Subgroups — #11741225

Order 896.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#131	C₂	1	1	yes	maximal, characteristic
2	#89912	C₂	1	8	yes	maximal, complemented
3	C₂×#6386	C₂	1	6	yes	maximal, complemented
4	C₂×C₇×#41	C₄	1	1	yes	characteristic
5	C₇×#131	C₄	1	4	yes	—
6	C₇×#131	C₂²	1	4	yes	—
7	C₂×C₇×M₁₆	C₄	1	3	yes	—
8	C₂²×C₇×C₈	C₂²	1	3	yes	—
9	#6386	C₂²	1	4	yes	—
10	#6386	C₂²	1	12	yes	complemented
11	C₂×C₇⋊C₁₆	C₂²	1	12	yes	complemented
12	#3407	—	7	1	no	maximal, Sylow
13	C₇×#41	C₈	1	4	yes	—
14	C₇×#41	C₂×C₄	1	4	yes	—
15	C₂×C₇×Q₈	C₈	1	1	yes	characteristic
16	C₂×C₇×D₄	C₈	1	3	yes	—
17	C₂²×C₄×C₇	C₂×C₄	1	3	yes	—
18	C₇×M₁₆	C₂×C₄	1	12	yes	—
19	C₂×C₇×C₈	C₂×C₄	1	3	yes	—
20	C₂×C₇×C₈	C₂×C₄	1	6	yes	—
21	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
22	C₂×C₇×C₈	C₂³	1	6	yes	—
23	C₇⋊C₁₆	C₂³	1	8	yes	—
24	C₂×#131	D₇	1	1	yes	characteristic
25	#504	—	7	8	no	—
26	C₂×M₃₂	—	7	6	no	—
27	C₂³×C₇	C₂×C₈	1	3	yes	—
28	C₇×Q₈	C₂×C₈	1	4	yes	—
29	C₇×D₄	C₂×C₈	1	12	yes	—
30	C₂×C₄×C₇	C₂×C₈	1	3	yes	—
31	C₂×C₄×C₇	C₂×C₈	1	6	yes	—
32	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₇	C₂²×C₄	1	6	yes	—
34	C₇×C₈	C₂²×C₄	1	1	yes	characteristic
35	C₇×C₈	C₂²×C₄	1	6	yes	—
36	C₇×C₈	C₂⁴	1	1	yes	characteristic
37	C₂×#41	Dic₇	1	1	yes	characteristic
38	#131	Dic₇	1	4	yes	—
39	#131	C₂×D₇	1	4	yes	—
40	C₂×M₁₆	Dic₇	1	3	yes	—
41	C₂²×C₈	C₂×D₇	1	3	yes	—
42	M₃₂	—	7	4	no	—
43	M₃₂	—	7	12	no	—
44	C₂×C₁₆	—	7	12	no	—
45	C₂²×C₇	—	4	3	no	—
46	C₂²×C₇	C₂²×C₈	1	1	yes	characteristic
47	C₂²×C₇	C₂²×C₈	1	6	yes	—
48	C₄×C₇	C₂²×C₈	1	1	yes	characteristic
49	C₄×C₇	C₂²×C₈	1	6	yes	—
50	C₄×C₇	C₂³×C₄	1	1	yes	characteristic
51	#41	C₇⋊C₈	1	4	yes	—
52	#41	C₂×Dic₇	1	4	yes	—
53	C₂×Q₈	C₇⋊C₈	1	1	yes	characteristic
54	C₂×D₄	C₇⋊C₈	1	3	yes	—
55	C₂²×C₄	C₂×Dic₇	1	3	yes	—
56	M₁₆	C₂×Dic₇	1	12	yes	—
57	C₂×C₈	C₂×Dic₇	1	3	yes	—
58	C₂×C₈	C₂×Dic₇	1	6	yes	—
59	C₂×C₈	C₂²×D₇	1	6	yes	—
60	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
61	C₁₆	—	7	8	no	—
62	C₂×C₇	—	2	1	no	—
63	C₂×C₇	—	2	6	no	—
64	C₂×C₇	C₂³×C₈	1	1	yes	characteristic
65	C₂³	C₂×C₇⋊C₈	1	3	yes	—
66	Q₈	C₂×C₇⋊C₈	1	4	yes	—
67	D₄	C₂×C₇⋊C₈	1	12	yes	—
68	C₂×C₄	C₂×C₇⋊C₈	1	3	yes	—
69	C₂×C₄	C₂×C₇⋊C₈	1	6	yes	—
70	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
71	C₂×C₄	C₂²×Dic₇	1	6	yes	—
72	C₈	C₂²×Dic₇	1	1	yes	characteristic
73	C₈	C₂²×Dic₇	1	6	yes	—
74	C₈	C₂³×D₇	1	1	yes	characteristic
75	C₇	#3407	1	1	yes	characteristic, complemented, Sylow
76	C₂²	—	4	3	no	—
77	C₂²	C₂²×C₇⋊C₈	1	1	yes	characteristic
78	C₂²	C₂²×C₇⋊C₈	1	6	yes	—
79	C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
80	C₄	C₂²×C₇⋊C₈	1	6	yes	—
81	C₄	C₂³×Dic₇	1	1	yes	characteristic
82	C₂	—	2	1	no	—
83	C₂	—	2	6	no	—
84	C₂	C₂³×C₇⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂³×D₇
commutator, socle	C₂×C₇	C₂³×C₈
Fitting	C₂×C₇×#131	C₂
prime radical	C₂×C₇×#41	C₄
cosocle	C₇×C₈	C₂⁴
hypercenter	C₂×#131	D₇
hypocenter	C₇	#3407
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #89912	C₂	semidirect
3. C₂×#6386	C₂	semidirect
10. #6386	C₂²	semidirect
11. C₂×C₇⋊C₁₆	C₂²	semidirect
75. C₇	#3407	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×#131	2
#89912	2
C₂×#6386	2
#3407	7