Subgroups — #5242

Order 192.

152 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
2	#951	C₂	1	2	yes	maximal, complemented
3	#936	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	#455	—	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×Dic₆	C₂²	1	1	yes	characteristic, complemented
13	#217	—	2	2	no	—
14	#216	—	2	2	no	—
15	#214	—	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
19	#137	—	3	2	no	—
20	#124	—	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	C₃×Q₈	D₄	1	2	yes	—
26	C₃×D₄	—	2	4	no	—
27	C₃×D₄	—	2	2	no	—
28	C₃×D₄	—	2	2	no	—
29	C₃×D₄	D₄	1	2	yes	—
30	C₂×C₃×C₄	—	2	2	no	—
31	C₂×C₃×C₄	—	2	2	no	—
32	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₂×Dic₃	—	2	1	no	—
34	C₂×Dic₃	—	2	1	no	—
35	Dic₆	—	2	2	no	—
36	C₃⋊C₈	—	2	2	no	—
37	#41	—	2	2	no	—
38	#41	C₂×D₃	1	2	yes	—
39	C₂×Q₈	—	3	1	no	—
40	C₂×D₄	—	2	2	no	—
41	C₂×D₄	—	2	2	no	—
42	C₂×D₄	C₂×D₃	1	1	yes	characteristic
43	Q₁₆	—	6	2	no	—
44	SD₁₆	—	6	2	no	—
45	M₁₆	—	3	2	no	—
46	C₂²⋊C₄	—	6	2	no	—
47	C₄²	—	3	1	no	—
48	C₂²×C₃	—	2	2	no	—
49	C₂²×C₃	—	2	2	no	—
50	C₂²×C₃	—	4	2	no	—
51	C₂²×C₃	—	8	2	no	—
52	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
53	C₃×C₄	—	2	2	no	—
54	C₃×C₄	C₂×D₄	1	2	yes	—
55	Dic₃	—	4	1	no	—
56	Dic₃	—	4	1	no	—
57	C₂³	—	2	2	no	—
58	C₂³	#67	1	2	yes	—
59	Q₈	—	6	2	no	—
60	Q₈	#67	1	2	yes	—
61	D₄	—	2	4	no	—
62	D₄	—	2	2	no	—
63	D₄	—	2	2	no	—
64	D₄	#67	1	2	yes	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	6	1	no	—
68	C₂×C₄	—	6	1	no	—
69	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
70	C₈	—	6	2	no	—
71	C₂×C₃	—	2	1	no	—
72	C₂×C₃	—	4	2	no	—
73	C₂×C₃	—	4	2	no	—
74	C₂×C₃	#120	1	1	yes	characteristic
75	C₂²	—	2	2	no	—
76	C₂²	—	2	2	no	—
77	C₂²	—	4	2	no	—
78	C₂²	—	8	2	no	—
79	C₂²	C₂×#67	1	1	yes	characteristic
80	C₄	—	2	2	no	—
81	C₄	—	12	1	no	—
82	C₄	—	12	1	no	—
83	C₄	C₂×#67	1	2	yes	—
84	C₃	#455	1	1	yes	characteristic, complemented, Sylow
85	C₂	—	2	1	no	—
86	C₂	—	4	2	no	—
87	C₂	—	4	2	no	—
88	C₂	#953	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#953
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting, prime radical	C₃×H(5,𝔽₂)	C₂
socle	C₂×C₃	#120
hypercenter	H(5,𝔽₂)	D₃
hypocenter	C₃	#455
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
2. #951	C₂	semidirect
3. #936	C₂	semidirect
4. #837	C₂	semidirect
5. #833	C₂	semidirect
12. C₂×Dic₆	C₂²	semidirect
16. C₄×Dic₃	C₂²	semidirect
17. #208	C₂²	semidirect
84. C₃	#455	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
#951	2
#936	2
#837	2
#833	2
#455	3