Subgroups — #28411531

Order 14656.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₂₉	C₂	1	1	yes	maximal, characteristic
2	#20300917	C₂	1	2	yes	maximal, complemented
3	C₂²⋊#14270080	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₂₂₉	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₂₂₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₂₉	C₄	1	1	yes	characteristic
7	C₂²×Dic₂₂₉	C₄	1	1	yes	characteristic
8	C₂×#14270080	C₂²	1	2	yes	—
9	C₄×Dic₂₂₉	C₄	1	1	yes	characteristic
10	C₄×Dic₂₂₉	C₄	1	1	yes	characteristic
11	C₄×Dic₂₂₉	C₂²	1	1	yes	characteristic
12	C₄×Dic₂₂₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₂₂₉	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₂₂₉	C₂²	1	2	yes	—
15	C₂×Dic₂₂₉	—	2	2	no	—
16	C₂×Dic₂₂₉	C₈	1	4	yes	complemented
17	C₂×Dic₂₂₉	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₂₂₉	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₂₂₉	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₂₂₉	C₂×C₄	1	1	yes	characteristic
21	#14270080	—	2	2	no	—
22	C₂³×C₂₂₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₂₉	—	2	2	no	—
24	C₂×C₄×C₂₂₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₂₉	C₂³	1	1	yes	characteristic
26	C₈×C₂₂₉	—	2	2	no	—
27	C₂²×C₂₂₉	—	2	1	no	—
28	C₂²×C₂₂₉	—	2	1	no	—
29	C₂²×C₂₂₉	C₂×C₈	1	2	yes	—
30	C₂²×C₂₂₉	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₂₂₉	—	2	1	no	—
32	C₄×C₂₂₉	#41	1	2	yes	—
33	Dic₂₂₉	—	2	1	no	—
34	Dic₂₂₉	C₂×C₈	1	2	yes	—
35	Dic₂₂₉	C₂×C₈	1	2	yes	—
36	Dic₂₂₉	M₁₆	1	2	yes	—
37	C₂×C₂₂₉	—	2	2	no	—
38	C₂×C₂₂₉	#117	1	1	yes	characteristic
39	C₂×C₂₂₉	C₂²×C₈	1	1	yes	characteristic
40	C₂×C₂₂₉	C₂×M₁₆	1	1	yes	characteristic
41	C₂₂₉	#431	1	1	yes	characteristic, complemented, Sylow
42	#431	—	229	1	no	maximal, Sylow
43	C₂×C₄²	—	229	1	no	—
44	C₄⋊C₈	—	229	2	no	—
45	C₂²⋊C₈	—	229	1	no	—
46	C₂²⋊C₈	D₂₂₉	1	1	yes	characteristic
47	C₄×C₈	—	229	2	no	—
48	C₂²×C₄	—	229	1	no	—
49	C₂²×C₄	—	229	1	no	—
50	C₂²×C₄	C₂×D₂₂₉	1	1	yes	characteristic
51	C₂×C₈	—	229	2	no	—
52	C₂×C₈	C₂×D₂₂₉	1	2	yes	—
53	C₄²	—	229	1	no	—
54	C₄²	—	229	1	no	—
55	C₄²	—	229	1	no	—
56	C₄²	—	229	1	no	—
57	C₂³	C₄×D₂₂₉	1	1	yes	characteristic
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	458	2	no	—
60	C₂×C₄	—	229	1	no	—
61	C₂×C₄	—	229	4	no	—
62	C₂×C₄	—	229	1	no	—
63	C₂×C₄	—	229	1	no	—
64	C₂×C₄	—	229	1	no	—
65	C₂×C₄	C₄×D₂₂₉	1	1	yes	characteristic
66	C₂×C₄	C₂²×D₂₂₉	1	1	yes	characteristic
67	C₈	—	2	2	no	—
68	C₈	—	458	2	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	C₈×D₂₂₉	1	2	yes	—
72	C₂²	C₂×C₄×D₂₂₉	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	458	1	no	—
75	C₄	—	229	2	no	—
76	C₄	—	229	2	no	—
77	C₄	—	229	2	no	—
78	C₄	#16814665	1	2	yes	—
79	C₂	—	2	2	no	—
80	C₂	#20300944	1	1	yes	characteristic
81	C₂	C₂×C₈×D₂₂₉	1	1	yes	characteristic
82	C₂	M₁₆×D₂₂₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₂₉
commutator	C₂×C₂₂₉	C₂²×C₈
Fitting	C₂²⋊C₈×C₂₂₉	C₂
socle	C₂²×C₂₂₉	C₂²×C₄
prime radical	C₂³×C₂₂₉	C₂×C₄
cosocle	C₂×C₄×C₂₂₉	C₂³
hypercenter	C₂²⋊C₈	D₂₂₉
hypocenter	C₂₂₉	#431
LCS	G ▷ C₂×C₂₂₉ ▷ C₂₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. #20300917	C₂	semidirect
4. C₈×Dic₂₂₉	C₂	semidirect
16. C₂×Dic₂₂₉	C₈	semidirect
41. C₂₂₉	#431	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₂₉	2
#20300917	2
C₂²⋊#14270080	2
C₈×Dic₂₂₉	2
C₂²⋊C₈×C₂₂₉	2
#431	229