Group #26019476 — subgroups

Subgroups — #26019476

Order 11392.

200 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 59 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	Q₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
2	#19151949	C₂	1	1	yes	maximal, characteristic
3	#19151883	C₂	1	1	yes	maximal, characteristic
4	#19151809	C₂	1	1	yes	maximal, characteristic
5	#19151346	C₂	1	1	yes	maximal, characteristic
6	#19151330	C₂	1	1	yes	maximal, characteristic
7	Q₈⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₅₆	—	2	1	no	—
9	#15668012	—	2	1	no	—
10	#15668012	C₂²	1	1	yes	characteristic
11	#15668012	C₂²	1	1	yes	characteristic
12	#15668011	—	2	2	no	—
13	#15668011	—	2	1	no	—
14	Q₈×Dic₈₉	—	2	1	no	—
15	C₄×Dic₁₇₈	—	2	1	no	—
16	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
17	#15667943	—	2	1	no	—
18	C₄⋊#10623494	C₂²	1	1	yes	characteristic
19	#15667937	—	2	2	no	—
20	#15667937	—	2	1	no	—
21	#15667936	—	2	1	no	—
22	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₇₈	—	2	1	no	—
26	C₂×Dic₁₇₈	—	2	2	no	—
27	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
28	C₂×Dic₁₇₈	D₄	1	2	yes	—
29	Dic₃₅₆	—	4	2	no	—
30	C₂×#10623494	—	2	1	no	—
31	C₄⋊Dic₈₉	—	2	2	no	—
32	C₄⋊Dic₈₉	—	2	1	no	—
33	C₄⋊Dic₈₉	—	2	1	no	—
34	C₄⋊Dic₈₉	—	2	1	no	—
35	C₄⋊Dic₈₉	—	2	1	no	—
36	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
37	#13083553	—	2	2	no	—
38	#13083553	—	2	1	no	—
39	C₄×Dic₈₉	—	2	2	no	—
40	C₄×Dic₈₉	—	2	1	no	—
41	C₂×Q₈×C₈₉	D₄	1	1	yes	characteristic
42	C₂×C₈×C₈₉	—	2	1	no	—
43	C₂×C₈×C₈₉	—	2	1	no	—
44	C₄⋊C₄×C₈₉	—	2	1	no	—
45	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
46	C₄²×C₈₉	—	2	1	no	—
47	C₄²×C₈₉	C₂³	1	1	yes	characteristic
48	Q₈×C₈₉	—	2	1	no	—
49	Q₈×C₈₉	Q₁₆	1	2	yes	complemented
50	C₂×C₄×C₈₉	—	2	1	no	—
51	C₂×C₄×C₈₉	—	2	1	no	—
52	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₈₉	—	2	1	no	—
56	C₂×Dic₈₉	—	2	2	no	—
57	C₂×Dic₈₉	—	2	1	no	—
58	C₂×Dic₈₉	—	4	1	no	—
59	Dic₁₇₈	—	2	1	no	—
60	Dic₁₇₈	—	4	2	no	—
61	Dic₁₇₈	—	4	1	no	—
62	Dic₁₇₈	—	4	1	no	—
63	Dic₁₇₈	—	8	2	no	—
64	Dic₁₇₈	SD₁₆	1	2	yes	—
65	C₈×C₈₉	—	4	1	no	—
66	C₈×C₈₉	—	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	1	no	—
71	C₄×C₈₉	—	4	1	no	—
72	C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
73	C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
74	C₄×C₈₉	#137	1	1	yes	characteristic
75	C₄×C₈₉	#137	1	1	yes	characteristic
76	Dic₈₉	—	2	2	no	—
77	Dic₈₉	—	4	2	no	—
78	Dic₈₉	—	4	1	no	—
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	—	89	1	no	—
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	D₈₉	1	1	yes	characteristic
92	C₂×Q₁₆	—	178	1	no	—
93	C₄⋊Q₈	—	89	1	no	—
94	C₄⋊Q₈	—	89	1	no	—
95	C₄⋊Q₈	—	178	1	no	—
96	C₄⋊Q₈	—	178	2	no	—
97	C₄⋊Q₈	—	178	1	no	—
98	C₄×Q₈	—	89	1	no	—
99	C₄×Q₈	—	178	1	no	—
100	C₄×Q₈	—	178	1	no	—
101	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
102	#106	—	178	1	no	—
103	C₄⋊C₈	—	89	1	no	—
104	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
105	#103	—	178	1	no	—
106	#103	—	178	2	no	—
107	#103	—	178	1	no	—
108	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
109	C₂×Q₈	—	89	1	no	—
110	C₂×Q₈	—	89	2	no	—
111	C₂×Q₈	—	178	2	no	—
112	C₂×Q₈	—	178	1	no	—
113	C₂×Q₈	D₃₅₆	1	1	yes	characteristic
114	Q₁₆	—	356	2	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	178	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	89	1	no	—
120	C₄⋊C₄	—	178	2	no	—
121	C₄⋊C₄	—	178	2	no	—
122	C₄⋊C₄	—	178	1	no	—
123	C₄⋊C₄	—	178	1	no	—
124	C₄⋊C₄	—	178	1	no	—
125	C₄⋊C₄	—	178	1	no	—
126	C₄⋊C₄	—	178	1	no	—
127	C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
128	C₄²	—	2	1	no	—
129	C₄²	—	178	2	no	—
130	C₄²	—	178	1	no	—
131	C₄²	C₂²×D₈₉	1	1	yes	characteristic
132	Q₈	—	2	1	no	—
133	Q₈	—	89	2	no	—
134	Q₈	—	356	2	no	—
135	Q₈	—	356	1	no	—
136	Q₈	—	356	1	no	—
137	Q₈	—	178	1	no	—
138	Q₈	—	712	2	no	—
139	Q₈	Dic₃₅₆	1	2	yes	complemented
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	356	1	no	—
143	C₂×C₄	—	178	1	no	—
144	C₂×C₄	—	178	2	no	—
145	C₂×C₄	—	178	1	no	—
146	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
147	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
148	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	4	1	no	—
151	C₈	—	356	1	no	—
152	C₂²	#15667982	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	1	no	—
156	C₄	—	356	2	no	—
157	C₄	—	356	1	no	—
158	C₄	—	178	2	no	—
159	C₄	—	712	1	no	—
160	C₄	SD₁₆×D₈₉	1	1	yes	characteristic
161	C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
162	C₄	#15668044	1	1	yes	characteristic
163	C₄	#15668045	1	1	yes	characteristic
164	C₂	#19151738	1	1	yes	characteristic
165	C₂	#19151756	1	1	yes	characteristic
166	C₂	#19151777	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₇₈

C₂

yes

maximal, characteristic

#19151949

C₂

yes

maximal, characteristic

#19151883

C₂

yes

maximal, characteristic

#19151809

C₂

yes

maximal, characteristic

#19151346

C₂

yes

maximal, characteristic

#19151330

C₂

yes

maximal, characteristic

Q₈⋊C₈×C₈₉

C₂

yes

maximal, characteristic

C₂×Dic₃₅₆

—

#15668012

—

#15668012

C₂²

yes

characteristic

#15668012

C₂²

yes

characteristic

#15668011

—

#15668011

—

Q₈×Dic₈₉

—

C₄×Dic₁₇₈

—

C₄×Dic₁₇₈

C₂²

yes

characteristic

#15667943

—

C₄⋊#10623494

C₂²

yes

—

—

—

C₄×Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

characteristic

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

D₄

yes

characteristic

C₂×Dic₁₇₈

D₄

yes

—

Dic₃₅₆

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

D₄

yes

characteristic

#13083553

—

#13083553

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₂×Q₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

Q₁₆

yes

complemented

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

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

SD₁₆

yes

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

#120

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₄×C₈₉

C₂×Q₁₆

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

Q₈²

—

—

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

D₈₉

yes

characteristic

C₂×Q₁₆

—

178

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

178

—

C₄⋊Q₈

—

178

—

C₄⋊Q₈

—

178

—

C₄×Q₈

—

C₄×Q₈

—

178

—

100

C₄×Q₈

—

178

—

101

C₄×Q₈

C₂×D₈₉

yes

characteristic

102

#106

—

178

—

103

C₄⋊C₈

—

104

C₄⋊C₈

C₂×D₈₉

yes

characteristic

105

#103

—

178

—

106

#103

—

178

—

107

#103

—

178

—

108

C₄×C₈

C₂×D₈₉

yes

characteristic

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

178

—

112

C₂×Q₈

—

178

—

113

C₂×Q₈

D₃₅₆

yes

characteristic

114

Q₁₆

—

356

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

178

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

178

—

121

C₄⋊C₄

—

178

—

122

C₄⋊C₄

—

178

—

123

C₄⋊C₄

—

178

—

124

C₄⋊C₄

—

178

—

125

C₄⋊C₄

—

178

—

126

C₄⋊C₄

—

178

—

127

C₄⋊C₄

D₃₅₆

yes

characteristic

128

C₄²

—

129

C₄²

—

178

—

130

C₄²

—

178

—

131

C₄²

C₂²×D₈₉

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

356

—

135

Q₈

—

356

—

136

Q₈

—

356

—

137

Q₈

—

178

—

138

Q₈

—

712

—

139

Q₈

Dic₃₅₆

yes

complemented

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

356

—

143

C₂×C₄

—

178

—

144

C₂×C₄

—

178

—

145

C₂×C₄

—

178

—

146

C₂×C₄

D₄×D₈₉

yes

characteristic

147

C₂×C₄

D₄×D₈₉

yes

characteristic

148

C₂×C₄

C₂×D₃₅₆

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

356

—

152

C₂²

#15667982

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

356

—

157

C₄

—

356

—

158

C₄

—

178

—

159

C₄

—

712

—

160

C₄

SD₁₆×D₈₉

yes

characteristic

161

C₄

C₂×Dic₃₅₆

yes

characteristic

162

C₄

#15668044

yes

characteristic

163

C₄

#15668045

yes

characteristic

164

C₂

#19151738

yes

characteristic

165

C₂

#19151756

yes

characteristic

166

C₂

#19151777

yes

characteristic

	subgroup	G/H
center	C₂²	#15667982
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	Q₈⋊C₈×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	#120
hypercenter	Q₈⋊C₈	D₈₉
hypocenter	C₈₉	#1645
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂²

#15667982

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

Q₈⋊C₈×C₈₉

C₂

socle, prime radical

C₂²×C₈₉

#120

hypercenter

Q₈⋊C₈

D₈₉

hypocenter

C₈₉

#1645

LCS

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
49. Q₈×C₈₉	Q₁₆	semidirect
84. C₈₉	#1645	semidirect
139. Q₈	Dic₃₅₆	semidirect

H ≅ G/N

type

49. Q₈×C₈₉

Q₁₆

semidirect

84. C₈₉

#1645

semidirect

139. Q₈

Dic₃₅₆

semidirect

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

subgroup	index
Q₈×Dic₁₇₈	2
#19151949	2
#19151883	2
#19151809	2
#19151346	2
#19151330	2
Q₈⋊C₈×C₈₉	2
#1645	89

subgroup

index

Q₈×Dic₁₇₈

Q₈⋊C₈×C₈₉

Subgroups — #26019476

Conjugacy classes of subgroups

Complements

Maximal subgroups