Subgroups — #90013

Order 448.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	#6512	C₂	1	2	yes	maximal, complemented
3	#6503	C₂	1	1	yes	maximal, characteristic, complemented
4	#6411	C₂	1	2	yes	maximal, complemented
5	#6407	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#41	—	2	2	no	—
7	C₇×#41	C₂²	1	2	yes	—
8	C₂×C₇×D₄	—	2	2	no	—
9	C₂×C₇×D₄	—	2	2	no	—
10	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₇	—	2	2	no	—
12	C₂×D₂₈	C₂²	1	1	yes	characteristic, complemented
13	#1143	—	2	2	no	—
14	#1141	—	2	2	no	—
15	C₄×Dic₇	C₂²	1	1	yes	characteristic, complemented
16	#1136	C₂²	1	2	yes	complemented
17	#453	—	7	1	no	maximal, Sylow
18	C₂³×C₇	—	2	2	no	—
19	C₂³×C₇	D₄	1	2	yes	—
20	C₂²×D₇	—	2	1	no	—
21	C₇×Q₈	D₄	1	2	yes	complemented
22	C₇×D₄	—	2	4	no	—
23	C₇×D₄	—	2	2	no	—
24	C₇×D₄	—	2	2	no	—
25	C₇×D₄	D₄	1	2	yes	complemented
26	C₂×C₄×C₇	—	2	2	no	—
27	C₂×C₄×C₇	—	2	2	no	—
28	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
29	C₂²⋊D₇	—	4	4	no	—
30	C₂×Dic₇	—	2	1	no	—
31	D₂₈	—	2	2	no	—
32	C₇⋊C₈	—	2	2	no	—
33	H(5,𝔽₂)	D₇	1	1	yes	characteristic, complemented
34	AGL(1,ℤ/8)	—	7	2	no	—
35	#127	—	7	1	no	—
36	#104	—	7	2	no	—
37	#100	—	7	1	no	—
38	C₂²×C₇	—	2	2	no	—
39	C₂²×C₇	—	2	2	no	—
40	C₂²×C₇	—	4	2	no	—
41	C₂²×C₇	—	8	2	no	—
42	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
43	C₂×D₇	—	4	2	no	—
44	C₂×D₇	—	4	1	no	—
45	C₄×C₇	—	2	2	no	—
46	C₄×C₇	C₂×D₄	1	2	yes	—
47	Dic₇	—	2	2	no	—
48	#41	—	2	2	no	—
49	#41	C₂×D₇	1	2	yes	—
50	C₂×D₄	—	2	2	no	—
51	C₂×D₄	—	2	2	no	—
52	C₂×D₄	—	7	1	no	—
53	C₂×D₄	—	14	2	no	—
54	C₂×D₄	C₂×D₇	1	1	yes	characteristic
55	SD₁₆	—	14	2	no	—
56	D₈	—	14	2	no	—
57	M₁₆	—	7	2	no	—
58	C₄²	—	7	1	no	—
59	C₂×C₇	—	2	1	no	—
60	C₂×C₇	—	4	2	no	—
61	C₂×C₇	—	4	2	no	—
62	C₂×C₇	#120	1	1	yes	characteristic
63	D₇	—	8	1	no	—
64	C₂³	—	2	2	no	—
65	C₂³	—	14	1	no	—
66	C₂³	C₂²⋊D₇	1	2	yes	—
67	Q₈	C₂²⋊D₇	1	2	yes	complemented
68	D₄	—	2	4	no	—
69	D₄	—	2	2	no	—
70	D₄	—	2	2	no	—
71	D₄	—	14	2	no	—
72	D₄	—	28	4	no	—
73	D₄	C₂²⋊D₇	1	2	yes	complemented
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	14	1	no	—
77	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
78	C₈	—	14	2	no	—
79	C₇	#453	1	1	yes	characteristic, complemented, Sylow
80	C₂²	—	2	2	no	—
81	C₂²	—	2	2	no	—
82	C₂²	—	4	2	no	—
83	C₂²	—	8	2	no	—
84	C₂²	—	28	2	no	—
85	C₂²	—	28	1	no	—
86	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	—	14	2	no	—
89	C₄	C₂×C₂²⋊D₇	1	2	yes	—
90	C₂	—	2	1	no	—
91	C₂	—	4	2	no	—
92	C₂	—	4	2	no	—
93	C₂	—	56	1	no	—
94	C₂	#6516	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6516
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×H(5,𝔽₂)	C₂
socle	C₂×C₇	#120
hypercenter	H(5,𝔽₂)	D₇
hypocenter	C₇	#453
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₇×H(5,𝔽₂)	C₂	semidirect
2. #6512	C₂	semidirect
3. #6503	C₂	semidirect
4. #6411	C₂	semidirect
5. #6407	C₂	semidirect
12. C₂×D₂₈	C₂²	semidirect
15. C₄×Dic₇	C₂²	semidirect
16. #1136	C₂²	semidirect
21. C₇×Q₈	D₄	semidirect
25. C₇×D₄	D₄	semidirect
33. H(5,𝔽₂)	D₇	semidirect
67. Q₈	C₂²⋊D₇	semidirect
73. D₄	C₂²⋊D₇	semidirect
79. C₇	#453	semidirect

Maximal subgroups

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

subgroup	index
C₇×H(5,𝔽₂)	2
#6512	2
#6503	2
#6411	2
#6407	2
#453	7