Subgroups — #4536

Order 192.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#120	C₂	1	1	yes	maximal, characteristic
2	#834	C₂	1	1	yes	maximal, characteristic, complemented
3	#833	C₂	1	1	yes	maximal, characteristic, complemented
4	AΣL(1,𝔽₁₆)	—	3	1	no	maximal, Sylow
5	C₂⁴×C₃	C₄	1	1	yes	characteristic, complemented
6	C₂×C₃×D₄	—	2	1	no	—
7	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
8	C₃×C₂²⋊C₄	—	2	1	no	—
9	C₃×C₂²⋊C₄	C₄	1	1	yes	characteristic, complemented
10	#217	—	2	1	no	—
11	#208	—	2	1	no	—
12	#120	D₃	1	1	yes	characteristic
13	#100	—	3	1	no	—
14	#99	—	3	1	no	—
15	C₂³×C₃	—	2	1	no	—
16	C₂³×C₃	—	4	2	no	—
17	C₂³×C₃	—	4	1	no	—
18	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₃	D₄	1	1	yes	characteristic
20	C₃×D₄	—	4	1	no	—
21	C₃×D₄	—	4	2	no	—
22	C₂×C₃×C₄	—	2	1	no	—
23	C₂×C₃×C₄	D₄	1	1	yes	characteristic
24	C₂×Dic₃	—	4	1	no	—
25	C₃⋊C₈	—	4	1	no	—
26	C₂⁴	Dic₃	1	1	yes	characteristic, complemented
27	C₂×D₄	—	2	1	no	—
28	C₂×D₄	C₂×D₃	1	1	yes	characteristic
29	M₁₆	—	6	1	no	—
30	C₂²⋊C₄	—	2	1	no	—
31	C₂²⋊C₄	—	6	1	no	—
32	C₂²⋊C₄	Dic₃	1	1	yes	characteristic, complemented
33	C₂²×C₃	—	2	2	no	—
34	C₂²×C₃	—	2	1	no	—
35	C₂²×C₃	—	2	1	no	—
36	C₂²×C₃	—	4	1	no	—
37	C₂²×C₃	—	4	1	no	—
38	C₂²×C₃	—	4	1	no	—
39	C₂²×C₃	—	4	2	no	—
40	C₂²×C₃	—	4	2	no	—
41	C₂²×C₃	—	4	1	no	—
42	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₃×C₄	—	2	1	no	—
44	C₃×C₄	—	4	1	no	—
45	Dic₃	—	8	1	no	—
46	C₂³	—	2	1	no	—
47	C₂³	—	4	2	no	—
48	C₂³	—	4	1	no	—
49	C₂³	C₂×Dic₃	1	1	yes	characteristic
50	C₂³	#67	1	1	yes	characteristic
51	D₄	—	4	1	no	—
52	D₄	—	4	2	no	—
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	12	1	no	—
55	C₂×C₄	#67	1	1	yes	characteristic
56	C₈	—	12	1	no	—
57	C₂×C₃	—	2	1	no	—
58	C₂×C₃	—	4	1	no	—
59	C₂×C₃	—	4	2	no	—
60	C₂×C₃	—	4	1	no	—
61	C₂×C₃	#99	1	1	yes	characteristic
62	C₂²	—	2	2	no	—
63	C₂²	—	2	1	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	4	1	no	—
68	C₂²	—	4	2	no	—
69	C₂²	—	4	2	no	—
70	C₂²	—	4	1	no	—
71	C₂²	#217	1	1	yes	characteristic
72	C₄	—	2	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	24	1	no	—
75	C₃	AΣL(1,𝔽₁₆)	1	1	yes	characteristic, complemented, Sylow
76	C₂	—	2	1	no	—
77	C₂	—	4	1	no	—
78	C₂	—	4	2	no	—
79	C₂	—	4	1	no	—
80	C₂	#834	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#834
commutator	C₂³×C₃	C₂×C₄
Frattini	C₂×D₄	C₂×D₃
Fitting, prime radical	C₃×#120	C₂
socle	C₂×C₃	#99
cosocle	C₂×C₃×D₄	C₂²
hypercenter	#120	D₃
hypocenter	C₃	AΣL(1,𝔽₁₆)
LCS	G ▷ C₂³×C₃ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #120

Complements

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

N	H ≅ G/N	type
2. #834	C₂	semidirect
3. #833	C₂	semidirect
5. C₂⁴×C₃	C₄	semidirect
9. C₃×C₂²⋊C₄	C₄	semidirect
26. C₂⁴	Dic₃	semidirect
32. C₂²⋊C₄	Dic₃	semidirect
75. C₃	AΣL(1,𝔽₁₆)	semidirect

Maximal subgroups

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

subgroup	index
C₃×#120	2
#834	2
#833	2
AΣL(1,𝔽₁₆)	3