Subgroups — #5753

Order 192.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1010	C₂	1	2	yes	maximal, complemented
2	C₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	#909	C₂	1	6	yes	maximal, complemented
4	C₂×Dic₁₂	C₂	1	3	yes	maximal, complemented
5	#904	C₂	1	3	yes	maximal, complemented
6	#578	—	3	1	no	maximal, Sylow
7	C₃×#41	C₂²	1	1	yes	characteristic
8	D₃×Q₈	—	2	2	no	—
9	#237	—	2	6	no	—
10	#235	C₂²	1	6	yes	—
11	C₂×Dic₆	C₂²	1	6	yes	—
12	C₃×M₁₆	C₂²	1	3	yes	—
13	C₂×C₃×C₈	C₂²	1	3	yes	—
14	Dic₁₂	C₂²	1	3	yes	—
15	Dic₁₂	C₂²	1	6	yes	—
16	D₂₄	C₂²	1	1	yes	characteristic
17	#204	C₂²	1	6	yes	—
18	2₋¹⁺⁴	—	3	2	no	—
19	#137	—	3	6	no	—
20	#135	—	3	3	no	—
21	C₂×Q₁₆	—	3	3	no	—
22	#131	D₃	1	1	yes	characteristic, complemented
23	C₃×Q₈	D₄	1	1	yes	characteristic
24	C₃×D₄	D₄	1	3	yes	—
25	C₂×C₃×C₄	C₂³	1	3	yes	—
26	#67	—	2	6	no	—
27	C₂×Dic₃	—	2	6	no	—
28	D₁₂	C₂³	1	2	yes	—
29	C₄×D₃	—	2	6	no	—
30	Dic₆	—	2	6	no	—
31	Dic₆	C₂³	1	6	yes	—
32	C₃×C₈	C₂³	1	3	yes	—
33	C₃×C₈	C₂³	1	1	yes	characteristic
34	#41	—	3	6	no	—
35	#41	—	6	6	no	—
36	#41	C₂×D₃	1	1	yes	characteristic
37	C₂×Q₈	—	3	6	no	—
38	C₂×Q₈	—	6	2	no	—
39	Q₁₆	—	3	3	no	—
40	Q₁₆	—	3	6	no	—
41	SD₁₆	—	3	6	no	—
42	D₈	—	3	1	no	—
43	M₁₆	C₂×D₃	1	3	yes	—
44	C₂×C₈	C₂×D₃	1	3	yes	—
45	C₂²×C₃	C₂×D₄	1	3	yes	—
46	C₂×D₃	—	2	2	no	—
47	C₃×C₄	C₂×D₄	1	3	yes	—
48	C₃×C₄	C₂⁴	1	1	yes	characteristic
49	Dic₃	—	2	6	no	—
50	Q₈	—	3	6	no	—
51	Q₈	—	6	6	no	—
52	Q₈	D₁₂	1	1	yes	characteristic
53	D₄	—	3	2	no	—
54	D₄	—	6	6	no	—
55	D₄	D₁₂	1	3	yes	—
56	C₂×C₄	—	6	6	no	—
57	C₂×C₄	—	6	6	no	—
58	C₂×C₄	C₂²×D₃	1	3	yes	—
59	C₈	C₂²×D₃	1	1	yes	characteristic
60	C₈	C₂²×D₃	1	3	yes	—
61	C₂×C₃	—	2	3	no	—
62	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
63	D₃	—	4	2	no	—
64	C₂²	—	6	2	no	—
65	C₂²	C₂×D₁₂	1	3	yes	—
66	C₄	—	6	6	no	—
67	C₄	C₂×D₁₂	1	3	yes	—
68	C₄	C₂³×D₃	1	1	yes	characteristic
69	C₃	#578	1	1	yes	characteristic, complemented, Sylow
70	C₂	—	2	3	no	—
71	C₂	—	12	2	no	—
72	C₂	C₂²×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₁₂
commutator, cosocle	C₃×C₄	C₂⁴
Frattini	C₄	C₂³×D₃
Fitting	C₃×#131	C₂
socle	C₂×C₃	C₂²×D₄
hypercenter	#131	D₃
hypocenter	C₃	#578
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #1010	C₂	semidirect
2. C₃×#131	C₂	semidirect
3. #909	C₂	semidirect
4. C₂×Dic₁₂	C₂	semidirect
5. #904	C₂	semidirect
22. #131	D₃	semidirect
69. C₃	#578	semidirect

Maximal subgroups

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

subgroup	index
#1010	2
C₃×#131	2
#909	2
C₂×Dic₁₂	2
#904	2
#578	3