Subgroups — #5234

Order 192.

158 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	1	yes	maximal, characteristic, complemented
3	C₂²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#830	C₂	1	4	yes	maximal, complemented
5	#408	—	3	1	no	maximal, Sylow
6	C₃×#41	—	2	2	no	—
7	C₃×#41	—	2	2	no	—
8	C₂×C₃×Q₈	C₄	1	1	yes	characteristic
9	C₂×C₃×D₄	C₄	1	2	yes	—
10	C₂×C₃×D₄	C₄	1	1	yes	characteristic
11	C₂²×C₃×C₄	C₂²	1	2	yes	—
12	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
13	#208	—	2	2	no	—
14	C₂×C₃⋊C₈	—	2	2	no	—
15	C₂×C₃⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃⋊C₈	C₂²	1	2	yes	—
17	C₂×#41	D₃	1	1	yes	characteristic
18	C₂×M₁₆	—	3	1	no	—
19	C₂²×C₈	—	3	1	no	—
20	C₂²⋊C₈	—	3	4	no	—
21	C₂³×C₃	C₂×C₄	1	2	yes	—
22	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
23	C₃×Q₈	—	2	2	no	—
24	C₃×D₄	—	2	4	no	—
25	C₃×D₄	—	2	2	no	—
26	C₂×C₃×C₄	—	2	2	no	—
27	C₂×C₃×C₄	—	2	2	no	—
28	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄	D₄	1	2	yes	—
31	C₂×C₃×C₄	D₄	1	2	yes	—
32	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₃⋊C₈	—	2	2	no	—
34	C₃⋊C₈	—	2	2	no	—
35	#41	—	2	2	no	—
36	#41	—	2	2	no	—
37	C₂×Q₈	Dic₃	1	1	yes	characteristic
38	C₂×D₄	Dic₃	1	2	yes	—
39	C₂×D₄	Dic₃	1	1	yes	characteristic
40	C₂²×C₄	C₂×D₃	1	2	yes	—
41	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
42	M₁₆	—	6	2	no	—
43	C₂×C₈	—	3	2	no	—
44	C₂×C₈	—	3	2	no	—
45	C₂×C₈	—	6	2	no	—
46	C₂²×C₃	—	2	2	no	—
47	C₂²×C₃	—	2	2	no	—
48	C₂²×C₃	—	2	4	no	—
49	C₂²×C₃	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
51	C₃×C₄	—	2	2	no	—
52	C₃×C₄	C₂²⋊C₄	1	2	yes	—
53	C₃×C₄	C₂×D₄	1	1	yes	characteristic
54	C₃×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂³	C₂×Dic₃	1	2	yes	—
56	C₂³	C₂×Dic₃	1	1	yes	characteristic
57	Q₈	—	2	2	no	—
58	D₄	—	2	4	no	—
59	D₄	—	2	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	C₂×Dic₃	1	2	yes	—
63	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
64	C₂×C₄	#67	1	2	yes	—
65	C₂×C₄	#67	1	2	yes	—
66	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
67	C₈	—	6	2	no	—
68	C₈	—	6	2	no	—
69	C₂×C₃	—	2	2	no	—
70	C₂×C₃	—	4	2	no	—
71	C₂×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₃	#131	1	2	yes	—
73	C₂²	—	2	2	no	—
74	C₂²	—	2	4	no	—
75	C₂²	—	2	2	no	—
76	C₂²	#217	1	2	yes	—
77	C₂²	C₂²×Dic₃	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	#217	1	2	yes	—
80	C₄	C₂×#67	1	1	yes	characteristic
81	C₄	C₂×#67	1	1	yes	characteristic
82	C₃	#408	1	1	yes	characteristic, complemented, Sylow
83	C₂	—	2	2	no	—
84	C₂	—	4	2	no	—
85	C₂	#948	1	2	yes	—
86	C₂	C₂×#217	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃
commutator, socle	C₂²×C₃	C₂²×C₄
Fitting, prime radical	C₂×C₃×#41	C₂
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂×#41	D₃
hypocenter	C₃	#408
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×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. C₂²×C₃⋊C₈	C₂	semidirect
4. #830	C₂	semidirect
82. C₃	#408	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
C₂²×C₃⋊C₈	2
#830	2
#408	3