Subgroups — #5240

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	#926	C₂	1	2	yes	maximal, complemented
3	#834	C₂	1	4	yes	maximal, complemented
4	#410	—	3	1	no	maximal, Sylow
5	C₃×#41	—	2	2	no	—
6	C₃×#41	—	2	2	no	—
7	C₂×C₃×Q₈	C₄	1	1	yes	characteristic, complemented
8	C₂×C₃×D₄	C₄	1	1	yes	characteristic, complemented
9	C₂×C₃×D₄	C₂²	1	2	yes	—
10	C₂²×C₃×C₄	C₄	1	2	yes	complemented
11	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
12	#217	C₂²	1	4	yes	—
13	C₄⋊Dic₃	—	2	2	no	—
14	C₄×Dic₃	—	2	2	no	—
15	C₂×#41	D₃	1	1	yes	characteristic
16	#117	—	3	2	no	—
17	#99	—	3	4	no	—
18	C₂³×C₃	C₂×C₄	1	2	yes	—
19	C₂³×C₃	C₂³	1	1	yes	characteristic
20	C₃×Q₈	—	2	2	no	—
21	C₃×D₄	—	2	4	no	—
22	C₃×D₄	—	2	2	no	—
23	C₂×C₃×C₄	—	2	2	no	—
24	C₂×C₃×C₄	—	2	2	no	—
25	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄	D₄	1	2	yes	—
29	C₂×C₃×C₄	D₄	1	2	yes	—
30	C₂×Dic₃	—	2	4	no	—
31	#41	—	2	2	no	—
32	#41	—	2	2	no	—
33	C₂×Q₈	Dic₃	1	1	yes	characteristic, complemented
34	C₂×D₄	Dic₃	1	1	yes	characteristic, complemented
35	C₂×D₄	C₂×D₃	1	2	yes	—
36	C₂²×C₄	Dic₃	1	2	yes	complemented
37	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
38	C₄⋊C₄	—	6	2	no	—
39	C₂²⋊C₄	—	3	4	no	—
40	C₄²	—	6	2	no	—
41	C₂²×C₃	—	2	2	no	—
42	C₂²×C₃	—	4	1	no	—
43	C₂²×C₃	—	4	2	no	—
44	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₃	C₂×D₄	1	2	yes	—
46	C₃×C₄	—	2	2	no	—
47	C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
48	C₃×C₄	C₂²⋊C₄	1	2	yes	—
49	C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
50	Dic₃	—	4	4	no	—
51	C₂³	C₂×Dic₃	1	2	yes	—
52	C₂³	C₂²×D₃	1	1	yes	characteristic
53	Q₈	—	2	2	no	—
54	D₄	—	2	4	no	—
55	D₄	—	2	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	6	4	no	—
59	C₂×C₄	C₂×Dic₃	1	2	yes	—
60	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
61	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
62	C₂×C₄	#67	1	2	yes	—
63	C₂×C₄	#67	1	2	yes	—
64	C₂×C₃	—	2	2	no	—
65	C₂×C₃	—	2	1	no	—
66	C₂×C₃	—	4	2	no	—
67	C₂×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂²	—	2	2	no	—
69	C₂²	—	4	1	no	—
70	C₂²	—	4	2	no	—
71	C₂²	C₂²×Dic₃	1	1	yes	characteristic
72	C₂²	C₂×#67	1	2	yes	—
73	C₄	—	2	2	no	—
74	C₄	—	12	4	no	—
75	C₄	#217	1	1	yes	characteristic
76	C₄	#217	1	2	yes	—
77	C₄	#217	1	1	yes	characteristic
78	C₃	#410	1	1	yes	characteristic, complemented, Sylow
79	C₂	—	2	2	no	—
80	C₂	—	2	1	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	C₂³	C₂²×D₃
Fitting, prime radical	C₂×C₃×#41	C₂
socle	C₂×C₃	C₂×C₂²⋊C₄
cosocle	C₂³×C₃	C₂³
hypercenter	C₂×#41	D₃
hypocenter	C₃	#410
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. #926	C₂	semidirect
3. #834	C₂	semidirect
7. C₂×C₃×Q₈	C₄	semidirect
8. C₂×C₃×D₄	C₄	semidirect
10. C₂²×C₃×C₄	C₄	semidirect
33. C₂×Q₈	Dic₃	semidirect
34. C₂×D₄	Dic₃	semidirect
36. C₂²×C₄	Dic₃	semidirect
78. C₃	#410	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
#926	2
#834	2
#410	3