Group #28399226 — subgroups

Subgroups — #28399226

Order 14464.

384 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 173 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×#14268647	C₂	1	1	yes	maximal, characteristic
2	#20298657	C₂	1	4	yes	maximal, complemented
3	#20298653	C₂	1	4	yes	maximal, complemented
4	#20298626	C₂	1	2	yes	maximal, complemented
5	#20298566	C₂	1	2	yes	maximal
6	C₂×#16813533	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×M₁₆×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14268647	C₂²	1	1	yes	characteristic
9	C₂²×#11744751	C₂²	1	2	yes	—
10	C₄×Dic₂₂₆	C₄	1	2	yes	—
11	C₄×D₄₅₂	C₄	1	2	yes	—
12	C₄×C₂²⋊D₁₁₃	C₄	1	4	yes	—
13	#16813581	C₂²	1	1	yes	characteristic
14	#16813579	C₂²	1	2	yes	—
15	C₄²×D₁₁₃	C₂²	1	2	yes	—
16	#16813551	C₂²	1	8	yes	—
17	#16813534	C₂²	1	8	yes	—
18	#16813533	C₂²	1	2	yes	—
19	#16813533	C₂²	1	2	yes	—
20	C₂×M₁₆×C₁₁₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
22	#97×C₁₁₃	C₂²	1	2	yes	—
23	C₄×C₈×C₁₁₃	C₂²	1	2	yes	—
24	#14268647	—	2	4	no	—
25	C₂×Dic₂₂₆	C₂×C₄	1	1	yes	characteristic
26	C₂×D₄₅₂	C₂×C₄	1	1	yes	characteristic
27	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₁₁₃	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
30	C₂×#11744751	—	2	4	no	—
31	C₂×#11744751	C₂³	1	4	yes	—
32	C₄⋊Dic₁₁₃	C₂×C₄	1	2	yes	—
33	#14268626	C₂×C₄	1	4	yes	—
34	C₂²⋊Dic₁₁₃	C₂×C₄	1	2	yes	—
35	#14268624	C₂×C₄	1	4	yes	—
36	C₄×Dic₁₁₃	C₂×C₄	1	4	yes	—
37	C₄×Dic₁₁₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
40	M₁₆×C₁₁₃	—	2	4	no	—
41	C₂×C₈×C₁₁₃	C₂³	1	4	yes	—
42	C₄²×C₁₁₃	C₂×C₄	1	2	yes	—
43	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
44	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
45	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×D₁₁₃	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₁₁₃	—	4	2	no	—
48	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₁₃	#41	1	4	yes	—
52	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
53	C₂²⋊D₁₁₃	—	2	4	no	—
54	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
55	C₂×Dic₁₁₃	C₂²×C₄	1	4	yes	—
56	D₄₅₂	—	2	2	no	—
57	C₄×D₁₁₃	—	2	4	no	—
58	C₄×D₁₁₃	—	4	4	no	—
59	Dic₂₂₆	—	2	2	no	—
60	C₈×C₁₁₃	—	2	4	no	—
61	#11744751	—	2	4	no	—
62	C₂²×C₁₁₃	—	2	2	no	—
63	C₂²×C₁₁₃	#117	1	2	yes	—
64	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₁₁₃	—	2	2	no	—
66	C₂×D₁₁₃	—	2	4	no	—
67	C₄×C₁₁₃	—	2	2	no	—
68	C₄×C₁₁₃	—	2	2	no	—
69	C₄×C₁₁₃	#117	1	2	yes	—
70	C₄×C₁₁₃	C₂×#41	1	2	yes	—
71	Dic₁₁₃	—	2	2	no	—
72	Dic₁₁₃	—	2	4	no	—
73	C₂×C₁₁₃	—	2	2	no	—
74	C₂×C₁₁₃	C₂×#117	1	1	yes	characteristic
75	C₂×C₁₁₃	#568	1	2	yes	—
76	D₁₁₃	—	4	2	no	—
77	#2923	—	113	1	no	maximal, Sylow
78	C₁₁₃	#2923	1	1	yes	characteristic, complemented, Sylow
79	C₄×#41	—	113	1	no	—
80	#433	—	113	4	no	—
81	#432	—	113	4	no	—
82	#424	—	113	2	no	—
83	#408	—	113	2	no	—
84	C₄×M₁₆	D₁₁₃	1	1	yes	characteristic, complemented
85	C₂×#97	—	113	1	no	—
86	C₂×#41	—	113	1	no	—
87	C₂×M₁₆	C₂×D₁₁₃	1	2	yes	—
88	C₂²×C₈	—	113	2	no	—
89	C₄×Q₈	—	113	2	no	—
90	C₄×D₄	—	113	4	no	—
91	C₄×D₄	—	113	2	no	—
92	#117	—	113	1	no	—
93	#117	—	113	2	no	—
94	C₂×C₄²	—	113	2	no	—
95	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
96	C₄⋊C₈	—	113	8	no	—
97	C₂²⋊C₈	—	113	8	no	—
98	#97	—	113	2	no	—
99	#97	—	113	2	no	—
100	#97	C₂×D₁₁₃	1	2	yes	—
101	C₄×C₈	C₂×D₁₁₃	1	2	yes	—
102	#41	—	226	4	no	—
103	C₂×Q₈	—	113	1	no	—
104	C₂×D₄	—	113	2	no	—
105	C₂×D₄	—	113	1	no	—
106	C₂²×C₄	—	113	4	no	—
107	C₂²×C₄	—	113	2	no	—
108	C₂²×C₄	C₄×D₁₁₃	1	2	yes	—
109	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
110	M₁₆	—	2	4	no	—
111	C₂×C₈	—	113	4	no	—
112	C₂×C₈	—	226	4	no	—
113	C₂×C₈	C₂²×D₁₁₃	1	4	yes	—
114	C₄⋊C₄	—	113	4	no	—
115	C₄⋊C₄	—	113	2	no	—
116	C₂²⋊C₄	—	113	2	no	—
117	C₂²⋊C₄	—	113	4	no	—
118	C₄²	—	113	4	no	—
119	C₄²	—	113	2	no	—
120	C₄²	C₄×D₁₁₃	1	2	yes	—
121	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
122	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
123	C₂³	—	113	2	no	—
124	C₂³	C₂×C₄×D₁₁₃	1	1	yes	characteristic
125	Q₈	—	226	2	no	—
126	D₄	—	226	4	no	—
127	D₄	—	226	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	113	4	no	—
130	C₂×C₄	—	113	2	no	—
131	C₂×C₄	—	452	4	no	—
132	C₂×C₄	—	226	4	no	—
133	C₂×C₄	C₂×C₄×D₁₁₃	1	2	yes	—
134	C₂×C₄	C₂×C₄×D₁₁₃	1	2	yes	—
135	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
136	C₂×C₄	#14268647	1	4	yes	—
137	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
138	C₈	—	2	4	no	—
139	C₈	—	226	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	226	2	no	—
142	C₂²	—	226	4	no	—
143	C₂²	#16813579	1	2	yes	—
144	C₂²	C₂²×C₄×D₁₁₃	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	226	2	no	—
148	C₄	—	226	4	no	—
149	C₄	#16813579	1	2	yes	—
150	C₄	C₂×#14268647	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	452	2	no	—
153	C₂	C₂×#16813579	1	1	yes	characteristic
154	C₂	#20299364	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14268647

C₂

yes

maximal, characteristic

#20298657

C₂

yes

maximal, complemented

#20298653

C₂

yes

maximal, complemented

#20298626

C₂

yes

maximal, complemented

#20298566

C₂

yes

maximal

C₂×#16813533

C₂

yes

maximal, characteristic, complemented

C₄×M₁₆×C₁₁₃

C₂

yes

maximal, characteristic, complemented

C₂×#14268647

C₂²

yes

characteristic

C₂²×#11744751

C₂²

yes

—

C₄×Dic₂₂₆

C₄

yes

—

C₄×D₄₅₂

C₄

yes

—

C₄×C₂²⋊D₁₁₃

C₄

yes

—

#16813581

C₂²

yes

characteristic

#16813579

C₂²

yes

—

C₄²×D₁₁₃

C₂²

yes

—

#16813551

C₂²

yes

—

#16813534

C₂²

yes

—

#16813533

C₂²

yes

—

#16813533

C₂²

yes

—

C₂×M₁₆×C₁₁₃

C₂²

yes

—

C₂×C₄²×C₁₁₃

C₂²

yes

characteristic

#97×C₁₁₃

C₂²

yes

—

C₄×C₈×C₁₁₃

C₂²

yes

—

#14268647

—

C₂×Dic₂₂₆

C₂×C₄

yes

characteristic

C₂×D₄₅₂

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₁₁₃

C₂×C₄

yes

—

C₂×C₄×D₁₁₃

C₂×C₄

yes

—

C₂×C₄×D₁₁₃

C₂³

yes

—

C₂×#11744751

—

C₂×#11744751

C₂³

yes

—

C₄⋊Dic₁₁₃

C₂×C₄

yes

—

#14268626

C₂×C₄

yes

—

C₂²⋊Dic₁₁₃

C₂×C₄

yes

—

#14268624

C₂×C₄

yes

—

C₄×Dic₁₁₃

C₂×C₄

yes

—

C₄×Dic₁₁₃

C₂³

yes

—

C₂²×C₄×C₁₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₁₃

C₂³

yes

characteristic

M₁₆×C₁₁₃

—

C₂×C₈×C₁₁₃

C₂³

yes

—

C₄²×C₁₁₃

C₂×C₄

yes

—

C₄²×C₁₁₃

C₂³

yes

characteristic

C₄²×C₁₁₃

C₂³

yes

characteristic

C₂³×C₁₁₃

C₂²×C₄

yes

characteristic

C₂²×D₁₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁₃

#41

yes

—

C₂×C₄×C₁₁₃

C₂⁴

yes

characteristic

C₂²⋊D₁₁₃

—

C₂×Dic₁₁₃

C₂²×C₄

yes

—

C₂×Dic₁₁₃

C₂²×C₄

yes

—

D₄₅₂

—

C₄×D₁₁₃

—

C₄×D₁₁₃

—

Dic₂₂₆

—

C₈×C₁₁₃

—

#11744751

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

#117

yes

—

C₂²×C₁₁₃

C₂³×C₄

yes

characteristic

C₂×D₁₁₃

—

C₂×D₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

#117

yes

—

C₄×C₁₁₃

C₂×#41

yes

—

Dic₁₁₃

—

Dic₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

C₂×#117

yes

characteristic

C₂×C₁₁₃

#568

yes

—

D₁₁₃

—

#2923

—

113

maximal, Sylow

C₁₁₃

#2923

yes

characteristic, complemented, Sylow

C₄×#41

—

113

—

#433

—

113

—

#432

—

113

—

#424

—

113

—

#408

—

113

—

C₄×M₁₆

D₁₁₃

yes

characteristic, complemented

C₂×#97

—

113

—

C₂×#41

—

113

—

C₂×M₁₆

C₂×D₁₁₃

yes

—

C₂²×C₈

—

113

—

C₄×Q₈

—

113

—

C₄×D₄

—

113

—

C₄×D₄

—

113

—

#117

—

113

—

#117

—

113

—

C₂×C₄²

—

113

—

C₂×C₄²

C₂×D₁₁₃

yes

characteristic

C₄⋊C₈

—

113

—

C₂²⋊C₈

—

113

—

#97

—

113

—

#97

—

113

—

100

#97

C₂×D₁₁₃

yes

—

101

C₄×C₈

C₂×D₁₁₃

yes

—

102

#41

—

226

—

103

C₂×Q₈

—

113

—

104

C₂×D₄

—

113

—

105

C₂×D₄

—

113

—

106

C₂²×C₄

—

113

—

107

C₂²×C₄

—

113

—

108

C₂²×C₄

C₄×D₁₁₃

yes

—

109

C₂²×C₄

C₂²×D₁₁₃

yes

characteristic

110

M₁₆

—

111

C₂×C₈

—

113

—

112

C₂×C₈

—

226

—

113

C₂×C₈

C₂²×D₁₁₃

yes

—

114

C₄⋊C₄

—

113

—

115

C₄⋊C₄

—

113

—

116

C₂²⋊C₄

—

113

—

117

C₂²⋊C₄

—

113

—

118

C₄²

—

113

—

119

C₄²

—

113

—

120

C₄²

C₄×D₁₁₃

yes

—

121

C₄²

C₂²×D₁₁₃

yes

characteristic

122

C₄²

C₂²×D₁₁₃

yes

characteristic

123

C₂³

—

113

—

124

C₂³

C₂×C₄×D₁₁₃

yes

characteristic

125

Q₈

—

226

—

126

D₄

—

226

—

127

D₄

—

226

—

128

C₂×C₄

—

129

C₂×C₄

—

113

—

130

C₂×C₄

—

113

—

131

C₂×C₄

—

452

—

132

C₂×C₄

—

226

—

133

C₂×C₄

C₂×C₄×D₁₁₃

yes

—

134

C₂×C₄

C₂×C₄×D₁₁₃

yes

—

135

C₂×C₄

C₂×C₄×D₁₁₃

yes

characteristic

136

C₂×C₄

#14268647

yes

—

137

C₂×C₄

C₂³×D₁₁₃

yes

characteristic

138

C₈

—

139

C₈

—

226

—

140

C₂²

—

141

C₂²

—

226

—

142

C₂²

—

226

—

143

C₂²

#16813579

yes

—

144

C₂²

C₂²×C₄×D₁₁₃

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

226

—

148

C₄

—

226

—

149

C₄

#16813579

yes

—

150

C₄

C₂×#14268647

yes

—

151

C₂

—

152

C₂

—

452

—

153

C₂

C₂×#16813579

yes

characteristic

154

C₂

#20299364

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₁₃
commutator, socle	C₂²×C₁₁₃	C₂³×C₄
Fitting	C₄×M₁₆×C₁₁₃	C₂
prime radical	C₂×#14268647	C₂²
cosocle	C₂×C₄×C₁₁₃	C₂⁴
hypercenter	C₄×M₁₆	D₁₁₃
hypocenter	C₁₁₃	#2923
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₁₃

commutator, socle

C₂²×C₁₁₃

C₂³×C₄

Fitting

C₄×M₁₆×C₁₁₃

C₂

prime radical

C₂×#14268647

C₂²

cosocle

C₂×C₄×C₁₁₃

C₂⁴

hypercenter

C₄×M₁₆

D₁₁₃

C₁₁₃

G ▷ C₂²×C₁₁₃ ▷ C₁₁₃

C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #20298657	C₂	semidirect
3. #20298653	C₂	semidirect
4. #20298626	C₂	semidirect
6. C₂×#16813533	C₂	semidirect
7. C₄×M₁₆×C₁₁₃	C₂	semidirect
78. C₁₁₃	#2923	semidirect
84. C₄×M₁₆	D₁₁₃	semidirect

H ≅ G/N

type

2. #20298657

C₂

semidirect

3. #20298653

C₂

semidirect

4. #20298626

C₂

semidirect

6. C₂×#16813533

C₂

semidirect

7. C₄×M₁₆×C₁₁₃

C₂

semidirect

78. C₁₁₃

#2923

semidirect

84. C₄×M₁₆

D₁₁₃

semidirect

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

subgroup	index
C₄×#14268647	2
#20298657	2
#20298653	2
#20298626	2
#20298566	2
C₂×#16813533	2
C₄×M₁₆×C₁₁₃	2
#2923	113

subgroup

index

C₄×M₁₆×C₁₁₃

113

Subgroups — #28399226

Conjugacy classes of subgroups

Complements

Maximal subgroups