Subgroups — #5068

Order 192.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#875	C₂	1	1	yes	maximal, characteristic
3	#810	C₂	1	2	yes	maximal, complemented
4	#809	C₂	1	2	yes	maximal
5	C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#488	—	3	1	no	maximal, Sylow
7	C₂×D₁₂	C₂²	1	1	yes	characteristic
8	C₂×Dic₆	C₂²	1	1	yes	characteristic
9	C₃×C₄⋊C₄	—	2	2	no	—
10	C₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₃×C₄²	C₂²	1	1	yes	characteristic
12	(C₂×D₃)⋊C₄	—	2	2	no	—
13	C₂×C₃⋊C₈	C₂²	1	2	yes	—
14	#125	D₃	1	1	yes	characteristic, complemented
15	#124	—	3	1	no	—
16	#103	—	3	2	no	—
17	#102	—	3	2	no	—
18	C₄×C₈	—	3	1	no	—
19	C₂²×D₃	—	2	1	no	—
20	C₂×C₃×C₄	—	2	2	no	—
21	C₂×C₃×C₄	D₄	1	2	yes	—
22	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
23	C₂×Dic₃	—	2	1	no	—
24	D₁₂	—	2	2	no	—
25	Dic₆	—	2	2	no	—
26	C₃⋊C₈	—	2	2	no	—
27	C₂×Q₈	—	3	1	no	—
28	C₂×D₄	—	3	1	no	—
29	C₂×C₈	—	3	2	no	—
30	C₄⋊C₄	—	2	2	no	—
31	C₄⋊C₄	C₂×D₃	1	2	yes	—
32	C₂²⋊C₄	—	6	2	no	—
33	C₄²	C₂×D₃	1	1	yes	characteristic
34	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
35	C₂×D₃	—	4	1	no	—
36	C₂×D₃	—	4	2	no	—
37	C₃×C₄	—	2	2	no	—
38	C₃×C₄	—	4	2	no	—
39	C₃×C₄	#41	1	2	yes	—
40	Dic₃	—	4	1	no	—
41	C₂³	—	6	1	no	—
42	Q₈	—	6	2	no	—
43	D₄	—	6	2	no	—
44	C₂×C₄	—	2	2	no	—
45	C₂×C₄	—	6	1	no	—
46	C₂×C₄	#67	1	2	yes	—
47	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
48	C₈	—	6	2	no	—
49	C₂×C₃	#124	1	1	yes	characteristic
50	C₂×C₃	#135	1	2	yes	—
51	D₃	—	8	1	no	—
52	C₂²	—	12	1	no	—
53	C₂²	—	12	2	no	—
54	C₂²	C₂×#67	1	1	yes	characteristic
55	C₄	—	2	2	no	—
56	C₄	—	4	2	no	—
57	C₄	—	12	1	no	—
58	C₄	#239	1	2	yes	—
59	C₃	#488	1	1	yes	characteristic, complemented, Sylow
60	C₂	—	24	1	no	—
61	C₂	#947	1	1	yes	characteristic
62	C₂	#950	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×#125	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂×D₁₂	C₂²
hypercenter	#125	D₃
hypocenter	C₃	#488
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
1. C₃×#125	C₂	semidirect
3. #810	C₂	semidirect
5. C₄×C₃⋊C₈	C₂	semidirect
14. #125	D₃	semidirect
59. C₃	#488	semidirect

Maximal subgroups

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

subgroup	index
C₃×#125	2
#875	2
#810	2
#809	2
C₄×C₃⋊C₈	2
#488	3