Subgroups — #89932

Order 448.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×D₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#6495	C₂	1	2	yes	maximal, complemented
3	#6464	C₂	1	1	yes	maximal, characteristic, complemented
4	#6407	C₂	1	2	yes	maximal, complemented
5	#6393	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×D₄	C₂²	1	2	yes	—
7	#1157	C₂²	1	1	yes	characteristic
8	C₇×D₈	—	2	1	no	—
9	C₇×D₈	—	2	1	no	—
10	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
11	#1142	—	2	2	no	—
12	#1141	—	2	2	no	—
13	#1136	C₂²	1	1	yes	characteristic
14	#1136	C₂²	1	2	yes	—
15	#1131	—	2	1	no	—
16	C₈×D₇	—	2	1	no	—
17	#472	—	7	1	no	maximal, Sylow
18	C₂³×C₇	—	2	2	no	—
19	C₇×D₄	—	2	2	no	—
20	C₇×D₄	—	2	2	no	—
21	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
22	C₂²⋊D₇	—	2	1	no	—
23	D₂₈	D₄	1	1	yes	characteristic
24	C₄×D₇	—	2	1	no	—
25	Dic₁₄	D₄	1	1	yes	characteristic
26	C₇×C₈	D₄	1	1	yes	characteristic
27	C₇×C₈	D₄	1	1	yes	characteristic
28	C₇⋊C₈	—	2	1	no	—
29	C₇⋊C₈	—	2	2	no	—
30	AGL(1,ℤ/8)	—	7	2	no	—
31	C₂×D₈	D₇	1	1	yes	characteristic, complemented
32	#131	—	7	1	no	—
33	#108	—	7	1	no	—
34	#100	—	7	2	no	—
35	C₂²×C₇	—	4	2	no	—
36	C₂²×C₇	—	8	2	no	—
37	C₂²×C₇	#41	1	1	yes	characteristic
38	C₂×D₇	—	2	1	no	—
39	C₄×C₇	C₂×D₄	1	1	yes	characteristic
40	C₄×C₇	C₂×D₄	1	1	yes	characteristic
41	Dic₇	—	2	1	no	—
42	#41	—	7	1	no	—
43	C₂×D₄	C₂×D₇	1	2	yes	—
44	SD₁₆	—	14	2	no	—
45	D₈	—	2	1	no	—
46	D₈	—	2	1	no	—
47	D₈	—	14	2	no	—
48	M₁₆	—	7	1	no	—
49	M₁₆	—	7	2	no	—
50	M₁₆	—	14	1	no	—
51	C₂×C₈	—	14	1	no	—
52	C₂×C₈	C₂×D₇	1	1	yes	characteristic
53	C₂×C₇	—	2	1	no	—
54	C₂×C₇	—	8	2	no	—
55	C₂×C₇	#121	1	1	yes	characteristic
56	D₇	—	4	1	no	—
57	C₂³	—	2	2	no	—
58	Q₈	—	7	1	no	—
59	D₄	—	2	2	no	—
60	D₄	—	2	2	no	—
61	D₄	—	7	1	no	—
62	D₄	—	14	1	no	—
63	C₂×C₄	—	14	1	no	—
64	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
65	C₈	—	14	2	no	—
66	C₈	—	14	1	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	—	4	2	no	—
71	C₂²	—	8	2	no	—
72	C₂²	—	14	1	no	—
73	C₂²	#1159	1	1	yes	characteristic
74	C₄	—	14	1	no	—
75	C₄	D₄×D₇	1	1	yes	characteristic
76	C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	8	2	no	—
79	C₂	—	28	1	no	—
80	C₂	#6501	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6501
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₂×C₇×D₈	C₂
socle	C₂×C₇	#121
hypercenter	C₂×D₈	D₇
hypocenter	C₇	#472
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×D₈	C₂	semidirect
2. #6495	C₂	semidirect
3. #6464	C₂	semidirect
4. #6407	C₂	semidirect
5. #6393	C₂	semidirect
31. C₂×D₈	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
C₂×C₇×D₈	2
#6495	2
#6464	2
#6407	2
#6393	2
#472	7