Group #28391544 — subgroups

Subgroups — #28391544

Order 14464.

260 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 113 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×#16813533	C₂	1	1	yes	maximal, characteristic
2	#20298468	C₂	1	1	yes	maximal, characteristic
3	#20298360	C₂	1	4	yes	maximal
4	C₂×#125×C₁₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11744751	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₁₃	—	2	2	no	—
7	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
8	#16813533	—	2	2	no	—
9	#125×C₁₁₃	—	2	2	no	—
10	#125×C₁₁₃	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₁₃	—	2	2	no	—
15	C₂×#11744751	—	2	2	no	—
16	C₂×#11744751	—	2	2	no	—
17	C₂×#11744751	Q₈	1	4	yes	—
18	C₄⋊Dic₁₁₃	—	2	4	no	—
19	C₄⋊Dic₁₁₃	—	4	4	no	—
20	C₂²×C₄×C₁₁₃	—	2	2	no	—
21	C₂²×C₄×C₁₁₃	D₄	1	2	yes	—
22	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₁₁₃	—	2	8	no	—
24	C₄⋊C₄×C₁₁₃	—	2	2	no	—
25	C₄⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
26	C₄²×C₁₁₃	C₂×C₄	1	2	yes	—
27	C₄²×C₁₁₃	D₄	1	2	yes	—
28	C₂³×C₁₁₃	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₁₃	—	2	2	no	—
30	C₂×C₄×C₁₁₃	—	2	2	no	—
31	C₂×C₄×C₁₁₃	—	2	4	no	—
32	C₂×C₄×C₁₁₃	—	4	2	no	—
33	C₂×C₄×C₁₁₃	—	4	2	no	—
34	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	4	yes	—
35	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₁₁₃	#41	1	2	yes	—
39	C₂×Dic₁₁₃	—	4	2	no	—
40	C₂×Dic₁₁₃	—	4	4	no	—
41	#11744751	—	2	4	no	—
42	C₂²×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₁₁₃	#124	1	1	yes	characteristic
44	C₂²×C₁₁₃	C₄⋊Q₈	1	1	yes	characteristic
45	C₂²×C₁₁₃	AGL(1,ℤ/8)	1	2	yes	—
46	C₂²×C₁₁₃	#137	1	2	yes	—
47	C₄×C₁₁₃	—	2	4	no	—
48	C₄×C₁₁₃	—	4	4	no	—
49	C₄×C₁₁₃	C₄×Q₈	1	2	yes	—
50	C₄×C₁₁₃	C₂²⋊Q₈	1	2	yes	—
51	Dic₁₁₃	—	8	2	no	—
52	C₂×C₁₁₃	#391	1	1	yes	characteristic
53	C₂×C₁₁₃	#417	1	2	yes	—
54	C₂×C₁₁₃	#490	1	1	yes	characteristic
55	C₂×C₁₁₃	#491	1	1	yes	characteristic
56	C₂×C₁₁₃	#501	1	2	yes	—
57	#1992	—	113	1	no	maximal, Sylow
58	C₁₁₃	#1992	1	1	yes	characteristic, complemented, Sylow
59	C₂×#125	D₁₁₃	1	1	yes	characteristic
60	C₂×#97	—	113	1	no	—
61	#384	—	113	1	no	—
62	#340	—	113	4	no	—
63	C₂²×C₈	—	113	2	no	—
64	#125	—	2	2	no	—
65	#125	Dic₁₁₃	1	4	yes	complemented
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	—	113	2	no	—
68	C₂×C₄⋊C₄	—	226	2	no	—
69	C₂×C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
70	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
71	#97	—	226	2	no	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	226	2	no	—
74	C₂²×C₄	C₂²⋊D₁₁₃	1	2	yes	—
75	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
76	C₂×C₈	—	113	4	no	—
77	C₂×C₈	—	226	2	no	—
78	C₂×C₈	—	226	2	no	—
79	C₄⋊C₄	—	2	8	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	452	4	no	—
82	C₄⋊C₄	—	226	4	no	—
83	C₄⋊C₄	C₂×Dic₁₁₃	1	4	yes	—
84	C₄²	C₂×Dic₁₁₃	1	2	yes	—
85	C₄²	C₂²⋊D₁₁₃	1	2	yes	—
86	C₂³	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	452	2	no	—
93	C₂×C₄	—	452	4	no	—
94	C₂×C₄	C₂²⋊Dic₁₁₃	1	4	yes	—
95	C₂×C₄	C₂²×Dic₁₁₃	1	1	yes	characteristic
96	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
97	C₂×C₄	Q₈×D₁₁₃	1	2	yes	—
98	C₂×C₄	#14268648	1	2	yes	—
99	C₈	—	226	4	no	—
100	C₂²	C₂×C₂²⋊Dic₁₁₃	1	1	yes	characteristic
101	C₂²	#16813611	1	1	yes	characteristic
102	C₂²	#16813620	1	1	yes	characteristic
103	C₂²	#16813647	1	2	yes	—
104	C₂²	#16813652	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	4	4	no	—
107	C₄	—	904	2	no	—
108	C₄	Q₈×Dic₁₁₃	1	2	yes	—
109	C₄	#16813598	1	2	yes	—
110	C₂	#20298505	1	1	yes	characteristic
111	C₂	#20298599	1	2	yes	—
112	C₂	#20298964	1	1	yes	characteristic
113	C₂	#20298968	1	1	yes	characteristic
114	C₂	#20299008	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16813533

C₂

yes

maximal, characteristic

#20298468

C₂

yes

maximal, characteristic

#20298360

C₂

yes

maximal

C₂×#125×C₁₁₃

C₂

yes

maximal, characteristic

C₂²×#11744751

C₂²

yes

—

C₂×C₄⋊Dic₁₁₃

—

C₂×C₄⋊Dic₁₁₃

C₂²

yes

—

#16813533

—

#125×C₁₁₃

—

#125×C₁₁₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₁₃

—

C₂×C₄⋊C₄×C₁₁₃

C₂²

yes

—

C₂×C₄²×C₁₁₃

C₂²

yes

characteristic

C₂²×Dic₁₁₃

—

C₂×#11744751

—

C₂×#11744751

—

C₂×#11744751

Q₈

yes

—

C₄⋊Dic₁₁₃

—

C₄⋊Dic₁₁₃

—

C₂²×C₄×C₁₁₃

—

C₂²×C₄×C₁₁₃

D₄

yes

—

C₂²×C₄×C₁₁₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₁₃

—

C₄⋊C₄×C₁₁₃

—

C₄⋊C₄×C₁₁₃

C₂×C₄

yes

—

C₄²×C₁₁₃

C₂×C₄

yes

—

C₄²×C₁₁₃

D₄

yes

—

C₂³×C₁₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₁₃

#41

yes

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

#11744751

—

C₂²×C₁₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₁₃

#124

yes

characteristic

C₂²×C₁₁₃

C₄⋊Q₈

yes

characteristic

C₂²×C₁₁₃

AGL(1,ℤ/8)

yes

—

C₂²×C₁₁₃

#137

yes

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

C₄×Q₈

yes

—

C₄×C₁₁₃

C₂²⋊Q₈

yes

—

Dic₁₁₃

—

C₂×C₁₁₃

#391

yes

characteristic

C₂×C₁₁₃

#417

yes

—

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

—

#1992

—

113

maximal, Sylow

C₁₁₃

#1992

yes

characteristic, complemented, Sylow

C₂×#125

D₁₁₃

yes

characteristic

C₂×#97

—

113

—

#384

—

113

—

#340

—

113

—

C₂²×C₈

—

113

—

#125

—

#125

Dic₁₁₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

113

—

C₂×C₄⋊C₄

—

226

—

C₂×C₄⋊C₄

C₂×D₁₁₃

yes

—

C₂×C₄²

C₂×D₁₁₃

yes

characteristic

#97

—

226

—

C₂²×C₄

—

C₂²×C₄

—

226

—

C₂²×C₄

C₂²⋊D₁₁₃

yes

—

C₂²×C₄

C₂²×D₁₁₃

yes

characteristic

C₂×C₈

—

113

—

C₂×C₈

—

226

—

C₂×C₈

—

226

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

452

—

C₄⋊C₄

—

226

—

C₄⋊C₄

C₂×Dic₁₁₃

yes

—

C₄²

C₂×Dic₁₁₃

yes

—

C₄²

C₂²⋊D₁₁₃

yes

—

C₂³

C₂×C₂²⋊D₁₁₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

452

—

C₂×C₄

—

452

—

C₂×C₄

C₂²⋊Dic₁₁₃

yes

—

C₂×C₄

C₂²×Dic₁₁₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₁₃

yes

characteristic

C₂×C₄

Q₈×D₁₁₃

yes

—

C₂×C₄

#14268648

yes

—

C₈

—

226

—

100

C₂²

C₂×C₂²⋊Dic₁₁₃

yes

characteristic

101

C₂²

#16813611

yes

characteristic

102

C₂²

#16813620

yes

characteristic

103

C₂²

#16813647

yes

—

104

C₂²

#16813652

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

904

—

108

C₄

Q₈×Dic₁₁₃

yes

—

109

C₄

#16813598

yes

—

110

C₂

#20298505

yes

characteristic

111

C₂

#20298599

yes

—

112

C₂

#20298964

yes

characteristic

113

C₂

#20298968

yes

characteristic

114

C₂

#20299008

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₁₃
commutator	C₂×C₄×C₁₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₁₃
Fitting	C₂×#125×C₁₁₃	C₂
socle, prime radical	C₂³×C₁₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₁₃	C₂³
hypercenter	C₂×#125	D₁₁₃
hypocenter	C₁₁₃	#1992
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₁₃

commutator

C₂×C₄×C₁₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₁₃

Fitting

C₂×#125×C₁₁₃

C₂

socle, prime radical

C₂³×C₁₁₃

C₂×D₄

cosocle

C₂²×C₄×C₁₁₃

C₂³

D₁₁₃

C₁₁₃

G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
10. #125×C₁₁₃	C₄	semidirect
58. C₁₁₃	#1992	semidirect
65. #125	Dic₁₁₃	semidirect

H ≅ G/N

type

10. #125×C₁₁₃

C₄

semidirect

58. C₁₁₃

#1992

semidirect

65. #125

Dic₁₁₃

semidirect

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

subgroup	index
C₂×#16813533	2
#20298468	2
#20298360	2
C₂×#125×C₁₁₃	2
#1992	113

subgroup

index

C₂×#125×C₁₁₃

113

Subgroups — #28391544

Conjugacy classes of subgroups

Complements

Maximal subgroups