Subgroups — #90285

Order 448.

260 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#120	C₂	1	1	yes	maximal, characteristic, complemented
2	#6516	C₂	1	2	yes	maximal, complemented
3	#6500	C₂	1	3	yes	maximal, complemented
4	#6499	C₂	1	3	yes	maximal, complemented
5	#6448	C₂	1	6	yes	maximal, complemented
6	C₂⁴×C₇	C₂²	1	1	yes	characteristic, complemented
7	C₂³×D₇	C₂²	1	1	yes	characteristic, complemented
8	C₂×C₇×D₄	C₂²	1	3	yes	—
9	C₂×C₂²⋊D₇	C₂²	1	6	yes	—
10	C₂×Dic₁₄	C₂²	1	3	yes	complemented
11	C₇×C₂²⋊C₄	C₂²	1	3	yes	complemented
12	C₂²⋊Dic₇	C₂²	1	6	yes	complemented
13	C₂²⋊Dic₇	C₂²	1	3	yes	complemented
14	(C₂×D₇)⋊C₄	C₂²	1	6	yes	complemented
15	C₄×Dic₇	C₂²	1	3	yes	complemented
16	H(3,𝔽₄)	—	7	1	no	maximal, Sylow
17	C₂³×C₇	—	4	3	no	—
18	C₂³×C₇	C₂³	1	1	yes	characteristic
19	C₂³×C₇	C₂³	1	3	yes	—
20	C₂²×D₇	—	4	3	no	—
21	C₂²×D₇	C₂³	1	2	yes	—
22	C₇×D₄	—	4	3	no	—
23	C₂×C₄×C₇	C₂³	1	3	yes	—
24	C₂²⋊D₇	—	4	6	no	—
25	C₂×Dic₇	C₂³	1	6	yes	—
26	Dic₁₄	—	4	3	no	—
27	#124	—	7	3	no	—
28	#124	—	7	6	no	—
29	#120	—	7	2	no	—
30	#120	—	7	3	no	—
31	#120	D₇	1	1	yes	characteristic, complemented
32	C₂²×C₇	—	2	6	no	—
33	C₂²×C₇	—	2	3	no	—
34	C₂²×C₇	—	2	3	no	—
35	C₂²×C₇	—	4	4	no	—
36	C₂²×C₇	C₂⁴	1	1	yes	characteristic
37	C₂×D₇	—	2	6	no	—
38	C₂×D₇	—	4	4	no	—
39	C₄×C₇	—	2	3	no	—
40	Dic₇	—	2	6	no	—
41	C₂⁴	—	7	1	no	—
42	C₂⁴	C₂×D₇	1	1	yes	characteristic, complemented
43	C₂×Q₈	—	7	3	no	—
44	C₂×D₄	—	7	6	no	—
45	C₂×D₄	C₂×D₇	1	3	yes	—
46	C₂²⋊C₄	—	7	3	no	—
47	C₂²⋊C₄	—	7	6	no	—
48	C₂²⋊C₄	—	7	6	no	—
49	C₂²⋊C₄	C₂×D₇	1	3	yes	complemented
50	C₄²	—	7	3	no	—
51	C₂×C₇	—	4	1	no	—
52	C₂×C₇	—	4	3	no	—
53	C₂×C₇	H(5,𝔽₂)	1	3	yes	—
54	D₇	—	4	2	no	—
55	C₂³	—	4	3	no	—
56	C₂³	—	7	2	no	—
57	C₂³	—	28	3	no	—
58	C₂³	C₂²×D₇	1	1	yes	characteristic
59	C₂³	C₂²×D₇	1	3	yes	—
60	Q₈	—	28	3	no	—
61	D₄	—	4	3	no	—
62	D₄	—	28	6	no	—
63	C₂×C₄	—	7	6	no	—
64	C₂×C₄	C₂²×D₇	1	3	yes	—
65	C₇	H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
66	C₂²	—	2	3	no	—
67	C₂²	—	2	3	no	—
68	C₂²	—	2	6	no	—
69	C₂²	—	4	4	no	—
70	C₂²	—	14	6	no	—
71	C₂²	—	28	4	no	—
72	C₂²	C₂³×D₇	1	1	yes	characteristic
73	C₄	—	2	3	no	—
74	C₄	—	14	6	no	—
75	C₂	—	4	3	no	—
76	C₂	—	4	1	no	—
77	C₂	—	28	2	no	—
78	C₂	#6548	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×#120	C₂
hypercenter	#120	D₇
hypocenter	C₇	H(3,𝔽₄)
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #120

Complements

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

N	H ≅ G/N	type
1. C₇×#120	C₂	semidirect
2. #6516	C₂	semidirect
3. #6500	C₂	semidirect
4. #6499	C₂	semidirect
5. #6448	C₂	semidirect
6. C₂⁴×C₇	C₂²	semidirect
7. C₂³×D₇	C₂²	semidirect
10. C₂×Dic₁₄	C₂²	semidirect
11. C₇×C₂²⋊C₄	C₂²	semidirect
12. C₂²⋊Dic₇	C₂²	semidirect
13. C₂²⋊Dic₇	C₂²	semidirect
14. (C₂×D₇)⋊C₄	C₂²	semidirect
15. C₄×Dic₇	C₂²	semidirect
31. #120	D₇	semidirect
42. C₂⁴	C₂×D₇	semidirect
49. C₂²⋊C₄	C₂×D₇	semidirect
65. C₇	H(3,𝔽₄)	semidirect

Maximal subgroups

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

subgroup	index
C₇×#120	2
#6516	2
#6500	2
#6499	2
#6448	2
H(3,𝔽₄)	7