Subgroups — C₂×#6516 #90531

Order 448.

662 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂²⋊D₇	C₂	1	3	yes	maximal, complemented
3	#6516	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂²⋊Dic₇	C₂	1	3	yes	maximal, complemented
5	C₂⁴×C₇	—	2	12	no	—
6	C₂⁴×C₇	C₂²	1	3	yes	—
7	C₂⁴×C₇	C₂²	1	4	yes	complemented
8	C₂³×D₇	C₂²	1	1	yes	characteristic, complemented
9	C₂×C₂²⋊D₇	—	2	12	no	—
10	C₂×C₂²⋊D₇	C₂²	1	12	yes	complemented
11	C₂²×Dic₇	C₂²	1	3	yes	complemented
12	C₂²⋊Dic₇	C₂²	1	12	yes	complemented
13	C₂×#120	—	7	1	no	maximal, Sylow
14	C₂³×C₇	—	2	32	no	—
15	C₂³×C₇	—	2	12	no	—
16	C₂³×C₇	—	2	24	no	—
17	C₂³×C₇	D₄	1	12	yes	complemented
18	C₂³×C₇	C₂³	1	6	yes	—
19	C₂³×C₇	C₂³	1	1	yes	characteristic
20	C₂²×D₇	—	2	6	no	—
21	C₂²×D₇	C₂³	1	2	yes	complemented
22	C₂²⋊D₇	—	2	24	no	—
23	C₂×Dic₇	—	2	6	no	—
24	C₂×Dic₇	C₂³	1	6	yes	complemented
25	C₂⁵	D₇	1	1	yes	characteristic, complemented
26	C₂²×D₄	—	7	3	no	—
27	#120	—	7	8	no	—
28	C₂×C₂²⋊C₄	—	7	3	no	—
29	C₂²×C₇	—	2	32	no	—
30	C₂²×C₇	—	2	24	no	—
31	C₂²×C₇	—	2	12	no	—
32	C₂²×C₇	C₂×D₄	1	6	yes	—
33	C₂²×C₇	C₂×D₄	1	12	yes	complemented
34	C₂²×C₇	C₂⁴	1	1	yes	characteristic
35	C₂×D₇	—	2	6	no	—
36	C₂×D₇	—	4	4	no	—
37	Dic₇	—	2	6	no	—
38	C₂⁴	—	2	12	no	—
39	C₂⁴	—	7	1	no	—
40	C₂⁴	C₂×D₇	1	3	yes	—
41	C₂⁴	C₂×D₇	1	4	yes	complemented
42	C₂×D₄	—	7	12	no	—
43	C₂×D₄	—	14	12	no	—
44	C₂²×C₄	—	7	3	no	—
45	C₂²⋊C₄	—	7	12	no	—
46	C₂×C₇	—	2	12	no	—
47	C₂×C₇	#120	1	4	yes	complemented
48	C₂×C₇	C₂²×D₄	1	3	yes	—
49	D₇	—	4	2	no	—
50	C₂³	—	2	32	no	—
51	C₂³	—	2	12	no	—
52	C₂³	—	2	24	no	—
53	C₂³	—	7	2	no	—
54	C₂³	—	14	6	no	—
55	C₂³	C₂²⋊D₇	1	12	yes	complemented
56	C₂³	C₂²×D₇	1	6	yes	—
57	C₂³	C₂²×D₇	1	1	yes	characteristic
58	D₄	—	14	24	no	—
59	C₂×C₄	—	7	6	no	—
60	C₂×C₄	—	14	6	no	—
61	C₇	C₂×#120	1	1	yes	characteristic, complemented, Sylow
62	C₂²	—	2	32	no	—
63	C₂²	—	2	24	no	—
64	C₂²	—	2	12	no	—
65	C₂²	—	14	6	no	—
66	C₂²	—	28	4	no	—
67	C₂²	C₂×C₂²⋊D₇	1	6	yes	—
68	C₂²	C₂×C₂²⋊D₇	1	12	yes	complemented
69	C₂²	C₂³×D₇	1	1	yes	characteristic
70	C₄	—	14	6	no	—
71	C₂	—	2	12	no	—
72	C₂	—	28	2	no	—
73	C₂	#6516	1	4	yes	direct product factor
74	C₂	C₂²×C₂²⋊D₇	1	3	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₇
commutator, cosocle	C₂²×C₇	C₂⁴
Frattini	C₂²	C₂³×D₇
Fitting	C₂⁵×C₇	C₂
socle	C₂³×C₇	C₂³
hypercenter	C₂⁵	D₇
hypocenter	C₇	C₂×#120
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂⁵

Complements

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

N	H ≅ G/N	type
1. C₂⁵×C₇	C₂	semidirect
2. C₂²×C₂²⋊D₇	C₂	semidirect
3. #6516	C₂	direct (N × H)
4. C₂×C₂²⋊Dic₇	C₂	semidirect
7. C₂⁴×C₇	C₂²	semidirect
8. C₂³×D₇	C₂²	semidirect
10. C₂×C₂²⋊D₇	C₂²	semidirect
11. C₂²×Dic₇	C₂²	semidirect
12. C₂²⋊Dic₇	C₂²	semidirect
17. C₂³×C₇	D₄	semidirect
21. C₂²×D₇	C₂³	semidirect
24. C₂×Dic₇	C₂³	semidirect
25. C₂⁵	D₇	semidirect
33. C₂²×C₇	C₂×D₄	semidirect
41. C₂⁴	C₂×D₇	semidirect
47. C₂×C₇	#120	semidirect
55. C₂³	C₂²⋊D₇	semidirect
61. C₇	C₂×#120	semidirect
68. C₂²	C₂×C₂²⋊D₇	semidirect
73. C₂	#6516	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₇	2
C₂²×C₂²⋊D₇	2
#6516	2
C₂×C₂²⋊Dic₇	2
C₂×#120	7

Subgroups — C2×#6516 #90531

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#6516 #90531