Group #28406684 — subgroups

Subgroups — #28406684

Order 14464.

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

normal subgroups only

This group has 200 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	C₂×#16813693	C₂	1	1	yes	maximal, characteristic
2	#20299605	C₂	1	2	yes	maximal
3	C₂×#16813676	C₂	1	1	yes	maximal, characteristic
4	#20299468	C₂	1	2	yes	maximal
5	#20298633	C₂	1	1	yes	maximal, characteristic
6	C₂×#14268657	C₂²	1	2	yes	—
7	C₂×#14268655	C₂²	1	1	yes	characteristic
8	C₂×C₄×#90673	C₂²	1	1	yes	characteristic
9	#16813693	—	2	2	no	—
10	#16813689	—	2	2	no	—
11	C₄⋊#11744753	—	2	2	no	—
12	#16813676	—	2	2	no	—
13	C₂×C₈×D₁₁₃	C₂²	1	1	yes	characteristic
14	#16813580	C₂²	1	2	yes	—
15	#16813555	C₄	1	2	yes	complemented
16	#16813554	C₄	1	1	yes	characteristic, complemented
17	#106×C₁₁₃	C₄	1	1	yes	characteristic, complemented
18	C₂²×#90673	—	2	1	no	—
19	#14268657	—	2	2	no	—
20	#14268657	—	2	2	no	—
21	#14268655	—	2	1	no	—
22	#14268655	—	2	1	no	—
23	C₂×#11744753	—	2	2	no	—
24	C₂×#11744753	D₄	1	1	yes	characteristic
25	C₂×#11744753	D₄	1	1	yes	characteristic
26	C₄×#90673	—	2	1	no	—
27	C₄×#90673	Q₈	1	2	yes	—
28	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
29	C₂×#11744751	C₂×C₄	1	1	yes	characteristic
30	C₈×D₁₁₃	D₄	1	2	yes	—
31	C₈×D₁₁₃	Q₈	1	2	yes	—
32	C₄⋊Dic₁₁₃	C₂×C₄	1	2	yes	—
33	#14268624	—	2	2	no	—
34	C₄×Dic₁₁₃	—	2	2	no	—
35	C₂×C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
37	C₂²×D₁₁₃	#41	1	1	yes	characteristic
38	C₂×#90673	—	2	2	no	—
39	C₂×#90673	—	4	2	no	—
40	C₂×C₄×C₁₁₃	—	2	2	no	—
41	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×Dic₁₁₃	—	2	2	no	—
43	C₂×Dic₁₁₃	C₂×D₄	1	1	yes	characteristic
44	C₄×D₁₁₃	C₂×D₄	1	1	yes	characteristic
45	C₄×D₁₁₃	C₂×Q₈	1	1	yes	characteristic
46	C₄×D₁₁₃	C₂×Q₈	1	1	yes	characteristic
47	C₄×D₁₁₃	#41	1	1	yes	characteristic
48	#11744753	—	2	1	no	—
49	#11744753	—	2	1	no	—
50	#11744753	—	4	2	no	—
51	C₈×C₁₁₃	C₄⋊C₄	1	2	yes	—
52	#11744751	C₄⋊C₄	1	2	yes	—
53	C₂²×C₁₁₃	C₄×D₄	1	1	yes	characteristic
54	C₂×D₁₁₃	—	2	2	no	—
55	C₂×D₁₁₃	C₂²⋊Q₈	1	1	yes	characteristic
56	C₂×D₁₁₃	#125	1	1	yes	characteristic
57	#90673	—	4	2	no	—
58	C₄×C₁₁₃	—	4	2	no	—
59	C₄×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₁₁₃	C₄×Q₈	1	1	yes	characteristic
61	Dic₁₁₃	—	4	2	no	—
62	Dic₁₁₃	#121	1	1	yes	characteristic
63	Dic₁₁₃	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₁₁₃	#389	1	1	yes	characteristic
65	C₂×C₁₁₃	#444	1	2	yes	—
66	D₁₁₃	—	2	1	no	—
67	D₁₁₃	—	2	1	no	—
68	#1946	—	113	1	no	maximal, Sylow
69	C₁₁₃	#1946	1	1	yes	characteristic, complemented, Sylow
70	C₂×#108	—	113	1	no	—
71	#427	—	113	1	no	—
72	#424	—	113	2	no	—
73	C₂×#97	—	113	1	no	—
74	#339	—	113	2	no	—
75	C₂×M₁₆	—	113	2	no	—
76	C₂²×C₈	—	113	1	no	—
77	C₂²×C₈	—	113	1	no	—
78	#117	—	113	2	no	—
79	C₂×C₄²	—	113	1	no	—
80	#108	—	226	2	no	—
81	#107	—	113	2	no	—
82	#106	—	113	1	no	—
83	#106	#90673	1	1	yes	characteristic, complemented
84	C₄⋊C₈	—	226	2	no	—
85	C₄⋊C₈	—	226	2	no	—
86	#97	—	226	2	no	—
87	C₂²×C₄	—	113	1	no	—
88	C₂²×C₄	—	226	1	no	—
89	M₁₆	—	226	2	no	—
90	M₁₆	—	226	2	no	—
91	C₂×C₈	—	113	1	no	—
92	C₂×C₈	—	113	2	no	—
93	C₂×C₈	—	113	2	no	—
94	C₂×C₈	—	113	1	no	—
95	C₂×C₈	—	113	1	no	—
96	C₂×C₈	—	226	2	no	—
97	C₂×C₈	—	226	1	no	—
98	C₂×C₈	—	226	1	no	—
99	C₂×C₈	C₂×#90673	1	1	yes	characteristic
100	C₄⋊C₄	—	113	2	no	—
101	C₄⋊C₄	C₂×#90673	1	2	yes	—
102	C₂²⋊C₄	—	226	2	no	—
103	C₄²	—	113	2	no	—
104	C₄²	—	226	2	no	—
105	C₄²	—	226	1	no	—
106	C₂³	—	113	1	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	113	1	no	—
109	C₂×C₄	—	113	1	no	—
110	C₂×C₄	—	113	1	no	—
111	C₂×C₄	—	113	1	no	—
112	C₂×C₄	—	113	1	no	—
113	C₂×C₄	—	452	2	no	—
114	C₂×C₄	—	226	2	no	—
115	C₂×C₄	—	226	2	no	—
116	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
117	C₈	—	113	2	no	—
118	C₈	—	452	2	no	—
119	C₈	—	226	1	no	—
120	C₈	—	226	1	no	—
121	C₈	C₄⋊#90673	1	2	yes	—
122	C₂²	—	113	1	no	—
123	C₂²	—	113	1	no	—
124	C₂²	—	226	2	no	—
125	C₂²	D₄×#90673	1	1	yes	characteristic
126	C₄	—	4	2	no	—
127	C₄	—	113	1	no	—
128	C₄	—	113	1	no	—
129	C₄	—	452	2	no	—
130	C₄	—	452	2	no	—
131	C₄	C₂×C₄⋊#90673	1	1	yes	characteristic
132	C₄	Q₈×#90673	1	1	yes	characteristic
133	C₂	—	226	1	no	—
134	C₂	—	226	1	no	—
135	C₂	#20299549	1	1	yes	characteristic
136	C₂	#20299643	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16813693

C₂

yes

maximal, characteristic

#20299605

C₂

yes

maximal

C₂×#16813676

C₂

yes

maximal, characteristic

#20299468

C₂

yes

maximal

#20298633

C₂

yes

maximal, characteristic

C₂×#14268657

C₂²

yes

—

C₂×#14268655

C₂²

yes

characteristic

C₂×C₄×#90673

C₂²

yes

—

—

—

—

C₂×C₈×D₁₁₃

C₂²

yes

characteristic

#16813580

C₂²

yes

—

#16813555

C₄

yes

complemented

#16813554

C₄

yes

characteristic, complemented

#106×C₁₁₃

C₄

yes

characteristic, complemented

C₂²×#90673

—

—

—

—

—

—

D₄

yes

characteristic

C₂×#11744753

D₄

yes

characteristic

C₄×#90673

—

C₄×#90673

Q₈

yes

—

C₂×C₄×D₁₁₃

C₂³

yes

characteristic

C₂×#11744751

C₂×C₄

yes

characteristic

C₈×D₁₁₃

D₄

yes

—

C₈×D₁₁₃

Q₈

yes

—

C₄⋊Dic₁₁₃

C₂×C₄

yes

—

#14268624

—

C₄×Dic₁₁₃

—

C₂×C₈×C₁₁₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₁₃

C₂×C₄

yes

—

C₂²×D₁₁₃

yes

—

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

C₂×D₄

yes

characteristic

C₄×D₁₁₃

C₂×D₄

yes

characteristic

C₄×D₁₁₃

C₂×Q₈

yes

characteristic

C₄×D₁₁₃

C₂×Q₈

yes

C₄×D₁₁₃

yes

—

—

—

C₈×C₁₁₃

C₄⋊C₄

yes

—

#11744751

C₄⋊C₄

yes

—

C₂²×C₁₁₃

C₄×D₄

yes

characteristic

C₂×D₁₁₃

—

C₂×D₁₁₃

C₂²⋊Q₈

yes

C₂×D₁₁₃

yes

—

C₄×C₁₁₃

—

C₄×C₁₁₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₁₃

C₄×Q₈

yes

characteristic

Dic₁₁₃

—

Dic₁₁₃

#121

yes

characteristic

Dic₁₁₃

C₄⋊Q₈

yes

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

—

D₁₁₃

—

D₁₁₃

—

#1946

—

113

maximal, Sylow

C₁₁₃

#1946

yes

characteristic, complemented, Sylow

C₂×#108

—

113

—

#427

—

113

—

#424

—

113

—

C₂×#97

—

113

—

#339

—

113

—

C₂×M₁₆

—

113

—

C₂²×C₈

—

113

—

C₂²×C₈

—

113

—

#117

—

113

—

C₂×C₄²

—

113

—

#108

—

226

—

#107

—

113

—

#106

—

113

—

#106

#90673

yes

characteristic, complemented

C₄⋊C₈

—

226

—

C₄⋊C₈

—

226

—

#97

—

226

—

C₂²×C₄

—

113

—

C₂²×C₄

—

226

—

M₁₆

—

226

—

M₁₆

—

226

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

226

—

C₂×C₈

—

226

—

C₂×C₈

—

226

—

C₂×C₈

C₂×#90673

yes

characteristic

100

C₄⋊C₄

—

113

—

101

C₄⋊C₄

C₂×#90673

yes

—

102

C₂²⋊C₄

—

226

—

103

C₄²

—

113

—

104

C₄²

—

226

—

105

C₄²

—

226

—

106

C₂³

—

113

—

107

C₂×C₄

—

108

C₂×C₄

—

113

—

109

C₂×C₄

—

113

—

110

C₂×C₄

—

113

—

111

C₂×C₄

—

113

—

112

C₂×C₄

—

113

—

113

C₂×C₄

—

452

—

114

C₂×C₄

—

226

—

115

C₂×C₄

—

226

—

116

C₂×C₄

C₂²×#90673

yes

characteristic

117

C₈

—

113

—

118

C₈

—

452

—

119

C₈

—

226

—

120

C₈

—

226

—

121

C₈

C₄⋊#90673

yes

—

122

C₂²

—

113

—

123

C₂²

—

113

—

124

C₂²

—

226

—

125

C₂²

D₄×#90673

yes

characteristic

126

C₄

—

127

C₄

—

113

—

128

C₄

—

113

—

129

C₄

—

452

—

130

C₄

—

452

—

131

C₄

C₂×C₄⋊#90673

yes

characteristic

132

C₄

Q₈×#90673

yes

characteristic

133

C₂

—

226

—

134

C₂

—

226

—

135

C₂

#20299549

yes

characteristic

136

C₂

#20299643

yes

—

	subgroup	G/H
center	C₂²	D₄×#90673
commutator	C₂×C₄×C₁₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#90673
Fitting	#106×C₁₁₃	C₄
socle	C₂²×C₁₁₃	C₄×D₄
prime radical	C₂²×D₁₁₃	#41
cosocle	C₂×C₄×D₁₁₃	C₂³
hypercenter	#106	#90673
hypocenter	C₁₁₃	#1946
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

subgroup

G/H

center

C₂²

D₄×#90673

commutator

C₂×C₄×C₁₁₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#90673

Fitting

#106×C₁₁₃

C₄

socle

C₂²×C₁₁₃

C₄×D₄

prime radical

C₂²×D₁₁₃

#41

cosocle

C₂×C₄×D₁₁₃

C₂³

C₁₁₃

G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

N	H ≅ G/N	type
15. #16813555	C₄	semidirect
16. #16813554	C₄	semidirect
17. #106×C₁₁₃	C₄	semidirect
69. C₁₁₃	#1946	semidirect
83. #106	#90673	semidirect

H ≅ G/N

type

15. #16813555

C₄

semidirect

16. #16813554

C₄

semidirect

17. #106×C₁₁₃

C₄

semidirect

69. C₁₁₃

#1946

semidirect

83. #106

#90673

semidirect

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

subgroup	index
C₂×#16813693	2
#20299605	2
C₂×#16813676	2
#20299468	2
#20298633	2
#1946	113

subgroup

index

113

Subgroups — #28406684

Conjugacy classes of subgroups

Complements

Maximal subgroups