Group #28406756 — subgroups

Subgroups — #28406756

Order 14464.

252 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 78 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	C₄×#14268655	C₂	1	1	yes	maximal, characteristic
2	#20299542	C₂	1	1	yes	maximal, characteristic
3	#20299471	C₂	1	4	yes	maximal
4	#125×D₁₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14268655	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	#16813614	C₄	1	1	yes	characteristic, complemented
10	#16813613	C₄	1	2	yes	complemented
11	#16813612	—	2	2	no	—
12	C₄⋊C₄×D₁₁₃	—	2	2	no	—
13	C₄⋊C₄×D₁₁₃	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	#14268655	—	2	2	no	—
18	#14268655	Q₈	1	4	yes	—
19	C₂×#11744753	—	2	2	no	—
20	C₄⋊#90673	—	2	4	no	—
21	C₂×C₄×D₁₁₃	—	2	2	no	—
22	C₂×C₄×D₁₁₃	D₄	1	2	yes	—
23	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
24	C₄⋊Dic₁₁₃	—	2	2	no	—
25	C₄⋊Dic₁₁₃	C₂×C₄	1	2	yes	—
26	#14268626	—	2	2	no	—
27	#14268626	—	2	2	no	—
28	#14268626	—	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	—	4	2	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	—	2	4	no	—
44	C₄×D₁₁₃	—	4	4	no	—
45	C₄×D₁₁₃	C₂×Q₈	1	2	yes	—
46	C₄×D₁₁₃	#41	1	2	yes	—
47	#11744753	—	2	4	no	—
48	C₂²×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂×D₁₁₃	#124	1	1	yes	characteristic
50	C₂×D₁₁₃	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×D₁₁₃	#135	1	4	yes	—
52	#90673	—	8	2	no	—
53	C₄×C₁₁₃	—	2	2	no	—
54	C₄×C₁₁₃	—	4	2	no	—
55	C₄×C₁₁₃	C₄×Q₈	1	2	yes	—
56	Dic₁₁₃	—	2	2	no	—
57	Dic₁₁₃	—	4	2	no	—
58	Dic₁₁₃	C₂²⋊Q₈	1	2	yes	—
59	C₂×C₁₁₃	#391	1	1	yes	characteristic
60	C₂×C₁₁₃	#416	1	2	yes	—
61	D₁₁₃	#488	1	2	yes	—
62	D₁₁₃	#499	1	2	yes	—
63	#1991	—	113	1	no	maximal, Sylow
64	C₁₁₃	#1991	1	1	yes	characteristic, complemented, Sylow
65	C₂×#125	—	113	1	no	—
66	C₂×C₄×C₈	—	113	1	no	—
67	#382	—	113	1	no	—
68	#340	—	113	4	no	—
69	C₂²×C₈	—	113	2	no	—
70	#125	—	113	2	no	—
71	#125	—	113	1	no	—
72	#125	—	226	2	no	—
73	#125	#90673	1	1	yes	characteristic, complemented
74	C₂×C₄⋊C₄	—	113	2	no	—
75	C₂×C₄⋊C₄	—	113	2	no	—
76	C₂×C₄⋊C₄	—	226	2	no	—
77	C₂×C₄²	—	113	1	no	—
78	C₄×C₈	—	226	2	no	—
79	#95	—	226	2	no	—
80	C₂²×C₄	—	113	1	no	—
81	C₂²×C₄	—	113	2	no	—
82	C₂²×C₄	—	226	2	no	—
83	C₂²×C₄	—	226	2	no	—
84	C₂×C₈	—	113	4	no	—
85	C₂×C₈	—	226	2	no	—
86	C₂×C₈	—	226	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	113	2	no	—
89	C₄⋊C₄	—	226	2	no	—
90	C₄⋊C₄	—	226	2	no	—
91	C₄⋊C₄	—	226	2	no	—
92	C₄⋊C₄	—	226	4	no	—
93	C₄⋊C₄	—	226	2	no	—
94	C₄⋊C₄	C₂×#90673	1	2	yes	—
95	C₄²	—	113	2	no	—
96	C₄²	—	113	1	no	—
97	C₄²	C₂×#90673	1	1	yes	characteristic
98	C₂³	—	113	1	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	4	no	—
105	C₂×C₄	—	452	2	no	—
106	C₂×C₄	—	452	4	no	—
107	C₂×C₄	—	226	4	no	—
108	C₂×C₄	—	226	2	no	—
109	C₂×C₄	C₂²⋊#90673	1	2	yes	—
110	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
111	C₈	—	226	4	no	—
112	C₂²	—	113	1	no	—
113	C₂²	—	113	1	no	—
114	C₂²	—	113	4	no	—
115	C₂²	C₂×C₂²⋊#90673	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	4	2	no	—
118	C₄	—	113	2	no	—
119	C₄	—	904	2	no	—
120	C₄	—	452	2	no	—
121	C₄	—	226	2	no	—
122	C₄	Q₈×#90673	1	2	yes	—
123	C₂	—	113	2	no	—
124	C₂	—	113	2	no	—
125	C₂	(C₂×#90673)⋊Q₈	1	1	yes	characteristic
126	C₂	#20299588	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14268655

C₂

yes

maximal, characteristic

#20299542

C₂

yes

maximal, characteristic

#20299471

C₂

yes

maximal

#125×D₁₁₃

C₂

yes

maximal, characteristic

C₂×#14268655

C₂²

yes

—

C₂×C₄⋊#90673

C₂²

yes

—

C₄×#11744753

—

(C₂×#90673)⋊C₄

—

#16813614

C₄

yes

characteristic, complemented

#16813613

C₄

yes

complemented

#16813612

—

C₄⋊C₄×D₁₁₃

—

C₄⋊C₄×D₁₁₃

C₂²

yes

—

C₄²×D₁₁₃

C₂²

yes

characteristic

#125×C₁₁₃

C₄

yes

characteristic, complemented

C₂²×#90673

—

#14268655

—

#14268655

Q₈

yes

—

C₂×#11744753

—

C₄⋊#90673

—

C₂×C₄×D₁₁₃

—

C₂×C₄×D₁₁₃

D₄

yes

—

C₂×C₄×D₁₁₃

C₂³

yes

characteristic

C₄⋊Dic₁₁₃

—

C₄⋊Dic₁₁₃

C₂×C₄

yes

—

#14268626

—

#14268626

—

#14268626

—

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

—

C₄×D₁₁₃

C₂×Q₈

yes

—

C₄×D₁₁₃

#41

yes

—

#11744753

—

C₂²×C₁₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₁₃

#124

yes

characteristic

C₂×D₁₁₃

C₄⋊Q₈

yes

characteristic

C₂×D₁₁₃

#135

yes

—

#90673

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

C₄×Q₈

yes

—

Dic₁₁₃

—

Dic₁₁₃

—

Dic₁₁₃

C₂²⋊Q₈

yes

—

C₂×C₁₁₃

#391

yes

characteristic

C₂×C₁₁₃

#416

yes

—

D₁₁₃

#488

yes

—

D₁₁₃

#499

yes

—

#1991

—

113

maximal, Sylow

C₁₁₃

#1991

yes

characteristic, complemented, Sylow

C₂×#125

—

113

—

C₂×C₄×C₈

—

113

—

#382

—

113

—

#340

—

113

—

C₂²×C₈

—

113

—

#125

—

113

—

#125

—

113

—

#125

—

226

—

#125

#90673

yes

characteristic, complemented

C₂×C₄⋊C₄

—

113

—

C₂×C₄⋊C₄

—

113

—

C₂×C₄⋊C₄

—

226

—

C₂×C₄²

—

113

—

C₄×C₈

—

226

—

#95

—

226

—

C₂²×C₄

—

113

—

C₂²×C₄

—

113

—

C₂²×C₄

—

226

—

C₂²×C₄

—

226

—

C₂×C₈

—

113

—

C₂×C₈

—

226

—

C₂×C₈

—

226

—

C₄⋊C₄

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

226

—

C₄⋊C₄

—

226

—

C₄⋊C₄

—

226

—

C₄⋊C₄

—

226

—

C₄⋊C₄

—

226

—

C₄⋊C₄

C₂×#90673

yes

—

C₄²

—

113

—

C₄²

—

113

—

C₄²

C₂×#90673

yes

characteristic

C₂³

—

113

—

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

—

452

—

107

C₂×C₄

—

226

—

108

C₂×C₄

—

226

—

109

C₂×C₄

C₂²⋊#90673

yes

—

110

C₂×C₄

C₂²×#90673

yes

characteristic

111

C₈

—

226

—

112

C₂²

—

113

—

113

C₂²

—

113

—

114

C₂²

—

113

—

115

C₂²

C₂×C₂²⋊#90673

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

113

—

119

C₄

—

904

—

120

C₄

—

452

—

121

C₄

—

226

—

122

C₄

Q₈×#90673

yes

—

123

C₂

—

113

—

124

C₂

—

113

—

125

C₂

(C₂×#90673)⋊Q₈

yes

characteristic

126

C₂

#20299588

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	#1991
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. #16813614	C₄	semidirect
10. #16813613	C₄	semidirect
15. #125×C₁₁₃	C₄	semidirect
64. C₁₁₃	#1991	semidirect
73. #125	#90673	semidirect

H ≅ G/N

type

9. #16813614

C₄

semidirect

10. #16813613

C₄

semidirect

15. #125×C₁₁₃

C₄

semidirect

64. C₁₁₃

#1991

semidirect

73. #125

#90673

semidirect

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

subgroup	index
C₄×#14268655	2
#20299542	2
#20299471	2
#125×D₁₁₃	2
#1991	113

subgroup

index

#125×D₁₁₃

113

Subgroups — #28406756

Conjugacy classes of subgroups

Complements

Maximal subgroups