Group #28406500 — subgroups

Subgroups — #28406500

Order 14464.

180 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299603	C₂	1	1	yes	maximal, characteristic
2	#20299602	C₂	1	1	yes	maximal, characteristic
3	#20299474	C₂	1	4	yes	maximal
4	#97×D₁₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14268657	C₂²	1	2	yes	—
6	C₂×#14268657	C₂²	1	2	yes	—
7	#16813689	—	2	2	no	—
8	C₄⋊#11744753	—	2	2	no	—
9	C₂×C₈×D₁₁₃	C₂²	1	2	yes	—
10	C₄²×D₁₁₃	C₂²	1	1	yes	characteristic
11	#16813533	C₄	1	1	yes	characteristic
12	#16813532	C₄	1	2	yes	—
13	#97×C₁₁₃	C₄	1	1	yes	characteristic
14	#14268657	—	2	4	no	—
15	#14268657	—	2	4	no	—
16	C₂×#11744753	—	2	2	no	—
17	C₂×#11744753	—	2	2	no	—
18	C₂×C₄×D₁₁₃	Q₈	1	2	yes	—
19	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
20	C₂×#11744751	C₂×C₄	1	2	yes	—
21	C₈×D₁₁₃	—	2	2	no	—
22	C₈×D₁₁₃	—	2	2	no	—
23	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₁₁₃	D₄	1	2	yes	—
25	C₂×C₈×C₁₁₃	C₂×C₄	1	2	yes	—
26	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×D₁₁₃	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₄×C₁₁₃	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×Dic₁₁₃	C₄⋊C₄	1	2	yes	—
31	C₂×Dic₁₁₃	C₂×D₄	1	1	yes	characteristic
32	C₄×D₁₁₃	—	4	2	no	—
33	C₄×D₁₁₃	#41	1	2	yes	—
34	C₄×D₁₁₃	#41	1	2	yes	—
35	#11744753	—	4	2	no	—
36	#11744753	—	4	2	no	—
37	C₈×C₁₁₃	—	2	2	no	—
38	#11744751	—	2	2	no	—
39	C₂²×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×D₁₁₃	—	2	2	no	—
41	C₂×D₁₁₃	#125	1	2	yes	—
42	C₄×C₁₁₃	—	2	2	no	—
43	C₄×C₁₁₃	#117	1	2	yes	—
44	Dic₁₁₃	—	2	2	no	—
45	Dic₁₁₃	#124	1	1	yes	characteristic
46	Dic₁₁₃	#124	1	1	yes	characteristic
47	C₂×C₁₁₃	#382	1	1	yes	characteristic
48	C₂×C₁₁₃	#430	1	2	yes	—
49	D₁₁₃	—	2	2	no	—
50	#1838	—	113	1	no	maximal, Sylow
51	C₁₁₃	#1838	1	1	yes	characteristic, complemented, Sylow
52	#423	—	113	1	no	—
53	#423	—	113	1	no	—
54	C₂×#97	—	113	1	no	—
55	#341	—	113	4	no	—
56	C₂×M₁₆	—	113	2	no	—
57	C₂×M₁₆	—	113	2	no	—
58	C₂²×C₈	—	113	2	no	—
59	C₂×C₄²	—	113	1	no	—
60	C₄⋊C₈	—	226	2	no	—
61	C₄⋊C₈	—	226	2	no	—
62	#97	—	113	2	no	—
63	#97	—	113	1	no	—
64	#97	#90673	1	1	yes	characteristic
65	C₂²×C₄	—	113	1	no	—
66	C₂²×C₄	—	113	2	no	—
67	M₁₆	—	226	4	no	—
68	M₁₆	—	226	4	no	—
69	C₂×C₈	—	113	2	no	—
70	C₂×C₈	—	226	2	no	—
71	C₂×C₈	—	226	2	no	—
72	C₂×C₈	—	226	2	no	—
73	C₂×C₈	—	226	2	no	—
74	C₂×C₈	C₂×#90673	1	2	yes	—
75	C₄²	—	113	2	no	—
76	C₄²	—	113	1	no	—
77	C₄²	C₂×#90673	1	1	yes	characteristic
78	C₂³	—	113	1	no	—
79	C₂×C₄	—	113	1	no	—
80	C₂×C₄	—	113	2	no	—
81	C₂×C₄	—	113	2	no	—
82	C₂×C₄	—	113	2	no	—
83	C₂×C₄	—	452	2	no	—
84	C₂×C₄	C₄⋊#90673	1	2	yes	—
85	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
86	C₈	—	2	2	no	—
87	C₈	—	452	2	no	—
88	C₈	—	452	2	no	—
89	C₈	—	226	2	no	—
90	C₂²	—	113	2	no	—
91	C₂²	—	226	2	no	—
92	C₂²	C₂×C₄⋊#90673	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	113	1	no	—
95	C₄	—	113	1	no	—
96	C₄	—	226	2	no	—
97	C₄	#16813701	1	2	yes	—
98	C₂	—	226	2	no	—
99	C₂	#20299542	1	1	yes	characteristic
100	C₂	#20299613	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20299603

C₂

yes

maximal, characteristic

#20299602

C₂

yes

maximal, characteristic

#20299474

C₂

yes

maximal

#97×D₁₁₃

C₂

yes

maximal, characteristic

C₂×#14268657

C₂²

yes

—

C₂×#14268657

C₂²

yes

—

#16813689

—

C₄⋊#11744753

—

C₂×C₈×D₁₁₃

C₂²

yes

—

C₄²×D₁₁₃

C₂²

yes

characteristic

#16813533

C₄

yes

characteristic

#16813532

C₄

yes

—

#97×C₁₁₃

C₄

yes

—

—

—

—

C₂×C₄×D₁₁₃

Q₈

yes

—

C₂×C₄×D₁₁₃

C₂³

yes

characteristic

C₂×#11744751

C₂×C₄

yes

—

C₈×D₁₁₃

—

C₈×D₁₁₃

—

C₄×Dic₁₁₃

C₂×C₄

yes

characteristic

C₄×Dic₁₁₃

D₄

yes

—

C₂×C₈×C₁₁₃

C₂×C₄

yes

—

C₄²×C₁₁₃

C₂×C₄

yes

characteristic

C₂²×D₁₁₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₁₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×Dic₁₁₃

C₄⋊C₄

yes

—

C₂×Dic₁₁₃

C₂×D₄

yes

characteristic

C₄×D₁₁₃

—

C₄×D₁₁₃

#41

yes

—

C₄×D₁₁₃

#41

yes

—

#11744753

—

#11744753

—

C₈×C₁₁₃

—

#11744751

—

C₂²×C₁₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₁₃

—

C₂×D₁₁₃

#125

yes

—

C₄×C₁₁₃

—

C₄×C₁₁₃

#117

yes

—

Dic₁₁₃

—

Dic₁₁₃

yes

Dic₁₁₃

yes

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

—

D₁₁₃

—

#1838

—

113

maximal, Sylow

C₁₁₃

#1838

yes

characteristic, complemented, Sylow

#423

—

113

—

#423

—

113

—

C₂×#97

—

113

—

#341

—

113

—

C₂×M₁₆

—

113

—

C₂×M₁₆

—

113

—

C₂²×C₈

—

113

—

C₂×C₄²

—

113

—

C₄⋊C₈

—

226

—

C₄⋊C₈

—

226

—

#97

—

113

—

#97

—

113

—

#97

#90673

yes

characteristic

C₂²×C₄

—

113

—

C₂²×C₄

—

113

—

M₁₆

—

226

—

M₁₆

—

226

—

C₂×C₈

—

113

—

C₂×C₈

—

226

—

C₂×C₈

—

226

—

C₂×C₈

—

226

—

C₂×C₈

—

226

—

C₂×C₈

C₂×#90673

yes

—

C₄²

—

113

—

C₄²

—

113

—

C₄²

C₂×#90673

yes

characteristic

C₂³

—

113

—

C₂×C₄

—

113

—

C₂×C₄

—

113

—

C₂×C₄

—

113

—

C₂×C₄

—

113

—

C₂×C₄

—

452

—

C₂×C₄

C₄⋊#90673

yes

—

C₂×C₄

C₂²×#90673

yes

characteristic

C₈

—

C₈

—

452

—

C₈

—

452

—

C₈

—

226

—

C₂²

—

113

—

C₂²

—

226

—

C₂²

C₂×C₄⋊#90673

yes

characteristic

C₄

—

C₄

—

113

—

C₄

—

113

—

C₄

—

226

—

C₄

#16813701

yes

—

C₂

—

226

—

C₂

#20299542

yes

characteristic

100

C₂

#20299613

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#90673
commutator	C₂×C₄×C₁₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#90673
Fitting	#97×C₁₁₃	C₄
socle	C₂²×C₁₁₃	C₂×C₄⋊C₄
prime radical	C₂²×D₁₁₃	C₂×Q₈
cosocle	C₂×C₄×D₁₁₃	C₂³
hypercenter	#97	#90673
hypocenter	C₁₁₃	#1838
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊#90673

commutator

C₂×C₄×C₁₁₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#90673

Fitting

#97×C₁₁₃

C₄

socle

C₂²×C₁₁₃

C₂×C₄⋊C₄

prime radical

C₂²×D₁₁₃

C₂×Q₈

cosocle

C₂×C₄×D₁₁₃

C₂³

C₁₁₃

G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
51. C₁₁₃	#1838	semidirect

H ≅ G/N

type

51. C₁₁₃

#1838

semidirect

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

subgroup	index
#20299603	2
#20299602	2
#20299474	2
#97×D₁₁₃	2
#1838	113

subgroup

index

#97×D₁₁₃

113

Subgroups — #28406500

Conjugacy classes of subgroups

Complements

Maximal subgroups