Subgroups — #28388546

Order 14464.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20298716	C₂	1	1	yes	maximal, characteristic
2	C₈×#11744751	C₂	1	1	yes	maximal, characteristic, complemented
3	#364×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#16813644	C₄	1	1	yes	characteristic
5	#16813628	—	2	1	no	—
6	#16813558	C₄	1	1	yes	characteristic
7	#16813550	—	2	1	no	—
8	C₄×#11744751	—	2	1	no	—
9	M₃₂×C₁₁₃	—	2	1	no	—
10	C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
11	#14268647	—	2	1	no	—
12	Dic₄₅₂	C₈	1	1	yes	characteristic, complemented
13	#14268636	—	2	1	no	—
14	D₉₀₄	C₈	1	1	yes	characteristic, complemented
15	#14268631	—	2	1	no	—
16	#14268630	—	4	1	no	—
17	C₂×#11744751	—	2	2	no	—
18	C₈×D₁₁₃	—	4	1	no	—
19	C₂×C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₁₁₃	D₄	1	1	yes	characteristic
21	C₄²×C₁₁₃	D₄	1	1	yes	characteristic
22	C₁₆×C₁₁₃	—	4	1	no	—
23	C₂×C₄×C₁₁₃	—	2	1	no	—
24	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²⋊D₁₁₃	—	4	1	no	—
26	D₄₅₂	—	2	1	no	—
27	C₄×D₁₁₃	—	4	1	no	—
28	Dic₂₂₆	—	2	1	no	—
29	C₈×C₁₁₃	C₂×C₈	1	1	yes	characteristic
30	C₈×C₁₁₃	M₁₆	1	1	yes	characteristic
31	C₈×C₁₁₃	D₈	1	1	yes	characteristic
32	C₈×C₁₁₃	SD₁₆	1	1	yes	characteristic
33	#11744751	—	2	4	no	—
34	#11744751	—	4	1	no	—
35	C₂²×C₁₁₃	#104	1	1	yes	characteristic
36	C₂×D₁₁₃	—	4	1	no	—
37	C₄×C₁₁₃	—	2	2	no	—
38	C₄×C₁₁₃	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₁₁₃	#102	1	1	yes	characteristic
40	Dic₁₁₃	—	4	1	no	—
41	C₂×C₁₁₃	—	2	1	no	—
42	C₂×C₁₁₃	D₄⋊C₈	1	1	yes	characteristic
43	D₁₁₃	—	8	1	no	—
44	#1335	—	113	1	no	maximal, Sylow
45	C₁₁₃	#1335	1	1	yes	characteristic, complemented, Sylow
46	#443	—	113	1	no	—
47	#364	D₁₁₃	1	1	yes	characteristic, complemented
48	C₈²	—	113	1	no	—
49	#135	—	113	1	no	—
50	#131	—	226	1	no	—
51	M₃₂	—	2	1	no	—
52	#108	—	113	1	no	—
53	#104	—	226	1	no	—
54	C₄×C₈	—	226	1	no	—
55	C₄×C₈	C₂×D₁₁₃	1	1	yes	characteristic
56	#41	—	226	1	no	—
57	Q₁₆	—	113	1	no	—
58	SD₁₆	—	226	1	no	—
59	D₈	—	113	1	no	—
60	M₁₆	—	452	1	no	—
61	M₁₆	—	226	1	no	—
62	C₂×C₈	—	452	1	no	—
63	C₂×C₈	—	226	2	no	—
64	C₂×C₈	C₄×D₁₁₃	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₁₁₃	1	1	yes	characteristic
66	C₄²	D₄₅₂	1	1	yes	characteristic
67	C₁₆	—	4	1	no	—
68	Q₈	—	226	1	no	—
69	D₄	—	452	1	no	—
70	D₄	—	226	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	452	1	no	—
73	C₂×C₄	#14268624	1	1	yes	characteristic
74	C₈	—	452	1	no	—
75	C₈	—	226	4	no	—
76	C₈	C₈×D₁₁₃	1	1	yes	characteristic
77	C₈	#14268630	1	1	yes	characteristic
78	C₈	#14268632	1	1	yes	characteristic
79	C₈	#14268634	1	1	yes	characteristic
80	C₂²	—	452	1	no	—
81	C₂²	#16813548	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	452	1	no	—
84	C₄	#16813534	1	1	yes	characteristic
85	C₄	#16813542	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	904	1	no	—
88	C₂	#20298325	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14268632
commutator	C₈×C₁₁₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₁₃
Fitting	#364×C₁₁₃	C₂
socle	C₂×C₁₁₃	D₄⋊C₈
prime radical	#16813644	C₄
cosocle	C₄×C₈×C₁₁₃	C₂²
hypercenter	#364	D₁₁₃
hypocenter	C₁₁₃	#1335
LCS	G ▷ C₈×C₁₁₃ ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. C₈×#11744751	C₂	semidirect
3. #364×C₁₁₃	C₂	semidirect
12. Dic₄₅₂	C₈	semidirect
14. D₉₀₄	C₈	semidirect
45. C₁₁₃	#1335	semidirect
47. #364	D₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
#20298716	2
C₈×#11744751	2
#364×C₁₁₃	2
#1335	113