Finite Group Databasebeta

← #26026125

Subgroups — #26026125

Order 11392.

512 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15668003C211yesmaximal, characteristic, complemented
2C2×#15667972C211yesmaximal, characteristic
3#19151535C211yesmaximal, characteristic, complemented
4#19151534C211yesmaximal, characteristic, complemented
5#19151532C212yesmaximal, complemented
6#19151524C211yesmaximal, characteristic, complemented
7#19151513C212yesmaximal
8#19151473C212yesmaximal, complemented
9#19151472C212yesmaximal, complemented
10#19151465C211yesmaximal, characteristic, complemented
11#397×C89C211yesmaximal, characteristic, complemented
12C22×C22⋊D89C2211yescharacteristic
13C23×Dic89C2211yescharacteristic
14C22×C4×D89C2211yescharacteristic
15#15668003—24no—
16#15667972—24no—
17C2×C4⋊Dic89C2212yes—
18C2×C4⋊Dic89C2211yescharacteristic
19C2×C4⋊Dic89C2212yes—
20C2×#13083553C2212yes—
21C2×#13083553C2211yescharacteristic
22C2×#13083553C2212yes—
23C2×C22⋊Dic89C2212yes—
24C2×#13083551C2212yes—
25C2×#13083551C2211yescharacteristic
26C2×#13083551C2212yes—
27C2×C4×Dic89C2211yescharacteristic
28#15667919C2212yes—
29#15667919C2212yes—
30#15667919C2211yescharacteristic
31#15667919C2212yes—
32C2×C4⋊C4×C89C2211yescharacteristic
33C2×C22⋊C4×C89C2212yes—
34C2×C22⋊C4×C89C2211yescharacteristic
35#95×C89C2211yescharacteristic
36#95×C89C2212yes—
37C23×D89C2311yescharacteristic
38C2×C22⋊D89—22no—
39C2×C22⋊D89—22no—
40C2×C22⋊D89—22no—
41C22×Dic89—22no—
42C22×Dic89—42no—
43C22×Dic89C2311yescharacteristic
44C22×Dic89C2312yes—
45C22×Dic89C2312yes—
46C22×Dic89C2311yescharacteristic
47C22×Dic89C2311yescharacteristic
48C2×C4×D89—22no—
49C2×C4×D89—42no—
50C4⋊Dic89—22no—
51C4⋊Dic89—24no—
52C4⋊Dic89—42no—
53#13083553—22no—
54#13083553—24no—
55#13083553—42no—
56C22⋊Dic89—42no—
57#13083551—24no—
58#13083551—22no—
59#13083551—42no—
60C4×Dic89D414yes—
61C24×C89C2311yescharacteristic
62C22×C4×C89C2311yescharacteristic
63C22×C4×C89C2312yes—
64C22×C4×C89C2312yes—
65C22×C4×C89C2311yescharacteristic
66C4⋊C4×C89—22no—
67C22⋊C4×C89—24no—
68C22⋊C4×C89—22no—
69C23×C89—21no—
70C23×C89—21no—
71C23×C89—22no—
72C23×C89—22no—
73C23×C89#4112yes—
74C23×C89C2411yescharacteristic
75C22×D89—21no—
76C22×D89—21no—
77C22×D89—22no—
78C22×D89—22no—
79C22×D89#4112yes—
80C2×C4×C89—22no—
81C2×C4×C89—21no—
82C2×C4×C89—22no—
83C2×C4×C89—22no—
84C2×C4×C89—22no—
85C2×C4×C89—22no—
86C2×C4×C89—22no—
87C2×C4×C89—22no—
88C2×C4×C89—22no—
89C2×C4×C89C2×D412yes—
90C22⋊D89—28no—
91C2×Dic89—21no—
92C2×Dic89—21no—
93C2×Dic89—22no—
94C2×Dic89—21no—
95C2×Dic89—22no—
96C2×Dic89—22no—
97C2×Dic89—22no—
98C2×Dic89—22no—
99C2×Dic89—22no—
100C2×Dic89—22no—
101C2×Dic89—22no—
102C2×Dic89—44no—
103C2×Dic89C2×D412yes—
104C2×Dic89C2×D412yes—
105C4×D89—44no—
106C22×C89—22no—
107C22×C89—24no—
108C22×C89—41no—
109C22×C89—41no—
110C22×C89—42no—
111C22×C89C22×D411yescharacteristic
112C22×C89C2×#4111yescharacteristic
113C22×C89C2×#4111yescharacteristic
114C22×C89H(5,𝔽2)12yes—
115C22×C892−1+412yes—
116C2×D89—22no—
117C2×D89—24no—
118C2×D89—41no—
119C2×D89—41no—
120C2×D89—42no—
121C4×C89—22no—
122C4×C89—42no—
123C4×C89—41no—
124C4×C89—42no—
125Dic89—22no—
126Dic89—22no—
127Dic89—42no—
128Dic89—41no—
129Dic89—42no—
130C2×C89—42no—
131C2×C89#52511yescharacteristic
132C2×C89#53511yescharacteristic
133C2×C89#53611yescharacteristic
134C2×C89#53912yes—
135C2×C89#53912yes—
136D89—42no—
137#2633—891nomaximal, Sylow
138C89#263311yescharacteristic, complemented, Sylow
139C2×#125—891no—
140C2×C4×D4—891no—
141#399—891no—
142#399—891no—
143#398—892no—
144#397—891no—
145#397D8911yescharacteristic, complemented
146#394—892no—
147#385—892no—
148#385—892no—
149#383—891no—
150C22×D4—891no—
151C23×C4—891no—
152C23×C4—891no—
153#125—1784no—
154C4×D4—1784no—
155C2×C4⋊C4—892no—
156C2×C4⋊C4—892no—
157C2×C4⋊C4—891no—
158C2×C4⋊C4—892no—
159C2×C4⋊C4—892no—
160C2×C4⋊C4—891no—
161C2×C4⋊C4C2×D8911yescharacteristic
162C2×C22⋊C4—892no—
163C2×C22⋊C4—892no—
164C2×C22⋊C4—891no—
165C2×C22⋊C4—892no—
166C2×C22⋊C4C2×D8911yescharacteristic
167C2×C22⋊C4C2×D8912yes—
168C2×C42—891no—
169#95—892no—
170#95—892no—
171#95—891no—
172#95—892no—
173#95C2×D8911yescharacteristic
174#95C2×D8912yes—
175C24—891no—
176C24C22×D8911yescharacteristic
177C2×D4—1782no—
178C2×D4—1782no—
179C2×D4—1782no—
180C22×C4—891no—
181C22×C4—891no—
182C22×C4—892no—
183C22×C4—891no—
184C22×C4—892no—
185C22×C4—3562no—
186C22×C4—3562no—
187C22×C4—1782no—
188C22×C4—1782no—
189C22×C4C22×D8912yes—
190C22×C4C22×D8912yes—
191C22×C4C22×D8911yescharacteristic
192C22×C4C22×D8911yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—3562no—
195C4⋊C4—3562no—
196C4⋊C4—1782no—
197C4⋊C4—1784no—
198C4⋊C4—1784no—
199C4⋊C4—1782no—
200C22⋊C4—24no—
201C22⋊C4—22no—
202C22⋊C4—3562no—
203C22⋊C4—3562no—
204C22⋊C4—1784no—
205C22⋊C4—1782no—
206C42—894no—
207C23—21no—
208C23—21no—
209C23—22no—
210C23—22no—
211C23—892no—
212C23—1781no—
213C23—1781no—
214C23—1782no—
215C23—1782no—
216C23#1308357312yes—
217C23C23×D8911yescharacteristic
218D4—1788no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—892no—
229C2×C4—892no—
230C2×C4—3564no—
231C2×C4—3564no—
232C2×C4—1781no—
233C2×C4—1782no—
234C2×C4—1782no—
235C2×C4—1782no—
236C2×C4—1782no—
237C2×C4—1782no—
238C2×C4—1782no—
239C2×C4—1782no—
240C2×C4—1781no—
241C2×C4—1781no—
242C2×C4—1782no—
243C2×C4D4×D8912yes—
244C22—24no—
245C22—22no—
246C22—41no—
247C22—41no—
248C22—42no—
249C22—3561no—
250C22—3561no—
251C22—3562no—
252C22—1784no—
253C22—1782no—
254C22C2×D4×D8911yescharacteristic
255C22#41×D8911yescharacteristic
256C22C2×#1308357311yescharacteristic
257C22#1566805812yes—
258C22#1566805912yes—
259C4—22no—
260C4—41no—
261C4—42no—
262C4—42no—
263C4—3561no—
264C4—3562no—
265C4—3562no—
266C4—1782no—
267C4—1782no—
268C2—42no—
269C2—3562no—
270C2#1915210811yescharacteristic
271C2#1915216611yescharacteristic
272C2#1915217011yescharacteristic
273C2#1915218912yes—
274C2#1915219212yes—
subgroupG/H
center, FrattiniC23C23×D89
commutator, socle, cosocleC23×C89C24
Fitting#397×C89C2
prime radicalC22×C22⋊D89C22
hypercenter#397D89
hypocenterC89#2633
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C23 ◁ #397

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#15668003C2semidirect
3. #19151535C2semidirect
4. #19151534C2semidirect
5. #19151532C2semidirect
6. #19151524C2semidirect
8. #19151473C2semidirect
9. #19151472C2semidirect
10. #19151465C2semidirect
11. #397×C89C2semidirect
138. C89#2633semidirect
145. #397D89semidirect

Maximal subgroups

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

subgroupindex
C2×#156680032
C2×#156679722
#191515352
#191515342
#191515322
#191515242
#191515132
#191514732
#191514722
#191514652
#397×C892
#263389