Group #26031474 — subgroups

Subgroups — #26031474

Order 11392.

362 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 113 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⋊Q₈×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152133	C₂	1	1	yes	maximal, characteristic
3	#19152008	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151960	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151958	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151930	C₂	1	2	yes	maximal, complemented
7	#19151913	C₂	1	2	yes	maximal
8	#19151892	C₂	1	2	yes	maximal, complemented
9	#19151887	C₂	1	2	yes	maximal, complemented
10	#19151653	C₂	1	1	yes	maximal, characteristic, complemented
11	#501×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₈₉	—	2	1	no	—
13	#15668012	C₂²	1	1	yes	characteristic
14	#15668011	—	2	1	no	—
15	#15668011	C₂²	1	2	yes	—
16	#15668010	—	2	1	no	—
17	#15668007	—	2	1	no	—
18	#15668000	C₂²	1	1	yes	characteristic
19	#15667997	—	2	1	no	—
20	#15667992	C₂²	1	1	yes	characteristic
21	#15667987	C₂²	1	1	yes	characteristic
22	#15667977	C₂²	1	1	yes	characteristic
23	#15667974	—	2	1	no	—
24	#15667974	C₂²	1	1	yes	characteristic
25	#15667970	—	2	1	no	—
26	C₄⋊C₄×D₈₉	—	2	1	no	—
27	C₄⋊C₄×D₈₉	C₂²	1	2	yes	—
28	C₄²×D₈₉	C₂²	1	1	yes	characteristic
29	#15667945	C₂²	1	2	yes	—
30	#15667943	C₂²	1	2	yes	—
31	#15667941	C₂²	1	2	yes	—
32	#15667937	C₂²	1	2	yes	—
33	#15667935	C₂²	1	2	yes	—
34	#15667934	C₂²	1	2	yes	—
35	#15667932	C₂²	1	2	yes	—
36	#15667924	C₂²	1	2	yes	—
37	#15667923	C₂²	1	1	yes	characteristic
38	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
39	#125×C₈₉	C₂²	1	1	yes	characteristic
40	#107×C₈₉	C₂²	1	2	yes	—
41	#106×C₈₉	C₂²	1	2	yes	—
42	#97×C₈₉	C₂²	1	1	yes	characteristic
43	C₂×Dic₁₇₈	—	2	1	no	—
44	C₂×Dic₁₇₈	—	2	2	no	—
45	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
46	Q₈×D₈₉	—	4	4	no	—
47	C₂×D₃₅₆	—	2	1	no	—
48	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
49	C₂×C₄×D₈₉	—	2	2	no	—
50	C₂×C₄×D₈₉	—	2	1	no	—
51	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
52	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
53	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
54	C₂×#10623494	C₂³	1	2	yes	—
55	C₄⋊Dic₈₉	—	2	1	no	—
56	C₄⋊Dic₈₉	—	2	1	no	—
57	C₄⋊Dic₈₉	C₂³	1	2	yes	—
58	#13083553	—	2	1	no	—
59	#13083553	—	2	2	no	—
60	#13083553	—	2	2	no	—
61	#13083551	—	2	1	no	—
62	#13083551	—	2	1	no	—
63	#13083551	—	2	1	no	—
64	#13083551	—	2	1	no	—
65	#13083551	—	2	1	no	—
66	C₄×Dic₈₉	—	2	1	no	—
67	C₄×Dic₈₉	D₄	1	1	yes	characteristic
68	C₄×Dic₈₉	D₄	1	1	yes	characteristic
69	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
70	C₂×Q₈×C₈₉	—	2	1	no	—
71	C₂×C₈×C₈₉	C₂³	1	2	yes	—
72	C₄⋊C₄×C₈₉	—	2	1	no	—
73	C₄⋊C₄×C₈₉	—	2	1	no	—
74	C₄⋊C₄×C₈₉	—	2	1	no	—
75	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
78	C₄²×C₈₉	C₂³	1	1	yes	characteristic
79	C₂²×D₈₉	—	2	1	no	—
80	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
81	Q₈×C₈₉	—	4	2	no	—
82	C₂×C₄×C₈₉	—	2	1	no	—
83	C₂×C₄×C₈₉	—	2	1	no	—
84	C₂×C₄×C₈₉	—	2	2	no	—
85	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
88	C₂×Dic₈₉	—	2	1	no	—
89	C₂×Dic₈₉	—	2	2	no	—
90	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
91	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
92	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
93	D₃₅₆	—	2	2	no	—
94	D₃₅₆	—	4	2	no	—
95	C₄×D₈₉	—	2	2	no	—
96	C₄×D₈₉	—	2	2	no	—
97	C₄×D₈₉	—	2	2	no	—
98	C₄×D₈₉	—	4	4	no	—
99	C₄×D₈₉	—	4	2	no	—
100	Dic₁₇₈	—	2	2	no	—
101	Dic₁₇₈	—	4	4	no	—
102	Dic₁₇₈	—	4	2	no	—
103	C₈×C₈₉	—	2	2	no	—
104	#10623494	—	2	2	no	—
105	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
106	C₂×D₈₉	—	2	1	no	—
107	C₂×D₈₉	—	2	1	no	—
108	C₂×D₈₉	—	4	1	no	—
109	C₂×D₈₉	—	4	1	no	—
110	C₂×D₈₉	—	4	1	no	—
111	C₂×D₈₉	AGL(1,ℤ/8)	1	2	yes	—
112	C₄×C₈₉	—	2	1	no	—
113	C₄×C₈₉	—	2	1	no	—
114	C₄×C₈₉	—	4	2	no	—
115	C₄×C₈₉	—	4	1	no	—
116	C₄×C₈₉	—	4	1	no	—
117	C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
118	Dic₈₉	—	2	1	no	—
119	Dic₈₉	—	2	1	no	—
120	Dic₈₉	—	4	1	no	—
121	Dic₈₉	—	4	2	no	—
122	Dic₈₉	#137	1	2	yes	—
123	C₂×C₈₉	#536	1	1	yes	characteristic
124	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
126	D₈₉	—	2	2	no	—
127	D₈₉	—	8	1	no	—
128	#3238	—	89	1	no	maximal, Sylow
129	C₈₉	#3238	1	1	yes	characteristic, complemented, Sylow
130	C₂×C₄⋊Q₈	—	89	1	no	—
131	#529	—	89	1	no	—
132	#501	—	89	1	no	—
133	#501	D₈₉	1	1	yes	characteristic, complemented
134	#489	—	89	1	no	—
135	#489	—	89	1	no	—
136	#483	—	89	2	no	—
137	#480	—	89	2	no	—
138	#477	—	89	2	no	—
139	#476	—	89	2	no	—
140	#432	—	89	1	no	—
141	C₂²×Q₈	—	178	1	no	—
142	C₄⋊Q₈	—	89	2	no	—
143	C₄⋊Q₈	—	89	1	no	—
144	C₄⋊Q₈	—	178	1	no	—
145	C₄⋊Q₈	—	178	1	no	—
146	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
147	#126	—	178	1	no	—
148	#125	C₂×D₈₉	1	1	yes	characteristic
149	#124	—	89	1	no	—
150	#123	—	178	1	no	—
151	C₂²⋊Q₈	—	89	1	no	—
152	#121	—	89	1	no	—
153	C₄×Q₈	—	89	1	no	—
154	C₄×D₄	—	89	1	no	—
155	C₄×D₄	—	178	1	no	—
156	#117	—	178	1	no	—
157	C₂×C₄⋊C₄	—	89	2	no	—
158	C₂×C₄⋊C₄	—	178	1	no	—
159	C₂×C₄²	—	89	1	no	—
160	#107	—	89	2	no	—
161	#107	C₂×D₈₉	1	2	yes	—
162	#106	—	89	2	no	—
163	#106	C₂×D₈₉	1	2	yes	—
164	C₄⋊C₈	—	89	2	no	—
165	#103	—	89	2	no	—
166	#103	—	89	2	no	—
167	#102	—	89	2	no	—
168	#102	—	89	2	no	—
169	C₂²⋊C₈	—	89	2	no	—
170	#97	—	89	1	no	—
171	#97	C₂×D₈₉	1	1	yes	characteristic
172	C₂×Q₈	—	2	1	no	—
173	C₂×Q₈	—	89	1	no	—
174	C₂×Q₈	—	356	4	no	—
175	C₂×Q₈	—	178	2	no	—
176	C₂×Q₈	—	178	1	no	—
177	C₂×D₄	—	89	1	no	—
178	C₂×D₄	—	178	1	no	—
179	C₂²×C₄	—	89	1	no	—
180	C₂²×C₄	—	89	1	no	—
181	C₂²×C₄	—	89	1	no	—
182	C₂²×C₄	—	178	1	no	—
183	C₂²×C₄	—	178	2	no	—
184	C₂×C₈	—	89	2	no	—
185	C₂×C₈	C₂²×D₈₉	1	2	yes	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	89	2	no	—
190	C₄⋊C₄	—	178	2	no	—
191	C₄⋊C₄	—	178	2	no	—
192	C₄⋊C₄	—	178	1	no	—
193	C₄⋊C₄	—	178	1	no	—
194	C₄⋊C₄	—	178	1	no	—
195	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
196	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
197	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
198	C₂²⋊C₄	—	178	1	no	—
199	C₂²⋊C₄	—	178	1	no	—
200	C₂²⋊C₄	—	178	1	no	—
201	C₂²⋊C₄	—	178	1	no	—
202	C₂²⋊C₄	—	178	1	no	—
203	C₄²	—	89	1	no	—
204	C₄²	—	89	1	no	—
205	C₄²	—	89	1	no	—
206	C₄²	—	178	1	no	—
207	C₄²	C₂²×D₈₉	1	1	yes	characteristic
208	C₂³	—	89	1	no	—
209	C₂³	—	178	1	no	—
210	Q₈	—	4	2	no	—
211	Q₈	—	356	2	no	—
212	Q₈	—	356	4	no	—
213	Q₈	—	178	2	no	—
214	D₄	—	356	2	no	—
215	D₄	—	178	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	89	1	no	—
220	C₂×C₄	—	89	1	no	—
221	C₂×C₄	—	89	1	no	—
222	C₂×C₄	—	356	2	no	—
223	C₂×C₄	—	356	4	no	—
224	C₂×C₄	—	178	2	no	—
225	C₂×C₄	—	178	2	no	—
226	C₂×C₄	—	178	1	no	—
227	C₂×C₄	—	178	2	no	—
228	C₂×C₄	—	178	2	no	—
229	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
230	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
231	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	—	178	2	no	—
234	C₂²	—	89	2	no	—
235	C₂²	—	356	1	no	—
236	C₂²	—	356	1	no	—
237	C₂²	—	356	1	no	—
238	C₂²	—	178	1	no	—
239	C₂²	—	178	1	no	—
240	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	4	1	no	—
244	C₄	—	4	2	no	—
245	C₄	—	4	1	no	—
246	C₄	—	89	2	no	—
247	C₄	—	356	1	no	—
248	C₄	—	356	2	no	—
249	C₄	—	178	1	no	—
250	C₄	—	178	1	no	—
251	C₄	#15668060	1	2	yes	—
252	C₂	—	178	2	no	—
253	C₂	—	712	1	no	—
254	C₂	#19152173	1	1	yes	characteristic
255	C₂	AGL(1,ℤ/8)×D₈₉	1	1	yes	characteristic
256	C₂	#137×D₈₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊Q₈×D₈₉

C₂

yes

maximal, characteristic, complemented

#19152133

C₂

yes

maximal, characteristic

#19152008

C₂

yes

maximal, characteristic, complemented

#19151960

C₂

yes

maximal, characteristic, complemented

#19151958

C₂

yes

maximal, characteristic, complemented

#19151930

C₂

yes

maximal, complemented

#19151913

C₂

yes

maximal

#19151892

C₂

yes

maximal, complemented

#19151887

C₂

yes

maximal, complemented

#19151653

C₂

yes

maximal, characteristic, complemented

#501×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₈₉

—

#15668012

C₂²

yes

characteristic

#15668011

—

#15668011

C₂²

yes

—

#15668010

—

#15668007

—

#15668000

C₂²

yes

characteristic

#15667997

—

#15667992

C₂²

yes

characteristic

#15667987

C₂²

yes

characteristic

#15667977

C₂²

yes

characteristic

#15667974

—

#15667974

C₂²

yes

characteristic

#15667970

—

C₄⋊C₄×D₈₉

—

C₄⋊C₄×D₈₉

C₂²

yes

—

C₄²×D₈₉

C₂²

yes

characteristic

#15667945

C₂²

yes

—

#15667943

C₂²

yes

—

#15667941

C₂²

yes

—

#15667937

C₂²

yes

—

#15667935

C₂²

yes

—

#15667934

C₂²

yes

—

#15667932

C₂²

yes

—

#15667924

C₂²

yes

—

#15667923

C₂²

yes

characteristic

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

#125×C₈₉

C₂²

yes

characteristic

#107×C₈₉

C₂²

yes

—

#106×C₈₉

C₂²

yes

—

#97×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

C₂³

yes

characteristic

Q₈×D₈₉

—

C₂×D₃₅₆

—

C₂×D₃₅₆

C₂³

yes

characteristic

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

D₄

yes

characteristic

C₂×C₄×D₈₉

D₄

yes

characteristic

C₂×C₄×D₈₉

C₂³

yes

characteristic

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

—

—

—

—

—

—

—

—

C₄×Dic₈₉

—

C₄×Dic₈₉

D₄

yes

characteristic

C₄×Dic₈₉

D₄

yes

characteristic

C₄×Dic₈₉

C₂³

yes

characteristic

C₂×Q₈×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

C₂³

yes

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

C₂×D₄

yes

characteristic

Q₈×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

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

C₂×D₄

yes

characteristic

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

100

Dic₁₇₈

—

101

Dic₁₇₈

—

102

Dic₁₇₈

—

103

C₈×C₈₉

—

104

#10623494

—

105

C₂²×C₈₉

C₂²×D₄

yes

characteristic

106

C₂×D₈₉

—

107

C₂×D₈₉

—

108

C₂×D₈₉

—

109

C₂×D₈₉

—

110

C₂×D₈₉

—

111

C₂×D₈₉

AGL(1,ℤ/8)

yes

—

112

C₄×C₈₉

—

113

C₄×C₈₉

—

114

C₄×C₈₉

—

115

C₄×C₈₉

—

116

C₄×C₈₉

—

117

C₄×C₈₉

2₋¹⁺⁴

yes

—

118

Dic₈₉

—

119

Dic₈₉

—

120

Dic₈₉

—

121

Dic₈₉

—

122

Dic₈₉

#137

yes

—

123

C₂×C₈₉

#536

yes

characteristic

124

C₂×C₈₉

C₂×AGL(1,ℤ/8)

yes

characteristic

125

C₂×C₈₉

C₂×#137

yes

characteristic

126

D₈₉

—

127

D₈₉

—

128

#3238

—

maximal, Sylow

129

C₈₉

#3238

yes

characteristic, complemented, Sylow

130

C₂×C₄⋊Q₈

—

131

#529

—

132

#501

—

133

#501

D₈₉

yes

characteristic, complemented

134

#489

—

135

#489

—

136

#483

—

137

#480

—

138

#477

—

139

#476

—

140

#432

—

141

C₂²×Q₈

—

178

—

142

C₄⋊Q₈

—

143

C₄⋊Q₈

—

144

C₄⋊Q₈

—

178

—

145

C₄⋊Q₈

—

178

—

146

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

147

#126

—

178

—

148

#125

C₂×D₈₉

yes

characteristic

149

#124

—

150

#123

—

178

—

151

C₂²⋊Q₈

—

152

#121

—

153

C₄×Q₈

—

154

C₄×D₄

—

155

C₄×D₄

—

178

—

156

#117

—

178

—

157

C₂×C₄⋊C₄

—

158

C₂×C₄⋊C₄

—

178

—

159

C₂×C₄²

—

160

#107

—

161

#107

C₂×D₈₉

yes

—

162

#106

—

163

#106

C₂×D₈₉

yes

—

164

C₄⋊C₈

—

165

#103

—

166

#103

—

167

#102

—

168

#102

—

169

C₂²⋊C₈

—

170

#97

—

171

#97

C₂×D₈₉

yes

characteristic

172

C₂×Q₈

—

173

C₂×Q₈

—

174

C₂×Q₈

—

356

—

175

C₂×Q₈

—

178

—

176

C₂×Q₈

—

178

—

177

C₂×D₄

—

178

C₂×D₄

—

178

—

179

C₂²×C₄

—

180

C₂²×C₄

—

181

C₂²×C₄

—

182

C₂²×C₄

—

178

—

183

C₂²×C₄

—

178

—

184

C₂×C₈

—

185

C₂×C₈

C₂²×D₈₉

yes

—

186

C₄⋊C₄

—

187

C₄⋊C₄

—

188

C₄⋊C₄

—

189

C₄⋊C₄

—

190

C₄⋊C₄

—

178

—

191

C₄⋊C₄

—

178

—

192

C₄⋊C₄

—

178

—

193

C₄⋊C₄

—

178

—

194

C₄⋊C₄

—

178

—

195

C₄⋊C₄

C₂²×D₈₉

yes

—

196

C₄⋊C₄

C₂²×D₈₉

yes

characteristic

197

C₄⋊C₄

C₂²×D₈₉

yes

characteristic

198

C₂²⋊C₄

—

178

—

199

C₂²⋊C₄

—

178

—

200

C₂²⋊C₄

—

178

—

201

C₂²⋊C₄

—

178

—

202

C₂²⋊C₄

—

178

—

203

C₄²

—

204

C₄²

—

205

C₄²

—

206

C₄²

—

178

—

207

C₄²

C₂²×D₈₉

yes

characteristic

208

C₂³

—

209

C₂³

—

178

—

210

Q₈

—

211

Q₈

—

356

—

212

Q₈

—

356

—

213

Q₈

—

178

—

214

D₄

—

356

—

215

D₄

—

178

—

216

C₂×C₄

—

217

C₂×C₄

—

218

C₂×C₄

—

219

C₂×C₄

—

220

C₂×C₄

—

221

C₂×C₄

—

222

C₂×C₄

—

356

—

223

C₂×C₄

—

356

—

224

C₂×C₄

—

178

—

225

C₂×C₄

—

178

—

226

C₂×C₄

—

178

—

227

C₂×C₄

—

178

—

228

C₂×C₄

—

178

—

229

C₂×C₄

D₄×D₈₉

yes

characteristic

230

C₂×C₄

D₄×D₈₉

yes

characteristic

231

C₂×C₄

C₂³×D₈₉

yes

characteristic

232

C₈

—

233

C₈

—

178

—

234

C₂²

—

235

C₂²

—

356

—

236

C₂²

—

356

—

237

C₂²

—

356

—

238

C₂²

—

178

—

239

C₂²

—

178

—

240

C₂²

C₂×D₄×D₈₉

yes

characteristic

241

C₄

—

242

C₄

—

243

C₄

—

244

C₄

—

245

C₄

—

246

C₄

—

247

C₄

—

356

—

248

C₄

—

356

—

249

C₄

—

178

—

250

C₄

—

178

—

251

C₄

#15668060

yes

—

252

C₂

—

178

—

253

C₂

—

712

—

254

C₂

#19152173

yes

characteristic

255

C₂

AGL(1,ℤ/8)×D₈₉

yes

characteristic

256

C₂

#137×D₈₉

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#501×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#15667987	C₂²
hypercenter	#501	D₈₉
hypocenter	C₈₉	#3238
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #501

subgroup

G/H

center

C₂²

C₂×D₄×D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

#501×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

C₂²

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #501

N	H ≅ G/N	type
1. C₄⋊Q₈×D₈₉	C₂	semidirect
3. #19152008	C₂	semidirect
4. #19151960	C₂	semidirect
5. #19151958	C₂	semidirect
6. #19151930	C₂	semidirect
8. #19151892	C₂	semidirect
9. #19151887	C₂	semidirect
10. #19151653	C₂	semidirect
11. #501×C₈₉	C₂	semidirect
129. C₈₉	#3238	semidirect
133. #501	D₈₉	semidirect

H ≅ G/N

type

1. C₄⋊Q₈×D₈₉

C₂

semidirect

3. #19152008

C₂

semidirect

4. #19151960

C₂

semidirect

5. #19151958

C₂

semidirect

6. #19151930

C₂

semidirect

8. #19151892

C₂

semidirect

9. #19151887

C₂

semidirect

10. #19151653

C₂

semidirect

11. #501×C₈₉

C₂

semidirect

129. C₈₉

#3238

semidirect

133. #501

D₈₉

semidirect

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

subgroup	index
C₄⋊Q₈×D₈₉	2
#19152133	2
#19152008	2
#19151960	2
#19151958	2
#19151930	2
#19151913	2
#19151892	2
#19151887	2
#19151653	2
#501×C₈₉	2
#3238	89

subgroup

index

C₄⋊Q₈×D₈₉

#501×C₈₉

Subgroups — #26031474

Conjugacy classes of subgroups

Complements

Maximal subgroups