Subgroups — #37548750

Order 23168.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#15668452	C₂	1	2	yes	maximal
2	C₈×#13089678	C₂	1	1	yes	maximal, characteristic
3	C₂×#15668452	—	2	2	no	—
4	C₄×#13089678	C₄	1	1	yes	characteristic
5	C₄×#13089678	C₂²	1	1	yes	characteristic
6	C₄×C₈×C₁₈₁	C₄	1	1	yes	characteristic
7	#15668452	—	4	2	no	—
8	C₂×#13089678	—	2	1	no	—
9	C₂×#13089678	D₄	1	1	yes	characteristic
10	C₂×#13089678	Q₈	1	1	yes	characteristic
11	C₂×C₈×C₁₈₁	C₈	1	1	yes	characteristic
12	C₂×C₈×C₁₈₁	C₈	1	1	yes	characteristic
13	C₄²×C₁₈₁	C₂×C₄	1	1	yes	characteristic
14	#13089678	—	2	2	no	—
15	#13089678	—	2	1	no	—
16	#13089678	D₈	1	1	yes	characteristic
17	#13089678	Q₁₆	1	1	yes	characteristic
18	C₂×C₄×C₁₈₁	C₄⋊C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₈₁	C₂×C₈	1	1	yes	characteristic
20	C₂×C₄×C₁₈₁	M₁₆	1	1	yes	characteristic
21	C₈×C₁₈₁	—	2	1	no	—
22	C₈×C₁₈₁	C₁₆	1	2	yes	complemented
23	C₂²×C₁₈₁	C₄⋊C₈	1	1	yes	characteristic
24	C₄×C₁₈₁	—	2	1	no	—
25	C₄×C₁₈₁	#107	1	1	yes	characteristic
26	C₄×C₁₈₁	#108	1	1	yes	characteristic
27	C₄×C₁₈₁	C₂×C₁₆	1	1	yes	characteristic
28	C₄×C₁₈₁	M₃₂	1	1	yes	characteristic
29	C₂×C₁₈₁	#335	1	1	yes	characteristic
30	C₂×C₁₈₁	C₄⋊C₁₆	1	1	yes	characteristic
31	C₂×C₁₈₁	#364	1	1	yes	characteristic
32	C₁₈₁	#1370	1	1	yes	characteristic, complemented, Sylow
33	#1370	—	181	1	no	maximal, Sylow
34	C₄⋊C₁₆	—	181	2	no	—
35	C₈²	—	181	1	no	—
36	C₂×C₁₆	—	362	2	no	—
37	C₄×C₈	—	181	1	no	—
38	C₄×C₈	—	181	1	no	—
39	C₄×C₈	#10624382	1	1	yes	characteristic
40	C₂×C₈	—	362	1	no	—
41	C₂×C₈	—	181	1	no	—
42	C₂×C₈	—	181	1	no	—
43	C₂×C₈	#13089685	1	1	yes	characteristic
44	C₂×C₈	#13089685	1	1	yes	characteristic
45	C₄²	C₂×#10624382	1	1	yes	characteristic
46	C₁₆	—	724	2	no	—
47	C₂×C₄	C₄⋊#10624382	1	1	yes	characteristic
48	C₂×C₄	C₂×#13089685	1	1	yes	characteristic
49	C₂×C₄	#15668459	1	1	yes	characteristic
50	C₈	—	2	1	no	—
51	C₈	—	362	2	no	—
52	C₈	—	362	1	no	—
53	C₈	—	181	1	no	—
54	C₈	—	181	1	no	—
55	C₈	#15668452	1	2	yes	complemented
56	C₂²	C₄⋊#13089685	1	1	yes	characteristic
57	C₄	—	2	1	no	—
58	C₄	#19153885	1	1	yes	characteristic
59	C₄	#19153887	1	1	yes	characteristic
60	C₄	C₂×#15668452	1	1	yes	characteristic
61	C₄	#19153890	1	1	yes	characteristic
62	C₂	#26042142	1	1	yes	characteristic
63	C₂	C₄⋊#15668452	1	1	yes	characteristic
64	C₂	#26042201	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#10624382
commutator	C₄×C₁₈₁	C₂×C₁₆
Frattini	C₄²	C₂×#10624382
Fitting	C₄×C₈×C₁₈₁	C₄
socle, prime radical	C₂²×C₁₈₁	C₄⋊C₈
cosocle	C₄×#13089678	C₂²
hypercenter	C₄×C₈	#10624382
hypocenter	C₁₈₁	#1370
LCS	G ▷ C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
22. C₈×C₁₈₁	C₁₆	semidirect
32. C₁₈₁	#1370	semidirect
55. C₈	#15668452	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#15668452	2
C₈×#13089678	2
#1370	181