Subgroups — #89914

Order 448.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#131	C₂	1	1	yes	maximal, characteristic, complemented
2	#6512	C₂	1	2	yes	maximal, complemented
3	C₂×D₅₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#6397	C₂	1	2	yes	maximal, complemented
5	#6393	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#41	C₂²	1	1	yes	characteristic
7	C₂×D₂₈	C₂²	1	2	yes	—
8	C₇×M₁₆	—	2	1	no	—
9	C₇×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₇×C₈	—	2	1	no	—
11	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
12	#1143	—	2	2	no	—
13	#1141	—	2	2	no	—
14	#1136	C₂²	1	2	yes	—
15	D₅₆	—	2	1	no	—
16	D₅₆	—	2	1	no	—
17	#472	—	7	1	no	maximal, Sylow
18	C₂²×D₇	—	2	2	no	—
19	C₇×Q₈	D₄	1	1	yes	characteristic
20	C₇×D₄	—	2	1	no	—
21	C₇×D₄	D₄	1	1	yes	characteristic
22	C₂×C₄×C₇	—	2	1	no	—
23	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
24	D₂₈	—	2	2	no	—
25	D₂₈	—	2	2	no	—
26	C₇×C₈	—	2	1	no	—
27	C₇×C₈	D₄	1	1	yes	characteristic
28	C₇×C₈	D₄	1	1	yes	characteristic
29	C₇⋊C₈	—	2	2	no	—
30	AGL(1,ℤ/8)	—	7	2	no	—
31	C₂×D₈	—	7	1	no	—
32	#131	D₇	1	1	yes	characteristic, complemented
33	#108	—	7	1	no	—
34	#100	—	7	2	no	—
35	C₂²×C₇	—	2	1	no	—
36	C₂²×C₇	#41	1	1	yes	characteristic
37	C₂×D₇	—	4	2	no	—
38	C₂×D₇	—	8	2	no	—
39	C₄×C₇	—	2	1	no	—
40	C₄×C₇	C₂×D₄	1	1	yes	characteristic
41	C₄×C₇	C₂×D₄	1	1	yes	characteristic
42	#41	C₂×D₇	1	1	yes	characteristic
43	C₂×D₄	—	7	2	no	—
44	SD₁₆	—	14	2	no	—
45	D₈	—	14	2	no	—
46	D₈	—	14	1	no	—
47	D₈	—	14	1	no	—
48	M₁₆	—	2	1	no	—
49	M₁₆	—	7	2	no	—
50	M₁₆	C₂×D₇	1	1	yes	characteristic
51	C₂×C₈	—	2	1	no	—
52	C₂×C₈	C₂×D₇	1	1	yes	characteristic
53	C₂×C₇	—	2	1	no	—
54	C₂×C₇	—	4	1	no	—
55	C₂×C₇	#121	1	1	yes	characteristic
56	D₇	—	8	2	no	—
57	C₂³	—	14	2	no	—
58	Q₈	D₂₈	1	1	yes	characteristic
59	D₄	—	2	1	no	—
60	D₄	—	14	2	no	—
61	D₄	—	14	2	no	—
62	D₄	D₂₈	1	1	yes	characteristic
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
65	C₈	—	2	1	no	—
66	C₈	—	14	2	no	—
67	C₈	C₂²⋊D₇	1	1	yes	characteristic
68	C₈	C₂²⋊D₇	1	1	yes	characteristic
69	C₇	#472	1	1	yes	characteristic, complemented, Sylow
70	C₂²	—	2	1	no	—
71	C₂²	—	28	2	no	—
72	C₂²	—	56	2	no	—
73	C₂²	#1157	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	C₂×D₂₈	1	1	yes	characteristic
76	C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	4	1	no	—
79	C₂	—	56	2	no	—
80	C₂	#6493	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6493
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#131	C₂
socle	C₂×C₇	#121
hypercenter	#131	D₇
hypocenter	C₇	#472
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131

Complements

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

N	H ≅ G/N	type
1. C₇×#131	C₂	semidirect
2. #6512	C₂	semidirect
3. C₂×D₅₆	C₂	semidirect
4. #6397	C₂	semidirect
5. #6393	C₂	semidirect
32. #131	D₇	semidirect
69. C₇	#472	semidirect

Maximal subgroups

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

subgroup	index
C₇×#131	2
#6512	2
C₂×D₅₆	2
#6397	2
#6393	2
#472	7