Subgroups — #4514

Order 192.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
2	C₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#812	C₂	1	1	yes	maximal, characteristic, complemented
4	#349	—	3	1	no	maximal, Sylow
5	C₂×C₄×D₃	C₄	1	1	yes	characteristic
6	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊C₈	C₄	1	1	yes	characteristic
8	ΓL(1,𝔽₂₅)	—	2	1	no	—
9	ΓL(1,𝔽₂₅)	—	2	1	no	—
10	C₃×C₁₆	—	2	1	no	—
11	C₃⋊C₁₆	—	2	1	no	—
12	C₂×M₁₆	—	3	1	no	—
13	M₃₂	—	3	1	no	—
14	M₃₂	D₃	1	1	yes	characteristic, complemented
15	C₂²×D₃	C₈	1	1	yes	characteristic
16	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₃	C₈	1	1	yes	characteristic
18	C₄×D₃	—	2	1	no	—
19	C₄×D₃	—	2	1	no	—
20	C₃×C₈	D₄	1	1	yes	characteristic
21	C₃×C₈	D₄	1	1	yes	characteristic
22	C₃⋊C₈	—	2	1	no	—
23	C₂²×C₄	—	3	1	no	—
24	M₁₆	—	6	1	no	—
25	M₁₆	—	6	1	no	—
26	C₂×C₈	—	3	1	no	—
27	C₂×C₈	C₂×D₃	1	1	yes	characteristic
28	C₁₆	—	2	1	no	—
29	C₁₆	—	6	1	no	—
30	C₂²×C₃	C₂×C₈	1	1	yes	characteristic
31	C₂×D₃	—	2	1	no	—
32	C₂×D₃	—	4	1	no	—
33	C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
34	C₃×C₄	M₁₆	1	1	yes	characteristic
35	Dic₃	—	2	1	no	—
36	C₂³	—	3	1	no	—
37	C₂×C₄	—	3	1	no	—
38	C₂×C₄	—	6	1	no	—
39	C₂×C₄	—	6	1	no	—
40	C₂×C₄	C₄×D₃	1	1	yes	characteristic
41	C₈	—	6	1	no	—
42	C₈	D₁₂	1	1	yes	characteristic
43	C₈	#67	1	1	yes	characteristic
44	C₂×C₃	—	2	1	no	—
45	C₂×C₃	C₂²⋊C₈	1	1	yes	characteristic
46	D₃	—	4	1	no	—
47	C₂²	—	6	1	no	—
48	C₂²	—	12	1	no	—
49	C₂²	D₃×C₈	1	1	yes	characteristic
50	C₄	—	6	1	no	—
51	C₄	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
52	C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
53	C₃	#349	1	1	yes	characteristic, complemented, Sylow
54	C₂	—	2	1	no	—
55	C₂	—	12	1	no	—
56	C₂	(C₂×D₃)⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	(C₂×D₃)⋊C₄
commutator	C₂²×C₃	C₂×C₈
Frattini	C₂×C₈	C₂×D₃
Fitting	C₃×M₃₂	C₂
socle	C₂×C₃	C₂²⋊C₈
prime radical	C₂²×D₃	C₈
cosocle	C₂×C₃×C₈	C₂²
hypercenter	M₃₂	D₃
hypocenter	C₃	#349
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₃×M₃₂	C₂	semidirect
3. #812	C₂	semidirect
14. M₃₂	D₃	semidirect
53. C₃	#349	semidirect

Maximal subgroups

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

subgroup	index
C₂×ΓL(1,𝔽₂₅)	2
C₃×M₃₂	2
#812	2
#349	3