Subgroups — #5067

Order 192.

110 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	#874	C₂	1	1	yes	maximal, characteristic
3	#809	C₂	1	4	yes	maximal, complemented
4	#803	C₂	1	1	yes	maximal, characteristic, complemented
5	#490	—	3	1	no	maximal, Sylow
6	C₂×D₁₂	—	2	2	no	—
7	C₂×D₁₂	C₂²	1	2	yes	—
8	C₃×C₄⋊C₄	—	2	2	no	—
9	C₃×C₄⋊C₄	C₂²	1	2	yes	—
10	C₃×C₄²	C₂²	1	1	yes	characteristic
11	C₂×C₃⋊C₈	C₂²	1	2	yes	—
12	#127	—	3	1	no	—
13	#125	D₃	1	1	yes	characteristic, complemented
14	#102	—	3	4	no	—
15	#97	—	3	1	no	—
16	C₂²×D₃	—	2	2	no	—
17	C₂×C₃×C₄	—	2	2	no	—
18	C₂×C₃×C₄	D₄	1	2	yes	—
19	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
20	D₁₂	—	2	4	no	—
21	D₁₂	—	4	4	no	—
22	C₃⋊C₈	—	2	2	no	—
23	C₂×D₄	—	3	2	no	—
24	C₂×D₄	—	6	2	no	—
25	C₂×C₈	—	3	2	no	—
26	C₄⋊C₄	—	2	2	no	—
27	C₄⋊C₄	C₂×D₃	1	2	yes	—
28	C₄²	C₂×D₃	1	1	yes	characteristic
29	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
30	C₂×D₃	—	4	2	no	—
31	C₂×D₃	—	4	4	no	—
32	C₃×C₄	—	2	2	no	—
33	C₃×C₄	—	4	2	no	—
34	C₃×C₄	#41	1	2	yes	—
35	C₂³	—	6	2	no	—
36	D₄	—	6	4	no	—
37	D₄	—	12	4	no	—
38	C₂×C₄	—	2	2	no	—
39	C₂×C₄	#67	1	2	yes	—
40	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
41	C₈	—	6	2	no	—
42	C₂×C₃	#124	1	1	yes	characteristic
43	C₂×C₃	AGL(1,ℤ/8)	1	2	yes	—
44	D₃	—	8	2	no	—
45	C₂²	—	12	2	no	—
46	C₂²	—	12	4	no	—
47	C₂²	C₂×#67	1	1	yes	characteristic
48	C₄	—	2	2	no	—
49	C₄	—	4	2	no	—
50	C₄	#239	1	2	yes	—
51	C₃	#490	1	1	yes	characteristic, complemented, Sylow
52	C₂	—	24	2	no	—
53	C₂	#947	1	1	yes	characteristic
54	C₂	#949	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	#874	C₂
hypercenter	#125	D₃
hypocenter	C₃	#490
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. #809	C₂	semidirect
4. #803	C₂	semidirect
13. #125	D₃	semidirect
51. C₃	#490	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
#874	2
#809	2
#803	2
#490	3