Subgroups — #4998

Order 192.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#117	C₂	1	1	yes	maximal, characteristic
2	#926	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#808	C₂	1	2	yes	maximal, complemented
5	#807	C₂	1	2	yes	maximal, complemented
6	#427	—	3	1	no	maximal, Sylow
7	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₃×C₄⋊C₄	C₂²	1	2	yes	—
9	C₃×C₂²⋊C₄	—	2	1	no	—
10	C₃×C₄²	—	2	1	no	—
11	#217	—	2	1	no	—
12	C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₄×Dic₃	—	2	1	no	—
14	C₂×C₃⋊C₈	C₄	1	4	yes	complemented
15	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₂²×C₈	—	3	1	no	—
18	#117	—	3	1	no	—
19	#117	D₃	1	1	yes	characteristic
20	#107	—	3	2	no	—
21	#106	—	3	2	no	—
22	C₂³×C₃	D₄	1	1	yes	characteristic
23	C₂×C₃×C₄	—	2	2	no	—
24	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₄	D₄	1	1	yes	characteristic
26	C₂×C₃×C₄	Q₈	1	2	yes	—
27	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
28	C₂×Dic₃	—	2	2	no	—
29	C₃⋊C₈	C₂×C₄	1	2	yes	—
30	C₃⋊C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
32	C₂×C₈	—	3	4	no	—
33	C₂×C₈	—	3	1	no	—
34	C₂×C₈	—	3	1	no	—
35	C₄⋊C₄	—	3	2	no	—
36	C₄⋊C₄	C₂×D₃	1	2	yes	—
37	C₂²⋊C₄	—	2	1	no	—
38	C₂²⋊C₄	—	6	1	no	—
39	C₄²	—	2	1	no	—
40	C₄²	—	6	1	no	—
41	C₂²×C₃	—	2	2	no	—
42	C₂²×C₃	C₄⋊C₄	1	2	yes	—
43	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
44	C₃×C₄	—	4	2	no	—
45	C₃×C₄	C₄⋊C₄	1	2	yes	—
46	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
47	C₃×C₄	C₂×Q₈	1	1	yes	characteristic
48	Dic₃	—	4	2	no	—
49	C₂³	#67	1	1	yes	characteristic
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	6	2	no	—
52	C₂×C₄	Dic₆	1	2	yes	—
53	C₂×C₄	C₄×D₃	1	2	yes	—
54	C₂×C₄	#67	1	1	yes	characteristic
55	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
56	C₈	—	3	2	no	—
57	C₈	—	3	2	no	—
58	C₂×C₃	—	2	2	no	—
59	C₂×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₃	#135	1	2	yes	—
61	C₂²	—	2	2	no	—
62	C₂²	Dic₃⋊C₄	1	2	yes	—
63	C₂²	C₂×#67	1	1	yes	characteristic
64	C₄	—	4	2	no	—
65	C₄	—	12	2	no	—
66	C₄	Dic₃⋊C₄	1	2	yes	—
67	C₄	C₂×Dic₆	1	1	yes	characteristic
68	C₄	C₂×C₄×D₃	1	1	yes	characteristic
69	C₃	#427	1	1	yes	characteristic, complemented, Sylow
70	C₂	—	2	2	no	—
71	C₂	C₂×Dic₃⋊C₄	1	1	yes	characteristic
72	C₂	#950	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#67
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×#117	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂³×C₃	D₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	#117	D₃
hypocenter	C₃	#427
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
4. #808	C₂	semidirect
5. #807	C₂	semidirect
14. C₂×C₃⋊C₈	C₄	semidirect
69. C₃	#427	semidirect

Maximal subgroups

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

subgroup	index
C₃×#117	2
#926	2
C₂²×C₃⋊C₈	2
#808	2
#807	2
#427	3