Subgroups — #27136159

Order 12352.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259959	C₂	1	1	yes	maximal, characteristic, complemented
2	#20259957	C₂	1	1	yes	maximal, characteristic, complemented
3	#20259868	C₂	1	1	yes	maximal, characteristic
4	#15690564	—	2	1	no	—
5	C₂²⋊#11715871	—	2	1	no	—
6	C₂×C₂²⋊D₁₉₃	—	2	1	no	—
7	C₂×C₂²⋊D₁₉₃	C₂²	1	1	yes	characteristic
8	C₂²⋊Dic₁₉₃	—	2	1	no	—
9	C₂²⋊Dic₁₉₃	C₄	1	1	yes	characteristic, complemented
10	C₂⁴×C₁₉₃	C₄	1	1	yes	characteristic, complemented
11	C₂×#11715871	—	4	1	no	—
12	#13095400	—	4	1	no	—
13	C₂²×D₁₉₃	D₄	1	1	yes	characteristic
14	C₂²⋊D₁₉₃	—	4	1	no	—
15	C₂²⋊D₁₉₃	—	4	2	no	—
16	C₂×Dic₁₉₃	—	2	1	no	—
17	C₂×Dic₁₉₃	D₄	1	1	yes	characteristic
18	C₂³×C₁₉₃	—	2	1	no	—
19	C₂³×C₁₉₃	—	4	2	no	—
20	C₂³×C₁₉₃	—	4	1	no	—
21	C₂³×C₁₉₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₁₉₃	—	2	2	no	—
23	C₂²×C₁₉₃	—	2	1	no	—
24	C₂²×C₁₉₃	—	4	1	no	—
25	C₂²×C₁₉₃	—	4	1	no	—
26	C₂²×C₁₉₃	—	4	2	no	—
27	C₂²×C₁₉₃	—	4	2	no	—
28	C₂²×C₁₉₃	—	4	1	no	—
29	C₂²×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×D₁₉₃	—	2	1	no	—
31	C₂×D₁₉₃	—	4	1	no	—
32	#11715871	—	8	1	no	—
33	Dic₁₉₃	—	2	1	no	—
34	Dic₁₉₃	—	4	1	no	—
35	C₂×C₁₉₃	—	2	1	no	—
36	C₂×C₁₉₃	—	4	2	no	—
37	C₂×C₁₉₃	—	4	1	no	—
38	C₂×C₁₉₃	#99	1	1	yes	characteristic
39	D₁₉₃	—	4	1	no	—
40	C₁₉₃	AΣL(1,𝔽₁₆)	1	1	yes	characteristic, complemented, Sylow
41	AΣL(1,𝔽₁₆)	—	193	1	no	maximal, Sylow
42	#120	—	193	1	no	—
43	#100	—	193	1	no	—
44	#99	—	193	1	no	—
45	C₂⁴	#11715871	1	1	yes	characteristic, complemented
46	C₂×D₄	—	386	1	no	—
47	C₂×D₄	—	193	1	no	—
48	M₁₆	—	386	1	no	—
49	C₂²⋊C₄	—	386	1	no	—
50	C₂²⋊C₄	—	386	1	no	—
51	C₂²⋊C₄	—	193	1	no	—
52	C₂³	—	2	1	no	—
53	C₂³	—	4	2	no	—
54	C₂³	—	4	1	no	—
55	C₂³	—	193	1	no	—
56	C₂³	C₂×#11715871	1	1	yes	characteristic
57	D₄	—	772	2	no	—
58	D₄	—	772	1	no	—
59	C₂×C₄	—	772	1	no	—
60	C₂×C₄	—	386	1	no	—
61	C₂×C₄	—	193	1	no	—
62	C₈	—	772	1	no	—
63	C₂²	—	2	2	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	4	2	no	—
68	C₂²	—	4	2	no	—
69	C₂²	—	4	1	no	—
70	C₂²	—	772	1	no	—
71	C₂²	—	386	1	no	—
72	C₂²	C₂²⋊#11715871	1	1	yes	characteristic
73	C₄	—	772	1	no	—
74	C₄	—	1544	1	no	—
75	C₄	—	386	1	no	—
76	C₂	—	2	1	no	—
77	C₂	—	4	2	no	—
78	C₂	—	4	1	no	—
79	C₂	—	772	1	no	—
80	C₂	#20259957	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20259957
commutator	C₂³×C₁₉₃	C₂×C₄
Frattini	C₂³	C₂×#11715871
Fitting	C₂⁴×C₁₉₃	C₄
socle	C₂×C₁₉₃	#99
prime radical	#20259868	C₂
cosocle	C₂×C₂²⋊D₁₉₃	C₂²
hypercenter	C₂⁴	#11715871
hypocenter	C₁₉₃	AΣL(1,𝔽₁₆)
LCS	G ▷ C₂³×C₁₉₃ ▷ C₂²×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂³ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. #20259959	C₂	semidirect
2. #20259957	C₂	semidirect
9. C₂²⋊Dic₁₉₃	C₄	semidirect
10. C₂⁴×C₁₉₃	C₄	semidirect
40. C₁₉₃	AΣL(1,𝔽₁₆)	semidirect
45. C₂⁴	#11715871	semidirect

Maximal subgroups

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

subgroup	index
#20259959	2
#20259957	2
#20259868	2
AΣL(1,𝔽₁₆)	193