Finite Group Databasebeta

← #26025200

Subgroups — #26025200

Order 11392.

676 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 171 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×#15668011C211yesmaximal, characteristic, complemented
2C2×#15667990C212yesmaximal, complemented
3C2×#15667989C212yesmaximal, complemented
4C2×#15667972C211yesmaximal, characteristic
5#19151535C211yesmaximal, characteristic, complemented
6#19151534C211yesmaximal, characteristic, complemented
7#19151533C211yesmaximal, characteristic, complemented
8#19151498C212yesmaximal
9#19151467C212yesmaximal, complemented
10#19151449C211yesmaximal, characteristic, complemented
11#399×C89C211yesmaximal, characteristic, complemented
12C22×Dic178C2212yes—
13C22×C22⋊D89C2211yescharacteristic
14C23×Dic89C2211yescharacteristic
15C22×C4×D89C2211yescharacteristic
16#15668011—24no—
17#15667990—28no—
18#15667989—28no—
19#15667972—24no—
20C2×C4⋊Dic89C2211yescharacteristic
21C2×C4⋊Dic89C2212yes—
22C2×#13083553C2212yes—
23C2×#13083553C2211yescharacteristic
24C2×#13083553C2212yes—
25C2×#13083553C2212yes—
26C2×#13083553C2212yes—
27C2×C22⋊Dic89C2212yes—
28C2×#13083551C2212yes—
29C2×#13083551C2212yes—
30C2×#13083551C2211yescharacteristic
31C2×C4×Dic89C2212yes—
32C2×C4×Dic89C2211yescharacteristic
33#15667919C2211yescharacteristic
34C2×C4⋊C4×C89C2211yescharacteristic
35C2×C4⋊C4×C89C2212yes—
36C2×C22⋊C4×C89C2212yes—
37C2×C22⋊C4×C89C2211yescharacteristic
38#95×C89C2211yescharacteristic
39C23×D89C2311yescharacteristic
40C2×Dic178—24no—
41C2×Dic178—24no—
42C2×Dic178—24no—
43C2×C22⋊D89—22no—
44C2×C22⋊D89—22no—
45C2×C22⋊D89Q814yescomplemented
46C22×Dic89—22no—
47C22×Dic89—24no—
48C22×Dic89C2311yescharacteristic
49C22×Dic89C2312yes—
50C22×Dic89C2312yes—
51C22×Dic89C2311yescharacteristic
52C22×Dic89C2311yescharacteristic
53C2×C4×D89—22no—
54C2×C4×D89—24no—
55C4⋊Dic89—24no—
56C4⋊Dic89—22no—
57#13083553—22no—
58#13083553—24no—
59#13083553—24no—
60#13083553—24no—
61#13083553D418yescomplemented
62C22⋊Dic89—24no—
63#13083551—22no—
64#13083551—24no—
65#13083551—24no—
66C4×Dic89—24no—
67C4×Dic89D414yes—
68C24×C89C2311yescharacteristic
69C22×C4×C89C2311yescharacteristic
70C22×C4×C89C2312yes—
71C22×C4×C89C2311yescharacteristic
72C22×C4×C89C2312yes—
73C4⋊C4×C89—24no—
74C4⋊C4×C89—22no—
75C22⋊C4×C89—22no—
76C22⋊C4×C89—24no—
77C23×C89—22no—
78C23×C89—21no—
79C23×C89—21no—
80C23×C89—22no—
81C23×C89C2×Q812yes—
82C23×C89C2411yescharacteristic
83C22×D89—22no—
84C22×D89—21no—
85C22×D89—21no—
86C22×D89—22no—
87C22×D89C2×Q812yes—
88C2×C4×C89—22no—
89C2×C4×C89—22no—
90C2×C4×C89—22no—
91C2×C4×C89—21no—
92C2×C4×C89—22no—
93C2×C4×C89—22no—
94C2×C4×C89—22no—
95C2×C4×C89—22no—
96C2×C4×C89C2×D412yes—
97C2×C4×C89C2×D414yes—
98C22⋊D89—28no—
99C2×Dic89—22no—
100C2×Dic89—22no—
101C2×Dic89—21no—
102C2×Dic89—22no—
103C2×Dic89—22no—
104C2×Dic89—22no—
105C2×Dic89—22no—
106C2×Dic89—44no—
107C2×Dic89C2×D414yes—
108C2×Dic89C2×D412yes—
109C2×Dic89C2×D412yes—
110C2×Dic89C2×D414yes—
111C2×Dic89C2×Q812yes—
112C2×Dic89#4112yes—
113C4×D89—44no—
114Dic178—48no—
115C22×C89—22no—
116C22×C89—24no—
117C22×C89—41no—
118C22×C89—41no—
119C22×C89—42no—
120C22×C89C22×D411yescharacteristic
121C22×C89C22×D412yes—
122C22×C89C22×Q811yescharacteristic
123C22×C89C2×#4111yescharacteristic
124C22×C892−1+412yes—
125C2×D89—22no—
126C2×D89—24no—
127C2×D89—41no—
128C2×D89—41no—
129C2×D89—42no—
130C4×C89—22no—
131C4×C89—24no—
132C4×C89—42no—
133C4×C89—41no—
134Dic89—24no—
135Dic89—42no—
136Dic89—41no—
137Dic89C22⋊Q814yescomplemented
138Dic89#12714yes—
139C2×C89—42no—
140C2×C89C2×C22⋊Q811yescharacteristic
141C2×C89C2×#12711yescharacteristic
142C2×C89#53611yescharacteristic
143C2×C89#54712yes—
144C2×C89D4×Q812yes—
145D89—42no—
146#2542—891nomaximal, Sylow
147C89#254211yescharacteristic, complemented, Sylow
148C2×C4⋊Q8—891no—
149C2×C22⋊Q8—892no—
150C2×C22⋊Q8—892no—
151C2×C4×D4—891no—
152#399—891no—
153#399—891no—
154#399—891no—
155#399D8911yescharacteristic, complemented
156#390—892no—
157#384—892no—
158C4×C4⋊C4—891no—
159C22×Q8—892no—
160C22×D4—891no—
161C23×C4—891no—
162C23×C4—891no—
163C4⋊Q8—1784no—
164C22⋊Q8—1788no—
165C22⋊Q8—1788no—
166C4×D4—1784no—
167C2×C4⋊C4—892no—
168C2×C4⋊C4—891no—
169C2×C4⋊C4—892no—
170C2×C4⋊C4—892no—
171C2×C4⋊C4—892no—
172C2×C4⋊C4—892no—
173C2×C4⋊C4—891no—
174C2×C4⋊C4C2×D8912yes—
175C2×C4⋊C4C2×D8911yescharacteristic
176C2×C22⋊C4—892no—
177C2×C22⋊C4—891no—
178C2×C22⋊C4—892no—
179C2×C22⋊C4—892no—
180C2×C22⋊C4C2×D8911yescharacteristic
181C2×C22⋊C4C2×D8912yes—
182C2×C42—891no—
183C2×C42—892no—
184#95—891no—
185#95C2×D8911yescharacteristic
186C24—891no—
187C24C22×D8911yescharacteristic
188C2×Q8—1784no—
189C2×Q8—1784no—
190C2×Q8—1784no—
191C2×D4—894no—
192C2×D4—1782no—
193C2×D4—1782no—
194C22×C4—891no—
195C22×C4—891no—
196C22×C4—892no—
197C22×C4—892no—
198C22×C4—891no—
199C22×C4—1782no—
200C22×C4—1784no—
201C22×C4—1782no—
202C22×C4—1784no—
203C22×C4C22×D8912yes—
204C22×C4C22×D8911yescharacteristic
205C22×C4C22×D8911yescharacteristic
206C22×C4C22×D8912yes—
207C4⋊C4—24no—
208C4⋊C4—22no—
209C4⋊C4—898no—
210C4⋊C4—1782no—
211C4⋊C4—1784no—
212C4⋊C4—1784no—
213C4⋊C4—1784no—
214C4⋊C4—1784no—
215C4⋊C4—1782no—
216C22⋊C4—22no—
217C22⋊C4—24no—
218C22⋊C4—1784no—
219C22⋊C4—1784no—
220C22⋊C4—1782no—
221C22⋊C4—1784no—
222C42—894no—
223C42—1784no—
224C23—22no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—892no—
229C23—1782no—
230C23—1781no—
231C23—1781no—
232C23—1782no—
233C23Q8×D8912yes—
234C23C23×D8911yescharacteristic
235Q8—3568no—
236D4—1788no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—892no—
246C2×C4—892no—
247C2×C4—894no—
248C2×C4—892no—
249C2×C4—892no—
250C2×C4—894no—
251C2×C4—3564no—
252C2×C4—3564no—
253C2×C4—1781no—
254C2×C4—1782no—
255C2×C4—1782no—
256C2×C4—1782no—
257C2×C4—1782no—
258C2×C4—1782no—
259C2×C4—1782no—
260C2×C4D4×D8912yes—
261C2×C4D4×D8914yes—
262C22—24no—
263C22—22no—
264C22—41no—
265C22—41no—
266C22—42no—
267C22—3561no—
268C22—3561no—
269C22—3562no—
270C22—1784no—
271C22—1782no—
272C22C2×D4×D8911yescharacteristic
273C22C2×D4×D8912yes—
274C22C2×Q8×D8911yescharacteristic
275C22#41×D8911yescharacteristic
276C22#1566805912yes—
277C4—22no—
278C4—24no—
279C4—41no—
280C4—42no—
281C4—894no—
282C4—894no—
283C4—3561no—
284C4—3562no—
285C4—1784no—
286C2—42no—
287C2—3562no—
288C2C22⋊Q8×D8911yescharacteristic
289C2#127×D8911yescharacteristic
290C2#1915217011yescharacteristic
291C2#1915224612yes—
292C2#1915226112yes—
subgroupG/H
center, FrattiniC23C23×D89
commutator, socle, cosocleC23×C89C24
Fitting#399×C89C2
prime radicalC22×C22⋊D89C22
hypercenter#399D89
hypocenterC89#2542
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#15668011C2semidirect
2. C2×#15667990C2semidirect
3. C2×#15667989C2semidirect
5. #19151535C2semidirect
6. #19151534C2semidirect
7. #19151533C2semidirect
9. #19151467C2semidirect
10. #19151449C2semidirect
11. #399×C89C2semidirect
45. C2×C22⋊D89Q8semidirect
61. #13083553D4semidirect
137. Dic89C22⋊Q8semidirect
147. C89#2542semidirect
155. #399D89semidirect

Maximal subgroups

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

subgroupindex
C2×#156680112
C2×#156679902
C2×#156679892
C2×#156679722
#191515352
#191515342
#191515332
#191514982
#191514672
#191514492
#399×C892
#254289