Subgroups — #89525

Order 448.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6551	C₂	1	1	yes	maximal, characteristic, complemented
2	#6510	C₂	1	1	yes	maximal, characteristic, complemented
3	#6471	C₂	1	2	yes	maximal, complemented
4	C₇×#101	C₂	1	1	yes	maximal, characteristic, complemented
5	#6399	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
7	#1161	—	2	2	no	—
8	Q₈×D₇	—	2	2	no	—
9	#1157	—	2	2	no	—
10	#1157	C₂²	1	2	yes	—
11	C₂×D₂₈	C₂²	1	1	yes	characteristic
12	C₇×M₁₆	C₂²	1	2	yes	—
13	(C₂×D₇)⋊C₄	—	2	2	no	—
14	C₄×Dic₇	C₂²	1	1	yes	characteristic
15	D₅₆	—	2	2	no	—
16	#1132	—	2	2	no	—
17	#454	—	7	1	no	maximal, Sylow
18	C₂²×D₇	—	2	1	no	—
19	C₇×Q₈	—	2	2	no	—
20	C₂×C₄×C₇	D₄	1	2	yes	—
21	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
22	C₂²⋊D₇	—	2	2	no	—
23	C₂×Dic₇	—	2	1	no	—
24	D₂₈	—	2	2	no	—
25	D₂₈	—	2	2	no	—
26	D₂₈	D₄	1	2	yes	—
27	C₄×D₇	—	2	4	no	—
28	C₄×D₇	—	2	2	no	—
29	Dic₁₄	—	2	2	no	—
30	Dic₁₄	D₄	1	2	yes	—
31	C₇×C₈	—	2	2	no	—
32	2₋¹⁺⁴	—	7	1	no	—
33	AGL(1,ℤ/8)	—	7	2	no	—
34	#124	—	7	1	no	—
35	#104	—	7	2	no	—
36	#101	D₇	1	1	yes	characteristic, complemented
37	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
38	C₂×D₇	—	2	2	no	—
39	C₂×D₇	—	4	2	no	—
40	C₂×D₇	—	4	1	no	—
41	C₄×C₇	—	2	2	no	—
42	C₄×C₇	C₂×D₄	1	2	yes	—
43	Dic₇	—	2	2	no	—
44	Dic₇	—	4	1	no	—
45	#41	—	7	2	no	—
46	#41	—	14	2	no	—
47	#41	—	14	2	no	—
48	C₂×Q₈	—	14	2	no	—
49	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
50	C₂×D₄	—	7	1	no	—
51	SD₁₆	—	14	2	no	—
52	D₈	—	14	2	no	—
53	M₁₆	C₂×D₇	1	2	yes	—
54	C₂²⋊C₄	—	14	2	no	—
55	C₄²	—	7	1	no	—
56	C₂×C₇	—	2	1	no	—
57	C₂×C₇	#120	1	1	yes	characteristic
58	D₇	—	4	2	no	—
59	D₇	—	8	1	no	—
60	C₂³	—	14	1	no	—
61	Q₈	—	2	2	no	—
62	Q₈	—	7	2	no	—
63	Q₈	—	14	2	no	—
64	D₄	—	7	2	no	—
65	D₄	—	14	2	no	—
66	D₄	—	14	2	no	—
67	D₄	—	14	2	no	—
68	C₂×C₄	—	14	4	no	—
69	C₂×C₄	—	14	2	no	—
70	C₂×C₄	—	14	1	no	—
71	C₂×C₄	D₂₈	1	2	yes	—
72	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
73	C₈	—	2	2	no	—
74	C₇	#454	1	1	yes	characteristic, complemented, Sylow
75	C₂²	—	14	2	no	—
76	C₂²	—	28	2	no	—
77	C₂²	—	28	1	no	—
78	C₂²	C₂×D₂₈	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	14	2	no	—
81	C₄	—	28	1	no	—
82	C₄	D₄×D₇	1	2	yes	—
83	C₂	—	2	1	no	—
84	C₂	—	28	2	no	—
85	C₂	—	56	1	no	—
86	C₂	#6445	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6445
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#101	C₂
socle	C₂×C₇	#120
hypercenter	#101	D₇
hypocenter	C₇	#454
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

Complements

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

N	H ≅ G/N	type
1. #6551	C₂	semidirect
2. #6510	C₂	semidirect
3. #6471	C₂	semidirect
4. C₇×#101	C₂	semidirect
5. #6399	C₂	semidirect
36. #101	D₇	semidirect
74. C₇	#454	semidirect

Maximal subgroups

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

subgroup	index
#6551	2
#6510	2
#6471	2
C₇×#101	2
#6399	2
#454	7