Subgroups — #28412453

Order 14656.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16814675	C₂	1	2	yes	maximal, complemented
2	#20301045	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄×D₂₂₉	C₂	1	1	yes	maximal, characteristic
4	#16814675	—	2	4	no	—
5	C₂×#14270087	C₂²	1	4	yes	complemented
6	C₂³×D₂₂₉	C₄	1	1	yes	characteristic
7	C₂²×Dic₂₂₉	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₂₂₉	—	2	4	no	—
9	C₂×C₄×D₂₂₉	C₄	1	2	yes	—
10	C₂×C₄×D₂₂₉	C₂²	1	2	yes	—
11	C₂²×C₄×C₂₂₉	C₄	1	1	yes	characteristic
12	#14270087	—	2	4	no	—
13	C₂²×D₂₂₉	—	2	2	no	—
14	C₂²×D₂₂₉	—	2	4	no	—
15	C₂²×D₂₂₉	C₂×C₄	1	2	yes	—
16	C₂×Dic₂₂₉	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₂₂₉	D₄	1	4	yes	—
18	C₂×Dic₂₂₉	C₂³	1	1	yes	characteristic
19	C₄×D₂₂₉	—	2	4	no	—
20	C₄×D₂₂₉	—	2	4	no	—
21	C₂³×C₂₂₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₂₉	—	2	2	no	—
23	C₂×C₄×C₂₂₉	C₂×C₄	1	2	yes	—
24	C₂²×C₂₂₉	—	2	2	no	—
25	C₂²×C₂₂₉	C₂²⋊C₄	1	2	yes	—
26	C₂²×C₂₂₉	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₂₂₉	—	2	8	no	—
28	C₂×D₂₂₉	—	2	2	no	—
29	C₂×D₂₂₉	—	2	2	no	—
30	C₂×D₂₂₉	M₁₆	1	4	yes	complemented
31	C₄×C₂₂₉	—	2	2	no	—
32	Dic₂₂₉	C₂²⋊C₄	1	2	yes	—
33	Dic₂₂₉	C₂×D₄	1	2	yes	—
34	C₂×C₂₂₉	—	2	2	no	—
35	C₂×C₂₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₂₂₉	C₂×M₁₆	1	2	yes	—
37	D₂₂₉	—	2	4	no	—
38	C₂₂₉	#407	1	1	yes	characteristic, complemented, Sylow
39	#407	—	229	1	no	maximal, Sylow
40	C₂³×C₄	—	229	1	no	—
41	C₂×M₁₆	—	229	2	no	—
42	C₂²⋊C₈	—	229	4	no	—
43	C₂⁴	—	229	1	no	—
44	C₂²×C₄	—	458	4	no	—
45	C₂²×C₄	—	229	2	no	—
46	C₂²×C₄	—	229	2	no	—
47	C₂²×C₄	—	229	1	no	—
48	C₂²×C₄	#11745021	1	1	yes	characteristic
49	M₁₆	—	458	4	no	—
50	C₂×C₈	—	229	4	no	—
51	C₂³	—	458	4	no	—
52	C₂³	—	458	2	no	—
53	C₂³	—	229	2	no	—
54	C₂³	C₂×#11745021	1	1	yes	characteristic
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	458	4	no	—
57	C₂×C₄	—	458	4	no	—
58	C₂×C₄	—	229	4	no	—
59	C₂×C₄	—	229	1	no	—
60	C₂×C₄	—	229	1	no	—
61	C₂×C₄	C₂×#11745021	1	2	yes	—
62	C₈	—	458	4	no	—
63	C₂²	—	2	2	no	—
64	C₂²	—	458	8	no	—
65	C₂²	—	458	2	no	—
66	C₂²	—	458	2	no	—
67	C₂²	—	229	4	no	—
68	C₂²	C₂²⋊#11745021	1	2	yes	—
69	C₂²	C₂²×#11745021	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	229	2	no	—
72	C₄	—	229	2	no	—
73	C₂	—	2	2	no	—
74	C₂	—	458	4	no	—
75	C₂	C₂×C₂²⋊#11745021	1	1	yes	characteristic
76	C₂	D₂₂₉⋊M₁₆	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11745021
commutator, socle	C₂²×C₂₂₉	C₂²×C₄
Fitting	C₂²×C₄×C₂₂₉	C₄
prime radical	C₂³×D₂₂₉	C₄
cosocle	C₂×Dic₂₂₉	C₂³
hypercenter	C₂²×C₄	#11745021
hypocenter	C₂₂₉	#407
LCS	G ▷ C₂²×C₂₂₉ ▷ C₂₂₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#16814675	C₂	semidirect
2. #20301045	C₂	semidirect
5. C₂×#14270087	C₂²	semidirect
30. C₂×D₂₂₉	M₁₆	semidirect
38. C₂₂₉	#407	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16814675	2
#20301045	2
C₂²×C₄×D₂₂₉	2
#407	229