Subgroups — #5241

Order 192.

168 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	#949	C₂	1	2	yes	maximal, complemented
3	#940	C₂	1	1	yes	maximal, characteristic, complemented
4	#837	C₂	1	2	yes	maximal, complemented
5	#833	C₂	1	1	yes	maximal, characteristic, complemented
6	#453	—	3	1	no	maximal, Sylow
7	C₃×#41	—	2	2	no	—
8	C₃×#41	C₂²	1	2	yes	—
9	C₂×C₃×D₄	—	2	2	no	—
10	C₂×C₃×D₄	—	2	2	no	—
11	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
12	C₂×#67	—	2	2	no	—
13	C₂×D₁₂	C₂²	1	1	yes	characteristic, complemented
14	#215	—	2	2	no	—
15	#213	—	2	2	no	—
16	C₄×Dic₃	C₂²	1	1	yes	characteristic, complemented
17	#208	C₂²	1	2	yes	complemented
18	H(5,𝔽₂)	D₃	1	1	yes	characteristic, complemented
19	AGL(1,ℤ/8)	—	3	2	no	—
20	#127	—	3	1	no	—
21	#104	—	3	2	no	—
22	#100	—	3	1	no	—
23	C₂³×C₃	—	2	2	no	—
24	C₂³×C₃	D₄	1	2	yes	—
25	C₂²×D₃	—	2	1	no	—
26	C₃×Q₈	D₄	1	2	yes	complemented
27	C₃×D₄	—	2	4	no	—
28	C₃×D₄	—	2	2	no	—
29	C₃×D₄	—	2	2	no	—
30	C₃×D₄	D₄	1	2	yes	complemented
31	C₂×C₃×C₄	—	2	2	no	—
32	C₂×C₃×C₄	—	2	2	no	—
33	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
34	#67	—	4	4	no	—
35	C₂×Dic₃	—	2	1	no	—
36	D₁₂	—	2	2	no	—
37	C₃⋊C₈	—	2	2	no	—
38	#41	—	2	2	no	—
39	#41	C₂×D₃	1	2	yes	—
40	C₂×D₄	—	2	2	no	—
41	C₂×D₄	—	2	2	no	—
42	C₂×D₄	—	3	1	no	—
43	C₂×D₄	—	6	2	no	—
44	C₂×D₄	C₂×D₃	1	1	yes	characteristic
45	SD₁₆	—	6	2	no	—
46	D₈	—	6	2	no	—
47	M₁₆	—	3	2	no	—
48	C₄²	—	3	1	no	—
49	C₂²×C₃	—	2	2	no	—
50	C₂²×C₃	—	2	2	no	—
51	C₂²×C₃	—	4	2	no	—
52	C₂²×C₃	—	8	2	no	—
53	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
54	C₂×D₃	—	4	2	no	—
55	C₂×D₃	—	4	1	no	—
56	C₃×C₄	—	2	2	no	—
57	C₃×C₄	C₂×D₄	1	2	yes	—
58	Dic₃	—	2	2	no	—
59	C₂³	—	2	2	no	—
60	C₂³	—	6	1	no	—
61	C₂³	#67	1	2	yes	—
62	Q₈	#67	1	2	yes	complemented
63	D₄	—	2	4	no	—
64	D₄	—	2	2	no	—
65	D₄	—	2	2	no	—
66	D₄	—	6	2	no	—
67	D₄	—	12	4	no	—
68	D₄	#67	1	2	yes	complemented
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	6	1	no	—
72	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
73	C₈	—	6	2	no	—
74	C₂×C₃	—	2	1	no	—
75	C₂×C₃	—	4	2	no	—
76	C₂×C₃	—	4	2	no	—
77	C₂×C₃	#120	1	1	yes	characteristic
78	D₃	—	8	1	no	—
79	C₂²	—	2	2	no	—
80	C₂²	—	2	2	no	—
81	C₂²	—	4	2	no	—
82	C₂²	—	8	2	no	—
83	C₂²	—	12	2	no	—
84	C₂²	—	12	1	no	—
85	C₂²	C₂×#67	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	6	2	no	—
88	C₄	C₂×#67	1	2	yes	—
89	C₃	#453	1	1	yes	characteristic, complemented, Sylow
90	C₂	—	2	1	no	—
91	C₂	—	4	2	no	—
92	C₂	—	4	2	no	—
93	C₂	—	24	1	no	—
94	C₂	#953	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#953
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×H(5,𝔽₂)	C₂
socle	C₂×C₃	#120
hypercenter	H(5,𝔽₂)	D₃
hypocenter	C₃	#453
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ 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. #949	C₂	semidirect
3. #940	C₂	semidirect
4. #837	C₂	semidirect
5. #833	C₂	semidirect
13. C₂×D₁₂	C₂²	semidirect
16. C₄×Dic₃	C₂²	semidirect
17. #208	C₂²	semidirect
18. H(5,𝔽₂)	D₃	semidirect
26. C₃×Q₈	D₄	semidirect
30. C₃×D₄	D₄	semidirect
62. Q₈	#67	semidirect
68. D₄	#67	semidirect
89. C₃	#453	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
#949	2
#940	2
#837	2
#833	2
#453	3