Subgroups — #28390175

Order 14464.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20298996	C₂	1	1	yes	maximal, characteristic, complemented
2	#20298971	C₂	1	1	yes	maximal, characteristic
3	#20298949	C₂	1	2	yes	maximal, complemented
4	#20298945	C₂	1	2	yes	maximal
5	C₈²×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14268636	—	2	2	no	—
7	#16813622	C₂²	1	2	yes	—
8	#16813622	C₂²	1	1	yes	characteristic
9	#16813619	C₂²	1	1	yes	characteristic
10	#16813554	—	2	2	no	—
11	#16813547	—	2	4	no	—
12	#16813544	—	2	4	no	—
13	C₄×C₈×C₁₁₃	C₂²	1	2	yes	—
14	C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₂₆	—	2	1	no	—
16	C₂×Dic₂₂₆	—	2	2	no	—
17	C₂×D₄₅₂	—	2	1	no	—
18	C₂×D₄₅₂	—	2	2	no	—
19	#14268636	—	4	4	no	—
20	C₄⋊Dic₁₁₃	—	2	2	no	—
21	C₄⋊Dic₁₁₃	—	2	2	no	—
22	C₄⋊Dic₁₁₃	—	2	2	no	—
23	C₂×C₈×C₁₁₃	D₄	1	4	yes	—
24	C₂×C₈×C₁₁₃	D₄	1	2	yes	—
25	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
26	C₂²×D₁₁₃	—	4	1	no	—
27	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
29	C₂×Dic₁₁₃	—	4	1	no	—
30	C₂×Dic₁₁₃	—	4	2	no	—
31	D₄₅₂	—	4	2	no	—
32	D₄₅₂	—	4	4	no	—
33	Dic₂₂₆	—	4	2	no	—
34	Dic₂₂₆	—	4	4	no	—
35	C₈×C₁₁₃	—	2	4	no	—
36	C₈×C₁₁₃	SD₁₆	1	4	yes	—
37	C₂²×C₁₁₃	#127	1	1	yes	characteristic
38	C₂×D₁₁₃	—	8	1	no	—
39	C₂×D₁₁₃	—	8	2	no	—
40	C₄×C₁₁₃	C₂×SD₁₆	1	2	yes	—
41	C₄×C₁₁₃	#135	1	4	yes	—
42	Dic₁₁₃	—	8	1	no	—
43	Dic₁₁₃	—	8	2	no	—
44	C₂×C₁₁₃	#492	1	1	yes	characteristic
45	C₂×C₁₁₃	#495	1	2	yes	—
46	D₁₁₃	—	16	1	no	—
47	#1708	—	113	1	no	maximal, Sylow
48	C₁₁₃	#1708	1	1	yes	characteristic, complemented, Sylow
49	#498	—	113	1	no	—
50	#492	—	113	1	no	—
51	#487	—	113	2	no	—
52	#486	—	113	2	no	—
53	C₈²	D₁₁₃	1	1	yes	characteristic, complemented
54	C₂×SD₁₆	—	226	2	no	—
55	C₄⋊Q₈	—	113	2	no	—
56	C₄⋊Q₈	—	113	1	no	—
57	#127	—	113	1	no	—
58	#106	—	226	2	no	—
59	#103	—	226	4	no	—
60	#102	—	226	4	no	—
61	C₄×C₈	C₂×D₁₁₃	1	2	yes	—
62	C₄×C₈	C₂×D₁₁₃	1	1	yes	characteristic
63	C₂×Q₈	—	226	2	no	—
64	C₂×Q₈	—	226	1	no	—
65	C₂×D₄	—	226	2	no	—
66	C₂×D₄	—	226	1	no	—
67	SD₁₆	—	452	4	no	—
68	C₂×C₈	D₄₅₂	1	4	yes	—
69	C₂×C₈	D₄₅₂	1	2	yes	—
70	C₄⋊C₄	—	226	2	no	—
71	C₄⋊C₄	—	226	2	no	—
72	C₄⋊C₄	—	226	2	no	—
73	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
74	C₂³	—	452	1	no	—
75	Q₈	—	452	2	no	—
76	Q₈	—	452	4	no	—
77	D₄	—	452	2	no	—
78	D₄	—	452	4	no	—
79	C₂×C₄	—	452	1	no	—
80	C₂×C₄	—	452	2	no	—
81	C₂×C₄	C₂×D₄₅₂	1	1	yes	characteristic
82	C₂×C₄	C₂×D₄₅₂	1	2	yes	—
83	C₈	—	2	4	no	—
84	C₈	#14268636	1	4	yes	—
85	C₂²	—	904	1	no	—
86	C₂²	—	904	2	no	—
87	C₂²	#16813619	1	1	yes	characteristic
88	C₄	—	904	1	no	—
89	C₄	—	904	2	no	—
90	C₄	C₂×#14268636	1	2	yes	—
91	C₄	#16813644	1	4	yes	—
92	C₂	—	1808	1	no	—
93	C₂	#20298971	1	1	yes	characteristic
94	C₂	#20298982	1	2	yes	—

	subgroup	G/H
center	C₂²	#16813619
commutator, cosocle	C₄²×C₁₁₃	C₂³
Frattini	C₄²	C₂²×D₁₁₃
Fitting	C₈²×C₁₁₃	C₂
socle	C₂²×C₁₁₃	#127
prime radical	#16813619	C₂²
hypercenter	C₈²	D₁₁₃
hypocenter	C₁₁₃	#1708
LCS	G ▷ C₄²×C₁₁₃ ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
1. #20298996	C₂	semidirect
3. #20298949	C₂	semidirect
5. C₈²×C₁₁₃	C₂	semidirect
48. C₁₁₃	#1708	semidirect
53. C₈²	D₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
#20298996	2
#20298971	2
#20298949	2
#20298945	2
C₈²×C₁₁₃	2
#1708	113