Group #28390475 — subgroups

Subgroups — #28390475

Order 14464.

228 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂×#16813557	C₂	1	2	yes	maximal
2	M₁₆×Dic₁₁₃	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic
4	#20298363	C₂	1	1	yes	maximal, characteristic
5	#341×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14268631	C₂²	1	2	yes	—
7	C₂²×#11744751	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
9	#16813557	C₄	1	8	yes	complemented
10	#16813532	—	2	2	no	—
11	C₈×Dic₁₁₃	—	2	2	no	—
12	C₈×Dic₁₁₃	C₄	1	4	yes	—
13	C₂×M₁₆×C₁₁₃	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₁₃	Q₈	1	2	yes	—
16	#14268631	—	2	2	no	—
17	#14268631	C₂×C₄	1	4	yes	—
18	C₂×#11744751	—	2	2	no	—
19	C₂×#11744751	C₂×C₄	1	2	yes	—
20	C₂×#11744751	C₂×C₄	1	4	yes	—
21	C₄×Dic₁₁₃	C₂×C₄	1	2	yes	—
22	C₄×Dic₁₁₃	D₄	1	1	yes	characteristic
23	C₄×Dic₁₁₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
25	M₁₆×C₁₁₃	—	2	2	no	—
26	M₁₆×C₁₁₃	C₂×C₄	1	4	yes	—
27	C₂×C₈×C₁₁₃	—	2	2	no	—
28	C₂×C₈×C₁₁₃	—	2	2	no	—
29	C₂×C₈×C₁₁₃	C₂×C₄	1	2	yes	—
30	C₂³×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
31	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
33	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₁₃	#41	1	2	yes	—
35	C₂×Dic₁₁₃	—	2	4	no	—
36	C₂×Dic₁₁₃	C₄⋊C₄	1	2	yes	—
37	C₂×Dic₁₁₃	C₄⋊C₄	1	2	yes	—
38	C₈×C₁₁₃	—	2	2	no	—
39	C₈×C₁₁₃	—	2	4	no	—
40	#11744751	—	2	4	no	—
41	#11744751	C₄²	1	4	yes	—
42	C₂²×C₁₁₃	—	2	2	no	—
43	C₂²×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₁₁₃	C₄×Q₈	1	2	yes	—
45	C₄×C₁₁₃	C₂×C₄²	1	1	yes	characteristic
46	C₄×C₁₁₃	#117	1	1	yes	characteristic
47	C₄×C₁₁₃	C₄×D₄	1	2	yes	—
48	Dic₁₁₃	—	2	2	no	—
49	Dic₁₁₃	#108	1	4	yes	complemented
50	C₂×C₁₁₃	—	2	2	no	—
51	C₂×C₁₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₁₁₃	C₂×#108	1	2	yes	—
53	C₄×#108	—	113	1	no	maximal, Sylow
54	C₁₁₃	C₄×#108	1	1	yes	characteristic, complemented, Sylow
55	C₂×#108	—	113	2	no	—
56	C₄×M₁₆	—	113	2	no	—
57	C₂×C₄×C₈	—	113	1	no	—
58	#341	—	113	1	no	—
59	#341	D₁₁₃	1	1	yes	characteristic
60	C₂×M₁₆	—	113	2	no	—
61	C₂×M₁₆	C₂×D₁₁₃	1	2	yes	—
62	C₂²×C₈	—	113	1	no	—
63	C₂²×C₈	C₂×D₁₁₃	1	1	yes	characteristic
64	C₂×C₄²	—	113	1	no	—
65	#108	—	113	8	no	—
66	#97	—	226	2	no	—
67	C₄×C₈	—	113	4	no	—
68	C₄×C₈	—	226	2	no	—
69	C₂²×C₄	—	113	2	no	—
70	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	113	4	no	—
73	M₁₆	—	226	2	no	—
74	M₁₆	C₄×D₁₁₃	1	4	yes	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	113	2	no	—
78	C₂×C₈	—	113	4	no	—
79	C₂×C₈	—	226	2	no	—
80	C₂×C₈	C₄×D₁₁₃	1	2	yes	—
81	C₄²	—	113	2	no	—
82	C₄²	—	113	1	no	—
83	C₄²	—	113	1	no	—
84	C₂³	Q₈×D₁₁₃	1	1	yes	characteristic
85	C₂×C₄	—	113	2	no	—
86	C₂×C₄	—	113	2	no	—
87	C₂×C₄	—	226	4	no	—
88	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
89	C₂×C₄	C₂×C₄×D₁₁₃	1	2	yes	—
90	C₂×C₄	D₄×D₁₁₃	1	1	yes	characteristic
91	C₂×C₄	#14268646	1	2	yes	—
92	C₈	—	2	2	no	—
93	C₈	—	2	4	no	—
94	C₈	—	113	4	no	—
95	C₈	—	226	4	no	—
96	C₂²	—	2	2	no	—
97	C₂²	C₄⋊C₄×D₁₁₃	1	1	yes	characteristic
98	C₂²	#16813587	1	2	yes	—
99	C₄	—	113	4	no	—
100	C₄	—	226	2	no	—
101	C₄	C₄²×D₁₁₃	1	1	yes	characteristic
102	C₄	#16813578	1	1	yes	characteristic
103	C₄	#16813582	1	2	yes	—
104	C₂	—	2	2	no	—
105	C₂	#20298449	1	1	yes	characteristic
106	C₂	#108×D₁₁₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16813557

C₂

yes

maximal

M₁₆×Dic₁₁₃

C₂

yes

maximal

C₂×C₈×Dic₁₁₃

C₂

yes

maximal, characteristic

#20298363

C₂

yes

maximal, characteristic

#341×C₁₁₃

C₂

yes

maximal, characteristic

C₂×#14268631

C₂²

yes

—

C₂²×#11744751

C₂²

yes

characteristic

C₂×C₄×Dic₁₁₃

C₂²

yes

characteristic

#16813557

C₄

yes

complemented

#16813532

—

C₈×Dic₁₁₃

—

C₈×Dic₁₁₃

C₄

yes

—

C₂×M₁₆×C₁₁₃

C₂²

yes

—

C₂²×C₈×C₁₁₃

C₂²

yes

characteristic

C₂²×Dic₁₁₃

Q₈

yes

—

#14268631

—

#14268631

C₂×C₄

yes

—

C₂×#11744751

—

C₂×#11744751

C₂×C₄

yes

—

C₂×#11744751

C₂×C₄

yes

—

C₄×Dic₁₁₃

C₂×C₄

yes

—

C₄×Dic₁₁₃

D₄

yes

characteristic

C₄×Dic₁₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₁₃

C₂³

yes

characteristic

M₁₆×C₁₁₃

—

M₁₆×C₁₁₃

C₂×C₄

yes

—

C₂×C₈×C₁₁₃

—

C₂×C₈×C₁₁₃

—

C₂×C₈×C₁₁₃

C₂×C₄

yes

—

C₂³×C₁₁₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁₃

#41

yes

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

C₄⋊C₄

yes

—

C₂×Dic₁₁₃

C₄⋊C₄

yes

—

C₈×C₁₁₃

—

C₈×C₁₁₃

—

#11744751

—

#11744751

C₄²

yes

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₁₃

C₄×Q₈

yes

—

C₄×C₁₁₃

C₂×C₄²

yes

characteristic

C₄×C₁₁₃

#117

yes

characteristic

C₄×C₁₁₃

C₄×D₄

yes

—

Dic₁₁₃

—

Dic₁₁₃

#108

yes

complemented

C₂×C₁₁₃

—

C₂×C₁₁₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₁₃

C₂×#108

yes

—

C₄×#108

—

113

maximal, Sylow

C₁₁₃

C₄×#108

yes

characteristic, complemented, Sylow

C₂×#108

—

113

—

C₄×M₁₆

—

113

—

C₂×C₄×C₈

—

113

—

#341

—

113

—

#341

D₁₁₃

yes

characteristic

C₂×M₁₆

—

113

—

C₂×M₁₆

C₂×D₁₁₃

yes

—

C₂²×C₈

—

113

—

C₂²×C₈

C₂×D₁₁₃

yes

characteristic

C₂×C₄²

—

113

—

#108

—

113

—

#97

—

226

—

C₄×C₈

—

113

—

C₄×C₈

—

226

—

C₂²×C₄

—

113

—

C₂²×C₄

C₂²×D₁₁₃

yes

characteristic

M₁₆

—

M₁₆

—

113

—

M₁₆

—

226

—

M₁₆

C₄×D₁₁₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

226

—

C₂×C₈

C₄×D₁₁₃

yes

—

C₄²

—

113

—

C₄²

—

113

—

C₄²

—

113

—

C₂³

Q₈×D₁₁₃

yes

characteristic

C₂×C₄

—

113

—

C₂×C₄

—

113

—

C₂×C₄

—

226

—

C₂×C₄

C₂×C₄×D₁₁₃

yes

characteristic

C₂×C₄

C₂×C₄×D₁₁₃

yes

—

C₂×C₄

D₄×D₁₁₃

yes

characteristic

C₂×C₄

#14268646

yes

—

C₈

—

C₈

—

C₈

—

113

—

C₈

—

226

—

C₂²

—

C₂²

C₄⋊C₄×D₁₁₃

yes

characteristic

C₂²

#16813587

yes

—

C₄

—

113

—

100

C₄

—

226

—

101

C₄

C₄²×D₁₁₃

yes

characteristic

102

C₄

#16813578

yes

characteristic

103

C₄

#16813582

yes

—

104

C₂

—

105

C₂

#20298449

yes

characteristic

106

C₂

#108×D₁₁₃

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₁₃
commutator	C₄×C₁₁₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₁₃
Fitting	#341×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×C₄⋊C₄
prime radical	C₂³×C₁₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₁₃	C₂³
hypercenter	#341	D₁₁₃
hypocenter	C₁₁₃	C₄×#108
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₁₁₃

commutator

C₄×C₁₁₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₁₃

Fitting

#341×C₁₁₃

C₂

socle

C₂²×C₁₁₃

C₂×C₄⋊C₄

prime radical

C₂³×C₁₁₃

C₂×Q₈

cosocle

C₂²×C₄×C₁₁₃

C₂³

D₁₁₃

C₁₁₃

G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
9. #16813557	C₄	semidirect
49. Dic₁₁₃	#108	semidirect
54. C₁₁₃	C₄×#108	semidirect

H ≅ G/N

type

9. #16813557

C₄

semidirect

49. Dic₁₁₃

#108

semidirect

54. C₁₁₃

C₄×#108

semidirect

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

subgroup	index
C₂×#16813557	2
M₁₆×Dic₁₁₃	2
C₂×C₈×Dic₁₁₃	2
#20298363	2
#341×C₁₁₃	2
C₄×#108	113

subgroup

index

C₂×#16813557

M₁₆×Dic₁₁₃

C₂×C₈×Dic₁₁₃

#20298363

#341×C₁₁₃

C₄×#108

113

Subgroups — #28390475

Conjugacy classes of subgroups

Complements

Maximal subgroups