Subgroups — #5000

Order 192.

96 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	#830	C₂	1	2	yes	maximal
3	C₄⋊(C₃⋊C₈)	C₂	1	2	yes	maximal, complemented
4	#803	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#433	—	3	1	no	maximal, Sylow
7	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₃×C₄⋊C₄	C₄	1	2	yes	—
9	C₃×C₂²⋊C₄	C₄	1	2	yes	—
10	C₃×C₄²	C₂²	1	1	yes	characteristic
11	C₃×C₄²	C₂²	1	1	yes	characteristic
12	C₂×C₃⋊C₈	C₂²	1	2	yes	—
13	C₂×C₃⋊C₈	C₂²	1	2	yes	—
14	#117	D₃	1	1	yes	characteristic
15	C₄⋊C₈	—	3	2	no	—
16	C₂²⋊C₈	—	3	2	no	—
17	#97	—	3	1	no	—
18	C₄×C₈	—	3	1	no	—
19	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₄	—	2	2	no	—
21	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
22	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
25	C₃⋊C₈	—	2	2	no	—
26	C₃⋊C₈	—	2	2	no	—
27	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
28	C₂×C₈	—	3	2	no	—
29	C₂×C₈	—	3	2	no	—
30	C₄⋊C₄	Dic₃	1	2	yes	—
31	C₂²⋊C₄	Dic₃	1	2	yes	—
32	C₄²	C₂×D₃	1	1	yes	characteristic
33	C₄²	C₂×D₃	1	1	yes	characteristic
34	C₂²×C₃	—	2	1	no	—
35	C₂²×C₃	—	2	2	no	—
36	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
37	C₃×C₄	—	2	1	no	—
38	C₃×C₄	—	2	2	no	—
39	C₃×C₄	—	2	2	no	—
40	C₃×C₄	#41	1	2	yes	—
41	C₂³	C₂×Dic₃	1	1	yes	characteristic
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	C₂×Dic₃	1	2	yes	—
44	C₂×C₄	C₂×Dic₃	1	2	yes	—
45	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
46	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
47	C₈	—	6	2	no	—
48	C₈	—	6	2	no	—
49	C₂×C₃	—	4	1	no	—
50	C₂×C₃	#117	1	1	yes	characteristic
51	C₂×C₃	#131	1	2	yes	—
52	C₂²	—	2	1	no	—
53	C₂²	—	2	2	no	—
54	C₂²	C₂²×Dic₃	1	1	yes	characteristic
55	C₄	—	2	1	no	—
56	C₄	—	2	2	no	—
57	C₄	—	2	2	no	—
58	C₄	#235	1	2	yes	—
59	C₃	#433	1	1	yes	characteristic, complemented, Sylow
60	C₂	—	4	1	no	—
61	C₂	#926	1	1	yes	characteristic
62	C₂	#948	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃
commutator, socle	C₂²×C₃	C₂²×C₄
Fitting	C₃×#117	C₂
prime radical	C₂³×C₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	#117	D₃
hypocenter	C₃	#433
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₃⋊C₈)	C₂	semidirect
4. #803	C₂	semidirect
5. C₄×C₃⋊C₈	C₂	semidirect
59. C₃	#433	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
#830	2
C₄⋊(C₃⋊C₈)	2
#803	2
C₄×C₃⋊C₈	2
#433	3