Subgroups — #15675647

Order 2944.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₆₈	C₂	1	4	yes	maximal, complemented
2	#13092794	C₂	1	2	yes	maximal, complemented
3	C₄×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₈₄	—	2	2	no	—
5	C₂×D₁₈₄	C₂²	1	4	yes	—
6	#10625045	—	2	2	no	—
7	C₂×C₁₆×C₂₃	C₂²	1	2	yes	—
8	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	D₃₆₈	—	2	8	no	—
10	C₂×D₉₂	—	2	4	no	—
11	C₂×D₉₂	—	4	4	no	—
12	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
13	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
14	C₄²×C₂₃	D₄	1	1	yes	characteristic
15	D₁₈₄	—	2	8	no	—
16	D₁₈₄	—	4	4	no	—
17	C₁₆×C₂₃	D₄	1	4	yes	complemented
18	C₂²×D₂₃	—	4	4	no	—
19	C₂×C₄×C₂₃	D₈	1	2	yes	—
20	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
21	D₉₂	—	4	4	no	—
22	D₉₂	—	4	4	no	—
23	D₉₂	—	8	8	no	—
24	C₈×C₂₃	D₈	1	2	yes	—
25	C₈×C₂₃	C₂×D₄	1	2	yes	—
26	#2246	—	23	1	no	maximal, Sylow
27	C₂²×C₂₃	C₂×D₈	1	1	yes	characteristic
28	C₂×D₂₃	—	8	4	no	—
29	C₂×D₂₃	—	8	8	no	—
30	C₄×C₂₃	D₁₆	1	4	yes	complemented
31	C₄×C₂₃	#127	1	1	yes	characteristic
32	C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
33	C₂×D₁₆	—	23	4	no	—
34	#493	—	23	2	no	—
35	C₄×C₁₆	D₂₃	1	1	yes	characteristic, complemented
36	C₂×C₂₃	#493	1	1	yes	characteristic
37	C₂×C₂₃	C₂×D₁₆	1	2	yes	—
38	D₂₃	—	16	4	no	—
39	C₂×D₈	—	23	4	no	—
40	C₂×D₈	—	46	2	no	—
41	#127	—	46	2	no	—
42	D₁₆	—	46	8	no	—
43	C₂×C₁₆	C₂×D₂₃	1	2	yes	—
44	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
45	C₂₃	#2246	1	1	yes	characteristic, complemented, Sylow
46	C₂×D₄	—	46	4	no	—
47	C₂×D₄	—	92	4	no	—
48	D₈	—	46	8	no	—
49	D₈	—	92	4	no	—
50	C₂×C₈	D₉₂	1	1	yes	characteristic
51	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
52	C₄²	D₉₂	1	1	yes	characteristic
53	C₁₆	D₉₂	1	4	yes	complemented
54	C₂³	—	92	4	no	—
55	D₄	—	92	4	no	—
56	D₄	—	92	4	no	—
57	D₄	—	184	8	no	—
58	C₂×C₄	D₁₈₄	1	2	yes	—
59	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
60	C₈	D₁₈₄	1	2	yes	—
61	C₈	C₂×D₉₂	1	2	yes	—
62	C₂²	—	184	4	no	—
63	C₂²	—	184	8	no	—
64	C₂²	C₂×D₁₈₄	1	1	yes	characteristic
65	C₄	D₃₆₈	1	4	yes	complemented
66	C₄	#10625045	1	1	yes	characteristic
67	C₄	C₂×D₁₈₄	1	1	yes	characteristic
68	C₂	—	368	4	no	—
69	C₂	#13092794	1	1	yes	characteristic
70	C₂	C₂×D₃₆₈	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₈₄
commutator, cosocle	C₂×C₈×C₂₃	C₂³
Frattini	C₂×C₈	C₂²×D₂₃
Fitting	C₄×C₁₆×C₂₃	C₂
socle	C₂²×C₂₃	C₂×D₈
hypercenter	C₄×C₁₆	D₂₃
hypocenter	C₂₃	#2246
LCS	G ▷ C₂×C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×D₃₆₈	C₂	semidirect
2. #13092794	C₂	semidirect
3. C₄×C₁₆×C₂₃	C₂	semidirect
17. C₁₆×C₂₃	D₄	semidirect
30. C₄×C₂₃	D₁₆	semidirect
35. C₄×C₁₆	D₂₃	semidirect
45. C₂₃	#2246	semidirect
53. C₁₆	D₉₂	semidirect
65. C₄	D₃₆₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃₆₈	2
#13092794	2
C₄×C₁₆×C₂₃	2
#2246	23