Subgroups — #28390205

Order 14464.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20298996	C₂	1	1	yes	maximal, characteristic
2	#20298994	C₂	1	2	yes	maximal
3	#20298950	C₂	1	2	yes	maximal
4	#20298319	C₂	1	1	yes	maximal, characteristic
5	#494×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	#16813622	C₂²	1	2	yes	—
7	#16813554	—	2	2	no	—
8	#16813553	—	2	4	no	—
9	#16813545	—	2	4	no	—
10	C₄×#11744751	C₂²	1	2	yes	—
11	C₂×Q₁₆×C₁₁₃	—	2	2	no	—
12	C₄⋊Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₂₆	—	2	2	no	—
16	C₂×#11744751	D₄	1	4	yes	—
17	C₄⋊Dic₁₁₃	—	2	2	no	—
18	C₄⋊Dic₁₁₃	—	2	2	no	—
19	C₂×Q₈×C₁₁₃	—	2	1	no	—
20	C₂×Q₈×C₁₁₃	—	2	1	no	—
21	Q₁₆×C₁₁₃	—	4	4	no	—
22	C₂×C₈×C₁₁₃	D₄	1	2	yes	—
23	C₄⋊C₄×C₁₁₃	—	2	2	no	—
24	C₄⋊C₄×C₁₁₃	—	2	2	no	—
25	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
26	Q₈×C₁₁₃	—	4	2	no	—
27	Q₈×C₁₁₃	—	4	2	no	—
28	C₂×C₄×C₁₁₃	—	4	1	no	—
29	C₂×C₄×C₁₁₃	—	4	1	no	—
30	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
31	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
32	C₂×Dic₁₁₃	—	4	2	no	—
33	Dic₂₂₆	—	4	4	no	—
34	C₈×C₁₁₃	SD₁₆	1	4	yes	—
35	#11744751	—	2	4	no	—
36	C₂²×C₁₁₃	#127	1	1	yes	characteristic
37	C₄×C₁₁₃	—	8	1	no	—
38	C₄×C₁₁₃	—	8	1	no	—
39	C₄×C₁₁₃	C₂×SD₁₆	1	2	yes	—
40	C₄×C₁₁₃	#137	1	4	yes	—
41	Dic₁₁₃	—	8	2	no	—
42	C₂×C₁₁₃	#492	1	1	yes	characteristic
43	C₂×C₁₁₃	#497	1	2	yes	—
44	#1716	—	113	1	no	maximal, Sylow
45	C₁₁₃	#1716	1	1	yes	characteristic, complemented, Sylow
46	#498	—	113	2	no	—
47	#498	—	113	1	no	—
48	#494	D₁₁₃	1	1	yes	characteristic
49	#487	—	113	2	no	—
50	#322	—	113	1	no	—
51	C₂×Q₁₆	—	2	2	no	—
52	C₄⋊Q₈	—	113	2	no	—
53	C₄⋊Q₈	C₂×D₁₁₃	1	1	yes	characteristic
54	C₄⋊Q₈	C₂×D₁₁₃	1	1	yes	characteristic
55	#106	—	226	4	no	—
56	#106	—	226	2	no	—
57	#103	—	226	4	no	—
58	C₄×C₈	—	113	2	no	—
59	C₄×C₈	C₂×D₁₁₃	1	1	yes	characteristic
60	C₂×Q₈	—	2	1	no	—
61	C₂×Q₈	—	2	1	no	—
62	C₂×Q₈	—	226	2	no	—
63	Q₁₆	—	4	4	no	—
64	C₂×C₈	—	113	4	no	—
65	C₂×C₈	C₂²⋊D₁₁₃	1	2	yes	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	226	2	no	—
69	C₄⋊C₄	—	226	2	no	—
70	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
71	Q₈	—	4	2	no	—
72	Q₈	—	4	2	no	—
73	Q₈	—	452	4	no	—
74	C₂×C₄	—	4	1	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	452	2	no	—
77	C₂×C₄	D₄×D₁₁₃	1	2	yes	—
78	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
79	C₈	—	226	4	no	—
80	C₈	#14268635	1	4	yes	—
81	C₂²	#16813618	1	1	yes	characteristic
82	C₄	—	8	1	no	—
83	C₄	—	8	1	no	—
84	C₄	—	904	2	no	—
85	C₄	C₂×#14268635	1	2	yes	—
86	C₄	#16813654	1	4	yes	—
87	C₂	#20298970	1	1	yes	characteristic
88	C₂	#20298991	1	2	yes	—

	subgroup	G/H
center	C₂²	#16813618
commutator, cosocle	C₄²×C₁₁₃	C₂³
Frattini	C₄²	C₂²×D₁₁₃
Fitting	#494×C₁₁₃	C₂
socle, prime radical	C₂²×C₁₁₃	#127
hypercenter	#494	D₁₁₃
hypocenter	C₁₁₃	#1716
LCS	G ▷ C₄²×C₁₁₃ ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
45. C₁₁₃	#1716	semidirect

Maximal subgroups

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

subgroup	index
#20298996	2
#20298994	2
#20298950	2
#20298319	2
#494×C₁₁₃	2
#1716	113