Subgroups — #11740630

Order 896.

1272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#6503	C₂	1	12	yes	maximal, complemented
3	C₂×#6437	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄²×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₇×D₄	C₂²	1	6	yes	—
6	C₂²×C₂²⋊D₇	C₂²	1	16	yes	—
7	C₂²×D₂₈	C₂²	1	6	yes	—
8	C₇×#127	—	2	4	no	—
9	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
10	#6503	—	2	48	no	—
11	C₂×C₄×Dic₇	C₂²	1	6	yes	—
12	#6437	—	2	4	no	—
13	#2867	—	7	1	no	maximal, Sylow
14	C₂⁴×C₇	C₂³	1	4	yes	—
15	C₂³×D₇	C₂³	1	4	yes	—
16	C₂×C₇×D₄	—	2	24	no	—
17	C₂×C₇×D₄	—	2	12	no	—
18	C₂²×C₄×C₇	C₂³	1	3	yes	—
19	C₂×C₂²⋊D₇	—	2	96	no	—
20	C₂²×Dic₇	C₂³	1	4	yes	—
21	C₂×D₂₈	—	2	12	no	—
22	C₂×D₂₈	—	2	24	no	—
23	C₄²×C₇	D₄	1	4	yes	complemented
24	C₄×Dic₇	D₄	1	24	yes	complemented
25	C₂×#127	—	7	12	no	—
26	C₂×#127	—	7	1	no	—
27	C₂×#127	D₇	1	1	yes	characteristic, complemented
28	C₄³	—	7	1	no	—
29	C₂³×C₇	—	2	24	no	—
30	C₂³×C₇	—	2	4	no	—
31	C₂³×C₇	C₂⁴	1	1	yes	characteristic
32	C₂²×D₇	—	2	4	no	—
33	C₂²×D₇	—	2	24	no	—
34	C₇×D₄	—	4	24	no	—
35	C₂×C₄×C₇	C₂×D₄	1	12	yes	—
36	C₂×C₄×C₇	C₂×D₄	1	6	yes	—
37	C₂²⋊D₇	—	4	64	no	—
38	C₂×Dic₇	C₂×D₄	1	24	yes	—
39	D₂₈	—	4	24	no	—
40	C₂²×D₄	—	7	16	no	—
41	C₂²×D₄	—	7	6	no	—
42	C₂²×D₄	C₂×D₇	1	6	yes	—
43	#127	—	2	4	no	—
44	#127	—	14	48	no	—
45	#127	—	14	4	no	—
46	C₂×C₄²	—	7	6	no	—
47	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
48	C₂²×C₇	—	4	16	no	—
49	C₂²×C₇	—	4	12	no	—
50	C₂²×C₇	C₂²×D₄	1	6	yes	—
51	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
52	C₂×D₇	—	4	12	no	—
53	C₂×D₇	—	4	16	no	—
54	C₄×C₇	#127	1	12	yes	complemented
55	Dic₇	#127	1	16	yes	complemented
56	C₂⁴	—	7	4	no	—
57	C₂⁴	C₂²×D₇	1	4	yes	—
58	C₂×D₄	—	2	24	no	—
59	C₂×D₄	—	2	12	no	—
60	C₂×D₄	—	14	96	no	—
61	C₂×D₄	—	14	24	no	—
62	C₂×D₄	—	14	12	no	—
63	C₂²×C₄	—	7	4	no	—
64	C₂²×C₄	C₂²×D₇	1	3	yes	—
65	C₄²	—	7	24	no	—
66	C₄²	C₂²⋊D₇	1	4	yes	complemented
67	C₂×C₇	—	8	4	no	—
68	C₂×C₇	C₂×#127	1	4	yes	—
69	C₂×C₇	C₂×#127	1	3	yes	—
70	D₇	—	8	4	no	—
71	C₂³	—	2	24	no	—
72	C₂³	—	2	4	no	—
73	C₂³	—	14	24	no	—
74	C₂³	—	14	4	no	—
75	C₂³	C₂³×D₇	1	1	yes	characteristic
76	D₄	—	4	24	no	—
77	D₄	—	28	64	no	—
78	D₄	—	28	24	no	—
79	C₂×C₄	—	7	24	no	—
80	C₂×C₄	D₄×D₇	1	12	yes	—
81	C₂×C₄	C₂×C₂²⋊D₇	1	6	yes	—
82	C₇	#2867	1	1	yes	characteristic, complemented, Sylow
83	C₂²	—	4	16	no	—
84	C₂²	—	4	12	no	—
85	C₂²	—	28	16	no	—
86	C₂²	—	28	12	no	—
87	C₂²	C₂×D₄×D₇	1	6	yes	—
88	C₂²	C₂²×C₂²⋊D₇	1	1	yes	characteristic
89	C₄	—	7	16	no	—
90	C₄	#6503	1	12	yes	complemented
91	C₂	—	8	4	no	—
92	C₂	—	56	4	no	—
93	C₂	D₇×#127	1	4	yes	—
94	C₂	C₂×#6503	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator, socle, cosocle	C₂³×C₇	C₂⁴
Fitting	C₂×C₇×#127	C₂
hypercenter	C₂×#127	D₇
hypocenter	C₇	#2867
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×#127

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#127	C₂	semidirect
2. C₂×#6503	C₂	semidirect
3. C₂×#6437	C₂	semidirect
4. C₄²×Dic₇	C₂	semidirect
23. C₄²×C₇	D₄	semidirect
24. C₄×Dic₇	D₄	semidirect
27. C₂×#127	D₇	semidirect
54. C₄×C₇	#127	semidirect
55. Dic₇	#127	semidirect
66. C₄²	C₂²⋊D₇	semidirect
82. C₇	#2867	semidirect
90. C₄	#6503	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#127	2
C₂×#6503	2
C₂×#6437	2
C₄²×Dic₇	2
#2867	7