Subgroups — D₄×Q₈×C₂₂₃ #28383261

Order 14272.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×C₂₂₃	C₂	1	2	yes	maximal, complemented
2	C₄⋊Q₈×C₂₂₃	C₂	1	3	yes	maximal, complemented
3	C₂²⋊Q₈×C₂₂₃	C₂	1	6	yes	maximal, complemented
4	C₄×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₄×C₂₂₃	C₂	1	3	yes	maximal
6	C₂×Q₈×C₂₂₃	—	2	8	no	—
7	C₂×Q₈×C₂₂₃	C₂²	1	6	yes	—
8	C₂×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
9	C₂×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₂₂₃	C₂²	1	6	yes	—
11	C₄⋊C₄×C₂₂₃	C₂²	1	3	yes	—
12	C₄⋊C₄×C₂₂₃	C₂²	1	3	yes	—
13	C₄⋊C₄×C₂₂₃	C₂²	1	6	yes	—
14	C₂²⋊C₄×C₂₂₃	C₂²	1	6	yes	—
15	C₄²×C₂₂₃	C₂²	1	3	yes	—
16	C₂³×C₂₂₃	C₂³	1	2	yes	—
17	Q₈×C₂₂₃	—	2	12	no	—
18	Q₈×C₂₂₃	D₄	1	4	yes	direct product factor
19	D₄×C₂₂₃	Q₈	1	4	yes	direct product factor
20	C₂×C₄×C₂₂₃	—	2	12	no	—
21	C₂×C₄×C₂₂₃	C₂³	1	3	yes	—
22	C₂×C₄×C₂₂₃	C₂³	1	6	yes	—
23	C₂×C₄×C₂₂₃	C₂³	1	3	yes	—
24	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
25	C₂²×C₂₂₃	—	2	2	no	—
26	C₂²×C₂₂₃	—	2	2	no	—
27	C₂²×C₂₂₃	C₂×Q₈	1	4	yes	complemented
28	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
29	C₄×C₂₂₃	—	2	6	no	—
30	C₄×C₂₂₃	—	2	3	no	—
31	C₄×C₂₂₃	C₂×D₄	1	6	yes	—
32	C₄×C₂₂₃	C₂×Q₈	1	2	yes	complemented
33	C₂×C₂₂₃	—	2	4	no	—
34	C₂×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
35	C₂×C₂₂₃	C₂²×Q₈	1	1	yes	characteristic
36	C₂×C₂₂₃	2₋¹⁺⁴	1	1	yes	characteristic
37	C₂₂₃	D₄×Q₈	1	1	yes	characteristic, direct product factor, Sylow
38	D₄×Q₈	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂²×Q₈	C₂×C₂₂₃	1	2	yes	complemented
40	C₄⋊Q₈	C₂×C₂₂₃	1	3	yes	complemented
41	C₂²⋊Q₈	C₂×C₂₂₃	1	6	yes	complemented
42	C₄×Q₈	C₂×C₂₂₃	1	1	yes	characteristic, complemented
43	C₄×D₄	C₂×C₂₂₃	1	3	yes	—
44	C₂×Q₈	—	2	8	no	—
45	C₂×Q₈	C₂²×C₂₂₃	1	6	yes	—
46	C₂×Q₈	C₂²×C₂₂₃	1	1	yes	characteristic
47	C₂×D₄	C₂²×C₂₂₃	1	1	yes	characteristic
48	C₂²×C₄	C₂²×C₂₂₃	1	6	yes	—
49	C₄⋊C₄	C₂²×C₂₂₃	1	3	yes	—
50	C₄⋊C₄	C₂²×C₂₂₃	1	3	yes	—
51	C₄⋊C₄	C₂²×C₂₂₃	1	6	yes	—
52	C₂²⋊C₄	C₂²×C₂₂₃	1	6	yes	—
53	C₄²	C₂²×C₂₂₃	1	3	yes	—
54	C₂³	C₂³×C₂₂₃	1	2	yes	—
55	Q₈	—	2	12	no	—
56	Q₈	D₄×C₂₂₃	1	4	yes	direct product factor
57	D₄	Q₈×C₂₂₃	1	4	yes	direct product factor
58	C₂×C₄	—	2	12	no	—
59	C₂×C₄	C₂³×C₂₂₃	1	3	yes	—
60	C₂×C₄	C₂³×C₂₂₃	1	6	yes	—
61	C₂×C₄	C₂³×C₂₂₃	1	3	yes	—
62	C₂×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
63	C₂²	—	2	2	no	—
64	C₂²	—	2	2	no	—
65	C₂²	C₂×Q₈×C₂₂₃	1	4	yes	complemented
66	C₂²	C₂⁴×C₂₂₃	1	1	yes	characteristic
67	C₄	—	2	6	no	—
68	C₄	—	2	3	no	—
69	C₄	C₂×D₄×C₂₂₃	1	6	yes	—
70	C₄	C₂×Q₈×C₂₂₃	1	2	yes	complemented
71	C₂	—	2	4	no	—
72	C₂	C₂²×D₄×C₂₂₃	1	1	yes	characteristic
73	C₂	C₂²×Q₈×C₂₂₃	1	1	yes	characteristic
74	C₂	2₋¹⁺⁴×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₂₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₂₂₃
prime radical	C₂×D₄×C₂₂₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₈×C₂₂₃	C₂	semidirect
2. C₄⋊Q₈×C₂₂₃	C₂	semidirect
3. C₂²⋊Q₈×C₂₂₃	C₂	semidirect
4. C₄×Q₈×C₂₂₃	C₂	semidirect
18. Q₈×C₂₂₃	D₄	direct (N × H)
19. D₄×C₂₂₃	Q₈	direct (N × H)
27. C₂²×C₂₂₃	C₂×Q₈	semidirect
32. C₄×C₂₂₃	C₂×Q₈	semidirect
37. C₂₂₃	D₄×Q₈	direct (N × H)
38. D₄×Q₈	C₂₂₃	direct (N × H)
39. C₂²×Q₈	C₂×C₂₂₃	semidirect
40. C₄⋊Q₈	C₂×C₂₂₃	semidirect
41. C₂²⋊Q₈	C₂×C₂₂₃	semidirect
42. C₄×Q₈	C₂×C₂₂₃	semidirect
56. Q₈	D₄×C₂₂₃	direct (N × H)
57. D₄	Q₈×C₂₂₃	direct (N × H)
65. C₂²	C₂×Q₈×C₂₂₃	semidirect
70. C₄	C₂×Q₈×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₈×C₂₂₃	2
C₄⋊Q₈×C₂₂₃	2
C₂²⋊Q₈×C₂₂₃	2
C₄×Q₈×C₂₂₃	2
C₄×D₄×C₂₂₃	2
D₄×Q₈	223

Subgroups — D4×Q8×C223 #28383261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×Q₈×C₂₂₃ #28383261