Subgroups — #5244

Order 192.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	#937	C₂	1	3	yes	maximal, complemented
3	#834	C₂	1	3	yes	maximal, complemented
4	B(4,𝔽₂)	—	3	1	no	maximal, Sylow
5	C₂³×D₃	C₂²	1	1	yes	characteristic, complemented
6	C₃×#41	—	2	3	no	—
7	C₂×C₃×D₄	—	2	3	no	—
8	C₂×C₃×D₄	C₂²	1	3	yes	—
9	C₂×#67	—	2	3	no	—
10	#217	C₂²	1	3	yes	complemented
11	(C₂×D₃)⋊C₄	—	2	3	no	—
12	H(5,𝔽₂)	D₃	1	1	yes	characteristic, complemented
13	#120	—	3	3	no	—
14	#99	—	3	3	no	—
15	C₂³×C₃	—	2	1	no	—
16	C₂³×C₃	D₄	1	3	yes	complemented
17	C₂³×C₃	C₂³	1	1	yes	characteristic
18	C₂²×D₃	—	2	3	no	—
19	C₂²×D₃	—	4	2	no	—
20	C₃×Q₈	—	2	1	no	—
21	C₃×D₄	—	2	3	no	—
22	C₃×D₄	—	2	6	no	—
23	C₂×C₃×C₄	—	2	3	no	—
24	C₂×C₃×C₄	D₄	1	3	yes	complemented
25	#67	—	4	6	no	—
26	C₂×Dic₃	—	2	3	no	—
27	C₂⁴	—	3	1	no	—
28	#41	—	2	3	no	—
29	C₂×D₄	—	2	3	no	—
30	C₂×D₄	—	6	3	no	—
31	C₂×D₄	C₂×D₃	1	3	yes	—
32	C₂²⋊C₄	—	3	3	no	—
33	C₂²⋊C₄	—	6	3	no	—
34	C₂²×C₃	—	2	3	no	—
35	C₂²×C₃	—	4	1	no	—
36	C₂²×C₃	—	4	3	no	—
37	C₂²×C₃	—	8	1	no	—
38	C₂²×C₃	C₂×D₄	1	3	yes	—
39	C₂×D₃	—	2	6	no	—
40	C₂×D₃	—	4	3	no	—
41	C₂×D₃	—	4	1	no	—
42	C₃×C₄	—	2	3	no	—
43	Dic₃	—	4	3	no	—
44	C₂³	—	2	1	no	—
45	C₂³	—	6	3	no	—
46	C₂³	—	12	2	no	—
47	C₂³	#67	1	3	yes	complemented
48	C₂³	C₂²×D₃	1	1	yes	characteristic
49	Q₈	—	2	1	no	—
50	D₄	—	2	3	no	—
51	D₄	—	2	6	no	—
52	D₄	—	12	6	no	—
53	C₂×C₄	—	2	3	no	—
54	C₂×C₄	—	6	3	no	—
55	C₂×C₄	#67	1	3	yes	complemented
56	C₂×C₃	—	2	3	no	—
57	C₂×C₃	—	4	3	no	—
58	C₂×C₃	#120	1	1	yes	characteristic
59	D₃	—	4	2	no	—
60	C₂²	—	2	3	no	—
61	C₂²	—	4	1	no	—
62	C₂²	—	4	3	no	—
63	C₂²	—	6	6	no	—
64	C₂²	—	8	1	no	—
65	C₂²	—	12	3	no	—
66	C₂²	—	12	1	no	—
67	C₂²	C₂×#67	1	3	yes	—
68	C₄	—	2	3	no	—
69	C₄	—	12	3	no	—
70	C₃	B(4,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
71	C₂	—	2	3	no	—
72	C₂	—	4	3	no	—
73	C₂	—	12	2	no	—
74	C₂	#953	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#953
commutator, cosocle	C₂³×C₃	C₂³
Frattini	C₂³	C₂²×D₃
Fitting	C₃×H(5,𝔽₂)	C₂
socle	C₂×C₃	#120
hypercenter	H(5,𝔽₂)	D₃
hypocenter	C₃	B(4,𝔽₂)
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₃×H(5,𝔽₂)	C₂	semidirect
2. #937	C₂	semidirect
3. #834	C₂	semidirect
5. C₂³×D₃	C₂²	semidirect
10. #217	C₂²	semidirect
12. H(5,𝔽₂)	D₃	semidirect
16. C₂³×C₃	D₄	semidirect
24. C₂×C₃×C₄	D₄	semidirect
47. C₂³	#67	semidirect
55. C₂×C₄	#67	semidirect
70. C₃	B(4,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
C₃×H(5,𝔽₂)	2
#937	2
#834	2
B(4,𝔽₂)	3