Subgroups — #28412381

Order 14656.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20301048	C₂	1	1	yes	maximal, characteristic, complemented
2	#20301046	C₂	1	1	yes	maximal, characteristic, complemented
3	#20300957	C₂	1	1	yes	maximal, characteristic
4	#16814675	—	2	1	no	—
5	C₂²⋊#11745021	—	2	1	no	—
6	C₂×C₂²⋊D₂₂₉	—	2	1	no	—
7	C₂×C₂²⋊D₂₂₉	C₂²	1	1	yes	characteristic
8	C₂²⋊Dic₂₂₉	—	2	1	no	—
9	C₂²⋊Dic₂₂₉	C₄	1	1	yes	characteristic, complemented
10	C₂⁴×C₂₂₉	C₄	1	1	yes	characteristic, complemented
11	C₂×#11745021	—	4	1	no	—
12	#14270087	—	4	1	no	—
13	C₂²×D₂₂₉	D₄	1	1	yes	characteristic
14	C₂²⋊D₂₂₉	—	4	1	no	—
15	C₂²⋊D₂₂₉	—	4	2	no	—
16	C₂×Dic₂₂₉	—	2	1	no	—
17	C₂×Dic₂₂₉	D₄	1	1	yes	characteristic
18	C₂³×C₂₂₉	—	2	1	no	—
19	C₂³×C₂₂₉	—	4	2	no	—
20	C₂³×C₂₂₉	—	4	1	no	—
21	C₂³×C₂₂₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₂₂₉	—	2	2	no	—
23	C₂²×C₂₂₉	—	2	1	no	—
24	C₂²×C₂₂₉	—	4	1	no	—
25	C₂²×C₂₂₉	—	4	1	no	—
26	C₂²×C₂₂₉	—	4	2	no	—
27	C₂²×C₂₂₉	—	4	2	no	—
28	C₂²×C₂₂₉	—	4	1	no	—
29	C₂²×C₂₂₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×D₂₂₉	—	2	1	no	—
31	C₂×D₂₂₉	—	4	1	no	—
32	#11745021	—	8	1	no	—
33	Dic₂₂₉	—	2	1	no	—
34	Dic₂₂₉	—	4	1	no	—
35	C₂×C₂₂₉	—	2	1	no	—
36	C₂×C₂₂₉	—	4	2	no	—
37	C₂×C₂₂₉	—	4	1	no	—
38	C₂×C₂₂₉	#99	1	1	yes	characteristic
39	D₂₂₉	—	4	1	no	—
40	C₂₂₉	AΣL(1,𝔽₁₆)	1	1	yes	characteristic, complemented, Sylow
41	AΣL(1,𝔽₁₆)	—	229	1	no	maximal, Sylow
42	#120	—	229	1	no	—
43	#100	—	229	1	no	—
44	#99	—	229	1	no	—
45	C₂⁴	#11745021	1	1	yes	characteristic, complemented
46	C₂×D₄	—	458	1	no	—
47	C₂×D₄	—	229	1	no	—
48	M₁₆	—	458	1	no	—
49	C₂²⋊C₄	—	458	1	no	—
50	C₂²⋊C₄	—	458	1	no	—
51	C₂²⋊C₄	—	229	1	no	—
52	C₂³	—	2	1	no	—
53	C₂³	—	4	2	no	—
54	C₂³	—	4	1	no	—
55	C₂³	—	229	1	no	—
56	C₂³	C₂×#11745021	1	1	yes	characteristic
57	D₄	—	916	2	no	—
58	D₄	—	916	1	no	—
59	C₂×C₄	—	916	1	no	—
60	C₂×C₄	—	458	1	no	—
61	C₂×C₄	—	229	1	no	—
62	C₈	—	916	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	—	916	1	no	—
71	C₂²	—	458	1	no	—
72	C₂²	C₂²⋊#11745021	1	1	yes	characteristic
73	C₄	—	1832	1	no	—
74	C₄	—	916	1	no	—
75	C₄	—	458	1	no	—
76	C₂	—	2	1	no	—
77	C₂	—	4	2	no	—
78	C₂	—	4	1	no	—
79	C₂	—	916	1	no	—
80	C₂	#20301046	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20301046
commutator	C₂³×C₂₂₉	C₂×C₄
Frattini	C₂³	C₂×#11745021
Fitting	C₂⁴×C₂₂₉	C₄
socle	C₂×C₂₂₉	#99
prime radical	#20300957	C₂
cosocle	C₂×C₂²⋊D₂₂₉	C₂²
hypercenter	C₂⁴	#11745021
hypocenter	C₂₂₉	AΣL(1,𝔽₁₆)
LCS	G ▷ C₂³×C₂₂₉ ▷ C₂²×C₂₂₉ ▷ C₂×C₂₂₉ ▷ C₂₂₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂³ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. #20301048	C₂	semidirect
2. #20301046	C₂	semidirect
9. C₂²⋊Dic₂₂₉	C₄	semidirect
10. C₂⁴×C₂₂₉	C₄	semidirect
40. C₂₂₉	AΣL(1,𝔽₁₆)	semidirect
45. C₂⁴	#11745021	semidirect

Maximal subgroups

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

subgroup	index
#20301048	2
#20301046	2
#20300957	2
AΣL(1,𝔽₁₆)	229