Subgroups — #28411689

Order 14656.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16814656	C₂	1	1	yes	maximal, characteristic
2	#20300986	C₂	1	1	yes	maximal, characteristic
3	Q₈×Dic₂₂₉	C₂	1	1	yes	maximal, characteristic
4	#20300917	C₂	1	1	yes	maximal, characteristic
5	#20300913	C₂	1	1	yes	maximal, characteristic
6	#20300912	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₁₆×C₂₂₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₄₅₈	C₂²	1	1	yes	characteristic
9	#16814656	—	2	2	no	—
10	C₂×#14270080	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₂₉	—	2	1	no	—
12	C₄⋊Dic₂₂₉	C₂²	1	1	yes	characteristic
13	#16814645	—	2	1	no	—
14	#16814645	—	2	1	no	—
15	C₄×Dic₂₂₉	—	2	1	no	—
16	C₄×Dic₂₂₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₂₂₉	C₂²	1	1	yes	characteristic
18	C₂×Q₈×C₂₂₉	C₂²	1	1	yes	characteristic
19	Q₁₆×C₂₂₉	—	2	2	no	—
20	C₂×C₈×C₂₂₉	C₂²	1	1	yes	characteristic
21	Dic₄₅₈	—	2	1	no	—
22	Dic₄₅₈	—	2	1	no	—
23	C₂×Dic₂₂₉	—	2	1	no	—
24	C₂×Dic₂₂₉	—	2	1	no	—
25	C₂×Dic₂₂₉	D₄	1	1	yes	characteristic
26	C₂×Dic₂₂₉	D₄	1	1	yes	characteristic
27	#14270080	—	2	1	no	—
28	Q₈×C₂₂₉	—	2	1	no	—
29	Q₈×C₂₂₉	—	2	1	no	—
30	Q₈×C₂₂₉	—	2	1	no	—
31	Q₈×C₂₂₉	D₄	1	2	yes	—
32	C₂×C₄×C₂₂₉	—	2	1	no	—
33	C₂×C₄×C₂₂₉	—	2	1	no	—
34	C₂×C₄×C₂₂₉	C₂³	1	1	yes	characteristic
35	C₈×C₂₂₉	—	2	1	no	—
36	C₂²×C₂₂₉	C₂×D₄	1	1	yes	characteristic
37	C₄×C₂₂₉	—	2	2	no	—
38	C₄×C₂₂₉	—	4	1	no	—
39	C₄×C₂₂₉	C₂×D₄	1	1	yes	characteristic
40	C₄×C₂₂₉	#41	1	1	yes	characteristic
41	Dic₂₂₉	—	2	1	no	—
42	Dic₂₂₉	—	4	1	no	—
43	Dic₂₂₉	—	4	1	no	—
44	Dic₂₂₉	Q₁₆	1	2	yes	complemented
45	C₂×C₂₂₉	#121	1	1	yes	characteristic
46	C₂×C₂₂₉	C₂×Q₁₆	1	1	yes	characteristic
47	C₂×C₂₂₉	#137	1	1	yes	characteristic
48	C₂₂₉	#462	1	1	yes	characteristic, complemented, Sylow
49	#462	—	229	1	no	maximal, Sylow
50	C₂×Q₁₆	—	229	1	no	—
51	C₂×Q₁₆	D₂₂₉	1	1	yes	characteristic
52	C₄⋊Q₈	—	229	1	no	—
53	C₄×Q₈	—	229	1	no	—
54	C₄⋊C₈	—	229	1	no	—
55	#103	—	229	1	no	—
56	#103	—	229	1	no	—
57	C₂×Q₈	—	229	1	no	—
58	C₂×Q₈	C₂×D₂₂₉	1	1	yes	characteristic
59	C₂×Q₈	C₂×D₂₂₉	1	1	yes	characteristic
60	Q₁₆	—	2	2	no	—
61	Q₁₆	—	458	2	no	—
62	C₂×C₈	—	229	1	no	—
63	C₂×C₈	C₂×D₂₂₉	1	1	yes	characteristic
64	C₄⋊C₄	—	458	1	no	—
65	C₄⋊C₄	—	458	1	no	—
66	C₄⋊C₄	—	458	1	no	—
67	C₄⋊C₄	—	229	1	no	—
68	C₄²	—	458	1	no	—
69	C₄²	—	229	1	no	—
70	Q₈	—	2	1	no	—
71	Q₈	—	2	1	no	—
72	Q₈	—	2	1	no	—
73	Q₈	—	458	1	no	—
74	Q₈	—	458	1	no	—
75	Q₈	C₂²⋊D₂₂₉	1	2	yes	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	458	1	no	—
79	C₂×C₄	—	458	1	no	—
80	C₂×C₄	—	229	1	no	—
81	C₂×C₄	—	229	1	no	—
82	C₂×C₄	C₂²×D₂₂₉	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	—	458	1	no	—
85	C₂²	D₄×D₂₂₉	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	4	1	no	—
88	C₄	—	916	1	no	—
89	C₄	—	916	1	no	—
90	C₄	—	458	1	no	—
91	C₄	—	229	2	no	—
92	C₄	C₂×C₂²⋊D₂₂₉	1	1	yes	characteristic
93	C₄	#16814665	1	1	yes	characteristic
94	C₂	#20300960	1	1	yes	characteristic
95	C₂	Q₁₆×D₂₂₉	1	1	yes	characteristic
96	C₂	#20301020	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₂₉
commutator, cosocle	C₂×C₄×C₂₂₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₉
Fitting	C₂×Q₁₆×C₂₂₉	C₂
socle, prime radical	C₂²×C₂₂₉	C₂×D₄
hypercenter	C₂×Q₁₆	D₂₂₉
hypocenter	C₂₂₉	#462
LCS	G ▷ C₂×C₄×C₂₂₉ ▷ C₂×C₂₂₉ ▷ C₂₂₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
44. Dic₂₂₉	Q₁₆	semidirect
48. C₂₂₉	#462	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16814656	2
#20300986	2
Q₈×Dic₂₂₉	2
#20300917	2
#20300913	2
#20300912	2
C₂×Q₁₆×C₂₂₉	2
#462	229