Subgroups — #11729263

Order 896.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#508	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₁₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#89316	C₂	1	2	yes	maximal, complemented
4	#89314	C₂	1	2	yes	maximal, complemented
5	#89294	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#135	—	2	1	no	—
7	C₂×D₅₆	—	2	1	no	—
8	C₇×Q₃₂	C₂²	1	1	yes	characteristic, complemented
9	C₇×SD₃₂	—	2	1	no	—
10	C₇×D₁₆	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₇×C₁₆	C₂²	1	1	yes	characteristic
12	D₁₁₂	—	2	1	no	—
13	D₁₁₂	C₂²	1	2	yes	—
14	C₇⋊C₃₂	C₂²	1	2	yes	complemented
15	#2263	—	7	1	no	maximal, Sylow
16	C₇×#41	—	4	1	no	—
17	C₂×D₂₈	—	4	1	no	—
18	C₇×Q₁₆	—	2	1	no	—
19	C₇×SD₁₆	—	4	1	no	—
20	C₇×D₈	—	2	1	no	—
21	C₂×C₇×C₈	D₄	1	1	yes	characteristic
22	D₅₆	—	2	2	no	—
23	D₅₆	—	4	1	no	—
24	C₇×C₁₆	D₄	1	1	yes	characteristic
25	C₇×C₁₆	C₂³	1	1	yes	characteristic
26	#508	D₇	1	1	yes	characteristic, complemented
27	C₂×D₁₆	—	7	1	no	—
28	SD₆₄	—	7	2	no	—
29	D₃₂	—	7	2	no	—
30	M₆₄	—	7	1	no	—
31	C₂²×D₇	—	8	1	no	—
32	C₇×Q₈	—	4	1	no	—
33	C₇×D₄	—	4	1	no	—
34	C₇×D₄	—	8	1	no	—
35	C₂×C₄×C₇	—	8	1	no	—
36	C₂×C₄×C₇	D₈	1	1	yes	characteristic
37	D₂₈	—	4	2	no	—
38	D₂₈	—	8	1	no	—
39	C₇×C₈	D₈	1	1	yes	characteristic
40	C₇×C₈	C₂×D₄	1	1	yes	characteristic
41	#135	—	2	1	no	—
42	C₂×D₈	—	14	1	no	—
43	Q₃₂	C₂×D₇	1	1	yes	characteristic, complemented
44	SD₃₂	—	2	1	no	—
45	D₁₆	—	7	2	no	—
46	D₁₆	—	14	1	no	—
47	D₁₆	C₂×D₇	1	1	yes	characteristic, complemented
48	C₂×C₁₆	C₂×D₇	1	1	yes	characteristic
49	C₃₂	—	7	2	no	—
50	C₂²×C₇	—	8	1	no	—
51	C₂²×C₇	D₁₆	1	1	yes	characteristic
52	C₂×D₇	—	8	2	no	—
53	C₂×D₇	—	16	2	no	—
54	C₄×C₇	—	8	1	no	—
55	C₄×C₇	D₁₆	1	1	yes	characteristic
56	C₄×C₇	C₂×D₈	1	1	yes	characteristic
57	#41	—	4	1	no	—
58	C₂×D₄	—	28	1	no	—
59	Q₁₆	—	2	1	no	—
60	SD₁₆	—	4	1	no	—
61	D₈	—	2	1	no	—
62	D₈	—	14	2	no	—
63	D₈	—	28	1	no	—
64	C₂×C₈	C₂²⋊D₇	1	1	yes	characteristic
65	C₁₆	C₂²⋊D₇	1	1	yes	characteristic
66	C₁₆	C₂²×D₇	1	1	yes	characteristic
67	C₂×C₇	—	2	1	no	—
68	C₂×C₇	—	16	1	no	—
69	C₂×C₇	C₂×D₁₆	1	1	yes	characteristic
70	D₇	—	16	2	no	—
71	C₂³	—	56	1	no	—
72	Q₈	—	4	1	no	—
73	D₄	—	4	1	no	—
74	D₄	—	8	1	no	—
75	D₄	—	28	2	no	—
76	D₄	—	56	1	no	—
77	C₂×C₄	—	8	1	no	—
78	C₂×C₄	#1141	1	1	yes	characteristic
79	C₈	#1141	1	1	yes	characteristic
80	C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
81	C₇	#2263	1	1	yes	characteristic, complemented, Sylow
82	C₂²	—	8	1	no	—
83	C₂²	—	56	2	no	—
84	C₂²	—	112	2	no	—
85	C₂²	#6400	1	1	yes	characteristic
86	C₄	—	8	1	no	—
87	C₄	#6400	1	1	yes	characteristic
88	C₄	C₂×#1141	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	16	1	no	—
91	C₂	—	112	2	no	—
92	C₂	C₂×#6400	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#6400
commutator, cosocle	C₇×C₁₆	C₂³
Frattini	C₁₆	C₂²×D₇
Fitting	C₇×#508	C₂
socle	C₂×C₇	C₂×D₁₆
hypercenter	#508	D₇
hypocenter	C₇	#2263
LCS	G ▷ C₇×C₁₆ ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #508

Complements

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

N	H ≅ G/N	type
1. C₇×#508	C₂	semidirect
2. C₂×D₁₁₂	C₂	semidirect
3. #89316	C₂	semidirect
4. #89314	C₂	semidirect
5. #89294	C₂	semidirect
8. C₇×Q₃₂	C₂²	semidirect
10. C₇×D₁₆	C₂²	semidirect
14. C₇⋊C₃₂	C₂²	semidirect
26. #508	D₇	semidirect
43. Q₃₂	C₂×D₇	semidirect
47. D₁₆	C₂×D₇	semidirect
81. C₇	#2263	semidirect

Maximal subgroups

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

subgroup	index
C₇×#508	2
C₂×D₁₁₂	2
#89316	2
#89314	2
#89294	2
#2263	7