Subgroups — #4525

Order 192.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
2	#830	C₂	1	2	yes	maximal, complemented
3	#323	—	3	1	no	maximal, Sylow
4	C₂⁴×C₃	C₄	1	1	yes	characteristic
5	C₂²×C₃×C₄	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₃×C₂²⋊C₄	—	2	2	no	—
8	C₂×C₃⋊C₈	—	2	2	no	—
9	C₂×C₂²⋊C₄	D₃	1	1	yes	characteristic
10	C₂²⋊C₈	—	3	2	no	—
11	C₂³×C₃	—	2	1	no	—
12	C₂³×C₃	—	2	1	no	—
13	C₂³×C₃	—	4	1	no	—
14	C₂³×C₃	—	4	1	no	—
15	C₂³×C₃	C₈	1	2	yes	complemented
16	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₃×C₄	—	2	1	no	—
18	C₂×C₃×C₄	—	4	1	no	—
19	C₂×C₃×C₄	—	4	1	no	—
20	C₂×C₃×C₄	D₄	1	2	yes	—
21	C₃⋊C₈	—	4	2	no	—
22	C₂⁴	Dic₃	1	1	yes	characteristic
23	C₂²×C₄	Dic₃	1	1	yes	characteristic
24	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
25	C₂×C₈	—	6	2	no	—
26	C₂²⋊C₄	—	2	2	no	—
27	C₂²×C₃	—	2	1	no	—
28	C₂²×C₃	—	2	1	no	—
29	C₂²×C₃	—	2	2	no	—
30	C₂²×C₃	—	4	2	no	—
31	C₂²×C₃	—	4	2	no	—
32	C₂²×C₃	—	4	1	no	—
33	C₂²×C₃	—	4	1	no	—
34	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₃	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₃	M₁₆	1	1	yes	characteristic
37	C₃×C₄	—	2	2	no	—
38	C₃×C₄	—	4	1	no	—
39	C₂³	—	2	1	no	—
40	C₂³	—	2	1	no	—
41	C₂³	—	4	1	no	—
42	C₂³	—	4	1	no	—
43	C₂³	C₃⋊C₈	1	2	yes	complemented
44	C₂³	C₂×Dic₃	1	1	yes	characteristic
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	4	1	no	—
47	C₂×C₄	—	4	1	no	—
48	C₂×C₄	#67	1	2	yes	—
49	C₈	—	12	2	no	—
50	C₂×C₃	—	2	1	no	—
51	C₂×C₃	—	2	1	no	—
52	C₂×C₃	—	4	2	no	—
53	C₂×C₃	C₂²⋊C₈	1	1	yes	characteristic
54	C₂×C₃	#99	1	1	yes	characteristic
55	C₂×C₃	#100	1	1	yes	characteristic
56	C₂²	—	2	1	no	—
57	C₂²	—	2	1	no	—
58	C₂²	—	2	2	no	—
59	C₂²	—	4	2	no	—
60	C₂²	—	4	2	no	—
61	C₂²	—	4	1	no	—
62	C₂²	—	4	1	no	—
63	C₂²	C₂×C₃⋊C₈	1	1	yes	characteristic
64	C₂²	#208	1	1	yes	characteristic
65	C₂²	#217	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	—	4	1	no	—
68	C₃	#323	1	1	yes	characteristic, complemented, Sylow
69	C₂	—	2	1	no	—
70	C₂	—	2	1	no	—
71	C₂	—	4	2	no	—
72	C₂	#830	1	1	yes	characteristic
73	C₂	#833	1	1	yes	characteristic
74	C₂	#834	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#217
commutator	C₂²×C₃	C₂×C₈
Frattini	C₂²×C₄	C₂×D₃
Fitting	C₂×C₃×C₂²⋊C₄	C₂
socle	C₂²×C₃	C₂²⋊C₄
prime radical	C₂⁴×C₃	C₄
cosocle	C₂²×C₃×C₄	C₂²
hypercenter	C₂×C₂²⋊C₄	D₃
hypocenter	C₃	#323
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. #830	C₂	semidirect
15. C₂³×C₃	C₈	semidirect
43. C₂³	C₃⋊C₈	semidirect
68. C₃	#323	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₂²⋊C₄	2
#830	2
#323	3