Subgroups — #37548876

Order 23168.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26042212	C₂	1	1	yes	maximal, characteristic, complemented
2	#26042208	C₂	1	1	yes	maximal, characteristic, complemented
3	D₁₆×D₁₈₁	C₂	1	1	yes	maximal, characteristic
4	#19153887	—	2	1	no	—
5	#19153866	—	2	1	no	—
6	D₈×D₁₈₁	—	2	1	no	—
7	D₂₈₉₆	C₄	1	1	yes	characteristic
8	#19153757	—	2	1	no	—
9	C₁₆×D₁₈₁	C₂²	1	1	yes	characteristic
10	D₁₆×C₁₈₁	C₄	1	1	yes	characteristic
11	#15668459	—	4	1	no	—
12	D₄×D₁₈₁	—	4	1	no	—
13	D₁₄₄₈	—	2	1	no	—
14	#15668434	—	4	1	no	—
15	C₈×D₁₈₁	D₄	1	1	yes	characteristic
16	#15668424	D₄	1	1	yes	characteristic
17	D₈×C₁₈₁	—	2	1	no	—
18	C₁₆×C₁₈₁	C₂×C₄	1	1	yes	characteristic
19	#13089685	—	8	1	no	—
20	C₂²×D₁₈₁	—	8	1	no	—
21	D₇₂₄	—	4	1	no	—
22	C₂²⋊D₁₈₁	—	8	1	no	—
23	C₄×D₁₈₁	D₈	1	1	yes	characteristic
24	#13089678	SD₁₆	1	1	yes	characteristic
25	D₄×C₁₈₁	—	4	1	no	—
26	C₈×C₁₈₁	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₁₈₁	—	8	1	no	—
28	C₂×D₁₈₁	—	8	1	no	—
29	C₂×D₁₈₁	—	16	2	no	—
30	C₂×D₁₈₁	SD₃₂	1	1	yes	characteristic
31	C₄×C₁₈₁	#102	1	1	yes	characteristic
32	Dic₁₈₁	D₁₆	1	1	yes	characteristic
33	C₂×C₁₈₁	—	16	1	no	—
34	C₂×C₁₈₁	#357	1	1	yes	characteristic
35	D₁₈₁	—	2	1	no	—
36	D₁₈₁	—	16	1	no	—
37	C₁₈₁	#1419	1	1	yes	characteristic, complemented, Sylow
38	#1419	—	181	1	no	maximal, Sylow
39	C₂×D₁₆	—	181	1	no	—
40	M₆₄	—	181	1	no	—
41	#368	—	181	1	no	—
42	C₂×D₈	—	362	1	no	—
43	D₁₆	—	362	1	no	—
44	D₁₆	—	181	1	no	—
45	D₁₆	#10624382	1	1	yes	characteristic
46	C₂×C₁₆	—	181	1	no	—
47	#108	—	362	1	no	—
48	C₃₂	—	362	1	no	—
49	C₂×D₄	—	724	1	no	—
50	D₈	—	2	1	no	—
51	D₈	—	362	1	no	—
52	D₈	—	724	1	no	—
53	M₁₆	—	724	1	no	—
54	C₂×C₈	—	181	1	no	—
55	C₁₆	—	181	1	no	—
56	C₁₆	C₂×#10624382	1	1	yes	characteristic
57	C₂³	—	1448	1	no	—
58	D₄	—	4	1	no	—
59	D₄	—	1448	1	no	—
60	D₄	—	724	1	no	—
61	C₂×C₄	—	181	1	no	—
62	C₈	—	1448	1	no	—
63	C₈	—	181	1	no	—
64	C₈	C₂²⋊#10624382	1	1	yes	characteristic
65	C₂²	—	8	1	no	—
66	C₂²	—	2896	2	no	—
67	C₂²	—	1448	1	no	—
68	C₂²	—	181	1	no	—
69	C₄	—	181	1	no	—
70	C₄	#19153878	1	1	yes	characteristic
71	C₂	—	16	1	no	—
72	C₂	—	2896	1	no	—
73	C₂	—	362	1	no	—
74	C₂	#26042191	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26042191
commutator	C₁₆×C₁₈₁	C₂×C₄
Frattini	C₁₆	C₂×#10624382
Fitting	D₁₆×C₁₈₁	C₄
socle	C₂×C₁₈₁	#357
prime radical	D₁₆×D₁₈₁	C₂
cosocle	C₁₆×D₁₈₁	C₂²
hypercenter	D₁₆	#10624382
hypocenter	C₁₈₁	#1419
LCS	G ▷ C₁₆×C₁₈₁ ▷ C₈×C₁₈₁ ▷ C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ D₁₆

Complements

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

N	H ≅ G/N	type
1. #26042212	C₂	semidirect
2. #26042208	C₂	semidirect
37. C₁₈₁	#1419	semidirect

Maximal subgroups

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

subgroup	index
#26042212	2
#26042208	2
D₁₆×D₁₈₁	2
#1419	181