Subgroups — C₂×#6400 #89918

Order 448.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×D₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#6400	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇×D₄	—	2	1	no	—
6	C₂×D₂₈	—	2	1	no	—
7	C₇×D₈	—	2	1	no	—
8	C₇×D₈	C₂²	1	2	yes	complemented
9	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
10	D₅₆	—	2	1	no	—
11	D₅₆	C₂²	1	2	yes	complemented
12	C₇⋊C₁₆	C₂²	1	2	yes	complemented
13	C₂×D₁₆	—	7	1	no	maximal, Sylow
14	C₂³×C₇	—	4	1	no	—
15	C₂²×D₇	—	4	1	no	—
16	C₇×D₄	—	2	2	no	—
17	C₇×D₄	—	4	1	no	—
18	C₂×C₄×C₇	D₄	1	1	yes	characteristic
19	D₂₈	—	2	2	no	—
20	D₂₈	—	4	1	no	—
21	C₇×C₈	D₄	1	1	yes	characteristic
22	C₇×C₈	C₂³	1	1	yes	characteristic
23	C₂×D₈	—	7	1	no	—
24	C₂×D₈	D₇	1	1	yes	characteristic, complemented
25	D₁₆	—	7	4	no	—
26	C₂×C₁₆	—	7	1	no	—
27	C₂²×C₇	—	4	2	no	—
28	C₂²×C₇	—	8	2	no	—
29	C₂²×C₇	D₈	1	1	yes	characteristic
30	C₂×D₇	—	4	2	no	—
31	C₂×D₇	—	8	2	no	—
32	C₄×C₇	D₈	1	1	yes	characteristic
33	C₄×C₇	C₂×D₄	1	1	yes	characteristic
34	C₂×D₄	—	2	1	no	—
35	C₂×D₄	—	14	1	no	—
36	D₈	—	2	1	no	—
37	D₈	—	7	2	no	—
38	D₈	—	14	1	no	—
39	D₈	C₂×D₇	1	2	yes	complemented
40	C₂×C₈	C₂×D₇	1	1	yes	characteristic
41	C₁₆	—	7	2	no	—
42	C₂×C₇	—	8	2	no	—
43	C₂×C₇	D₁₆	1	2	yes	complemented
44	C₂×C₇	C₂×D₈	1	1	yes	characteristic
45	D₇	—	8	2	no	—
46	C₂³	—	4	1	no	—
47	C₂³	—	28	1	no	—
48	D₄	—	2	2	no	—
49	D₄	—	4	1	no	—
50	D₄	—	14	2	no	—
51	D₄	—	28	1	no	—
52	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
53	C₈	C₂²⋊D₇	1	1	yes	characteristic
54	C₈	C₂²×D₇	1	1	yes	characteristic
55	C₇	C₂×D₁₆	1	1	yes	characteristic, complemented, Sylow
56	C₂²	—	4	2	no	—
57	C₂²	—	8	2	no	—
58	C₂²	—	28	2	no	—
59	C₂²	—	56	2	no	—
60	C₂²	#1141	1	1	yes	characteristic
61	C₄	#1141	1	1	yes	characteristic
62	C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
63	C₂	—	8	2	no	—
64	C₂	—	56	2	no	—
65	C₂	#6400	1	2	yes	direct product factor
66	C₂	C₂×#1141	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#1141
commutator, cosocle	C₇×C₈	C₂³
Frattini	C₈	C₂²×D₇
Fitting	C₂×C₇×D₈	C₂
socle	C₂²×C₇	D₈
hypercenter	C₂×D₈	D₇
hypocenter	C₇	C₂×D₁₆
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×D₈	C₂	semidirect
2. C₂×D₅₆	C₂	semidirect
3. #6400	C₂	direct (N × H)
4. C₂×C₇⋊C₁₆	C₂	semidirect
8. C₇×D₈	C₂²	semidirect
11. D₅₆	C₂²	semidirect
12. C₇⋊C₁₆	C₂²	semidirect
24. C₂×D₈	D₇	semidirect
39. D₈	C₂×D₇	semidirect
43. C₂×C₇	D₁₆	semidirect
55. C₇	C₂×D₁₆	semidirect
65. C₂	#6400	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×D₈	2
C₂×D₅₆	2
#6400	2
C₂×C₇⋊C₁₆	2
C₂×D₁₆	7

Subgroups — C2×#6400 #89918

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#6400 #89918