Subgroups — #4475

Order 192.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#937	C₂	1	1	yes	maximal, characteristic
2	C₃×#100	C₂	1	1	yes	maximal, characteristic, complemented
3	#834	C₂	1	1	yes	maximal, characteristic, complemented
4	AΣL(1,𝔽₁₆)	—	3	1	no	maximal, Sylow
5	C₂³×D₃	C₄	1	1	yes	characteristic, complemented
6	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
7	C₂×#67	—	2	1	no	—
8	C₃×M₁₆	—	2	1	no	—
9	#217	—	2	1	no	—
10	#217	C₄	1	1	yes	characteristic, complemented
11	(C₂×D₃)⋊C₄	—	2	1	no	—
12	#120	—	3	1	no	—
13	#100	D₃	1	1	yes	characteristic, complemented
14	#99	—	3	1	no	—
15	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₃	D₄	1	1	yes	characteristic
17	C₂²×D₃	—	2	1	no	—
18	C₂²×D₃	—	4	2	no	—
19	C₂²×D₃	—	4	1	no	—
20	C₃×D₄	—	4	1	no	—
21	C₂×C₃×C₄	D₄	1	1	yes	characteristic
22	#67	—	4	2	no	—
23	C₂×Dic₃	—	2	1	no	—
24	C₂×Dic₃	—	4	1	no	—
25	C₃×C₈	—	4	1	no	—
26	C₂⁴	—	3	1	no	—
27	C₂×D₄	—	6	1	no	—
28	C₂×D₄	C₂×D₃	1	1	yes	characteristic
29	M₁₆	—	2	1	no	—
30	C₂²⋊C₄	—	3	1	no	—
31	C₂²⋊C₄	—	6	1	no	—
32	C₂²⋊C₄	—	6	1	no	—
33	C₂²×C₃	—	2	1	no	—
34	C₂²×C₃	—	2	1	no	—
35	C₂²×C₃	—	4	1	no	—
36	C₂²×C₃	—	4	1	no	—
37	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×D₃	—	2	2	no	—
39	C₂×D₃	—	4	1	no	—
40	C₂×D₃	—	4	2	no	—
41	C₂×D₃	—	4	1	no	—
42	C₂×D₃	—	4	2	no	—
43	C₃×C₄	—	2	1	no	—
44	Dic₃	—	4	1	no	—
45	Dic₃	—	8	1	no	—
46	C₂³	—	6	1	no	—
47	C₂³	—	12	1	no	—
48	C₂³	—	12	2	no	—
49	C₂³	C₄×D₃	1	1	yes	characteristic
50	C₂³	D₁₂	1	1	yes	characteristic
51	D₄	—	4	1	no	—
52	D₄	—	12	2	no	—
53	C₂×C₄	—	6	1	no	—
54	C₂×C₄	—	12	1	no	—
55	C₂×C₄	#67	1	1	yes	characteristic
56	C₈	—	4	1	no	—
57	C₂×C₃	—	2	1	no	—
58	C₂×C₃	—	4	1	no	—
59	C₂×C₃	—	4	1	no	—
60	C₂×C₃	#99	1	1	yes	characteristic
61	D₃	—	4	2	no	—
62	C₂²	—	2	1	no	—
63	C₂²	—	2	1	no	—
64	C₂²	—	4	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	6	2	no	—
67	C₂²	—	12	1	no	—
68	C₂²	—	12	2	no	—
69	C₂²	—	12	2	no	—
70	C₂²	—	12	1	no	—
71	C₂²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
72	C₄	—	2	1	no	—
73	C₄	—	12	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	1	no	—
79	C₂	—	12	2	no	—
80	C₂	#806	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#806
commutator	C₂³×C₃	C₂×C₄
Frattini	C₂×D₄	C₂×D₃
Fitting	C₃×#100	C₂
socle	C₂×C₃	#99
prime radical	#937	C₂
cosocle	C₂×C₃×D₄	C₂²
hypercenter	#100	D₃
hypocenter	C₃	AΣL(1,𝔽₁₆)
LCS	G ▷ C₂³×C₃ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #100

Complements

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

N	H ≅ G/N	type
2. C₃×#100	C₂	semidirect
3. #834	C₂	semidirect
5. C₂³×D₃	C₄	semidirect
10. #217	C₄	semidirect
13. #100	D₃	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
#937	2
C₃×#100	2
#834	2
AΣL(1,𝔽₁₆)	3