Subgroups — Q₈×D₂₈ #90266

Order 448.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₇	C₂	1	2	yes	maximal, complemented
2	C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#6460	C₂	1	6	yes	maximal, complemented
4	C₄×D₂₈	C₂	1	3	yes	maximal
5	#6432	C₂	1	3	yes	maximal, complemented
6	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
7	Q₈×D₇	—	2	8	no	—
8	C₂×D₂₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₇	C₂²	1	6	yes	—
10	C₂×Dic₁₄	C₂²	1	6	yes	—
11	C₇×C₄⋊C₄	C₂²	1	3	yes	—
12	C₄²×C₇	C₂²	1	3	yes	—
13	(C₂×D₇)⋊C₄	C₂²	1	6	yes	—
14	C₄⋊Dic₇	C₂²	1	3	yes	—
15	C₄⋊Dic₇	C₂²	1	6	yes	—
16	D₄×Q₈	—	7	1	no	maximal, Sylow
17	C₂²×D₇	C₂³	1	2	yes	—
18	C₇×Q₈	D₄	1	4	yes	complemented
19	C₂×C₄×C₇	C₂³	1	3	yes	—
20	C₂×C₄×C₇	C₂³	1	3	yes	—
21	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
22	C₂×Dic₇	C₂³	1	6	yes	—
23	D₂₈	Q₈	1	4	yes	direct product factor
24	C₄×D₇	—	2	12	no	—
25	Dic₁₄	—	2	12	no	—
26	C₂²×Q₈	—	7	2	no	—
27	C₄⋊Q₈	—	7	3	no	—
28	C₂²⋊Q₈	—	7	6	no	—
29	C₄×Q₈	D₇	1	1	yes	characteristic, complemented
30	C₄×D₄	—	7	3	no	—
31	C₂²×C₇	C₂⁴	1	1	yes	characteristic
32	C₂×D₇	—	2	2	no	—
33	C₂×D₇	—	2	2	no	—
34	C₂×D₇	C₂×Q₈	1	4	yes	complemented
35	C₄×C₇	—	2	3	no	—
36	C₄×C₇	C₂×D₄	1	6	yes	—
37	C₄×C₇	C₂×Q₈	1	2	yes	complemented
38	Dic₇	—	2	6	no	—
39	C₂×Q₈	—	7	6	no	—
40	C₂×Q₈	—	14	8	no	—
41	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
42	C₂×D₄	—	7	1	no	—
43	C₂²×C₄	—	7	6	no	—
44	C₄⋊C₄	—	7	3	no	—
45	C₄⋊C₄	—	7	6	no	—
46	C₄⋊C₄	C₂×D₇	1	3	yes	—
47	C₂²⋊C₄	—	7	6	no	—
48	C₄²	C₂×D₇	1	3	yes	—
49	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
50	C₂×C₇	C₂²×Q₈	1	1	yes	characteristic
51	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
52	D₇	—	2	4	no	—
53	C₂³	—	7	2	no	—
54	Q₈	—	14	12	no	—
55	Q₈	D₂₈	1	4	yes	direct product factor
56	D₄	—	7	4	no	—
57	C₂×C₄	—	7	6	no	—
58	C₂×C₄	—	14	12	no	—
59	C₂×C₄	C₂²×D₇	1	3	yes	—
60	C₂×C₄	C₂²×D₇	1	3	yes	—
61	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
62	C₇	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
63	C₂²	—	7	4	no	—
64	C₂²	—	14	2	no	—
65	C₂²	—	14	2	no	—
66	C₂²	C₂³×D₇	1	1	yes	characteristic
67	C₄	—	2	3	no	—
68	C₄	—	14	6	no	—
69	C₄	C₂×D₂₈	1	6	yes	—
70	C₄	Q₈×D₇	1	2	yes	complemented
71	C₂	—	14	4	no	—
72	C₂	C₂²×D₂₈	1	1	yes	characteristic
73	C₂	C₂×Q₈×D₇	1	1	yes	characteristic
74	C₂	#6554	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₄×C₇×Q₈	C₂
prime radical	C₂×D₂₈	C₂²
hypercenter	C₄×Q₈	D₇
hypocenter	C₇	D₄×Q₈
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×D₇	C₂	semidirect
2. C₄×C₇×Q₈	C₂	semidirect
3. #6460	C₂	semidirect
5. #6432	C₂	semidirect
18. C₇×Q₈	D₄	semidirect
23. D₂₈	Q₈	direct (N × H)
29. C₄×Q₈	D₇	semidirect
34. C₂×D₇	C₂×Q₈	semidirect
37. C₄×C₇	C₂×Q₈	semidirect
55. Q₈	D₂₈	direct (N × H)
62. C₇	D₄×Q₈	semidirect
70. C₄	Q₈×D₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×D₇	2
C₄×C₇×Q₈	2
#6460	2
C₄×D₂₈	2
#6432	2
D₄×Q₈	7

Subgroups — Q8×D28 #90266

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×D₂₈ #90266