Finite Group Databasebeta

← #26026405

Subgroups — #26026405

Order 11392.

498 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 133 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×#15667994C211yesmaximal, characteristic
2C2×#15667992C211yesmaximal, characteristic, complemented
3#19151536C211yesmaximal, characteristic, complemented
4#19151524C212yesmaximal, complemented
5#19151519C211yesmaximal, characteristic, complemented
6#19151514C212yesmaximal
7#19151510C211yesmaximal, characteristic, complemented
8#19151497C212yesmaximal, complemented
9#19151491C212yesmaximal, complemented
10#19151445C211yesmaximal, characteristic, complemented
11#399×C89C211yesmaximal, characteristic, complemented
12C22×Dic178C2211yescharacteristic
13C22×C22⋊D89C2211yescharacteristic
14C22×C4×D89C2211yescharacteristic
15#15667994—24no—
16#15667992—24no—
17C2×C4⋊Dic89C2212yes—
18C2×C4⋊Dic89C2211yescharacteristic
19C2×C4⋊Dic89C2212yes—
20C2×#13083553C2211yescharacteristic
21C2×C22⋊Dic89C2211yescharacteristic
22C2×C22⋊Dic89C2212yes—
23C2×#13083551C2212yes—
24C2×#13083551C2212yes—
25C2×#13083551C2211yescharacteristic
26C2×C4×Dic89C2212yes—
27#15667919C2211yescharacteristic
28#15667919C2212yes—
29#15667919C2212yes—
30#15667919C2212yes—
31#15667919C2212yes—
32C2×C4⋊C4×C89C2212yes—
33C2×C4⋊C4×C89C2211yescharacteristic
34C2×C22⋊C4×C89C2212yes—
35C2×C22⋊C4×C89C2211yescharacteristic
36#95×C89C2211yescharacteristic
37C23×D89C2311yescharacteristic
38C2×Dic178—22no—
39C2×Dic178—42no—
40C2×C22⋊D89—22no—
41C2×C22⋊D89—42no—
42C22×Dic89C2311yescharacteristic
43C22×Dic89C2312yes—
44C22×Dic89C2312yes—
45C22×Dic89C2311yescharacteristic
46C22×Dic89C2311yescharacteristic
47C2×C4×D89—42no—
48C2×C4×D89D414yes—
49C4⋊Dic89—24no—
50C4⋊Dic89—22no—
51C4⋊Dic89—42no—
52#13083553—22no—
53C22⋊Dic89—22no—
54C22⋊Dic89—42no—
55#13083551—24no—
56#13083551—22no—
57#13083551—24no—
58C4×Dic89—42no—
59C24×C89C2311yescharacteristic
60C22×C4×C89C2311yescharacteristic
61C22×C4×C89C2312yes—
62C22×C4×C89C2311yescharacteristic
63C22×C4×C89C2312yes—
64C4⋊C4×C89—22no—
65C4⋊C4×C89—24no—
66C22⋊C4×C89—22no—
67C22⋊C4×C89—24no—
68C23×C89—21no—
69C23×C89—22no—
70C23×C89—22no—
71C23×C89—21no—
72C23×C89—21no—
73C23×C89C2411yescharacteristic
74C22×D89—22no—
75C22×D89—22no—
76C22×D89—21no—
77C22×D89—21no—
78C22×D89C2×D412yes—
79C2×C4×C89—22no—
80C2×C4×C89—22no—
81C2×C4×C89—22no—
82C2×C4×C89—21no—
83C2×C4×C89—22no—
84C2×C4×C89—22no—
85C2×C4×C89—22no—
86C2×C4×C89—22no—
87C2×C4×C89C2×D412yes—
88C2×C4×C89#4114yes—
89C22⋊D89—44no—
90C2×Dic89—22no—
91C2×Dic89—21no—
92C2×Dic89—22no—
93C2×Dic89—21no—
94C2×Dic89—22no—
95C2×Dic89—22no—
96C2×Dic89—22no—
97C2×Dic89—22no—
98C2×Dic89—22no—
99C2×Dic89—21no—
100C2×Dic89—21no—
101C2×Dic89—22no—
102C2×Dic89C2×D412yes—
103C4×D89—44no—
104Dic178—44no—
105C22×C89—41no—
106C22×C89—41no—
107C22×C89—41no—
108C22×C89—42no—
109C22×C89—42no—
110C22×C89C22×D411yescharacteristic
111C22×C89C2×#4112yes—
112C22×C89H(5,𝔽2)12yes—
113C22×C892−1+411yescharacteristic
114C22×C892−1+411yescharacteristic
115C2×D89—22no—
116C2×D89—22no—
117C2×D89—22no—
118C2×D89—42no—
119C2×D89—42no—
120C4×C89—22no—
121C4×C89—24no—
122C4×C89—41no—
123C4×C89—42no—
124Dic89—22no—
125Dic89—42no—
126Dic89—41no—
127Dic89—41no—
128Dic89—42no—
129C2×C89—81no—
130C2×C89#53211yescharacteristic
131C2×C89#53411yescharacteristic
132C2×C89#53611yescharacteristic
133C2×C89#53912yes—
134C2×C89#54212yes—
135D89—42no—
136#2663—891nomaximal, Sylow
137C89#266311yescharacteristic, complemented, Sylow
138C2×#123—891no—
139C2×C22⋊Q8—891no—
140#400—891no—
141#399D8911yescharacteristic, complemented
142#397—892no—
143#395—891no—
144#394—892no—
145#393—891no—
146#389—892no—
147#388—892no—
148C4×C22⋊C4—891no—
149C22×Q8—891no—
150C22×D4—891no—
151C23×C4—891no—
152#123—1784no—
153C22⋊Q8—1784no—
154C2×C4⋊C4—892no—
155C2×C4⋊C4—892no—
156C2×C4⋊C4—891no—
157C2×C4⋊C4—891no—
158C2×C4⋊C4C2×D8911yescharacteristic
159C2×C4⋊C4C2×D8912yes—
160C2×C22⋊C4—892no—
161C2×C22⋊C4—891no—
162C2×C22⋊C4—892no—
163C2×C22⋊C4—891no—
164C2×C22⋊C4—892no—
165C2×C22⋊C4C2×D8911yescharacteristic
166C2×C22⋊C4C2×D8912yes—
167C2×C42—892no—
168#95—891no—
169#95—892no—
170#95—892no—
171#95—892no—
172#95—892no—
173#95C2×D8911yescharacteristic
174C24—891no—
175C24C22×D8911yescharacteristic
176C2×Q8—3562no—
177C2×Q8—1782no—
178C2×D4—3562no—
179C2×D4—1782no—
180C22×C4—891no—
181C22×C4—894no—
182C22×C4—892no—
183C22×C4—891no—
184C22×C4—892no—
185C22×C4—891no—
186C22×C4—3562no—
187C22×C4C22×D8911yescharacteristic
188C22×C4C22×D8912yes—
189C22×C4C22×D8911yescharacteristic
190C22×C4C22×D8912yes—
191C4⋊C4—22no—
192C4⋊C4—24no—
193C4⋊C4—3562no—
194C4⋊C4—1784no—
195C4⋊C4—1782no—
196C4⋊C4—1782no—
197C22⋊C4—22no—
198C22⋊C4—24no—
199C22⋊C4—3562no—
200C22⋊C4—1784no—
201C22⋊C4—1782no—
202C22⋊C4—1782no—
203C22⋊C4—1784no—
204C42—3562no—
205C23—21no—
206C23—22no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—892no—
211C23—1782no—
212C23—1782no—
213C23—1781no—
214C23—1781no—
215C23C23×D8911yescharacteristic
216Q8—3564no—
217D4—3564no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—892no—
227C2×C4—3564no—
228C2×C4—1781no—
229C2×C4—1782no—
230C2×C4—1781no—
231C2×C4—1782no—
232C2×C4—1782no—
233C2×C4—1782no—
234C2×C4—1782no—
235C2×C4—1782no—
236C2×C4—1781no—
237C2×C4—1781no—
238C2×C4—1782no—
239C2×C4—1782no—
240C2×C4D4×D8912yes—
241C2×C4#1308357314yes—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—42no—
246C22—42no—
247C22—3562no—
248C22—3562no—
249C22—1782no—
250C22—1782no—
251C22—1782no—
252C22C2×D4×D8911yescharacteristic
253C22C2×#1308357312yes—
254C22#1566805812yes—
255C22#1566805911yescharacteristic
256C22#1566806011yescharacteristic
257C4—22no—
258C4—24no—
259C4—41no—
260C4—42no—
261C4—3561no—
262C4—3562no—
263C4—3561no—
264C4—3562no—
265C4—1782no—
266C2—81no—
267C2—3562no—
268C2#1915214611yescharacteristic
269C2#1915215911yescharacteristic
270C2#1915217111yescharacteristic
271C2#1915219212yes—
272C2#1915222112yes—
subgroupG/H
center, FrattiniC23C23×D89
commutator, socle, cosocleC23×C89C24
Fitting#399×C89C2
prime radicalC22×C22⋊D89C22
hypercenter#399D89
hypocenterC89#2663
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
2. C2×#15667992C2semidirect
3. #19151536C2semidirect
4. #19151524C2semidirect
5. #19151519C2semidirect
7. #19151510C2semidirect
8. #19151497C2semidirect
9. #19151491C2semidirect
10. #19151445C2semidirect
11. #399×C89C2semidirect
137. C89#2663semidirect
141. #399D89semidirect

Maximal subgroups

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

subgroupindex
C2×#156679942
C2×#156679922
#191515362
#191515242
#191515192
#191515142
#191515102
#191514972
#191514912
#191514452
#399×C892
#266389