Subgroups — #90408

Order 448.

428 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×D₇	C₂	1	4	yes	maximal, complemented
2	C₇×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	#6501	C₂	1	4	yes	maximal, complemented
4	#6500	C₂	1	4	yes	maximal, complemented
5	C₄×D₂₈	C₂	1	2	yes	maximal, complemented
6	C₂³×D₇	C₂²	1	4	yes	complemented
7	C₂×C₇×D₄	C₂²	1	4	yes	complemented
8	C₂×C₇×D₄	C₂²	1	2	yes	complemented
9	C₂×C₂²⋊D₇	C₂²	1	8	yes	complemented
10	D₄×D₇	—	2	16	no	—
11	C₂×D₂₈	C₂²	1	2	yes	—
12	C₂×C₄×D₇	C₂²	1	4	yes	complemented
13	C₄²×C₇	C₂²	1	1	yes	characteristic, complemented
14	C₂²⋊Dic₇	C₂²	1	4	yes	complemented
15	(C₂×D₇)⋊C₄	C₂²	1	4	yes	complemented
16	C₄⋊Dic₇	C₂²	1	2	yes	complemented
17	D₄²	—	7	1	no	maximal, Sylow
18	C₂³×C₇	C₂³	1	4	yes	—
19	C₂²×D₇	—	2	16	no	—
20	C₂²×D₇	—	4	4	no	—
21	C₂²×D₇	C₂³	1	4	yes	—
22	C₇×D₄	—	2	8	no	—
23	C₇×D₄	—	4	2	no	—
24	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₇	C₂³	1	2	yes	—
26	C₂²⋊D₇	—	2	16	no	—
27	C₂×Dic₇	C₂³	1	4	yes	—
28	D₂₈	D₄	1	8	yes	complemented
29	C₄×D₇	—	2	8	no	—
30	C₂²×D₄	—	7	4	no	—
31	#127	D₇	1	1	yes	characteristic, complemented
32	#121	—	7	4	no	—
33	#120	—	7	4	no	—
34	C₄×D₄	—	7	2	no	—
35	C₂²×C₇	—	2	8	no	—
36	C₂²×C₇	—	2	4	no	—
37	C₂²×C₇	C₂⁴	1	1	yes	characteristic
38	C₂×D₇	—	2	4	no	—
39	C₂×D₇	—	2	4	no	—
40	C₂×D₇	—	4	16	no	—
41	C₂×D₇	C₂×D₄	1	8	yes	complemented
42	C₄×C₇	—	2	1	no	—
43	C₄×C₇	C₂×D₄	1	4	yes	complemented
44	Dic₇	—	2	4	no	—
45	C₂⁴	—	7	4	no	—
46	C₂×D₄	—	7	8	no	—
47	C₂×D₄	—	7	2	no	—
48	C₂×D₄	—	14	16	no	—
49	C₂×D₄	C₂×D₇	1	4	yes	complemented
50	C₂×D₄	C₂×D₇	1	2	yes	complemented
51	C₂²×C₄	—	7	4	no	—
52	C₄⋊C₄	—	7	2	no	—
53	C₂²⋊C₄	—	7	4	no	—
54	C₂²⋊C₄	—	7	4	no	—
55	C₄²	C₂×D₇	1	1	yes	characteristic, complemented
56	C₂×C₇	—	4	4	no	—
57	C₂×C₇	C₂²×D₄	1	2	yes	—
58	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
59	D₇	—	2	8	no	—
60	C₂³	—	7	4	no	—
61	C₂³	—	14	16	no	—
62	C₂³	—	28	4	no	—
63	C₂³	C₂²×D₇	1	4	yes	—
64	D₄	—	2	8	no	—
65	D₄	—	4	2	no	—
66	D₄	—	7	8	no	—
67	D₄	—	14	16	no	—
68	C₂×C₄	—	7	4	no	—
69	C₂×C₄	—	14	8	no	—
70	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₇	1	2	yes	—
72	C₇	D₄²	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	2	8	no	—
74	C₂²	—	2	4	no	—
75	C₂²	—	7	8	no	—
76	C₂²	—	14	4	no	—
77	C₂²	—	14	4	no	—
78	C₂²	—	28	16	no	—
79	C₂²	C₂³×D₇	1	1	yes	characteristic
80	C₄	—	2	1	no	—
81	C₄	—	14	4	no	—
82	C₄	D₄×D₇	1	4	yes	complemented
83	C₂	—	4	4	no	—
84	C₂	—	14	8	no	—
85	C₂	C₂×D₄×D₇	1	2	yes	—
86	C₂	#6548	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×#127	C₂
hypercenter	#127	D₇
hypocenter	C₇	D₄²
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×D₇	C₂	semidirect
2. C₇×#127	C₂	semidirect
3. #6501	C₂	semidirect
4. #6500	C₂	semidirect
5. C₄×D₂₈	C₂	semidirect
6. C₂³×D₇	C₂²	semidirect
7. C₂×C₇×D₄	C₂²	semidirect
8. C₂×C₇×D₄	C₂²	semidirect
9. C₂×C₂²⋊D₇	C₂²	semidirect
12. C₂×C₄×D₇	C₂²	semidirect
13. C₄²×C₇	C₂²	semidirect
14. C₂²⋊Dic₇	C₂²	semidirect
15. (C₂×D₇)⋊C₄	C₂²	semidirect
16. C₄⋊Dic₇	C₂²	semidirect
28. D₂₈	D₄	semidirect
31. #127	D₇	semidirect
41. C₂×D₇	C₂×D₄	semidirect
43. C₄×C₇	C₂×D₄	semidirect
49. C₂×D₄	C₂×D₇	semidirect
50. C₂×D₄	C₂×D₇	semidirect
55. C₄²	C₂×D₇	semidirect
72. C₇	D₄²	semidirect
82. C₄	D₄×D₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×D₇	2
C₇×#127	2
#6501	2
#6500	2
C₄×D₂₈	2
D₄²	7