Subgroups — #5189

Order 192.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#946	C₂	1	2	yes	maximal
3	C₂×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
4	#835	C₂	1	2	yes	maximal, complemented
5	#817	C₂	1	1	yes	maximal, characteristic, complemented
6	#470	—	3	1	no	maximal, Sylow
7	C₂×C₃×Q₈	C₂²	1	2	yes	—
8	C₂×C₄×D₃	C₂²	1	1	yes	characteristic
9	C₃×Q₁₆	—	2	2	no	—
10	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
11	(C₂×D₃)⋊C₄	—	2	2	no	—
12	C₄⋊Dic₃	C₂²	1	2	yes	—
13	Dic₃⋊C₄	—	2	2	no	—
14	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	ΓL(1,𝔽₂₅)	—	2	2	no	—
16	C₂×Q₁₆	D₃	1	1	yes	characteristic, complemented
17	C₂×M₁₆	—	3	1	no	—
18	C₂²⋊Q₈	—	3	2	no	—
19	#106	—	3	1	no	—
20	#103	—	3	2	no	—
21	C₂²×D₃	D₄	1	1	yes	characteristic
22	C₃×Q₈	—	2	2	no	—
23	C₃×Q₈	—	2	2	no	—
24	C₂×C₃×C₄	—	2	2	no	—
25	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
26	C₂×Dic₃	—	2	2	no	—
27	C₂×Dic₃	D₄	1	1	yes	characteristic
28	C₄×D₃	—	2	1	no	—
29	C₄×D₃	—	2	1	no	—
30	C₃×C₈	D₄	1	2	yes	—
31	C₃⋊C₈	—	2	1	no	—
32	C₂×Q₈	C₂×D₃	1	2	yes	—
33	C₂²×C₄	—	3	1	no	—
34	Q₁₆	—	2	2	no	—
35	M₁₆	—	6	2	no	—
36	C₂×C₈	—	3	1	no	—
37	C₂×C₈	C₂×D₃	1	1	yes	characteristic
38	C₄⋊C₄	—	3	2	no	—
39	C₄⋊C₄	—	6	2	no	—
40	C₂²⋊C₄	—	6	2	no	—
41	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
42	C₂×D₃	—	2	1	no	—
43	C₂×D₃	—	2	2	no	—
44	C₃×C₄	—	4	2	no	—
45	C₃×C₄	C₂×D₄	1	1	yes	characteristic
46	C₃×C₄	#41	1	1	yes	characteristic
47	Dic₃	—	2	1	no	—
48	Dic₃	—	4	2	no	—
49	C₂³	—	3	1	no	—
50	Q₈	—	2	2	no	—
51	Q₈	—	2	2	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	3	1	no	—
54	C₂×C₄	—	6	1	no	—
55	C₂×C₄	—	6	1	no	—
56	C₂×C₄	—	6	2	no	—
57	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
58	C₈	—	6	1	no	—
59	C₈	#67	1	2	yes	—
60	C₂×C₃	#121	1	1	yes	characteristic
61	C₂×C₃	#137	1	2	yes	—
62	D₃	—	4	1	no	—
63	C₂²	—	6	2	no	—
64	C₂²	—	6	1	no	—
65	C₂²	D₃×D₄	1	1	yes	characteristic
66	C₄	—	4	2	no	—
67	C₄	—	6	1	no	—
68	C₄	—	12	2	no	—
69	C₄	#237	1	1	yes	characteristic
70	C₄	C₂×#67	1	1	yes	characteristic
71	C₃	#470	1	1	yes	characteristic, complemented, Sylow
72	C₂	—	12	1	no	—
73	C₂	#918	1	2	yes	—
74	C₂	#938	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₂×C₃×Q₁₆	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂²×D₃	D₄
hypercenter	C₂×Q₁₆	D₃
hypocenter	C₃	#470
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×Q₁₆	C₂	semidirect
4. #835	C₂	semidirect
5. #817	C₂	semidirect
16. C₂×Q₁₆	D₃	semidirect
71. C₃	#470	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×Q₁₆	2
#946	2
C₂×ΓL(1,𝔽₂₅)	2
#835	2
#817	2
#470	3