Subgroups — #4545

Order 192.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#127	C₂	1	1	yes	maximal, characteristic
2	#834	C₂	1	2	yes	maximal, complemented
3	#353	—	3	1	no	maximal, Sylow
4	C₂×C₃×D₄	—	2	2	no	—
5	C₂×C₃×D₄	C₄	1	1	yes	characteristic, complemented
6	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
7	C₃×C₄²	C₄	1	1	yes	characteristic, complemented
8	#217	—	2	2	no	—
9	#127	D₃	1	1	yes	characteristic
10	#99	—	3	2	no	—
11	C₂³×C₃	—	2	1	no	—
12	C₂³×C₃	D₄	1	2	yes	—
13	C₃×D₄	—	4	1	no	—
14	C₃×D₄	—	4	1	no	—
15	C₃×D₄	—	4	4	no	—
16	C₂×C₃×C₄	—	2	1	no	—
17	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₃	—	4	2	no	—
19	C₂×D₄	—	2	2	no	—
20	C₂×D₄	Dic₃	1	1	yes	characteristic, complemented
21	C₂×D₄	C₂×D₃	1	1	yes	characteristic
22	C₂²⋊C₄	—	6	2	no	—
23	C₄²	Dic₃	1	1	yes	characteristic, complemented
24	C₂²×C₃	—	2	2	no	—
25	C₂²×C₃	—	4	2	no	—
26	C₂²×C₃	—	4	2	no	—
27	C₂²×C₃	—	4	1	no	—
28	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₃×C₄	—	2	2	no	—
30	C₃×C₄	—	2	1	no	—
31	Dic₃	—	8	2	no	—
32	C₂³	—	2	1	no	—
33	C₂³	#67	1	2	yes	—
34	D₄	—	4	1	no	—
35	D₄	—	4	1	no	—
36	D₄	—	4	4	no	—
37	C₂×C₄	—	2	1	no	—
38	C₂×C₄	—	12	2	no	—
39	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
40	C₂×C₃	—	2	1	no	—
41	C₂×C₃	—	4	2	no	—
42	C₂×C₃	—	8	1	no	—
43	C₂×C₃	#99	1	1	yes	characteristic
44	C₂²	—	2	2	no	—
45	C₂²	—	4	2	no	—
46	C₂²	—	4	2	no	—
47	C₂²	—	4	1	no	—
48	C₂²	#217	1	1	yes	characteristic
49	C₄	—	2	2	no	—
50	C₄	—	2	1	no	—
51	C₄	—	24	2	no	—
52	C₃	#353	1	1	yes	characteristic, complemented, Sylow
53	C₂	—	2	1	no	—
54	C₂	—	4	2	no	—
55	C₂	—	8	1	no	—
56	C₂	#834	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#834
commutator	C₂×C₃×C₄	C₂×C₄
Frattini	C₂×D₄	C₂×D₃
Fitting, prime radical	C₃×#127	C₂
socle	C₂×C₃	#99
cosocle	C₂×C₃×D₄	C₂²
hypercenter	#127	D₃
hypocenter	C₃	#353
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #127

Complements

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

N	H ≅ G/N	type
2. #834	C₂	semidirect
5. C₂×C₃×D₄	C₄	semidirect
7. C₃×C₄²	C₄	semidirect
20. C₂×D₄	Dic₃	semidirect
23. C₄²	Dic₃	semidirect
52. C₃	#353	semidirect

Maximal subgroups

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

subgroup	index
C₃×#127	2
#834	2
#353	3