Subgroups — #89655

Order 448.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×D₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#6458	C₂	1	2	yes	maximal, complemented
4	C₇×#107	C₂	1	1	yes	maximal, characteristic, complemented
5	#6383	C₂	1	2	yes	maximal, complemented
6	C₂×D₂₈	—	2	2	no	—
7	C₂×D₂₈	C₂²	1	2	yes	—
8	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
9	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₇×C₄⋊C₄	C₂²	1	2	yes	—
11	(C₂×D₇)⋊C₄	—	2	2	no	—
12	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
13	D₅₆	—	2	2	no	—
14	C₈×D₇	—	2	2	no	—
15	#466	—	7	1	no	maximal, Sylow
16	C₂²×D₇	—	2	2	no	—
17	C₂²×D₇	D₄	1	1	yes	characteristic
18	C₂×C₄×C₇	—	2	2	no	—
19	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₂×Dic₇	D₄	1	1	yes	characteristic
21	D₂₈	—	2	2	no	—
22	D₂₈	—	2	2	no	—
23	D₂₈	—	4	4	no	—
24	C₄×D₇	—	2	1	no	—
25	C₄×D₇	—	2	1	no	—
26	C₇×C₈	D₄	1	2	yes	complemented
27	C₇⋊C₈	—	2	1	no	—
28	C₂×D₈	—	7	1	no	—
29	C₂²×C₈	—	7	1	no	—
30	#121	—	7	2	no	—
31	#107	D₇	1	1	yes	characteristic, complemented
32	#102	—	7	2	no	—
33	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
34	C₂×D₇	—	2	1	no	—
35	C₂×D₇	—	2	1	no	—
36	C₂×D₇	—	4	2	no	—
37	C₂×D₇	—	4	2	no	—
38	C₂×D₇	—	4	2	no	—
39	C₂×D₇	D₈	1	2	yes	complemented
40	C₄×C₇	—	4	2	no	—
41	C₄×C₇	C₂×D₄	1	1	yes	characteristic
42	C₄×C₇	#41	1	1	yes	characteristic
43	Dic₇	—	2	1	no	—
44	C₂×D₄	—	7	2	no	—
45	C₂×D₄	—	14	2	no	—
46	C₂²×C₄	—	7	1	no	—
47	D₈	—	14	2	no	—
48	C₂×C₈	—	7	1	no	—
49	C₂×C₈	—	14	2	no	—
50	C₂×C₈	C₂×D₇	1	1	yes	characteristic
51	C₄⋊C₄	C₂×D₇	1	2	yes	—
52	C₂²⋊C₄	—	14	2	no	—
53	C₂×C₇	#121	1	1	yes	characteristic
54	C₂×C₇	C₂×D₈	1	1	yes	characteristic
55	C₂×C₇	#135	1	1	yes	characteristic
56	D₇	—	2	2	no	—
57	D₇	—	8	2	no	—
58	C₂³	—	7	1	no	—
59	C₂³	—	14	2	no	—
60	D₄	—	14	2	no	—
61	D₄	—	14	2	no	—
62	D₄	—	28	4	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	7	1	no	—
65	C₂×C₄	—	14	1	no	—
66	C₂×C₄	—	14	1	no	—
67	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
68	C₈	—	14	1	no	—
69	C₈	D₂₈	1	2	yes	complemented
70	C₇	#466	1	1	yes	characteristic, complemented, Sylow
71	C₂²	—	7	2	no	—
72	C₂²	—	14	1	no	—
73	C₂²	—	14	1	no	—
74	C₂²	—	28	2	no	—
75	C₂²	—	28	2	no	—
76	C₂²	—	28	2	no	—
77	C₂²	D₄×D₇	1	1	yes	characteristic
78	C₄	—	4	2	no	—
79	C₄	—	14	1	no	—
80	C₄	C₂×D₂₈	1	1	yes	characteristic
81	C₄	#1161	1	1	yes	characteristic
82	C₂	—	14	2	no	—
83	C₂	—	56	2	no	—
84	C₂	#6458	1	1	yes	characteristic
85	C₂	D₇×D₈	1	1	yes	characteristic
86	C₂	#6482	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#107	C₂
socle	C₂²×C₇	C₂×D₄
hypercenter	#107	D₇
hypocenter	C₇	#466
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
1. C₂×D₅₆	C₂	semidirect
2. C₂×C₈×D₇	C₂	semidirect
3. #6458	C₂	semidirect
4. C₇×#107	C₂	semidirect
5. #6383	C₂	semidirect
26. C₇×C₈	D₄	semidirect
31. #107	D₇	semidirect
39. C₂×D₇	D₈	semidirect
69. C₈	D₂₈	semidirect
70. C₇	#466	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅₆	2
C₂×C₈×D₇	2
#6458	2
C₇×#107	2
#6383	2
#466	7