Subgroups — #5207

Order 192.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#208	C₂	1	2	yes	maximal, complemented
3	#830	C₂	1	4	yes	maximal, complemented
4	#407	—	3	1	no	maximal, Sylow
5	C₂⁴×C₃	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₄	—	2	4	no	—
7	C₂²×C₃×C₄	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₄	C₄	1	2	yes	—
9	C₂²×C₃×C₄	C₂²	1	2	yes	—
10	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
11	#208	—	2	4	no	—
12	C₂×C₃⋊C₈	C₂²	1	4	yes	complemented
13	C₂³×C₄	D₃	1	1	yes	characteristic
14	C₂×M₁₆	—	3	2	no	—
15	C₂²⋊C₈	—	3	4	no	—
16	C₂³×C₃	—	2	2	no	—
17	C₂³×C₃	—	2	4	no	—
18	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₃	C₂×C₄	1	2	yes	—
20	C₂×C₃×C₄	—	2	2	no	—
21	C₂×C₃×C₄	—	2	4	no	—
22	C₂×C₃×C₄	—	2	4	no	—
23	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄	D₄	1	4	yes	—
26	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
27	C₃⋊C₈	—	2	4	no	—
28	C₂⁴	Dic₃	1	1	yes	characteristic
29	C₂²×C₄	—	2	4	no	—
30	C₂²×C₄	Dic₃	1	1	yes	characteristic
31	C₂²×C₄	Dic₃	1	2	yes	—
32	C₂²×C₄	C₂×D₃	1	2	yes	—
33	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
34	M₁₆	—	6	4	no	—
35	C₂×C₈	—	3	4	no	—
36	C₂²×C₃	—	2	8	no	—
37	C₂²×C₃	—	2	2	no	—
38	C₂²×C₃	—	2	2	no	—
39	C₂²×C₃	—	2	2	no	—
40	C₂²×C₃	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₃	M₁₆	1	4	yes	complemented
42	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
43	C₃×C₄	—	2	2	no	—
44	C₃×C₄	C₂²⋊C₄	1	2	yes	—
45	C₃×C₄	C₂×D₄	1	2	yes	—
46	C₂³	—	2	4	no	—
47	C₂³	—	2	2	no	—
48	C₂³	C₂×Dic₃	1	2	yes	—
49	C₂³	C₂×Dic₃	1	1	yes	characteristic
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	2	4	no	—
52	C₂×C₄	—	2	4	no	—
53	C₂×C₄	C₂×Dic₃	1	2	yes	—
54	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
55	C₂×C₄	#67	1	4	yes	—
56	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
57	C₈	—	6	4	no	—
58	C₂×C₃	—	2	2	no	—
59	C₂×C₃	—	2	4	no	—
60	C₂×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₃	C₂×M₁₆	1	2	yes	—
62	C₂²	—	2	8	no	—
63	C₂²	—	2	2	no	—
64	C₂²	—	2	2	no	—
65	C₂²	—	2	2	no	—
66	C₂²	#208	1	4	yes	complemented
67	C₂²	#217	1	2	yes	—
68	C₂²	C₂²×Dic₃	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	#217	1	2	yes	—
71	C₄	C₂×#67	1	2	yes	—
72	C₃	#407	1	1	yes	characteristic, complemented, Sylow
73	C₂	—	2	4	no	—
74	C₂	—	2	2	no	—
75	C₂	C₂×#208	1	2	yes	—
76	C₂	C₂×#217	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃
commutator, socle	C₂²×C₃	C₂²×C₄
Fitting	C₂³×C₃×C₄	C₂
prime radical	C₂⁴×C₃	C₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂³×C₄	D₃
hypocenter	C₃	#407
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#208	C₂	semidirect
3. #830	C₂	semidirect
12. C₂×C₃⋊C₈	C₂²	semidirect
41. C₂²×C₃	M₁₆	semidirect
66. C₂²	#208	semidirect
72. C₃	#407	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄	2
C₂×#208	2
#830	2
#407	3