Group #26021371 — subgroups

Subgroups — #26021371

Order 11392.

342 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 75 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×#15668043	C₂	1	2	yes	maximal
2	C₂×#15667931	C₂	1	2	yes	maximal, complemented
3	#19151564	C₂	1	1	yes	maximal, characteristic
4	#19151363	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×Q₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13083574	C₂²	1	1	yes	characteristic
7	#15668043	—	2	4	no	—
8	#15668043	—	2	4	no	—
9	C₂×#13083564	—	2	2	no	—
10	C₂×#13083561	—	2	2	no	—
11	C₂×#13083558	C₂²	1	1	yes	characteristic
12	C₂×#13083558	C₂²	1	2	yes	—
13	#15667931	—	2	4	no	—
14	#15667924	—	2	2	no	—
15	C₂²×Q₈×C₈₉	C₂²	1	2	yes	—
16	C₂×Q₁₆×C₈₉	—	2	4	no	—
17	C₂×Q₁₆×C₈₉	—	2	2	no	—
18	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	#13083574	—	2	2	no	—
20	#13083574	—	2	2	no	—
21	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
22	C₂×D₃₅₆	D₄	1	1	yes	characteristic
23	C₂×C₂²⋊D₈₉	—	2	1	no	—
24	C₂×C₄×D₈₉	—	2	1	no	—
25	#13083564	—	2	8	no	—
26	#13083561	—	2	8	no	—
27	#13083558	—	2	2	no	—
28	#13083558	—	2	2	no	—
29	#13083558	D₄	1	4	yes	—
30	C₂×#10623494	—	2	2	no	—
31	C₂×#10623494	—	2	1	no	—
32	C₂×Q₈×C₈₉	—	2	2	no	—
33	C₂×Q₈×C₈₉	—	2	2	no	—
34	C₂×Q₈×C₈₉	—	2	2	no	—
35	C₂×Q₈×C₈₉	—	2	2	no	—
36	C₂×Q₈×C₈₉	—	2	2	no	—
37	C₂×Q₈×C₈₉	D₄	1	4	yes	—
38	C₂²×C₄×C₈₉	—	2	2	no	—
39	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
40	Q₁₆×C₈₉	—	2	8	no	—
41	C₂×C₈×C₈₉	—	2	2	no	—
42	C₂×C₈×C₈₉	D₄	1	2	yes	—
43	C₂³×C₈₉	#41	1	1	yes	characteristic
44	C₂²×D₈₉	—	2	1	no	—
45	Q₈×C₈₉	—	2	4	no	—
46	Q₈×C₈₉	—	2	4	no	—
47	Q₈×C₈₉	—	4	2	no	—
48	Q₈×C₈₉	—	4	2	no	—
49	C₂×C₄×C₈₉	—	2	4	no	—
50	C₂×C₄×C₈₉	—	4	2	no	—
51	C₂×C₄×C₈₉	—	4	2	no	—
52	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₂²⋊D₈₉	—	4	2	no	—
57	C₂×Dic₈₉	—	2	1	no	—
58	D₃₅₆	—	2	2	no	—
59	C₄×D₈₉	—	4	1	no	—
60	C₄×D₈₉	—	4	1	no	—
61	Dic₁₇₈	—	2	2	no	—
62	C₈×C₈₉	—	2	2	no	—
63	#10623494	—	2	4	no	—
64	#10623494	—	4	1	no	—
65	C₂²×C₈₉	—	2	2	no	—
66	C₂²×C₈₉	#121	1	2	yes	—
67	C₂²×C₈₉	#121	1	1	yes	characteristic
68	C₂×D₈₉	—	4	1	no	—
69	C₂×D₈₉	—	4	2	no	—
70	C₄×C₈₉	—	4	4	no	—
71	C₄×C₈₉	#120	1	2	yes	—
72	C₄×C₈₉	#120	1	1	yes	characteristic
73	C₄×C₈₉	#127	1	1	yes	characteristic
74	Dic₈₉	—	4	1	no	—
75	C₂×C₈₉	—	2	2	no	—
76	C₂×C₈₉	#392	1	1	yes	characteristic
77	C₂×C₈₉	#473	1	2	yes	—
78	D₈₉	—	8	1	no	—
79	#2020	—	89	1	no	maximal, Sylow
80	C₈₉	#2020	1	1	yes	characteristic, complemented, Sylow
81	C₂×#137	—	89	2	no	—
82	C₂²×Q₁₆	D₈₉	1	1	yes	characteristic, complemented
83	C₂×#101	—	89	2	no	—
84	#408	—	89	1	no	—
85	#341	—	89	1	no	—
86	C₂×#41	—	89	1	no	—
87	C₂²×Q₈	C₂×D₈₉	1	2	yes	—
88	#137	—	178	4	no	—
89	#137	—	178	4	no	—
90	C₂×Q₁₆	—	2	4	no	—
91	C₂×Q₁₆	—	2	2	no	—
92	C₂×Q₁₆	—	178	2	no	—
93	C₂×SD₁₆	—	178	2	no	—
94	C₂×M₁₆	—	89	2	no	—
95	C₂×M₁₆	—	89	1	no	—
96	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
97	#101	—	178	4	no	—
98	C₂²⋊C₈	—	178	2	no	—
99	#41	—	178	2	no	—
100	#41	—	178	2	no	—
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	89	1	no	—
107	C₂×Q₈	C₂²⋊D₈₉	1	4	yes	—
108	C₂×D₄	—	89	1	no	—
109	C₂×D₄	—	178	1	no	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	—	178	1	no	—
112	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
113	Q₁₆	—	2	8	no	—
114	Q₁₆	—	178	8	no	—
115	SD₁₆	—	178	8	no	—
116	M₁₆	—	89	4	no	—
117	M₁₆	—	178	2	no	—
118	M₁₆	—	178	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	178	1	no	—
121	C₂×C₈	—	178	2	no	—
122	C₂×C₈	C₂²⋊D₈₉	1	2	yes	—
123	C₂³	—	178	1	no	—
124	C₂³	#13083573	1	1	yes	characteristic
125	Q₈	—	2	4	no	—
126	Q₈	—	2	4	no	—
127	Q₈	—	4	2	no	—
128	Q₈	—	4	2	no	—
129	Q₈	—	178	2	no	—
130	D₄	—	356	2	no	—
131	D₄	—	178	2	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	356	1	no	—
136	C₂×C₄	—	356	1	no	—
137	C₂×C₄	—	178	1	no	—
138	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
139	C₂×C₄	D₄×D₈₉	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	356	1	no	—
144	C₈	—	178	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	356	1	no	—
147	C₂²	—	356	2	no	—
148	C₂²	#15667983	1	1	yes	characteristic
149	C₂²	#15667984	1	2	yes	—
150	C₄	—	4	4	no	—
151	C₄	—	356	1	no	—
152	C₄	#15667980	1	2	yes	—
153	C₄	#15667981	1	1	yes	characteristic
154	C₄	#15668008	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	712	1	no	—
157	C₂	#19151506	1	1	yes	characteristic
158	C₂	#19151870	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668043

C₂

yes

maximal

C₂×#15667931

C₂

yes

maximal, complemented

#19151564

C₂

yes

maximal, characteristic

#19151363

C₂

yes

maximal, characteristic, complemented

C₂²×Q₁₆×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083574

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×#13083558

C₂²

yes

—

#15667931

—

#15667924

—

C₂²×Q₈×C₈₉

C₂²

yes

—

C₂×Q₁₆×C₈₉

—

C₂×Q₁₆×C₈₉

—

C₂²×C₈×C₈₉

C₂²

yes

characteristic

#13083574

—

#13083574

—

C₂×Dic₁₇₈

D₄

yes

characteristic

C₂×D₃₅₆

D₄

yes

characteristic

C₂×C₂²⋊D₈₉

—

C₂×C₄×D₈₉

—

—

—

—

—

D₄

yes

—

C₂×#10623494

—

C₂×#10623494

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₂³×C₈₉

#41

yes

characteristic

C₂²×D₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#121

yes

—

C₂²×C₈₉

#121

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#120

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#392

yes

characteristic

C₂×C₈₉

#473

yes

—

D₈₉

—

#2020

—

maximal, Sylow

C₈₉

#2020

yes

characteristic, complemented, Sylow

C₂×#137

—

C₂²×Q₁₆

D₈₉

yes

characteristic, complemented

C₂×#101

—

#408

—

#341

—

C₂×#41

—

C₂²×Q₈

C₂×D₈₉

yes

—

#137

—

178

—

#137

—

178

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

178

—

C₂×SD₁₆

—

178

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₈₉

yes

characteristic

#101

—

178

—

C₂²⋊C₈

—

178

—

#41

—

178

—

100

#41

—

178

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

C₂²⋊D₈₉

yes

—

108

C₂×D₄

—

109

C₂×D₄

—

178

—

110

C₂²×C₄

—

111

C₂²×C₄

—

178

—

112

C₂²×C₄

C₂²×D₈₉

yes

characteristic

113

Q₁₆

—

114

Q₁₆

—

178

—

115

SD₁₆

—

178

—

116

M₁₆

—

117

M₁₆

—

178

—

118

M₁₆

—

178

—

119

C₂×C₈

—

120

C₂×C₈

—

178

—

121

C₂×C₈

—

178

—

122

C₂×C₈

C₂²⋊D₈₉

yes

—

123

C₂³

—

178

—

124

C₂³

#13083573

yes

characteristic

125

Q₈

—

126

Q₈

—

127

Q₈

—

128

Q₈

—

129

Q₈

—

178

—

130

D₄

—

356

—

131

D₄

—

178

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

356

—

136

C₂×C₄

—

356

—

137

C₂×C₄

—

178

—

138

C₂×C₄

D₄×D₈₉

yes

characteristic

139

C₂×C₄

D₄×D₈₉

yes

—

140

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

141

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

142

C₈

—

143

C₈

—

356

—

144

C₈

—

178

—

145

C₂²

—

146

C₂²

—

356

—

147

C₂²

—

356

—

148

C₂²

#15667983

yes

characteristic

149

C₂²

#15667984

yes

—

150

C₄

—

151

C₄

—

356

—

152

C₄

#15667980

yes

—

153

C₄

#15667981

yes

characteristic

154

C₄

#15668008

yes

characteristic

155

C₂

—

156

C₂

—

712

—

157

C₂

#19151506

yes

characteristic

158

C₂

#19151870

yes

—

	subgroup	G/H
center	C₂²	#15667983
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	C₂²×Q₁₆×C₈₉	C₂
socle	C₂²×C₈₉	#121
prime radical	C₂×#13083574	C₂²
hypercenter	C₂²×Q₁₆	D₈₉
hypocenter	C₈₉	#2020
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

#15667983

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

C₂²×Q₁₆×C₈₉

C₂

C₂²×C₈₉

C₂²

C₂²×Q₁₆

D₈₉

hypocenter

C₈₉

#2020

LCS

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
2. C₂×#15667931	C₂	semidirect
4. #19151363	C₂	semidirect
5. C₂²×Q₁₆×C₈₉	C₂	semidirect
80. C₈₉	#2020	semidirect
82. C₂²×Q₁₆	D₈₉	semidirect

H ≅ G/N

type

2. C₂×#15667931

C₂

semidirect

4. #19151363

C₂

semidirect

5. C₂²×Q₁₆×C₈₉

C₂

semidirect

80. C₈₉

#2020

semidirect

82. C₂²×Q₁₆

D₈₉

semidirect

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

subgroup	index
C₂×#15668043	2
C₂×#15667931	2
#19151564	2
#19151363	2
C₂²×Q₁₆×C₈₉	2
#2020	89

subgroup

index

C₂²×Q₁₆×C₈₉

Subgroups — #26021371

Conjugacy classes of subgroups

Complements

Maximal subgroups