Subgroups — #28388557

Order 14464.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20298728	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#14268622	C₂	1	1	yes	maximal, characteristic
3	#346×C₁₁₃	C₂	1	1	yes	maximal, characteristic
4	C₄×Dic₂₂₆	C₄	1	1	yes	characteristic
5	C₂×#14268622	—	2	1	no	—
6	C₄⋊#11744751	C₄	1	1	yes	characteristic
7	#16813551	—	2	1	no	—
8	#16813532	—	2	1	no	—
9	C₂×C₁₆×C₁₁₃	—	2	1	no	—
10	C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₂₆	C₈	1	1	yes	characteristic
12	C₂×#11744751	—	2	1	no	—
13	C₄⋊Dic₁₁₃	C₈	1	1	yes	characteristic
14	#14268626	—	2	1	no	—
15	C₄×Dic₁₁₃	—	2	1	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	—	2	2	no	—
21	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₁₃	C₂×C₈	1	1	yes	characteristic
23	C₂×C₄×C₁₁₃	M₁₆	1	1	yes	characteristic
24	C₂×Dic₁₁₃	—	2	1	no	—
25	C₂×Dic₁₁₃	—	2	1	no	—
26	Dic₂₂₆	—	4	1	no	—
27	C₈×C₁₁₃	—	2	1	no	—
28	C₈×C₁₁₃	SD₁₆	1	1	yes	characteristic
29	C₈×C₁₁₃	Q₁₆	1	1	yes	characteristic
30	#11744751	—	4	1	no	—
31	C₂²×C₁₁₃	C₂²⋊C₈	1	1	yes	characteristic
32	C₄×C₁₁₃	—	2	1	no	—
33	C₄×C₁₁₃	—	2	1	no	—
34	C₄×C₁₁₃	#103	1	1	yes	characteristic
35	C₄×C₁₁₃	#104	1	1	yes	characteristic
36	Dic₁₁₃	—	4	1	no	—
37	Dic₁₁₃	—	4	1	no	—
38	C₂×C₁₁₃	Q₈⋊C₈	1	1	yes	characteristic
39	C₂×C₁₁₃	#349	1	1	yes	characteristic
40	C₂×C₁₁₃	#350	1	1	yes	characteristic
41	#1338	—	113	1	no	maximal, Sylow
42	C₁₁₃	#1338	1	1	yes	characteristic, complemented, Sylow
43	#446	—	113	1	no	—
44	C₄⋊C₁₆	—	113	1	no	—
45	#346	D₁₁₃	1	1	yes	characteristic
46	C₄×Q₈	—	113	1	no	—
47	C₂×C₁₆	—	2	1	no	—
48	C₂×C₁₆	—	226	1	no	—
49	C₄⋊C₈	—	113	1	no	—
50	C₄⋊C₈	—	226	1	no	—
51	#97	—	226	1	no	—
52	C₄×C₈	C₂×D₁₁₃	1	1	yes	characteristic
53	C₂×Q₈	—	113	1	no	—
54	C₂×C₈	—	226	1	no	—
55	C₂×C₈	D₄₅₂	1	1	yes	characteristic
56	C₂×C₈	C₂²⋊D₁₁₃	1	1	yes	characteristic
57	C₄⋊C₄	—	113	1	no	—
58	C₄⋊C₄	—	226	1	no	—
59	C₄²	—	226	1	no	—
60	C₄²	C₄×D₁₁₃	1	1	yes	characteristic
61	C₁₆	—	2	2	no	—
62	C₁₆	—	452	1	no	—
63	Q₈	—	452	1	no	—
64	C₂×C₄	—	226	1	no	—
65	C₂×C₄	—	226	1	no	—
66	C₂×C₄	#14268624	1	1	yes	characteristic
67	C₂×C₄	C₈×D₁₁₃	1	1	yes	characteristic
68	C₂×C₄	#14268630	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	—	452	1	no	—
71	C₈	#14268636	1	1	yes	characteristic
72	C₈	Dic₄₅₂	1	1	yes	characteristic
73	C₂²	#16813534	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	—	452	1	no	—
77	C₄	—	452	1	no	—
78	C₄	#16813547	1	1	yes	characteristic
79	C₄	#16813550	1	1	yes	characteristic
80	C₂	#20298330	1	1	yes	characteristic
81	C₂	#20298379	1	1	yes	characteristic
82	C₂	#20298383	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14268624
commutator	C₂×C₄×C₁₁₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₁₃
Fitting	#346×C₁₁₃	C₂
socle, prime radical	C₂²×C₁₁₃	C₂²⋊C₈
cosocle	C₄×C₈×C₁₁₃	C₂²
hypercenter	#346	D₁₁₃
hypocenter	C₁₁₃	#1338
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
42. C₁₁₃	#1338	semidirect

Maximal subgroups

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

subgroup	index
#20298728	2
C₄⋊#14268622	2
#346×C₁₁₃	2
#1338	113