Group #28390497 — subgroups

Subgroups — #28390497

Order 14464.

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

normal subgroups only

This group has 564 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	C₂³×Dic₂₂₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16813540	C₂	1	4	yes	maximal, complemented
3	#20298562	C₂	1	1	yes	maximal, characteristic
4	#407×C₁₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₂₆	—	2	8	no	—
6	C₂²×Dic₂₂₆	—	2	2	no	—
7	C₂²×Dic₂₂₆	—	2	2	no	—
8	C₂²×Dic₂₂₆	C₄	1	2	yes	—
9	C₂²×Dic₂₂₆	C₂²	1	2	yes	—
10	C₂³×Dic₁₁₃	C₄	1	2	yes	—
11	C₂×#14268631	C₂²	1	2	yes	—
12	#16813540	—	2	8	no	—
13	C₂²⋊#11744751	—	2	2	no	—
14	C₂³×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₁₁₃	C₂²	1	2	yes	—
16	C₂²⋊C₈×C₁₁₃	—	2	2	no	—
17	C₂×Dic₂₂₆	—	2	16	no	—
18	C₂×Dic₂₂₆	—	2	16	no	—
19	C₂×Dic₂₂₆	—	2	4	no	—
20	C₂×Dic₂₂₆	—	2	4	no	—
21	C₂×Dic₂₂₆	—	2	2	no	—
22	C₂×Dic₂₂₆	—	2	2	no	—
23	C₂×Dic₂₂₆	—	2	8	no	—
24	C₂×Dic₂₂₆	D₄	1	8	yes	—
25	C₂²×Dic₁₁₃	—	2	4	no	—
26	C₂²×Dic₁₁₃	—	2	8	no	—
27	C₂²×Dic₁₁₃	C₂×C₄	1	4	yes	—
28	#14268631	—	2	4	no	—
29	C₂×#11744751	—	2	2	no	—
30	C₂⁴×C₁₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₁₃	—	2	4	no	—
32	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₁₃	D₄	1	2	yes	—
34	C₂²×C₄×C₁₁₃	D₄	1	2	yes	—
35	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
36	M₁₆×C₁₁₃	—	2	4	no	—
37	C₂×C₈×C₁₁₃	—	2	2	no	—
38	C₂³×C₁₁₃	—	2	4	no	—
39	C₂³×C₁₁₃	—	2	2	no	—
40	C₂³×C₁₁₃	C₂²⋊C₄	1	2	yes	—
41	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₁₃	—	2	2	no	—
43	C₂×C₄×C₁₁₃	—	2	4	no	—
44	C₂×C₄×C₁₁₃	—	2	4	no	—
45	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
47	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₁₁₃	—	2	16	no	—
50	C₂×Dic₁₁₃	—	2	4	no	—
51	C₂×Dic₁₁₃	—	2	4	no	—
52	C₂×Dic₁₁₃	C₂²⋊C₄	1	8	yes	—
53	Dic₂₂₆	—	2	16	no	—
54	Dic₂₂₆	—	2	8	no	—
55	Dic₂₂₆	—	2	8	no	—
56	C₈×C₁₁₃	—	4	2	no	—
57	#11744751	—	4	2	no	—
58	C₂²×C₁₁₃	—	2	8	no	—
59	C₂²×C₁₁₃	—	2	2	no	—
60	C₂²×C₁₁₃	—	2	2	no	—
61	C₂²×C₁₁₃	—	2	2	no	—
62	C₂²×C₁₁₃	#101	1	4	yes	complemented
63	C₂²×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
64	C₂²×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₁₁₃	—	2	2	no	—
66	C₄×C₁₁₃	#120	1	2	yes	—
67	C₄×C₁₁₃	#120	1	2	yes	—
68	Dic₁₁₃	—	2	8	no	—
69	C₂×C₁₁₃	—	2	4	no	—
70	C₂×C₁₁₃	—	2	2	no	—
71	C₂×C₁₁₃	#379	1	1	yes	characteristic
72	C₂×C₁₁₃	C₂×#101	1	2	yes	—
73	#1784	—	113	1	no	maximal, Sylow
74	C₁₁₃	#1784	1	1	yes	characteristic, complemented, Sylow
75	C₂³×Q₈	—	113	1	no	—
76	C₂×#101	—	113	4	no	—
77	#407	—	113	1	no	—
78	#407	D₁₁₃	1	1	yes	characteristic
79	C₂²×Q₈	—	113	2	no	—
80	C₂²×Q₈	—	113	2	no	—
81	C₂²×Q₈	—	226	8	no	—
82	C₂²×Q₈	—	226	2	no	—
83	C₂²×Q₈	—	226	2	no	—
84	C₂³×C₄	—	113	2	no	—
85	C₂³×C₄	C₂×D₁₁₃	1	1	yes	characteristic
86	C₂×M₁₆	—	113	2	no	—
87	C₂×M₁₆	C₂×D₁₁₃	1	2	yes	—
88	#101	—	226	8	no	—
89	C₂²⋊C₈	—	2	2	no	—
90	C₂²⋊C₈	—	226	2	no	—
91	C₂⁴	C₄×D₁₁₃	1	1	yes	characteristic
92	C₂×Q₈	—	113	8	no	—
93	C₂×Q₈	—	226	8	no	—
94	C₂×Q₈	—	226	16	no	—
95	C₂×Q₈	—	226	16	no	—
96	C₂×Q₈	—	226	4	no	—
97	C₂×Q₈	—	226	4	no	—
98	C₂×Q₈	—	226	2	no	—
99	C₂×Q₈	—	226	2	no	—
100	C₂²×C₄	—	2	4	no	—
101	C₂²×C₄	—	113	4	no	—
102	C₂²×C₄	—	226	8	no	—
103	C₂²×C₄	—	226	4	no	—
104	C₂²×C₄	C₄×D₁₁₃	1	1	yes	characteristic
105	C₂²×C₄	D₄₅₂	1	2	yes	—
106	C₂²×C₄	C₂²⋊D₁₁₃	1	2	yes	—
107	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
108	M₁₆	—	2	4	no	—
109	M₁₆	—	226	4	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	226	2	no	—
112	C₂³	—	2	2	no	—
113	C₂³	—	2	4	no	—
114	C₂³	#14268624	1	2	yes	—
115	C₂³	C₂×C₄×D₁₁₃	1	1	yes	characteristic
116	Q₈	—	226	16	no	—
117	Q₈	—	226	8	no	—
118	Q₈	—	226	8	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	113	8	no	—
123	C₂×C₄	—	226	16	no	—
124	C₂×C₄	—	226	4	no	—
125	C₂×C₄	—	226	4	no	—
126	C₂×C₄	#14268624	1	2	yes	—
127	C₂×C₄	D₄×D₁₁₃	1	4	yes	—
128	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
129	C₂×C₄	C₂×D₄₅₂	1	1	yes	characteristic
130	C₈	—	4	2	no	—
131	C₈	—	452	2	no	—
132	C₂²	—	2	8	no	—
133	C₂²	—	2	2	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	#16813540	1	4	yes	complemented
137	C₂²	C₂²⋊C₄×D₁₁₃	1	2	yes	—
138	C₂²	C₂×#14268624	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	226	8	no	—
141	C₄	#16813590	1	2	yes	—
142	C₄	#16813592	1	2	yes	—
143	C₂	—	2	2	no	—
144	C₂	—	2	4	no	—
145	C₂	#20298453	1	1	yes	characteristic
146	C₂	C₂×#16813540	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Dic₂₂₆

C₂

yes

maximal, characteristic

C₂×#16813540

C₂

yes

maximal, complemented

#20298562

C₂

yes

maximal, characteristic

#407×C₁₁₃

C₂

yes

maximal, characteristic

C₂²×Dic₂₂₆

—

C₂²×Dic₂₂₆

—

C₂²×Dic₂₂₆

—

C₂²×Dic₂₂₆

C₄

yes

—

C₂²×Dic₂₂₆

C₂²

yes

—

C₂³×Dic₁₁₃

C₄

yes

—

C₂×#14268631

C₂²

yes

—

#16813540

—

C₂²⋊#11744751

—

C₂³×C₄×C₁₁₃

C₂²

yes

characteristic

C₂×M₁₆×C₁₁₃

C₂²

yes

—

C₂²⋊C₈×C₁₁₃

—

C₂×Dic₂₂₆

—

C₂×Dic₂₂₆

—

C₂×Dic₂₂₆

—

C₂×Dic₂₂₆

—

C₂×Dic₂₂₆

—

C₂×Dic₂₂₆

—

C₂×Dic₂₂₆

—

C₂×Dic₂₂₆

D₄

yes

—

C₂²×Dic₁₁₃

—

C₂²×Dic₁₁₃

—

C₂²×Dic₁₁₃

C₂×C₄

yes

—

#14268631

—

C₂×#11744751

—

C₂⁴×C₁₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₁₃

—

C₂²×C₄×C₁₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₁₃

D₄

yes

—

C₂²×C₄×C₁₁₃

D₄

yes

—

C₂²×C₄×C₁₁₃

C₂³

yes

characteristic

M₁₆×C₁₁₃

—

C₂×C₈×C₁₁₃

—

C₂³×C₁₁₃

—

C₂³×C₁₁₃

—

C₂³×C₁₁₃

C₂²⋊C₄

yes

—

C₂³×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂×D₄

yes

characteristic

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

C₂²⋊C₄

yes

—

Dic₂₂₆

—

Dic₂₂₆

—

Dic₂₂₆

—

C₈×C₁₁₃

—

#11744751

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

#101

yes

complemented

C₂²×C₁₁₃

C₂×C₂²⋊C₄

yes

—

C₂²×C₁₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₁₃

—

C₄×C₁₁₃

#120

yes

—

C₄×C₁₁₃

#120

yes

—

Dic₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

#379

yes

characteristic

C₂×C₁₁₃

C₂×#101

yes

—

#1784

—

113

maximal, Sylow

C₁₁₃

#1784

yes

characteristic, complemented, Sylow

C₂³×Q₈

—

113

—

C₂×#101

—

113

—

#407

—

113

—

#407

D₁₁₃

yes

characteristic

C₂²×Q₈

—

113

—

C₂²×Q₈

—

113

—

C₂²×Q₈

—

226

—

C₂²×Q₈

—

226

—

C₂²×Q₈

—

226

—

C₂³×C₄

—

113

—

C₂³×C₄

C₂×D₁₁₃

yes

characteristic

C₂×M₁₆

—

113

—

C₂×M₁₆

C₂×D₁₁₃

yes

—

#101

—

226

—

C₂²⋊C₈

—

C₂²⋊C₈

—

226

—

C₂⁴

C₄×D₁₁₃

yes

characteristic

C₂×Q₈

—

113

—

C₂×Q₈

—

226

—

C₂×Q₈

—

226

—

C₂×Q₈

—

226

—

C₂×Q₈

—

226

—

C₂×Q₈

—

226

—

C₂×Q₈

—

226

—

C₂×Q₈

—

226

—

100

C₂²×C₄

—

101

C₂²×C₄

—

113

—

102

C₂²×C₄

—

226

—

103

C₂²×C₄

—

226

—

104

C₂²×C₄

C₄×D₁₁₃

yes

characteristic

105

C₂²×C₄

D₄₅₂

yes

—

106

C₂²×C₄

C₂²⋊D₁₁₃

yes

—

107

C₂²×C₄

C₂²×D₁₁₃

yes

characteristic

108

M₁₆

—

109

M₁₆

—

226

—

110

C₂×C₈

—

111

C₂×C₈

—

226

—

112

C₂³

—

113

C₂³

—

114

C₂³

#14268624

yes

—

115

C₂³

C₂×C₄×D₁₁₃

yes

characteristic

116

Q₈

—

226

—

117

Q₈

—

226

—

118

Q₈

—

226

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

113

—

123

C₂×C₄

—

226

—

124

C₂×C₄

—

226

—

125

C₂×C₄

—

226

—

126

C₂×C₄

#14268624

yes

—

127

C₂×C₄

D₄×D₁₁₃

yes

—

128

C₂×C₄

C₂×C₂²⋊D₁₁₃

yes

characteristic

129

C₂×C₄

C₂×D₄₅₂

yes

characteristic

130

C₈

—

131

C₈

—

452

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

#16813540

yes

complemented

137

C₂²

C₂²⋊C₄×D₁₁₃

yes

—

138

C₂²

C₂×#14268624

yes

characteristic

139

C₄

—

140

C₄

—

226

—

141

C₄

#16813590

yes

—

142

C₄

#16813592

yes

—

143

C₂

—

144

C₂

—

145

C₂

#20298453

yes

characteristic

146

C₂

C₂×#16813540

yes

—

	subgroup	G/H
center	C₂²	C₂×#14268624
commutator	C₂³×C₁₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₁₃
Fitting	#407×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₁₃	C₂×C₄
cosocle	C₂²×C₄×C₁₁₃	C₂³
hypercenter	#407	D₁₁₃
hypocenter	C₁₁₃	#1784
LCS	G ▷ C₂³×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

C₂×#14268624

commutator

C₂³×C₁₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₁₃

Fitting

#407×C₁₁₃

C₂

socle

C₂²×C₁₁₃

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₁₁₃

C₂×C₄

cosocle

C₂²×C₄×C₁₁₃

C₂³

D₁₁₃

C₁₁₃

G ▷ C₂³×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
2. C₂×#16813540	C₂	semidirect
62. C₂²×C₁₁₃	#101	semidirect
74. C₁₁₃	#1784	semidirect
136. C₂²	#16813540	semidirect

H ≅ G/N

type

2. C₂×#16813540

C₂

semidirect

62. C₂²×C₁₁₃

#101

semidirect

74. C₁₁₃

#1784

semidirect

136. C₂²

#16813540

semidirect

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

subgroup	index
C₂³×Dic₂₂₆	2
C₂×#16813540	2
#20298562	2
#407×C₁₁₃	2
#1784	113

subgroup

index

C₂³×Dic₂₂₆

C₂×#16813540

#20298562

#407×C₁₁₃

#1784

113

Subgroups — #28390497

Conjugacy classes of subgroups

Complements

Maximal subgroups