Subgroups — #89887

Order 448.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#6493	C₂	1	2	yes	maximal, complemented
3	C₂×D₅₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#6395	C₂	1	2	yes	maximal, complemented
5	#6392	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊D₇	—	2	2	no	—
8	C₂×D₂₈	C₂²	1	2	yes	—
9	C₂×C₇×C₈	—	2	2	no	—
10	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
12	(C₂×D₇)⋊C₄	—	2	2	no	—
13	C₄⋊Dic₇	C₂²	1	2	yes	—
14	D₅₆	—	2	2	no	—
15	#466	—	7	1	no	maximal, Sylow
16	C₂³×C₇	D₄	1	1	yes	characteristic
17	C₂²×D₇	—	2	2	no	—
18	C₂×C₄×C₇	—	2	1	no	—
19	C₂×C₄×C₇	—	2	1	no	—
20	C₂×C₄×C₇	D₄	1	1	yes	characteristic
21	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
22	C₂²⋊D₇	—	4	4	no	—
23	C₂×Dic₇	—	2	2	no	—
24	D₂₈	—	2	2	no	—
25	D₂₈	—	2	2	no	—
26	C₇×C₈	—	2	1	no	—
27	C₇×C₈	D₄	1	2	yes	complemented
28	C₂×D₈	—	7	1	no	—
29	C₂²×C₈	D₇	1	1	yes	characteristic, complemented
30	#121	—	7	2	no	—
31	#107	—	7	1	no	—
32	#102	—	7	2	no	—
33	C₂²×C₇	—	2	1	no	—
34	C₂²×C₇	—	2	1	no	—
35	C₂²×C₇	D₈	1	2	yes	complemented
36	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
37	C₂×D₇	—	4	2	no	—
38	C₂×D₇	—	4	2	no	—
39	C₂×D₇	—	4	2	no	—
40	C₄×C₇	—	2	1	no	—
41	C₄×C₇	C₂×D₄	1	1	yes	characteristic
42	C₄×C₇	#41	1	1	yes	characteristic
43	Dic₇	—	4	2	no	—
44	C₂×D₄	—	7	2	no	—
45	C₂×D₄	—	14	2	no	—
46	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
47	D₈	—	14	2	no	—
48	C₂×C₈	—	2	2	no	—
49	C₂×C₈	C₂×D₇	1	1	yes	characteristic
50	C₂×C₈	C₂×D₇	1	1	yes	characteristic
51	C₄⋊C₄	—	7	2	no	—
52	C₂²⋊C₄	—	14	2	no	—
53	C₂×C₇	—	2	2	no	—
54	C₂×C₇	#121	1	1	yes	characteristic
55	C₂×C₇	C₂×D₈	1	1	yes	characteristic
56	C₂×C₇	#135	1	1	yes	characteristic
57	D₇	—	8	2	no	—
58	C₂³	—	14	2	no	—
59	C₂³	D₂₈	1	1	yes	characteristic
60	D₄	—	14	2	no	—
61	D₄	—	14	2	no	—
62	D₄	—	28	4	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	14	2	no	—
66	C₂×C₄	D₂₈	1	1	yes	characteristic
67	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
68	C₈	—	2	1	no	—
69	C₈	C₂²⋊D₇	1	2	yes	complemented
70	C₇	#466	1	1	yes	characteristic, complemented, Sylow
71	C₂²	—	2	1	no	—
72	C₂²	—	2	1	no	—
73	C₂²	—	28	2	no	—
74	C₂²	—	28	2	no	—
75	C₂²	—	28	2	no	—
76	C₂²	D₅₆	1	2	yes	complemented
77	C₂²	C₂×D₂₈	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	28	2	no	—
80	C₄	#1157	1	1	yes	characteristic
81	C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
82	C₂	—	2	2	no	—
83	C₂	—	56	2	no	—
84	C₂	C₂×D₅₆	1	1	yes	characteristic
85	C₂	#6467	1	1	yes	characteristic
86	C₂	#6493	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₂₈
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₂²×C₇×C₈	C₂
socle	C₂²×C₇	C₂×D₄
hypercenter	C₂²×C₈	D₇
hypocenter	C₇	#466
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×C₈	C₂	semidirect
2. #6493	C₂	semidirect
3. C₂×D₅₆	C₂	semidirect
4. #6395	C₂	semidirect
5. #6392	C₂	semidirect
27. C₇×C₈	D₄	semidirect
29. C₂²×C₈	D₇	semidirect
35. C₂²×C₇	D₈	semidirect
69. C₈	C₂²⋊D₇	semidirect
70. C₇	#466	semidirect
76. C₂²	D₅₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×C₈	2
#6493	2
C₂×D₅₆	2
#6395	2
#6392	2
#466	7