Subgroups — #4537

Order 192.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#121	C₂	1	1	yes	maximal, characteristic
2	C₃⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
3	#830	C₂	1	1	yes	maximal, characteristic, complemented
4	#327	—	3	1	no	maximal, Sylow
5	C₂×C₃×D₄	—	2	1	no	—
6	C₂×C₃×D₄	C₄	1	1	yes	characteristic, complemented
7	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₂²×Dic₃	—	2	1	no	—
9	C₃×C₄⋊C₄	C₄	1	1	yes	characteristic, complemented
10	C₃×C₂²⋊C₄	—	2	1	no	—
11	C₂×C₃⋊C₈	—	2	1	no	—
12	#121	D₃	1	1	yes	characteristic
13	C₂²⋊C₈	—	3	1	no	—
14	#95	—	3	1	no	—
15	C₂³×C₃	—	2	1	no	—
16	C₂³×C₃	D₄	1	1	yes	characteristic
17	C₃×D₄	—	4	1	no	—
18	C₃×D₄	—	4	1	no	—
19	C₃×D₄	—	4	1	no	—
20	C₂×C₃×C₄	—	2	1	no	—
21	C₂×C₃×C₄	—	4	1	no	—
22	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₄	D₄	1	1	yes	characteristic
24	C₂×Dic₃	—	2	2	no	—
25	C₂×Dic₃	—	4	1	no	—
26	C₂×Dic₃	—	4	1	no	—
27	C₃⋊C₈	—	4	1	no	—
28	C₂×D₄	—	2	1	no	—
29	C₂×D₄	Dic₃	1	1	yes	characteristic, complemented
30	C₂²×C₄	—	6	1	no	—
31	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
32	C₂×C₈	—	6	1	no	—
33	C₄⋊C₄	Dic₃	1	1	yes	characteristic, complemented
34	C₂²⋊C₄	—	2	1	no	—
35	C₂²×C₃	—	2	1	no	—
36	C₂²×C₃	—	2	1	no	—
37	C₂²×C₃	—	4	1	no	—
38	C₂²×C₃	—	4	1	no	—
39	C₂²×C₃	—	4	1	no	—
40	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₃	D₈	1	1	yes	characteristic
42	C₂²×C₃	SD₁₆	1	1	yes	characteristic
43	C₃×C₄	—	2	1	no	—
44	C₃×C₄	—	2	1	no	—
45	C₃×C₄	—	4	1	no	—
46	Dic₃	—	4	2	no	—
47	C₂³	—	2	1	no	—
48	C₂³	#67	1	1	yes	characteristic
49	D₄	—	4	1	no	—
50	D₄	—	4	1	no	—
51	D₄	—	4	1	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	6	2	no	—
55	C₂×C₄	—	12	1	no	—
56	C₂×C₄	—	12	1	no	—
57	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
58	C₂×C₄	#67	1	1	yes	characteristic
59	C₈	—	12	1	no	—
60	C₂×C₃	—	2	1	no	—
61	C₂×C₃	—	2	1	no	—
62	C₂×C₃	—	8	1	no	—
63	C₂×C₃	#99	1	1	yes	characteristic
64	C₂×C₃	#102	1	1	yes	characteristic
65	C₂×C₃	#104	1	1	yes	characteristic
66	C₂²	—	2	1	no	—
67	C₂²	—	2	1	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	4	1	no	—
70	C₂²	—	4	1	no	—
71	C₂²	#213	1	1	yes	characteristic
72	C₂²	#214	1	1	yes	characteristic
73	C₂²	#217	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	12	2	no	—
78	C₃	#327	1	1	yes	characteristic, complemented, Sylow
79	C₂	—	2	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	8	1	no	—
82	C₂	#832	1	1	yes	characteristic
83	C₂	#834	1	1	yes	characteristic
84	C₂	#837	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#217
commutator	C₂×C₃×C₄	C₂×C₄
Frattini	C₂²×C₄	C₂×D₃
Fitting, prime radical	C₃×#121	C₂
socle	C₂²×C₃	C₂²⋊C₄
cosocle	C₂²×C₃×C₄	C₂²
hypercenter	#121	D₃
hypocenter	C₃	#327
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

Complements

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

N	H ≅ G/N	type
2. C₃⋊#95	C₂	semidirect
3. #830	C₂	semidirect
6. C₂×C₃×D₄	C₄	semidirect
9. C₃×C₄⋊C₄	C₄	semidirect
29. C₂×D₄	Dic₃	semidirect
33. C₄⋊C₄	Dic₃	semidirect
78. C₃	#327	semidirect

Maximal subgroups

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

subgroup	index
C₃×#121	2
C₃⋊#95	2
#830	2
#327	3