Subgroups — #4838

Order 192.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×ΓL(1,𝔽₂₅)	C₂	1	2	yes	maximal, complemented
2	C₄×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₃)⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#436	—	3	1	no	maximal, Sylow
7	C₂×D₁₂	C₄	1	1	yes	characteristic
8	C₂×C₄×D₃	C₂²	1	2	yes	—
9	C₂×C₃×C₈	C₂²	1	2	yes	—
10	C₃×C₄²	C₂²	1	1	yes	characteristic
11	(C₂×D₃)⋊C₄	C₄	1	2	yes	—
12	C₄⋊Dic₃	C₄	1	1	yes	characteristic
13	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	ΓL(1,𝔽₂₅)	—	2	4	no	—
16	C₂×M₁₆	—	3	2	no	—
17	C₄×D₄	—	3	1	no	—
18	C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
19	C₂²⋊C₈	—	3	2	no	—
20	C₄×C₈	—	3	1	no	—
21	C₂²×D₃	C₂×C₄	1	2	yes	—
22	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
25	C₂×Dic₃	C₂×C₄	1	2	yes	—
26	D₁₂	—	2	2	no	—
27	C₄×D₃	—	2	2	no	—
28	C₄×D₃	—	2	2	no	—
29	C₃×C₈	—	2	2	no	—
30	C₃⋊C₈	—	2	1	no	—
31	C₃⋊C₈	D₄	1	2	yes	complemented
32	C₂×D₄	—	3	1	no	—
33	C₂²×C₄	—	3	2	no	—
34	M₁₆	—	6	4	no	—
35	C₂×C₈	—	3	1	no	—
36	C₂×C₈	—	3	1	no	—
37	C₂×C₈	C₂×D₃	1	2	yes	—
38	C₄⋊C₄	—	3	1	no	—
39	C₂²⋊C₄	—	3	2	no	—
40	C₄²	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×D₃	—	2	2	no	—
43	C₂×D₃	—	2	2	no	—
44	C₂×D₃	—	2	2	no	—
45	C₃×C₄	—	2	1	no	—
46	C₃×C₄	M₁₆	1	2	yes	complemented
47	C₃×C₄	C₂×D₄	1	1	yes	characteristic
48	C₃×C₄	#41	1	1	yes	characteristic
49	Dic₃	—	2	2	no	—
50	C₂³	—	3	2	no	—
51	D₄	—	6	2	no	—
52	C₂×C₄	—	3	2	no	—
53	C₂×C₄	—	6	2	no	—
54	C₂×C₄	—	6	2	no	—
55	C₂×C₄	C₄×D₃	1	1	yes	characteristic
56	C₂×C₄	C₄×D₃	1	1	yes	characteristic
57	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₈	—	3	2	no	—
60	C₈	—	6	1	no	—
61	C₂×C₃	C₄×D₄	1	1	yes	characteristic
62	C₂×C₃	C₂×M₁₆	1	1	yes	characteristic
63	C₂×C₃	#131	1	1	yes	characteristic
64	D₃	—	4	2	no	—
65	C₂²	—	6	2	no	—
66	C₂²	—	6	2	no	—
67	C₂²	—	6	2	no	—
68	C₂²	C₂×C₄×D₃	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	6	2	no	—
71	C₄	ΓL(1,𝔽₂₅)	1	2	yes	complemented
72	C₄	D₃×D₄	1	1	yes	characteristic
73	C₄	#239	1	1	yes	characteristic
74	C₃	#436	1	1	yes	characteristic, complemented, Sylow
75	C₂	—	12	2	no	—
76	C₂	#893	1	1	yes	characteristic
77	C₂	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
78	C₂	#907	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃
commutator, socle	C₂²×C₃	C₂²×C₄
Fitting	C₃×C₄⋊C₈	C₂
prime radical	C₂×D₁₂	C₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₄⋊C₈	D₃
hypocenter	C₃	#436
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×ΓL(1,𝔽₂₅)	C₂	semidirect
3. C₃×C₄⋊C₈	C₂	semidirect
4. (C₂×D₃)⋊C₈	C₂	semidirect
5. C₄×C₃⋊C₈	C₂	semidirect
18. C₄⋊C₈	D₃	semidirect
31. C₃⋊C₈	D₄	semidirect
46. C₃×C₄	M₁₆	semidirect
71. C₄	ΓL(1,𝔽₂₅)	semidirect
74. C₃	#436	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
C₄×D₁₂	2
C₃×C₄⋊C₈	2
(C₂×D₃)⋊C₈	2
C₄×C₃⋊C₈	2
#436	3