Group #28406761 — subgroups

Subgroups — #28406761

Order 14464.

244 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 70 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	#20299602	C₂	1	1	yes	maximal, characteristic
2	#20299534	C₂	1	1	yes	maximal, characteristic
3	#20299468	C₂	1	4	yes	maximal
4	#20299154	C₂	1	1	yes	maximal, characteristic
5	C₂×#14268657	C₂²	1	2	yes	—
6	C₂×C₄×#90673	C₂²	1	2	yes	—
7	C₄⋊#11744753	—	2	2	no	—
8	(C₂×#90673)⋊C₄	—	2	2	no	—
9	#16813621	C₄	1	2	yes	complemented
10	#16813614	C₄	1	1	yes	characteristic, complemented
11	#16813600	—	2	2	no	—
12	#16813587	—	2	2	no	—
13	#16813580	C₂²	1	2	yes	—
14	C₄²×D₁₁₃	C₂²	1	1	yes	characteristic
15	#125×C₁₁₃	C₄	1	1	yes	characteristic, complemented
16	C₂²×#90673	—	2	2	no	—
17	#14268657	—	2	4	no	—
18	C₂×#11744753	—	2	2	no	—
19	C₄×#90673	—	2	2	no	—
20	C₄×#90673	Q₈	1	4	yes	complemented
21	C₂×Dic₂₂₆	—	2	2	no	—
22	C₂×C₄×D₁₁₃	D₄	1	2	yes	—
23	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
24	C₄⋊Dic₁₁₃	C₂×C₄	1	2	yes	—
25	#14268626	—	2	2	no	—
26	#14268626	—	2	2	no	—
27	#14268624	—	2	2	no	—
28	C₄×Dic₁₁₃	—	2	2	no	—
29	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₁₁₃	D₄	1	2	yes	—
31	C₄⋊C₄×C₁₁₃	—	2	2	no	—
32	C₄⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
33	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×D₁₁₃	C₂×D₄	1	1	yes	characteristic
35	C₂×#90673	—	2	4	no	—
36	C₂×#90673	—	4	4	no	—
37	C₂×C₄×C₁₁₃	—	2	2	no	—
38	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₁₁₃	—	2	2	no	—
41	C₂×Dic₁₁₃	C₂²⋊C₄	1	2	yes	—
42	C₂×Dic₁₁₃	C₂×D₄	1	1	yes	characteristic
43	C₄×D₁₁₃	—	4	2	no	—
44	C₄×D₁₁₃	C₂×Q₈	1	2	yes	—
45	C₄×D₁₁₃	#41	1	2	yes	—
46	Dic₂₂₆	—	4	4	no	—
47	#11744753	—	4	2	no	—
48	C₂²×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂×D₁₁₃	—	2	2	no	—
50	C₂×D₁₁₃	C₂²⋊Q₈	1	2	yes	—
51	#90673	—	4	4	no	—
52	C₄×C₁₁₃	—	2	2	no	—
53	C₄×C₁₁₃	—	4	2	no	—
54	C₄×C₁₁₃	C₄×Q₈	1	2	yes	—
55	Dic₁₁₃	—	2	2	no	—
56	Dic₁₁₃	—	4	2	no	—
57	Dic₁₁₃	#124	1	1	yes	characteristic
58	Dic₁₁₃	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₁₁₃	#391	1	1	yes	characteristic
60	C₂×C₁₁₃	#421	1	2	yes	—
61	D₁₁₃	—	2	2	no	—
62	#1995	—	113	1	no	maximal, Sylow
63	C₁₁₃	#1995	1	1	yes	characteristic, complemented, Sylow
64	#533	—	113	1	no	—
65	#423	—	113	1	no	—
66	#376	—	113	1	no	—
67	#339	—	113	4	no	—
68	C₂×M₁₆	—	113	2	no	—
69	C₄⋊Q₈	—	113	2	no	—
70	#125	—	113	1	no	—
71	#125	#90673	1	1	yes	characteristic, complemented
72	C₂²⋊Q₈	—	226	2	no	—
73	C₄×Q₈	—	226	2	no	—
74	#117	—	113	2	no	—
75	C₂×C₄²	—	113	1	no	—
76	C₂×C₄²	—	113	2	no	—
77	C₄⋊C₈	—	226	2	no	—
78	#95	—	226	2	no	—
79	C₂×Q₈	—	226	2	no	—
80	C₂²×C₄	—	113	1	no	—
81	C₂²×C₄	—	113	2	no	—
82	C₂²×C₄	—	226	2	no	—
83	M₁₆	—	226	4	no	—
84	C₂×C₈	—	226	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	113	2	no	—
87	C₄⋊C₄	—	226	2	no	—
88	C₄⋊C₄	—	226	2	no	—
89	C₄⋊C₄	C₂×#90673	1	2	yes	—
90	C₂²⋊C₄	—	226	2	no	—
91	C₄²	—	113	2	no	—
92	C₄²	—	113	1	no	—
93	C₄²	—	113	4	no	—
94	C₄²	—	226	2	no	—
95	C₄²	—	226	2	no	—
96	C₄²	C₂×#90673	1	1	yes	characteristic
97	C₂³	—	113	1	no	—
98	Q₈	—	452	4	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	113	1	no	—
101	C₂×C₄	—	113	2	no	—
102	C₂×C₄	—	113	2	no	—
103	C₂×C₄	—	113	2	no	—
104	C₂×C₄	—	452	2	no	—
105	C₂×C₄	—	452	4	no	—
106	C₂×C₄	—	226	4	no	—
107	C₂×C₄	—	226	2	no	—
108	C₂×C₄	C₂²⋊#90673	1	2	yes	—
109	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
110	C₈	—	452	2	no	—
111	C₂²	—	113	2	no	—
112	C₂²	—	226	2	no	—
113	C₂²	C₂×C₂²⋊#90673	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	2	no	—
116	C₄	—	113	1	no	—
117	C₄	—	113	1	no	—
118	C₄	—	452	4	no	—
119	C₄	—	452	2	no	—
120	C₄	—	226	2	no	—
121	C₄	Q₈×#90673	1	2	yes	—
122	C₂	—	226	2	no	—
123	C₂	(C₂×#90673)⋊Q₈	1	1	yes	characteristic
124	C₂	#20299595	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20299602

C₂

yes

maximal, characteristic

#20299534

C₂

yes

maximal, characteristic

#20299468

C₂

yes

maximal

#20299154

C₂

yes

maximal, characteristic

C₂×#14268657

C₂²

yes

—

C₂×C₄×#90673

C₂²

yes

—

C₄⋊#11744753

—

(C₂×#90673)⋊C₄

—

#16813621

C₄

yes

complemented

#16813614

C₄

yes

characteristic, complemented

#16813600

—

#16813587

—

#16813580

C₂²

yes

—

C₄²×D₁₁₃

C₂²

yes

characteristic

#125×C₁₁₃

C₄

yes

characteristic, complemented

C₂²×#90673

—

—

—

—

Q₈

yes

complemented

C₂×Dic₂₂₆

—

C₂×C₄×D₁₁₃

D₄

yes

—

C₂×C₄×D₁₁₃

C₂³

yes

characteristic

C₄⋊Dic₁₁₃

C₂×C₄

yes

—

#14268626

—

#14268626

—

#14268624

—

C₄×Dic₁₁₃

—

C₄×Dic₁₁₃

C₂×C₄

yes

characteristic

C₄×Dic₁₁₃

D₄

yes

—

C₄⋊C₄×C₁₁₃

—

C₄⋊C₄×C₁₁₃

C₂×C₄

yes

—

C₄²×C₁₁₃

C₂×C₄

yes

characteristic

C₂²×D₁₁₃

C₂×D₄

yes

characteristic

C₂×#90673

—

C₂×#90673

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

C₂²⋊C₄

yes

—

C₂×Dic₁₁₃

C₂×D₄

yes

characteristic

C₄×D₁₁₃

—

C₄×D₁₁₃

C₂×Q₈

yes

—

C₄×D₁₁₃

#41

yes

—

Dic₂₂₆

—

#11744753

—

C₂²×C₁₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₁₃

—

C₂×D₁₁₃

C₂²⋊Q₈

yes

—

#90673

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

C₄×Q₈

yes

—

Dic₁₁₃

—

Dic₁₁₃

—

Dic₁₁₃

#124

yes

characteristic

Dic₁₁₃

C₄⋊Q₈

yes

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

—

D₁₁₃

—

#1995

—

113

maximal, Sylow

C₁₁₃

#1995

yes

characteristic, complemented, Sylow

#533

—

113

—

#423

—

113

—

#376

—

113

—

#339

—

113

—

C₂×M₁₆

—

113

—

C₄⋊Q₈

—

113

—

#125

—

113

—

#125

#90673

yes

characteristic, complemented

C₂²⋊Q₈

—

226

—

C₄×Q₈

—

226

—

#117

—

113

—

C₂×C₄²

—

113

—

C₂×C₄²

—

113

—

C₄⋊C₈

—

226

—

#95

—

226

—

C₂×Q₈

—

226

—

C₂²×C₄

—

113

—

C₂²×C₄

—

113

—

C₂²×C₄

—

226

—

M₁₆

—

226

—

C₂×C₈

—

226

—

C₄⋊C₄

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

226

—

C₄⋊C₄

—

226

—

C₄⋊C₄

C₂×#90673

yes

—

C₂²⋊C₄

—

226

—

C₄²

—

113

—

C₄²

—

113

—

C₄²

—

113

—

C₄²

—

226

—

C₄²

—

226

—

C₄²

C₂×#90673

yes

characteristic

C₂³

—

113

—

Q₈

—

452

—

C₂×C₄

—

100

C₂×C₄

—

113

—

101

C₂×C₄

—

113

—

102

C₂×C₄

—

113

—

103

C₂×C₄

—

113

—

104

C₂×C₄

—

452

—

105

C₂×C₄

—

452

—

106

C₂×C₄

—

226

—

107

C₂×C₄

—

226

—

108

C₂×C₄

C₂²⋊#90673

yes

—

109

C₂×C₄

C₂²×#90673

yes

characteristic

110

C₈

—

452

—

111

C₂²

—

113

—

112

C₂²

—

226

—

113

C₂²

C₂×C₂²⋊#90673

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

113

—

117

C₄

—

113

—

118

C₄

—

452

—

119

C₄

—

452

—

120

C₄

—

226

—

121

C₄

Q₈×#90673

yes

—

122

C₂

—

226

—

123

C₂

(C₂×#90673)⋊Q₈

yes

characteristic

124

C₂

#20299595

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#90673
commutator	C₂×C₄×C₁₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#90673
Fitting	#125×C₁₁₃	C₄
socle	C₂²×C₁₁₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₁₃	C₂×D₄
cosocle	C₂×C₄×D₁₁₃	C₂³
hypercenter	#125	#90673
hypocenter	C₁₁₃	#1995
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

subgroup

G/H

center

C₂²

C₂×C₂²⋊#90673

commutator

C₂×C₄×C₁₁₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#90673

Fitting

#125×C₁₁₃

C₄

socle

C₂²×C₁₁₃

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₁₃

C₂×D₄

cosocle

C₂×C₄×D₁₁₃

C₂³

C₁₁₃

G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #125

N	H ≅ G/N	type
9. #16813621	C₄	semidirect
10. #16813614	C₄	semidirect
15. #125×C₁₁₃	C₄	semidirect
20. C₄×#90673	Q₈	semidirect
63. C₁₁₃	#1995	semidirect
71. #125	#90673	semidirect

H ≅ G/N

type

9. #16813621

C₄

semidirect

10. #16813614

C₄

semidirect

15. #125×C₁₁₃

C₄

semidirect

20. C₄×#90673

Q₈

semidirect

63. C₁₁₃

#1995

semidirect

71. #125

#90673

semidirect

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

subgroup	index
#20299602	2
#20299534	2
#20299468	2
#20299154	2
#1995	113

subgroup

index

113

Subgroups — #28406761

Conjugacy classes of subgroups

Complements

Maximal subgroups