Subgroups — #28411691

Order 14656.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16814656	C₂	1	2	yes	maximal
2	#20300988	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#14270080	C₂	1	1	yes	maximal, characteristic
4	#20300911	C₂	1	2	yes	maximal
5	C₄×Q₈×C₂₂₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₅₈	C₂²	1	2	yes	—
7	#16814656	—	2	4	no	—
8	C₂×#14270080	C₂²	1	2	yes	—
9	C₄⋊Dic₂₂₉	—	2	1	no	—
10	C₄⋊Dic₂₂₉	—	2	1	no	—
11	C₂×Q₈×C₂₂₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₂₂₉	—	2	1	no	—
13	C₄⋊C₄×C₂₂₉	C₂²	1	1	yes	characteristic
14	C₄²×C₂₂₉	—	2	1	no	—
15	C₄²×C₂₂₉	C₂²	1	1	yes	characteristic
16	Dic₄₅₈	—	2	2	no	—
17	Dic₄₅₈	—	2	2	no	—
18	C₂×Dic₂₂₉	—	2	2	no	—
19	#14270080	—	2	2	no	—
20	Q₈×C₂₂₉	—	2	1	no	—
21	Q₈×C₂₂₉	D₄	1	2	yes	—
22	C₂×C₄×C₂₂₉	—	2	1	no	—
23	C₂×C₄×C₂₂₉	—	2	1	no	—
24	C₂×C₄×C₂₂₉	D₄	1	1	yes	characteristic
25	C₂×C₄×C₂₂₉	D₄	1	1	yes	characteristic
26	C₂×C₄×C₂₂₉	C₂³	1	1	yes	characteristic
27	C₂²×C₂₂₉	C₂×D₄	1	1	yes	characteristic
28	C₄×C₂₂₉	—	2	2	no	—
29	C₄×C₂₂₉	—	2	1	no	—
30	C₄×C₂₂₉	—	4	1	no	—
31	C₄×C₂₂₉	Q₁₆	1	2	yes	complemented
32	C₄×C₂₂₉	C₂×D₄	1	1	yes	characteristic
33	C₄×C₂₂₉	#41	1	1	yes	characteristic
34	Dic₂₂₉	—	4	2	no	—
35	C₂×C₂₂₉	#121	1	1	yes	characteristic
36	C₂×C₂₂₉	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₂₂₉	#137	1	1	yes	characteristic
38	C₂₂₉	#462	1	1	yes	characteristic, complemented, Sylow
39	#462	—	229	1	no	maximal, Sylow
40	C₂×Q₁₆	—	229	2	no	—
41	C₄⋊Q₈	—	229	1	no	—
42	C₄×Q₈	D₂₂₉	1	1	yes	characteristic
43	C₄⋊C₈	—	229	1	no	—
44	#103	—	229	2	no	—
45	C₂×Q₈	—	229	2	no	—
46	C₂×Q₈	C₂×D₂₂₉	1	1	yes	characteristic
47	Q₁₆	—	458	4	no	—
48	C₂×C₈	—	229	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	—	458	1	no	—
51	C₄⋊C₄	—	458	1	no	—
52	C₄⋊C₄	C₂×D₂₂₉	1	1	yes	characteristic
53	C₄²	—	2	1	no	—
54	C₄²	C₂×D₂₂₉	1	1	yes	characteristic
55	Q₈	—	2	1	no	—
56	Q₈	—	458	2	no	—
57	Q₈	—	458	2	no	—
58	Q₈	D₉₁₆	1	2	yes	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	458	2	no	—
62	C₂×C₄	C₂²⋊D₂₂₉	1	1	yes	characteristic
63	C₂×C₄	C₂²⋊D₂₂₉	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₂₂₉	1	1	yes	characteristic
65	C₈	—	458	2	no	—
66	C₂²	C₂×C₂²⋊D₂₂₉	1	1	yes	characteristic
67	C₄	—	2	2	no	—
68	C₄	—	2	1	no	—
69	C₄	—	4	1	no	—
70	C₄	—	916	2	no	—
71	C₄	#16814656	1	2	yes	complemented
72	C₄	C₂×D₉₁₆	1	1	yes	characteristic
73	C₄	#16814666	1	1	yes	characteristic
74	C₂	#20300962	1	1	yes	characteristic
75	C₂	C₂×#16814656	1	1	yes	characteristic
76	C₂	#20301018	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₂₉
commutator, cosocle	C₂×C₄×C₂₂₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₉
Fitting	C₄×Q₈×C₂₂₉	C₂
socle, prime radical	C₂²×C₂₂₉	C₂×D₄
hypercenter	C₄×Q₈	D₂₂₉
hypocenter	C₂₂₉	#462
LCS	G ▷ C₂×C₄×C₂₂₉ ▷ C₂×C₂₂₉ ▷ C₂₂₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
31. C₄×C₂₂₉	Q₁₆	semidirect
38. C₂₂₉	#462	semidirect
71. C₄	#16814656	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
#20300988	2
C₄⋊#14270080	2
#20300911	2
C₄×Q₈×C₂₂₉	2
#462	229