Subgroups — #28388612

Order 14464.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20298978	C₂	1	1	yes	maximal, characteristic
2	#20298419	C₂	1	1	yes	maximal, characteristic
3	#364×C₁₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₅₂	—	2	1	no	—
5	C₂×Dic₄₅₂	—	2	1	no	—
6	#16813622	C₄	1	2	yes	—
7	#16813562	—	2	1	no	—
8	M₃₂×C₁₁₃	—	2	1	no	—
9	C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₂₆	—	2	2	no	—
11	Dic₄₅₂	—	4	1	no	—
12	Dic₄₅₂	—	4	1	no	—
13	Dic₄₅₂	—	4	1	no	—
14	Dic₄₅₂	—	4	1	no	—
15	C₄⋊Dic₁₁₃	—	4	2	no	—
16	#14268622	—	4	1	no	—
17	C₂×C₈×C₁₁₃	D₄	1	1	yes	characteristic
18	C₂×C₈×C₁₁₃	D₄	1	1	yes	characteristic
19	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₁₁₃	—	4	1	no	—
21	C₂×C₄×C₁₁₃	—	2	1	no	—
22	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×Dic₁₁₃	—	4	2	no	—
24	Dic₂₂₆	—	4	2	no	—
25	Dic₂₂₆	—	4	2	no	—
26	C₈×C₁₁₃	D₈	1	1	yes	characteristic
27	C₈×C₁₁₃	D₈	1	1	yes	characteristic
28	C₈×C₁₁₃	SD₁₆	1	1	yes	characteristic
29	C₈×C₁₁₃	SD₁₆	1	1	yes	characteristic
30	C₂²×C₁₁₃	#100	1	1	yes	characteristic
31	C₄×C₁₁₃	—	2	2	no	—
32	C₄×C₁₁₃	#102	1	1	yes	characteristic
33	C₄×C₁₁₃	#102	1	1	yes	characteristic
34	Dic₁₁₃	—	8	2	no	—
35	C₂×C₁₁₃	—	2	1	no	—
36	C₂×C₁₁₃	#331	1	1	yes	characteristic
37	#1358	—	113	1	no	maximal, Sylow
38	C₁₁₃	#1358	1	1	yes	characteristic, complemented, Sylow
39	#494	—	113	1	no	—
40	#364	—	113	1	no	—
41	#364	D₁₁₃	1	1	yes	characteristic
42	C₂×Q₁₆	—	226	1	no	—
43	C₂×Q₁₆	—	226	1	no	—
44	C₄⋊Q₈	—	113	2	no	—
45	M₃₂	—	2	1	no	—
46	M₃₂	—	226	1	no	—
47	C₄×C₈	C₂×D₁₁₃	1	1	yes	characteristic
48	C₂×Q₈	—	226	2	no	—
49	Q₁₆	—	452	1	no	—
50	Q₁₆	—	452	1	no	—
51	Q₁₆	—	452	1	no	—
52	Q₁₆	—	452	1	no	—
53	C₂×C₈	D₄₅₂	1	1	yes	characteristic
54	C₂×C₈	C₂²⋊D₁₁₃	1	1	yes	characteristic
55	C₄⋊C₄	—	452	2	no	—
56	C₄²	C₄×D₁₁₃	1	1	yes	characteristic
57	C₁₆	—	4	1	no	—
58	C₁₆	—	452	1	no	—
59	Q₈	—	452	2	no	—
60	Q₈	—	452	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	452	2	no	—
63	C₂×C₄	#14268624	1	1	yes	characteristic
64	C₈	#14268632	1	1	yes	characteristic
65	C₈	D₉₀₄	1	1	yes	characteristic
66	C₈	#14268634	1	1	yes	characteristic
67	C₈	#14268636	1	1	yes	characteristic
68	C₂²	#16813538	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	904	2	no	—
71	C₄	#16813542	1	1	yes	characteristic
72	C₄	#16813544	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	#20298341	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20298341
commutator	C₄²×C₁₁₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₁₃
Fitting	#364×C₁₁₃	C₂
socle	C₂×C₁₁₃	#331
prime radical	C₂²×C₁₁₃	#100
cosocle	C₄×C₈×C₁₁₃	C₂²
hypercenter	#364	D₁₁₃
hypocenter	C₁₁₃	#1358
LCS	G ▷ C₄²×C₁₁₃ ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
38. C₁₁₃	#1358	semidirect

Maximal subgroups

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

subgroup	index
#20298978	2
#20298419	2
#364×C₁₁₃	2
#1358	113