Subgroups — #28391740

Order 14464.

290 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20298459	C₂	1	3	yes	maximal
2	#20298365	C₂	1	3	yes	maximal
3	#543×C₁₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₁₁₃	C₂²	1	1	yes	characteristic
5	C₂×C₂²⋊Dic₁₁₃	C₂²	1	3	yes	—
6	#16813529	—	2	6	no	—
7	C₂²⋊Q₈×C₁₁₃	—	2	6	no	—
8	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	3	yes	—
9	C₂²×Dic₁₁₃	—	2	3	no	—
10	C₂²×Dic₁₁₃	—	2	3	no	—
11	C₂²×Dic₁₁₃	—	2	3	no	—
12	C₂²×Dic₁₁₃	Q₈	1	2	yes	complemented
13	C₂²⋊Dic₁₁₃	—	2	6	no	—
14	C₂⁴×C₁₁₃	C₂³	1	1	yes	characteristic
15	C₂×Q₈×C₁₁₃	—	2	2	no	—
16	C₂²×C₄×C₁₁₃	D₄	1	6	yes	—
17	C₄⋊C₄×C₁₁₃	—	2	6	no	—
18	C₂²⋊C₄×C₁₁₃	—	2	6	no	—
19	C₂³×C₁₁₃	—	4	2	no	—
20	C₂³×C₁₁₃	C₂×D₄	1	3	yes	—
21	C₂³×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
22	C₂³×C₁₁₃	#41	1	3	yes	—
23	Q₈×C₁₁₃	—	8	2	no	—
24	C₂×C₄×C₁₁₃	—	2	6	no	—
25	C₂×C₄×C₁₁₃	—	4	6	no	—
26	C₂×Dic₁₁₃	—	2	6	no	—
27	C₂×Dic₁₁₃	—	4	1	no	—
28	C₂×Dic₁₁₃	—	4	3	no	—
29	C₂×Dic₁₁₃	—	4	6	no	—
30	C₂×Dic₁₁₃	—	4	3	no	—
31	C₂²×C₁₁₃	—	2	6	no	—
32	C₂²×C₁₁₃	—	4	1	no	—
33	C₂²×C₁₁₃	—	4	3	no	—
34	C₂²×C₁₁₃	#120	1	1	yes	characteristic
35	C₂²×C₁₁₃	C₂²⋊Q₈	1	3	yes	—
36	C₂²×C₁₁₃	#124	1	3	yes	—
37	C₄×C₁₁₃	—	4	6	no	—
38	Dic₁₁₃	—	4	2	no	—
39	Dic₁₁₃	—	8	3	no	—
40	C₂×C₁₁₃	—	2	3	no	—
41	C₂×C₁₁₃	—	2	3	no	—
42	C₂×C₁₁₃	#393	1	1	yes	characteristic
43	C₂×C₁₁₃	#458	1	2	yes	—
44	#2030	—	113	1	no	maximal, Sylow
45	C₁₁₃	#2030	1	1	yes	characteristic, complemented, Sylow
46	#543	D₁₁₃	1	1	yes	characteristic
47	#381	—	113	3	no	—
48	#342	—	113	3	no	—
49	C₂³×C₄	—	113	1	no	—
50	C₂²⋊Q₈	—	2	6	no	—
51	C₂×C₂²⋊C₄	—	113	3	no	—
52	C₂×C₂²⋊C₄	C₂×D₁₁₃	1	3	yes	—
53	#95	—	226	6	no	—
54	C₂⁴	C₂²×D₁₁₃	1	1	yes	characteristic
55	C₂×Q₈	—	2	2	no	—
56	C₂²×C₄	—	113	2	no	—
57	C₂²×C₄	—	226	3	no	—
58	C₂²×C₄	—	226	3	no	—
59	C₂²×C₄	—	226	3	no	—
60	C₂²×C₄	C₂²⋊D₁₁₃	1	6	yes	—
61	C₄⋊C₄	—	2	6	no	—
62	C₂²⋊C₄	—	2	6	no	—
63	C₂²⋊C₄	—	226	6	no	—
64	C₂³	—	4	2	no	—
65	C₂³	C₂×C₂²⋊D₁₁₃	1	3	yes	—
66	C₂³	Q₈×D₁₁₃	1	1	yes	characteristic
67	C₂³	#14268648	1	3	yes	—
68	Q₈	—	8	2	no	—
69	C₂×C₄	—	2	6	no	—
70	C₂×C₄	—	4	6	no	—
71	C₂×C₄	—	452	6	no	—
72	C₂×C₄	—	452	3	no	—
73	C₂×C₄	—	452	1	no	—
74	C₂×C₄	—	452	3	no	—
75	C₂×C₄	—	226	6	no	—
76	C₂²	—	2	6	no	—
77	C₂²	—	4	1	no	—
78	C₂²	—	4	3	no	—
79	C₂²	#16813591	1	1	yes	characteristic
80	C₂²	#16813598	1	3	yes	—
81	C₂²	#16813611	1	3	yes	—
82	C₄	—	4	6	no	—
83	C₄	—	904	3	no	—
84	C₄	—	452	2	no	—
85	C₂	—	2	3	no	—
86	C₂	—	2	3	no	—
87	C₂	#20298509	1	1	yes	characteristic
88	C₂	#20298788	1	2	yes	—

	subgroup	G/H
center	C₂²	#16813591
commutator, prime radical, cosocle	C₂⁴×C₁₁₃	C₂³
Frattini	C₂⁴	C₂²×D₁₁₃
Fitting	#543×C₁₁₃	C₂
socle	C₂²×C₁₁₃	#120
hypercenter	#543	D₁₁₃
hypocenter	C₁₁₃	#2030
LCS	G ▷ C₂⁴×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #543

Complements

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

N	H ≅ G/N	type
12. C₂²×Dic₁₁₃	Q₈	semidirect
45. C₁₁₃	#2030	semidirect

Maximal subgroups

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

subgroup	index
#20298459	2
#20298365	2
#543×C₁₁₃	2
#2030	113