Subgroups — #89461

Order 448.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₇	C₂	1	1	yes	maximal, characteristic
3	#6435	C₂	1	1	yes	maximal, characteristic
4	C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#6389	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#6377	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
9	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₄²×C₇	C₂²	1	1	yes	characteristic
12	(C₂×D₇)⋊C₄	C₄	1	2	yes	—
13	Dic₇⋊C₄	C₄	1	2	yes	—
14	C₄×Dic₇	C₂²	1	1	yes	characteristic
15	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
17	#1131	C₄	1	4	yes	complemented
18	C₈×D₇	C₄	1	4	yes	complemented
19	#405	—	7	1	no	maximal, Sylow
20	C₂²×D₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₂×Dic₇	C₂×C₄	1	2	yes	—
24	C₂×Dic₇	C₂×C₄	1	1	yes	characteristic
25	C₄×D₇	C₂×C₄	1	2	yes	—
26	C₄×D₇	C₂×C₄	1	2	yes	—
27	C₇×C₈	C₂×C₄	1	2	yes	—
28	C₇×C₈	C₂×C₄	1	2	yes	—
29	C₇⋊C₈	C₂×C₄	1	2	yes	—
30	C₇⋊C₈	C₂×C₄	1	2	yes	—
31	C₂×M₁₆	—	7	1	no	—
32	C₂²×C₈	—	7	1	no	—
33	#117	—	7	1	no	—
34	#97	—	7	1	no	—
35	#97	—	7	1	no	—
36	C₄×C₈	—	7	1	no	—
37	C₄×C₈	D₇	1	1	yes	characteristic, complemented
38	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×D₇	—	2	2	no	—
40	C₂×D₇	C₄²	1	2	yes	—
41	C₄×C₇	—	2	2	no	—
42	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
43	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
44	Dic₇	—	2	2	no	—
45	Dic₇	C₄²	1	2	yes	—
46	C₂²×C₄	—	7	1	no	—
47	M₁₆	—	7	4	no	—
48	C₂×C₈	—	7	1	no	—
49	C₂×C₈	—	7	1	no	—
50	C₂×C₈	—	7	4	no	—
51	C₂×C₈	C₂×D₇	1	1	yes	characteristic
52	C₂×C₈	C₂×D₇	1	1	yes	characteristic
53	C₄⋊C₄	—	7	2	no	—
54	C₂²⋊C₄	—	7	2	no	—
55	C₄²	—	7	1	no	—
56	C₄²	C₂×D₇	1	1	yes	characteristic
57	C₂×C₇	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₇	#131	1	2	yes	—
59	D₇	—	2	2	no	—
60	C₂³	—	7	1	no	—
61	C₂×C₄	—	7	1	no	—
62	C₂×C₄	—	7	2	no	—
63	C₂×C₄	—	7	2	no	—
64	C₂×C₄	—	7	2	no	—
65	C₂×C₄	C₄×D₇	1	2	yes	—
66	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
67	C₈	—	7	2	no	—
68	C₈	—	7	2	no	—
69	C₈	C₄×D₇	1	2	yes	—
70	C₈	C₄×D₇	1	2	yes	—
71	C₇	#405	1	1	yes	characteristic, complemented, Sylow
72	C₂²	—	7	2	no	—
73	C₂²	—	14	2	no	—
74	C₂²	C₂×C₄×D₇	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	7	2	no	—
77	C₄	—	14	2	no	—
78	C₄	C₂×C₄×D₇	1	1	yes	characteristic
79	C₄	C₂×C₄×D₇	1	1	yes	characteristic
80	C₂	—	14	2	no	—
81	C₂	C₄²×D₇	1	1	yes	characteristic
82	C₂	#6464	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₇
commutator	C₂×C₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₄×C₇×C₈	C₂
socle	C₂²×C₇	C₂²×C₄
prime radical	C₂²×D₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₄×C₈	D₇
hypocenter	C₇	#405
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
4. C₄×C₇×C₈	C₂	semidirect
5. #6389	C₂	semidirect
6. C₈×Dic₇	C₂	semidirect
7. #6377	C₂	semidirect
17. #1131	C₄	semidirect
18. C₈×D₇	C₄	semidirect
37. C₄×C₈	D₇	semidirect
71. C₇	#405	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1131	2
C₂×C₈×D₇	2
#6435	2
C₄×C₇×C₈	2
#6389	2
C₈×Dic₇	2
#6377	2
#405	7