Subgroups — #28390245

Order 14464.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299003	C₂	1	2	yes	maximal, complemented
2	#20298978	C₂	1	1	yes	maximal, characteristic, complemented
3	#20298971	C₂	1	1	yes	maximal, characteristic
4	#20298945	C₂	1	2	yes	maximal
5	#322×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₄₅₂	—	2	2	no	—
7	C₂×#14268636	—	2	2	no	—
8	#16813622	C₂²	1	2	yes	—
9	#16813622	C₂²	1	1	yes	characteristic
10	#16813619	C₂²	1	1	yes	characteristic
11	#16813556	—	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	Dic₄₅₂	—	4	4	no	—
20	#14268636	—	4	4	no	—
21	C₄⋊Dic₁₁₃	—	2	2	no	—
22	C₄⋊Dic₁₁₃	—	2	2	no	—
23	C₄⋊Dic₁₁₃	—	2	2	no	—
24	C₂×C₈×C₁₁₃	D₄	1	4	yes	—
25	C₂×C₈×C₁₁₃	D₄	1	2	yes	—
26	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
27	C₂²×D₁₁₃	—	4	1	no	—
28	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
30	C₂×Dic₁₁₃	—	4	2	no	—
31	C₂×Dic₁₁₃	—	4	1	no	—
32	D₄₅₂	—	4	2	no	—
33	D₄₅₂	—	4	4	no	—
34	Dic₂₂₆	—	4	2	no	—
35	Dic₂₂₆	—	4	4	no	—
36	C₈×C₁₁₃	—	2	4	no	—
37	C₈×C₁₁₃	D₈	1	4	yes	—
38	C₂²×C₁₁₃	#127	1	1	yes	characteristic
39	C₂×D₁₁₃	—	8	1	no	—
40	C₂×D₁₁₃	—	8	2	no	—
41	C₄×C₁₁₃	C₂×D₈	1	2	yes	—
42	C₄×C₁₁₃	#137	1	4	yes	—
43	Dic₁₁₃	—	8	1	no	—
44	Dic₁₁₃	—	8	2	no	—
45	C₂×C₁₁₃	#493	1	1	yes	characteristic
46	C₂×C₁₁₃	#497	1	2	yes	—
47	D₁₁₃	—	16	1	no	—
48	#1722	—	113	1	no	maximal, Sylow
49	C₁₁₃	#1722	1	1	yes	characteristic, complemented, Sylow
50	#500	—	113	2	no	—
51	#494	—	113	1	no	—
52	#492	—	113	1	no	—
53	#486	—	113	2	no	—
54	#322	D₁₁₃	1	1	yes	characteristic, complemented
55	C₂×Q₁₆	—	226	2	no	—
56	C₂×SD₁₆	—	226	2	no	—
57	C₄⋊Q₈	—	113	2	no	—
58	C₄⋊Q₈	—	113	1	no	—
59	#127	—	113	1	no	—
60	#107	—	226	4	no	—
61	#102	—	226	4	no	—
62	C₄×C₈	C₂×D₁₁₃	1	2	yes	—
63	C₄×C₈	C₂×D₁₁₃	1	1	yes	characteristic
64	C₂×Q₈	—	226	1	no	—
65	C₂×Q₈	—	226	2	no	—
66	C₂×D₄	—	226	2	no	—
67	C₂×D₄	—	226	1	no	—
68	Q₁₆	—	452	4	no	—
69	SD₁₆	—	452	4	no	—
70	C₂×C₈	D₄₅₂	1	4	yes	—
71	C₂×C₈	D₄₅₂	1	2	yes	—
72	C₄⋊C₄	—	226	2	no	—
73	C₄⋊C₄	—	226	2	no	—
74	C₄⋊C₄	—	226	2	no	—
75	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
76	C₂³	—	452	1	no	—
77	Q₈	—	452	2	no	—
78	Q₈	—	452	4	no	—
79	D₄	—	452	2	no	—
80	D₄	—	452	4	no	—
81	C₂×C₄	—	452	2	no	—
82	C₂×C₄	—	452	1	no	—
83	C₂×C₄	C₂×D₄₅₂	1	1	yes	characteristic
84	C₂×C₄	C₂×D₄₅₂	1	2	yes	—
85	C₈	—	2	4	no	—
86	C₈	D₉₀₄	1	4	yes	—
87	C₂²	—	904	1	no	—
88	C₂²	—	904	2	no	—
89	C₂²	#16813619	1	1	yes	characteristic
90	C₄	—	904	1	no	—
91	C₄	—	904	2	no	—
92	C₄	C₂×D₉₀₄	1	2	yes	—
93	C₄	#16813655	1	4	yes	—
94	C₂	—	1808	1	no	—
95	C₂	#20298975	1	1	yes	characteristic
96	C₂	#20298992	1	2	yes	—

	subgroup	G/H
center	C₂²	#16813619
commutator, cosocle	C₄²×C₁₁₃	C₂³
Frattini	C₄²	C₂²×D₁₁₃
Fitting	#322×C₁₁₃	C₂
socle	C₂²×C₁₁₃	#127
prime radical	#16813619	C₂²
hypercenter	#322	D₁₁₃
hypocenter	C₁₁₃	#1722
LCS	G ▷ C₄²×C₁₁₃ ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
1. #20299003	C₂	semidirect
2. #20298978	C₂	semidirect
5. #322×C₁₁₃	C₂	semidirect
49. C₁₁₃	#1722	semidirect
54. #322	D₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
#20299003	2
#20298978	2
#20298971	2
#20298945	2
#322×C₁₁₃	2
#1722	113