄	—	2	2	no	—
49	C₂×C₇×D₄	—	2	1	no	—
50	C₂×C₇×D₄	—	2	1	no	—
51	C₂×C₇×D₄	C₂³	1	1	yes	characteristic
52	C₂×C₇×D₄	C₂³	1	2	yes	—
53	C₂²×C₄×C₇	—	2	1	no	—
54	C₂²×C₄×C₇	—	2	2	no	—
55	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
58	C₇×SD₁₆	—	2	4	no	—
59	C₇×D₈	—	2	4	no	—
60	C₇×M₁₆	—	2	4	no	—
61	C₂×C₇×C₈	C₂³	1	2	yes	—
62	C₂×C₇×C₈	C₂³	1	2	yes	—
63	C₇×C₄⋊C₄	—	2	1	no	—
64	C₇×C₄⋊C₄	—	2	1	no	—
65	C₇×C₄⋊C₄	—	2	1	no	—
66	C₇×C₄⋊C₄	—	2	1	no	—
67	C₇×C₄⋊C₄	—	2	1	no	—
68	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
69	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
70	C₇×C₄⋊C₄	C₂³	1	2	yes	—
71	C₇×C₂²⋊C₄	—	2	1	no	—
72	C₇×C₂²⋊C₄	—	2	1	no	—
73	C₇×C₂²⋊C₄	—	2	1	no	—
74	C₇×C₂²⋊C₄	—	2	2	no	—
75	C₇×C₂²⋊C₄	—	2	1	no	—
76	C₇×C₂²⋊C₄	—	2	2	no	—
77	C₇×C₂²⋊C₄	—	2	1	no	—
78	C₄²×C₇	—	2	1	no	—
79	C₄²×C₇	D₄	1	1	yes	characteristic
80	C₄²×C₇	D₄	1	1	yes	characteristic
81	C₄²×C₇	C₂³	1	1	yes	characteristic
82	C₄²×C₇	C₂³	1	1	yes	characteristic
83	#532	C₂×C₇	1	1	yes	characteristic, complemented
84	#529	C₂×C₇	1	1	yes	characteristic, complemented
85	#501	C₂×C₇	1	1	yes	characteristic, complemented
86	#496	C₂×C₇	1	1	yes	characteristic, complemented
87	#490	C₂×C₇	1	1	yes	characteristic, complemented
88	#489	C₂×C₇	1	1	yes	characteristic, complemented
89	#469	C₂×C₇	1	2	yes	complemented
90	#468	C₂×C₇	1	2	yes	complemented
91	C₄×SD₁₆	C₂×C₇	1	2	yes	complemented
92	C₄×D₈	C₂×C₇	1	2	yes	complemented
93	C₄×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
94	C₂³×C₇	—	2	1	no	—
95	C₂³×C₇	—	2	2	no	—
96	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
97	C₇×Q₈	—	2	2	no	—
98	C₇×Q₈	—	4	2	no	—
99	C₇×D₄	—	2	2	no	—
100	C₇×D₄	—	2	2	no	—
101	C₇×D₄	—	2	2	no	—
102	C₇×D₄	—	4	2	no	—
103	C₇×D₄	—	4	2	no	—
104	C₇×D₄	—	4	2	no	—
105	C₇×D₄	—	4	4	no	—
106	C₂×C₄×C₇	—	2	2	no	—
107	C₂×C₄×C₇	—	2	2	no	—
108	C₂×C₄×C₇	—	2	1	no	—
109	C₂×C₄×C₇	—	2	1	no	—
110	C₂×C₄×C₇	—	2	1	no	—
111	C₂×C₄×C₇	—	2	2	no	—
112	C₂×C₄×C₇	—	4	1	no	—
113	C₂×C₄×C₇	—	4	2	no	—
114	C₂×C₄×C₇	—	4	4	no	—
115	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
121	C₇×C₈	—	2	2	no	—
122	C₇×C₈	#41	1	4	yes	complemented
123	C₂×#41	—	2	1	no	—
124	C₂×SD₁₆	C₂²×C₇	1	2	yes	—
125	C₂×D₈	C₂²×C₇	1	2	yes	—
126	C₂×M₁₆	C₂²×C₇	1	2	yes	—
127	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
128	#127	C₂²×C₇	1	1	yes	characteristic
129	#126	—	2	1	no	—
130	#125	C₂²×C₇	1	1	yes	characteristic
131	#124	—	2	1	no	—
132	#124	C₂²×C₇	1	1	yes	characteristic
133	#123	—	2	1	no	—
134	C₂²⋊Q₈	C₂²×C₇	1	1	yes	characteristic
135	#121	—	2	1	no	—
136	#121	C₂²×C₇	1	1	yes	characteristic
137	#121	C₂²×C₇	1	2	yes	—
138	C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
139	C₄×D₄	—	2	1	no	—
140	C₄×D₄	—	2	1	no	—
141	C₄×D₄	C₂²×C₇	1	1	yes	characteristic
142	C₄×D₄	C₂²×C₇	1	2	yes	—
143	#117	—	2	1	no	—
144	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
145	#107	C₂²×C₇	1	2	yes	—
146	#106	C₂²×C₇	1	2	yes	—
147	#103	C₂²×C₇	1	2	yes	—
148	#102	C₂²×C₇	1	2	yes	—
149	#102	C₂²×C₇	1	2	yes	—
150	#102	C₂²×C₇	1	2	yes	—
151	#97	C₂²×C₇	1	1	yes	characteristic
152	#97	C₂²×C₇	1	1	yes	characteristic
153	C₄×C₈	C₂²×C₇	1	2	yes	—
154	C₂²×C₇	—	2	1	no	—
155	C₂²×C₇	—	2	1	no	—
156	C₂²×C₇	—	2	1	no	—
157	C₂²×C₇	—	4	1	no	—
158	C₂²×C₇	—	4	2	no	—
159	C₂²×C₇	—	4	2	no	—
160	C₂²×C₇	—	4	1	no	—
161	C₂²×C₇	—	4	2	no	—
162	C₂²×C₇	—	4	1	no	—
163	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
164	C₄×C₇	—	2	1	no	—
165	C₄×C₇	—	2	1	no	—
166	C₄×C₇	—	2	1	no	—
167	C₄×C₇	—	2	1	no	—
168	C₄×C₇	—	4	1	no	—
169	C₄×C₇	—	4	1	no	—
170	C₄×C₇	—	4	2	no	—
171	C₄×C₇	—	4	1	no	—
172	C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
173	C₄×C₇	C₂×#41	1	2	yes	—
174	#41	—	4	4	no	—
175	C₂×Q₈	—	2	1	no	—
176	C₂×Q₈	C₂³×C₇	1	1	yes	characteristic
177	C₂×D₄	—	2	1	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	C₂³×C₇	1	1	yes	characteristic
182	C₂×D₄	C₂³×C₇	1	2	yes	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	C₇×D₄	1	1	yes	characteristic
186	C₂²×C₄	C₇×D₄	1	1	yes	characteristic
187	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
188	SD₁₆	—	2	4	no	—
189	D₈	—	2	4	no	—
190	M₁₆	—	2	4	no	—
191	C₂×C₈	C₂³×C₇	1	2	yes	—
192	C₂×C₈	C₂³×C₇	1	2	yes	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
199	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
200	C₄⋊C₄	C₂³×C₇	1	2	yes	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₂²⋊C₄	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	C₇×D₄	1	1	yes	characteristic
210	C₄²	C₇×D₄	1	1	yes	characteristic
211	C₄²	C₂³×C₇	1	1	yes	characteristic
212	C₄²	C₂³×C₇	1	1	yes	characteristic
213	C₂×C₇	—	4	1	no	—
214	C₂×C₇	—	8	2	no	—
215	C₂×C₇	—	8	1	no	—
216	C₂×C₇	#532	1	1	yes	characteristic
217	C₂×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
218	C₂×C₇	#575	1	1	yes	characteristic
219	C₂³	—	2	1	no	—
220	C₂³	—	2	2	no	—
221	C₂³	C₂×C₇×D₄	1	1	yes	characteristic
222	Q₈	—	2	2	no	—
223	Q₈	—	4	2	no	—
224	D₄	—	2	2	no	—
225	D₄	—	2	2	no	—
226	D₄	—	2	2	no	—
227	D₄	—	4	2	no	—
228	D₄	—	4	2	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
241	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
242	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
243	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
244	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
245	C₂×C₄	C₂⁴×C₇	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	C₇×#41	1	4	yes	complemented
248	C₇	#3171	1	1	yes	characteristic, direct product factor, Sylow
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	4	1	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	1	no	—
258	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	4	1	no	—
264	C₄	—	4	2	no	—
265	C₄	—	4	1	no	—
266	C₄	—	4	1	no	—
267	C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
268	C₄	C₂×C₇×#41	1	2	yes	—
269	C₂	—	4	1	no	—
270	C₂	—	8	2	no	—
271	C₂	—	8	1	no	—
272	C₂	C₇×#532	1	1	yes	characteristic
273	C₂	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
274	C₂	C₇×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇
socle	C₂²×C₇	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#532	C₂	semidirect
2. C₇×#529	C₂	semidirect
3. C₇×#501	C₂	semidirect
4. C₇×#496	C₂	semidirect
5. C₇×#490	C₂	semidirect
6. C₇×#489	C₂	semidirect
7. C₇×#469	C₂	semidirect
8. C₇×#468	C₂	semidirect
9. C₄×C₇×SD₁₆	C₂	semidirect
10. C₄×C₇×D₈	C₂	semidirect
11. C₄×C₇×M₁₆	C₂	semidirect
43. #3171	C₇	direct (N × H)
83. #532	C₂×C₇	semidirect
84. #529	C₂×C₇	semidirect
85. #501	C₂×C₇	semidirect
86. #496	C₂×C₇	semidirect
87. #490	C₂×C₇	semidirect
88. #489	C₂×C₇	semidirect
89. #469	C₂×C₇	semidirect
90. #468	C₂×C₇	semidirect
91. C₄×SD₁₆	C₂×C₇	semidirect
92. C₄×D₈	C₂×C₇	semidirect
93. C₄×M₁₆	C₂×C₇	semidirect
122. C₇×C₈	#41	semidirect
247. C₈	C₇×#41	semidirect
248. C₇	#3171	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×#532	2
C₇×#529	2
C₇×#501	2
C₇×#496	2
C₇×#490	2
C₇×#489	2
C₇×#469	2
C₇×#468	2
C₄×C₇×SD₁₆	2
C₄×C₇×D₈	2
C₄×C₇×M₁₆	2
#3171	7

Subgroups — C7×#3171 #11742480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#3171 #11742480