Group #26021351 — subgroups

Subgroups — #26021351

Order 11392.

370 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂²×#13083564	C₂	1	2	yes	maximal
2	C₂×#15668012	C₂	1	1	yes	maximal, characteristic
3	C₂×#15667935	C₂	1	2	yes	maximal
4	#19151353	C₂	1	1	yes	maximal, characteristic
5	#391×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
7	C₂×#13083564	—	2	4	no	—
8	C₂×#13083564	—	2	4	no	—
9	C₂×#13083564	—	2	4	no	—
10	C₂²×#10623494	C₂²	1	2	yes	—
11	#15668012	—	2	4	no	—
12	C₂×C₄⋊Dic₈₉	—	2	2	no	—
13	#15667935	—	2	4	no	—
14	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
17	#95×C₈₉	—	2	2	no	—
18	C₂×Dic₁₇₈	—	2	2	no	—
19	C₂×Dic₁₇₈	—	2	2	no	—
20	C₂×Dic₁₇₈	—	2	2	no	—
21	C₂×Dic₁₇₈	—	2	2	no	—
22	C₂×Dic₁₇₈	—	2	2	no	—
23	C₂×Dic₁₇₈	D₄	1	4	yes	—
24	C₂²×Dic₈₉	—	2	2	no	—
25	#13083564	—	2	16	no	—
26	C₂×#10623494	—	2	2	no	—
27	C₂×#10623494	—	2	2	no	—
28	C₂×#10623494	D₄	1	4	yes	—
29	C₄⋊Dic₈₉	—	2	8	no	—
30	C₂×Q₈×C₈₉	—	2	1	no	—
31	C₂×Q₈×C₈₉	—	2	2	no	—
32	C₂×Q₈×C₈₉	—	2	2	no	—
33	C₂×Q₈×C₈₉	D₄	1	2	yes	—
34	C₂²×C₄×C₈₉	—	2	1	no	—
35	C₂²×C₄×C₈₉	—	2	1	no	—
36	C₂²×C₄×C₈₉	D₄	1	2	yes	—
37	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₈₉	—	2	2	no	—
39	C₄²×C₈₉	—	2	2	no	—
40	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
41	Q₈×C₈₉	—	2	4	no	—
42	Q₈×C₈₉	—	4	2	no	—
43	C₂×C₄×C₈₉	—	2	2	no	—
44	C₂×C₄×C₈₉	—	2	2	no	—
45	C₂×C₄×C₈₉	—	2	2	no	—
46	C₂×C₄×C₈₉	—	4	1	no	—
47	C₂×C₄×C₈₉	—	4	1	no	—
48	C₂×C₄×C₈₉	—	4	1	no	—
49	C₂×C₄×C₈₉	—	4	2	no	—
50	C₂×C₄×C₈₉	Q₁₆	1	4	yes	complemented
51	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
55	C₂×Dic₈₉	—	2	4	no	—
56	C₂×Dic₈₉	—	4	2	no	—
57	C₂×Dic₈₉	—	4	2	no	—
58	Dic₁₇₈	—	2	4	no	—
59	Dic₁₇₈	—	2	4	no	—
60	Dic₁₇₈	—	4	2	no	—
61	Dic₁₇₈	—	4	2	no	—
62	#10623494	—	2	4	no	—
63	C₂²×C₈₉	#120	1	1	yes	characteristic
64	C₂²×C₈₉	#121	1	1	yes	characteristic
65	C₂²×C₈₉	#127	1	1	yes	characteristic
66	C₂²×C₈₉	C₂×Q₁₆	1	2	yes	—
67	C₂²×C₈₉	#137	1	2	yes	—
68	C₄×C₈₉	—	2	4	no	—
69	C₄×C₈₉	—	4	2	no	—
70	C₄×C₈₉	—	8	1	no	—
71	C₄×C₈₉	#120	1	2	yes	—
72	C₄×C₈₉	#121	1	2	yes	—
73	Dic₈₉	—	4	4	no	—
74	C₂×C₈₉	#392	1	1	yes	characteristic
75	C₂×C₈₉	#451	1	2	yes	—
76	C₂×C₈₉	#462	1	2	yes	—
77	C₂×C₈₉	#494	1	1	yes	characteristic
78	C₂×C₈₉	#497	1	1	yes	characteristic
79	#2016	—	89	1	no	maximal, Sylow
80	C₈₉	#2016	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₁₆	—	89	2	no	—
82	C₂×C₄⋊Q₈	—	89	1	no	—
83	C₂×#103	—	89	2	no	—
84	#391	D₈₉	1	1	yes	characteristic
85	#336	—	89	1	no	—
86	C₂²×Q₈	—	89	2	no	—
87	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
88	C₂×Q₁₆	—	178	4	no	—
89	C₂×Q₁₆	—	178	4	no	—
90	C₂×Q₁₆	—	178	4	no	—
91	C₂²×C₈	—	89	2	no	—
92	C₄⋊Q₈	—	178	4	no	—
93	C₂×C₄⋊C₄	—	178	2	no	—
94	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
95	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
96	#103	—	178	4	no	—
97	#95	—	2	2	no	—
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	—	89	4	no	—
102	C₂×Q₈	—	178	2	no	—
103	C₂×Q₈	—	178	2	no	—
104	C₂×Q₈	—	178	2	no	—
105	C₂×Q₈	—	178	2	no	—
106	C₂×Q₈	—	178	2	no	—
107	C₂×Q₈	D₃₅₆	1	2	yes	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	—	178	2	no	—
111	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
112	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
113	Q₁₆	—	178	16	no	—
114	C₂×C₈	—	89	4	no	—
115	C₂×C₈	—	178	2	no	—
116	C₂×C₈	—	178	2	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	178	8	no	—
119	C₄²	—	2	2	no	—
120	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
121	Q₈	—	2	4	no	—
122	Q₈	—	4	2	no	—
123	Q₈	—	356	2	no	—
124	Q₈	—	356	2	no	—
125	Q₈	—	178	4	no	—
126	Q₈	—	178	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	356	2	no	—
135	C₂×C₄	—	356	2	no	—
136	C₂×C₄	—	178	4	no	—
137	C₂×C₄	#13083564	1	4	yes	complemented
138	C₂×C₄	D₄×D₈₉	1	2	yes	—
139	C₂×C₄	D₄×D₈₉	1	2	yes	—
140	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
141	C₂×C₄	#13083574	1	1	yes	characteristic
142	C₈	—	178	4	no	—
143	C₂²	#15667980	1	1	yes	characteristic
144	C₂²	#15667986	1	1	yes	characteristic
145	C₂²	#15668008	1	1	yes	characteristic
146	C₂²	C₂×#13083564	1	2	yes	—
147	C₂²	#15668042	1	2	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	4	2	no	—
150	C₄	—	8	1	no	—
151	C₄	—	356	4	no	—
152	C₄	#15667982	1	2	yes	—
153	C₄	#15667985	1	2	yes	—
154	C₂	#19151507	1	1	yes	characteristic
155	C₂	#19151753	1	2	yes	—
156	C₂	#19151808	1	2	yes	—
157	C₂	#19151977	1	1	yes	characteristic
158	C₂	#19151993	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13083564

C₂

yes

maximal

C₂×#15668012

C₂

yes

maximal, characteristic

C₂×#15667935

C₂

yes

maximal

#19151353

C₂

yes

maximal, characteristic

#391×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

—

—

—

C₂²

yes

—

#15668012

—

C₂×C₄⋊Dic₈₉

—

#15667935

—

C₂²×Q₈×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#95×C₈₉

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

D₄

yes

—

C₂²×Dic₈₉

—

—

—

—

D₄

yes

—

C₄⋊Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

Q₁₆

yes

complemented

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

#10623494

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

C₂×Q₁₆

yes

—

C₂²×C₈₉

#137

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#120

yes

—

C₄×C₈₉

#121

yes

—

Dic₈₉

—

C₂×C₈₉

#392

yes

characteristic

C₂×C₈₉

#451

yes

—

C₂×C₈₉

#462

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×#103

—

#391

D₈₉

yes

characteristic

#336

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

178

—

C₂×Q₁₆

—

178

—

C₂×Q₁₆

—

178

—

C₂²×C₈

—

C₄⋊Q₈

—

178

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

C₂×C₄²

C₂×D₈₉

yes

characteristic

#103

—

178

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

178

—

103

C₂×Q₈

—

178

—

104

C₂×Q₈

—

178

—

105

C₂×Q₈

—

178

—

106

C₂×Q₈

—

178

—

107

C₂×Q₈

D₃₅₆

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

178

—

111

C₂²×C₄

C₂²⋊D₈₉

yes

—

112

C₂²×C₄

C₂²×D₈₉

yes

characteristic

113

Q₁₆

—

178

—

114

C₂×C₈

—

115

C₂×C₈

—

178

—

116

C₂×C₈

—

178

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

178

—

119

C₄²

—

120

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

356

—

124

Q₈

—

356

—

125

Q₈

—

178

—

126

Q₈

—

178

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

356

—

135

C₂×C₄

—

356

—

136

C₂×C₄

—

178

—

137

C₂×C₄

#13083564

yes

complemented

138

C₂×C₄

D₄×D₈₉

yes

—

139

C₂×C₄

D₄×D₈₉

yes

—

140

C₂×C₄

C₂×D₃₅₆

yes

characteristic

141

C₂×C₄

#13083574

yes

characteristic

142

C₈

—

178

—

143

C₂²

#15667980

yes

characteristic

144

C₂²

#15667986

yes

characteristic

145

C₂²

#15668008

yes

characteristic

146

C₂²

C₂×#13083564

yes

—

147

C₂²

#15668042

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

356

—

152

C₄

#15667982

yes

—

153

C₄

#15667985

yes

—

154

C₂

#19151507

yes

characteristic

155

C₂

#19151753

yes

—

156

C₂

#19151808

yes

—

157

C₂

#19151977

yes

characteristic

158

C₂

#19151993

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₉
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#391×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	#391	D₈₉
hypocenter	C₈₉	#2016
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₉

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#391×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₄

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
50. C₂×C₄×C₈₉	Q₁₆	semidirect
80. C₈₉	#2016	semidirect
137. C₂×C₄	#13083564	semidirect

H ≅ G/N

type

50. C₂×C₄×C₈₉

Q₁₆

semidirect

80. C₈₉

#2016

semidirect

137. C₂×C₄

#13083564

semidirect

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

subgroup	index
C₂²×#13083564	2
C₂×#15668012	2
C₂×#15667935	2
#19151353	2
#391×C₈₉	2
#2016	89

subgroup

index

#391×C₈₉

Subgroups — #26021351

Conjugacy classes of subgroups

Complements

Maximal subgroups