Subgroups — C₇×D₄×Q₈ #90570

Order 448.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₂²×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₇
prime radical	C₂×C₇×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×Q₈	C₂	semidirect
2. C₇×C₄⋊Q₈	C₂	semidirect
3. C₇×C₂²⋊Q₈	C₂	semidirect
4. C₄×C₇×Q₈	C₂	semidirect
16. D₄×Q₈	C₇	direct (N × H)
19. C₇×Q₈	D₄	direct (N × H)
20. C₇×D₄	Q₈	direct (N × H)
26. C₂²×Q₈	C₂×C₇	semidirect
27. C₄⋊Q₈	C₂×C₇	semidirect
28. C₂²⋊Q₈	C₂×C₇	semidirect
29. C₄×Q₈	C₂×C₇	semidirect
33. C₂²×C₇	C₂×Q₈	semidirect
38. C₄×C₇	C₂×Q₈	semidirect
55. Q₈	C₇×D₄	direct (N × H)
56. D₄	C₇×Q₈	direct (N × H)
62. C₇	D₄×Q₈	direct (N × H)
65. C₂²	C₂×C₇×Q₈	semidirect
70. C₄	C₂×C₇×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×Q₈	2
C₇×C₄⋊Q₈	2
C₇×C₂²⋊Q₈	2
C₄×C₇×Q₈	2
C₄×C₇×D₄	2
D₄×Q₈	7

Subgroups — C7×D4×Q8 #90570

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×D₄×Q₈ #90570