Subgroups — #4708

Order 192.

130 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	C₂×D₃×C₈	C₂	1	1	yes	maximal, characteristic
3	#872	C₂	1	1	yes	maximal, characteristic
4	C₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
5	#815	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#405	—	3	1	no	maximal, Sylow
9	C₂×C₄×D₃	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄²	C₂²	1	1	yes	characteristic
13	(C₂×D₃)⋊C₄	C₄	1	2	yes	—
14	Dic₃⋊C₄	C₄	1	2	yes	—
15	C₄×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	ΓL(1,𝔽₂₅)	C₄	1	4	yes	complemented
19	D₃×C₈	C₄	1	4	yes	complemented
20	C₂×M₁₆	—	3	1	no	—
21	C₂²×C₈	—	3	1	no	—
22	#117	—	3	1	no	—
23	#97	—	3	1	no	—
24	#97	D₃	1	1	yes	characteristic, complemented
25	C₄×C₈	—	3	1	no	—
26	C₄×C₈	—	3	1	no	—
27	C₂²×D₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
30	C₂×Dic₃	C₂×C₄	1	2	yes	—
31	C₂×Dic₃	C₂×C₄	1	1	yes	characteristic
32	C₄×D₃	C₂×C₄	1	2	yes	—
33	C₄×D₃	C₂×C₄	1	2	yes	—
34	C₃×C₈	C₂×C₄	1	2	yes	—
35	C₃×C₈	C₂×C₄	1	2	yes	—
36	C₃⋊C₈	C₂×C₄	1	2	yes	—
37	C₃⋊C₈	C₂×C₄	1	2	yes	—
38	C₂²×C₄	—	3	1	no	—
39	M₁₆	—	3	4	no	—
40	C₂×C₈	—	3	1	no	—
41	C₂×C₈	—	3	1	no	—
42	C₂×C₈	—	3	4	no	—
43	C₂×C₈	C₂×D₃	1	1	yes	characteristic
44	C₂×C₈	C₂×D₃	1	1	yes	characteristic
45	C₄⋊C₄	—	3	2	no	—
46	C₂²⋊C₄	—	3	2	no	—
47	C₄²	—	3	1	no	—
48	C₄²	C₂×D₃	1	1	yes	characteristic
49	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×D₃	—	2	2	no	—
51	C₂×D₃	C₄²	1	2	yes	—
52	C₃×C₄	—	2	2	no	—
53	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
54	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
55	Dic₃	—	2	2	no	—
56	Dic₃	C₄²	1	2	yes	—
57	C₂³	—	3	1	no	—
58	C₂×C₄	—	3	1	no	—
59	C₂×C₄	—	3	2	no	—
60	C₂×C₄	—	3	2	no	—
61	C₂×C₄	—	3	2	no	—
62	C₂×C₄	C₄×D₃	1	2	yes	—
63	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
64	C₈	—	3	2	no	—
65	C₈	—	3	2	no	—
66	C₈	C₄×D₃	1	2	yes	—
67	C₈	C₄×D₃	1	2	yes	—
68	C₂×C₃	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₃	#131	1	2	yes	—
70	D₃	—	2	2	no	—
71	C₂²	—	3	2	no	—
72	C₂²	—	6	2	no	—
73	C₂²	C₂×C₄×D₃	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	3	2	no	—
76	C₄	—	6	2	no	—
77	C₄	C₂×C₄×D₃	1	1	yes	characteristic
78	C₄	C₂×C₄×D₃	1	1	yes	characteristic
79	C₃	#405	1	1	yes	characteristic, complemented, Sylow
80	C₂	—	6	2	no	—
81	C₂	C₄²×D₃	1	1	yes	characteristic
82	C₂	#907	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃
commutator	C₂×C₃	C₂×C₄²
Fitting	C₃×#97	C₂
socle	C₂²×C₃	C₂²×C₄
prime radical	C₂²×D₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	#97	D₃
hypocenter	C₃	#405
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
4. C₃×#97	C₂	semidirect
5. #815	C₂	semidirect
6. C₈×Dic₃	C₂	semidirect
7. C₄×C₃⋊C₈	C₂	semidirect
18. ΓL(1,𝔽₂₅)	C₄	semidirect
19. D₃×C₈	C₄	semidirect
24. #97	D₃	semidirect
79. C₃	#405	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
C₂×D₃×C₈	2
#872	2
C₃×#97	2
#815	2
C₈×Dic₃	2
C₄×C₃⋊C₈	2
#405	3