Subgroups — #5063

Order 192.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#125	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
3	#810	C₂	1	1	yes	maximal, characteristic
4	#810	C₂	1	1	yes	maximal, characteristic
5	#808	C₂	1	1	yes	maximal, characteristic
6	#807	C₂	1	1	yes	maximal, characteristic
7	C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
8	#479	—	3	1	no	maximal, Sylow
9	C₂×Dic₆	C₂²	1	1	yes	characteristic
10	C₃×C₄⋊C₄	—	2	1	no	—
11	C₃×C₄⋊C₄	—	2	1	no	—
12	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₃×C₄²	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
16	Dic₃⋊C₄	—	2	1	no	—
17	C₄×Dic₃	—	2	1	no	—
18	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	#125	D₃	1	1	yes	characteristic
21	C₄×Q₈	—	3	1	no	—
22	#107	—	3	1	no	—
23	#106	—	3	1	no	—
24	C₄⋊C₈	—	3	1	no	—
25	#103	—	3	1	no	—
26	#103	—	3	1	no	—
27	C₂×C₃×C₄	—	2	1	no	—
28	C₂×C₃×C₄	—	2	1	no	—
29	C₂×C₃×C₄	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄	D₄	1	1	yes	characteristic
31	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₂×Dic₃	—	2	1	no	—
33	C₂×Dic₃	—	2	1	no	—
34	Dic₆	—	2	1	no	—
35	Dic₆	Q₈	1	2	yes	—
36	C₃⋊C₈	—	2	1	no	—
37	C₃⋊C₈	—	2	1	no	—
38	C₂×Q₈	—	3	1	no	—
39	C₂×C₈	—	3	1	no	—
40	C₂×C₈	—	3	1	no	—
41	C₄⋊C₄	—	2	1	no	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	3	1	no	—
44	C₄⋊C₄	—	6	1	no	—
45	C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
46	C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
47	C₄²	—	6	1	no	—
48	C₄²	C₂×D₃	1	1	yes	characteristic
49	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
50	C₃×C₄	—	2	1	no	—
51	C₃×C₄	—	2	1	no	—
52	C₃×C₄	—	4	1	no	—
53	C₃×C₄	—	4	1	no	—
54	C₃×C₄	C₂×Q₈	1	1	yes	characteristic
55	C₃×C₄	#41	1	1	yes	characteristic
56	Dic₃	—	2	2	no	—
57	Dic₃	—	4	1	no	—
58	Q₈	—	3	2	no	—
59	Q₈	—	6	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	6	1	no	—
63	C₂×C₄	—	6	1	no	—
64	C₂×C₄	#67	1	1	yes	characteristic
65	C₂×C₄	#67	1	1	yes	characteristic
66	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
67	C₈	—	6	1	no	—
68	C₈	—	6	1	no	—
69	C₂×C₃	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂×C₃	#135	1	1	yes	characteristic
71	C₂×C₃	#137	1	1	yes	characteristic
72	C₂²	C₂×#67	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	2	1	no	—
75	C₄	—	4	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	6	2	no	—
78	C₄	—	12	1	no	—
79	C₄	D₃×Q₈	1	1	yes	characteristic
80	C₄	#239	1	1	yes	characteristic
81	C₃	#479	1	1	yes	characteristic, complemented, Sylow
82	C₂	#946	1	1	yes	characteristic
83	C₂	#950	1	1	yes	characteristic
84	C₂	#951	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×#125	C₂
socle, prime radical	C₂²×C₃	C₂×D₄
hypercenter	#125	D₃
hypocenter	C₃	#479
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
81. C₃	#479	semidirect

Maximal subgroups

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

subgroup	index
C₃×#125	2
C₄×Dic₆	2
#810	2
#808	2
#807	2
C₄⋊(C₃⋊C₈)	2
#479	3