Subgroups — C₂²×D₃₆₈ #15686503

Order 2944.

440 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₈₄	C₂	1	2	yes	maximal, complemented
2	C₂×D₃₆₈	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₉₂	—	2	2	no	—
5	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×D₁₈₄	—	2	6	no	—
7	C₂×D₁₈₄	C₂²	1	12	yes	complemented
8	C₂×C₁₆×C₂₃	C₂²	1	6	yes	complemented
9	D₃₆₈	C₂²	1	16	yes	direct product factor
10	C₂³×D₂₃	—	4	2	no	—
11	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
12	C₂×D₉₂	—	2	12	no	—
13	C₂×D₉₂	—	4	6	no	—
14	C₂×C₈×C₂₃	D₄	1	3	yes	—
15	C₂×C₈×C₂₃	C₂³	1	3	yes	—
16	D₁₈₄	—	2	12	no	—
17	D₁₈₄	C₂³	1	8	yes	complemented
18	C₁₆×C₂₃	C₂³	1	4	yes	complemented
19	C₂³×C₂₃	D₈	1	1	yes	characteristic
20	C₂²×D₂₃	—	4	12	no	—
21	C₂²×D₂₃	—	8	8	no	—
22	C₂×C₄×C₂₃	D₈	1	3	yes	—
23	C₂×C₄×C₂₃	C₂×D₄	1	3	yes	—
24	D₉₂	—	2	8	no	—
25	D₉₂	—	4	12	no	—
26	C₈×C₂₃	C₂×D₄	1	3	yes	—
27	C₈×C₂₃	C₂⁴	1	1	yes	characteristic
28	C₂²×D₁₆	—	23	1	no	maximal, Sylow
29	C₂²×C₂₃	D₁₆	1	4	yes	complemented
30	C₂²×C₂₃	C₂×D₈	1	3	yes	—
31	C₂×D₂₃	—	4	8	no	—
32	C₂×D₂₃	—	8	24	no	—
33	C₄×C₂₃	C₂×D₈	1	3	yes	—
34	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
35	C₂²×D₈	—	23	2	no	—
36	C₂×D₁₆	—	23	12	no	—
37	C₂²×C₁₆	D₂₃	1	1	yes	characteristic, complemented
38	C₂×C₂₃	C₂×D₁₆	1	6	yes	complemented
39	C₂×C₂₃	C₂²×D₈	1	1	yes	characteristic
40	D₂₃	—	8	8	no	—
41	C₂²×D₄	—	46	2	no	—
42	C₂×D₈	—	23	12	no	—
43	C₂×D₈	—	46	6	no	—
44	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
45	D₁₆	—	23	16	no	—
46	C₂×C₁₆	C₂×D₂₃	1	6	yes	complemented
47	C₂₃	C₂²×D₁₆	1	1	yes	characteristic, complemented, Sylow
48	C₂⁴	—	92	2	no	—
49	C₂×D₄	—	46	12	no	—
50	C₂×D₄	—	92	6	no	—
51	C₂²×C₄	D₉₂	1	1	yes	characteristic
52	D₈	—	23	8	no	—
53	D₈	—	46	12	no	—
54	C₂×C₈	D₉₂	1	3	yes	—
55	C₂×C₈	C₂²×D₂₃	1	3	yes	—
56	C₁₆	C₂²×D₂₃	1	4	yes	complemented
57	C₂³	—	92	12	no	—
58	C₂³	—	184	8	no	—
59	C₂³	D₁₈₄	1	1	yes	characteristic
60	D₄	—	46	8	no	—
61	D₄	—	92	12	no	—
62	C₂×C₄	D₁₈₄	1	3	yes	—
63	C₂×C₄	C₂×D₉₂	1	3	yes	—
64	C₈	C₂×D₉₂	1	3	yes	—
65	C₈	C₂³×D₂₃	1	1	yes	characteristic
66	C₂²	—	92	8	no	—
67	C₂²	—	184	24	no	—
68	C₂²	D₃₆₈	1	4	yes	direct product factor
69	C₂²	C₂×D₁₈₄	1	3	yes	—
70	C₄	C₂×D₁₈₄	1	3	yes	—
71	C₄	C₂²×D₉₂	1	1	yes	characteristic
72	C₂	—	184	8	no	—
73	C₂	C₂×D₃₆₈	1	6	yes	direct product factor
74	C₂	C₂²×D₁₈₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₁₈₄
commutator, cosocle	C₈×C₂₃	C₂⁴
Frattini	C₈	C₂³×D₂₃
Fitting	C₂²×C₁₆×C₂₃	C₂
socle	C₂³×C₂₃	D₈
hypercenter	C₂²×C₁₆	D₂₃
hypocenter	C₂₃	C₂²×D₁₆
LCS	G ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₈₄	C₂	semidirect
2. C₂×D₃₆₈	C₂	direct (N × H)
3. C₂²×C₁₆×C₂₃	C₂	semidirect
7. C₂×D₁₈₄	C₂²	semidirect
8. C₂×C₁₆×C₂₃	C₂²	semidirect
9. D₃₆₈	C₂²	direct (N × H)
17. D₁₈₄	C₂³	semidirect
18. C₁₆×C₂₃	C₂³	semidirect
29. C₂²×C₂₃	D₁₆	semidirect
37. C₂²×C₁₆	D₂₃	semidirect
38. C₂×C₂₃	C₂×D₁₆	semidirect
46. C₂×C₁₆	C₂×D₂₃	semidirect
47. C₂₃	C₂²×D₁₆	semidirect
56. C₁₆	C₂²×D₂₃	semidirect
68. C₂²	D₃₆₈	direct (N × H)
73. C₂	C₂×D₃₆₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₁₈₄	2
C₂×D₃₆₈	2
C₂²×C₁₆×C₂₃	2
C₂²×D₁₆	23

Subgroups — C22×D368 #15686503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₃₆₈ #15686503