Group #26019473 — subgroups

Subgroups — #26019473

Order 11392.

200 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 65 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	#19151948	C₂	1	1	yes	maximal, characteristic
2	#19151883	C₂	1	1	yes	maximal, characteristic
3	#19151808	C₂	1	1	yes	maximal, characteristic
4	#19151347	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊#10623494	C₂	1	1	yes	maximal, characteristic
6	Q₈⋊#10623494	C₂	1	1	yes	maximal, characteristic
7	Q₈²×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13083564	—	2	1	no	—
9	#15668012	C₂²	1	1	yes	characteristic
10	#15667943	—	2	1	no	—
11	C₄⋊#10623494	C₂²	1	1	yes	characteristic
12	C₄⋊#10623494	C₂²	1	1	yes	characteristic
13	#15667936	—	2	1	no	—
14	#15667936	—	2	2	no	—
15	#15667935	—	2	1	no	—
16	C₄×#10623494	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₈₉	—	2	2	no	—
18	C₄⋊Q₈×C₈₉	—	2	1	no	—
19	C₄⋊Q₈×C₈₉	—	2	1	no	—
20	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₈₉	—	2	1	no	—
22	C₄×Q₈×C₈₉	—	2	1	no	—
23	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₇₈	—	2	1	no	—
26	#13083564	—	4	2	no	—
27	C₂×#10623494	—	2	1	no	—
28	C₂×#10623494	—	2	1	no	—
29	C₂×#10623494	—	2	1	no	—
30	C₄⋊Dic₈₉	—	2	1	no	—
31	C₄⋊Dic₈₉	—	2	1	no	—
32	C₂×Q₈×C₈₉	—	2	2	no	—
33	C₂×Q₈×C₈₉	D₄	1	1	yes	characteristic
34	C₂×Q₈×C₈₉	D₄	1	1	yes	characteristic
35	C₂×Q₈×C₈₉	D₄	1	2	yes	—
36	C₄⋊C₄×C₈₉	—	2	2	no	—
37	C₄⋊C₄×C₈₉	—	2	2	no	—
38	C₄⋊C₄×C₈₉	—	2	1	no	—
39	C₄⋊C₄×C₈₉	—	2	1	no	—
40	C₄⋊C₄×C₈₉	—	2	1	no	—
41	C₄⋊C₄×C₈₉	—	2	1	no	—
42	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
43	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
44	C₄²×C₈₉	—	2	2	no	—
45	C₄²×C₈₉	—	2	1	no	—
46	C₄²×C₈₉	—	2	1	no	—
47	C₄²×C₈₉	C₂³	1	1	yes	characteristic
48	Q₈×C₈₉	—	2	1	no	—
49	Q₈×C₈₉	—	2	1	no	—
50	Q₈×C₈₉	—	4	2	no	—
51	Q₈×C₈₉	—	8	2	no	—
52	Q₈×C₈₉	SD₁₆	1	2	yes	—
53	Q₈×C₈₉	Q₁₆	1	2	yes	complemented
54	C₂×C₄×C₈₉	—	2	1	no	—
55	C₂×C₄×C₈₉	—	2	1	no	—
56	C₂×C₄×C₈₉	—	2	2	no	—
57	C₂×C₄×C₈₉	—	2	1	no	—
58	C₂×C₄×C₈₉	—	2	1	no	—
59	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₂×Dic₈₉	—	4	1	no	—
63	Dic₁₇₈	—	4	1	no	—
64	Dic₁₇₈	—	4	1	no	—
65	#10623494	—	4	1	no	—
66	#10623494	—	4	1	no	—
67	#10623494	—	4	1	no	—
68	C₂²×C₈₉	#120	1	1	yes	characteristic
69	C₄×C₈₉	—	2	2	no	—
70	C₄×C₈₉	—	2	2	no	—
71	C₄×C₈₉	—	2	1	no	—
72	C₄×C₈₉	—	4	2	no	—
73	C₄×C₈₉	—	4	1	no	—
74	C₄×C₈₉	—	4	1	no	—
75	C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
76	C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
77	C₄×C₈₉	#137	1	1	yes	characteristic
78	C₄×C₈₉	#137	1	1	yes	characteristic
79	Dic₈₉	—	8	1	no	—
80	C₂×C₈₉	#448	1	1	yes	characteristic
81	C₂×C₈₉	#451	1	1	yes	characteristic
82	C₂×C₈₉	#455	1	1	yes	characteristic
83	#1645	—	89	1	no	maximal, Sylow
84	C₈₉	#1645	1	1	yes	characteristic, complemented, Sylow
85	Q₈²	D₈₉	1	1	yes	characteristic
86	#487	—	89	1	no	—
87	#475	—	89	1	no	—
88	#462	—	89	1	no	—
89	#333	—	89	1	no	—
90	Q₈⋊C₈	—	89	1	no	—
91	Q₈⋊C₈	—	89	1	no	—
92	C₂×Q₁₆	—	178	1	no	—
93	C₄⋊Q₈	—	2	2	no	—
94	C₄⋊Q₈	—	2	1	no	—
95	C₄⋊Q₈	—	2	1	no	—
96	C₄⋊Q₈	—	89	1	no	—
97	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
98	C₄×Q₈	—	2	1	no	—
99	C₄×Q₈	—	2	1	no	—
100	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
101	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
102	#106	—	178	1	no	—
103	C₄⋊C₈	—	89	1	no	—
104	C₄⋊C₈	—	89	1	no	—
105	#103	—	178	1	no	—
106	#103	—	178	2	no	—
107	#103	—	178	1	no	—
108	C₄×C₈	—	89	1	no	—
109	C₂×Q₈	—	2	2	no	—
110	C₂×Q₈	—	178	1	no	—
111	C₂×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
112	C₂×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
113	C₂×Q₈	C₂²⋊D₈₉	1	2	yes	—
114	Q₁₆	—	356	2	no	—
115	C₂×C₈	—	178	1	no	—
116	C₂×C₈	—	178	1	no	—
117	C₂×C₈	—	178	1	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	2	1	no	—
123	C₄⋊C₄	—	2	1	no	—
124	C₄⋊C₄	—	178	1	no	—
125	C₄⋊C₄	—	178	1	no	—
126	C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
127	C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
128	C₄²	—	2	2	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	2	1	no	—
131	C₄²	C₂²×D₈₉	1	1	yes	characteristic
132	Q₈	—	2	1	no	—
133	Q₈	—	2	1	no	—
134	Q₈	—	4	2	no	—
135	Q₈	—	8	2	no	—
136	Q₈	—	356	1	no	—
137	Q₈	—	356	1	no	—
138	Q₈	#13083561	1	2	yes	—
139	Q₈	#13083564	1	2	yes	complemented
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	356	1	no	—
146	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
147	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
148	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
149	C₈	—	356	1	no	—
150	C₈	—	356	1	no	—
151	C₈	—	356	1	no	—
152	C₂²	#15667981	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	2	1	no	—
156	C₄	—	4	2	no	—
157	C₄	—	4	1	no	—
158	C₄	—	4	1	no	—
159	C₄	—	712	1	no	—
160	C₄	C₂×#13083561	1	1	yes	characteristic
161	C₄	C₂×#13083564	1	1	yes	characteristic
162	C₄	#15668043	1	1	yes	characteristic
163	C₄	#15668043	1	1	yes	characteristic
164	C₂	#19151737	1	1	yes	characteristic
165	C₂	#19151754	1	1	yes	characteristic
166	C₂	#19151776	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151948

C₂

yes

maximal, characteristic

#19151883

C₂

yes

maximal, characteristic

#19151808

C₂

yes

maximal, characteristic

#19151347

C₂

yes

maximal, characteristic

Q₈⋊#10623494

C₂

yes

maximal, characteristic

Q₈⋊#10623494

C₂

yes

maximal, characteristic

Q₈²×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083564

—

#15668012

C₂²

yes

characteristic

#15667943

—

C₄⋊#10623494

C₂²

yes

characteristic

C₄⋊#10623494

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

C₄×Q₈×C₈₉

—

C₄×Q₈×C₈₉

—

C₄×Q₈×C₈₉

C₂²

yes

characteristic

C₄×Q₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

—

—

—

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

D₄

yes

characteristic

C₂×Q₈×C₈₉

D₄

yes

characteristic

C₂×Q₈×C₈₉

D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

D₄

yes

characteristic

C₄⋊C₄×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

SD₁₆

yes

—

Q₈×C₈₉

Q₁₆

yes

complemented

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

#10623494

—

#10623494

—

#10623494

—

C₂²×C₈₉

#120

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₄×C₈₉

C₂×Q₁₆

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

Q₈²

D₈₉

yes

—

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₂×Q₁₆

—

178

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

100

C₄×Q₈

C₂×D₈₉

yes

characteristic

101

C₄×Q₈

C₂×D₈₉

yes

characteristic

102

#106

—

178

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

#103

—

178

—

106

#103

—

178

—

107

#103

—

178

—

108

C₄×C₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

178

—

111

C₂×Q₈

C₂²⋊D₈₉

yes

characteristic

112

C₂×Q₈

C₂²⋊D₈₉

yes

characteristic

113

C₂×Q₈

C₂²⋊D₈₉

yes

—

114

Q₁₆

—

356

—

115

C₂×C₈

—

178

—

116

C₂×C₈

—

178

—

117

C₂×C₈

—

178

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

178

—

125

C₄⋊C₄

—

178

—

126

C₄⋊C₄

C₂²⋊D₈₉

yes

characteristic

127

C₄⋊C₄

C₂²⋊D₈₉

yes

characteristic

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

C₂²×D₈₉

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

356

—

137

Q₈

—

356

—

138

Q₈

#13083561

yes

—

139

Q₈

#13083564

yes

complemented

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

356

—

146

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

147

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

148

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

149

C₈

—

356

—

150

C₈

—

356

—

151

C₈

—

356

—

152

C₂²

#15667981

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

712

—

160

C₄

C₂×#13083561

yes

characteristic

161

C₄

C₂×#13083564

yes

characteristic

162

C₄

#15668043

yes

characteristic

163

C₄

#15668043

yes

characteristic

164

C₂

#19151737

yes

characteristic

165

C₂

#19151754

yes

characteristic

166

C₂

#19151776

yes

characteristic

	subgroup	G/H
center	C₂²	#15667981
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	Q₈²×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	#120
hypercenter	Q₈²	D₈₉
hypocenter	C₈₉	#1645
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#15667981

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

Q₈²×C₈₉

C₂

socle, prime radical

C₂²×C₈₉

#120

hypercenter

Q₈²

D₈₉

hypocenter

C₈₉

#1645

LCS

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
53. Q₈×C₈₉	Q₁₆	semidirect
84. C₈₉	#1645	semidirect
139. Q₈	#13083564	semidirect

H ≅ G/N

type

53. Q₈×C₈₉

Q₁₆

semidirect

84. C₈₉

#1645

semidirect

139. Q₈

#13083564

semidirect

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

subgroup	index
#19151948	2
#19151883	2
#19151808	2
#19151347	2
Q₈⋊#10623494	2
Q₈²×C₈₉	2
#1645	89

subgroup

index

Q₈²×C₈₉

Subgroups — #26019473

Conjugacy classes of subgroups

Complements

Maximal subgroups