Subgroups — #5237

Order 192.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×#208	C₂	1	2	yes	maximal, complemented
3	#836	C₂	1	2	yes	maximal, complemented
4	#833	C₂	1	2	yes	maximal, complemented
5	#413	—	3	1	no	maximal, Sylow
6	C₃×#41	—	2	2	no	—
7	C₃×#41	—	2	2	no	—
8	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×D₄	C₄	1	2	yes	—
10	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄	C₄	1	2	yes	—
12	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
13	#208	—	2	2	no	—
14	#208	C₂²	1	4	yes	—
15	C₂×C₃⋊C₈	—	2	2	no	—
16	C₂×#41	D₃	1	1	yes	characteristic
17	C₂×M₁₆	—	3	2	no	—
18	#101	—	3	2	no	—
19	#100	—	3	2	no	—
20	C₂³×C₃	C₂×C₄	1	2	yes	—
21	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
22	C₃×Q₈	—	2	2	no	—
23	C₃×D₄	—	2	4	no	—
24	C₃×D₄	—	2	2	no	—
25	C₂×C₃×C₄	—	2	2	no	—
26	C₂×C₃×C₄	—	2	2	no	—
27	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄	D₄	1	2	yes	—
30	C₂×C₃×C₄	D₄	1	2	yes	—
31	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₃⋊C₈	—	2	4	no	—
33	#41	—	2	2	no	—
34	#41	—	2	2	no	—
35	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
36	C₂×D₄	Dic₃	1	2	yes	—
37	C₂×D₄	C₂×D₃	1	1	yes	characteristic
38	C₂²×C₄	Dic₃	1	2	yes	—
39	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
40	M₁₆	—	3	4	no	—
41	M₁₆	—	6	2	no	—
42	C₂×C₈	—	6	2	no	—
43	C₂²×C₃	—	2	2	no	—
44	C₂²×C₃	—	4	1	no	—
45	C₂²×C₃	—	4	2	no	—
46	C₂²×C₃	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
48	C₃×C₄	—	2	2	no	—
49	C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
50	C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
51	C₃×C₄	C₂×D₄	1	2	yes	—
52	C₂³	C₂×Dic₃	1	2	yes	—
53	C₂³	C₂×Dic₃	1	1	yes	characteristic
54	Q₈	—	2	2	no	—
55	D₄	—	2	4	no	—
56	D₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	C₂×Dic₃	1	2	yes	—
60	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
61	C₂×C₄	#67	1	2	yes	—
62	C₂×C₄	#67	1	2	yes	—
63	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
64	C₈	—	6	4	no	—
65	C₂×C₃	—	2	1	no	—
66	C₂×C₃	—	2	2	no	—
67	C₂×C₃	—	4	2	no	—
68	C₂×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²	—	2	2	no	—
70	C₂²	—	4	1	no	—
71	C₂²	—	4	2	no	—
72	C₂²	#217	1	2	yes	—
73	C₂²	C₂²×Dic₃	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	#217	1	1	yes	characteristic
76	C₄	#217	1	1	yes	characteristic
77	C₄	C₂×#67	1	2	yes	—
78	C₃	#413	1	1	yes	characteristic, complemented, Sylow
79	C₂	—	2	1	no	—
80	C₂	—	2	2	no	—
81	C₂	—	4	2	no	—
82	C₂	C₂×#217	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#217
commutator	C₂²×C₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting, prime radical	C₂×C₃×#41	C₂
socle	C₂×C₃	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂×#41	D₃
hypocenter	C₃	#413
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
2. C₂×#208	C₂	semidirect
3. #836	C₂	semidirect
4. #833	C₂	semidirect
78. C₃	#413	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#41	2
C₂×#208	2
#836	2
#833	2
#413	3