Finite Group Databasebeta

← #26023919

Subgroups — #26023919

Order 11392.

780 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 161 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×#15667990C212yesmaximal, complemented
2C2×C22⋊C4×D89C212yesmaximal
3#19151521C212yesmaximal, complemented
4#19151510C211yesmaximal, characteristic, complemented
5#19151472C212yesmaximal, complemented
6#19151469C212yesmaximal, complemented
7#19151455C212yesmaximal, complemented
8#19151453C211yesmaximal, characteristic
9#399×C89C211yesmaximal, characteristic, complemented
10C24×D89C2211yescharacteristic
11C22×Dic178C2211yescharacteristic, complemented
12C22×C4×D89C2212yes—
13C22×C4×D89C2212yes—
14#15667990—28no—
15C2×C4⋊Dic89C2212yescomplemented
16C2×#13083553C2212yescomplemented
17C2×#13083553C2212yescomplemented
18C2×C22⋊Dic89C2212yes—
19C2×C22⋊Dic89C2211yescharacteristic
20C2×#13083551C2212yes—
21C2×#13083551C2212yes—
22C2×#13083551C2212yes—
23C2×#13083551C2212yes—
24C22⋊C4×D89—28no—
25#15667919C2211yescharacteristic, complemented
26#15667919C2212yescomplemented
27#15667919C2212yescomplemented
28C2×C4⋊C4×C89C2212yescomplemented
29C2×C4⋊C4×C89C2211yescharacteristic, complemented
30C2×C22⋊C4×C89C2212yes—
31C2×C22⋊C4×C89C2211yescharacteristic
32#95×C89C2211yescharacteristic, complemented
33C23×D89—24no—
34C23×D89—24no—
35C23×D89—41no—
36C23×D89—41no—
37C23×D89Q814yescomplemented
38C23×D89C2312yes—
39C2×Dic178—24no—
40C2×Dic178—41no—
41C22×Dic89C2312yes—
42C22×Dic89C2312yes—
43C22×Dic89C2312yes—
44C2×C4×D89—24no—
45C2×C4×D89—24no—
46C2×C4×D89—24no—
47C2×C4×D89—24no—
48C2×C4×D89—42no—
49C2×C4×D89—42no—
50C4⋊Dic89—42no—
51#13083553—24no—
52#13083553—24no—
53C22⋊Dic89—24no—
54C22⋊Dic89—41no—
55#13083551—24no—
56#13083551—24no—
57#13083551—24no—
58#13083551D418yes—
59C24×C89C2311yescharacteristic
60C22×C4×C89C2312yes—
61C22×C4×C89C2311yescharacteristic
62C22×C4×C89C2311yescharacteristic
63C22×C4×C89C2312yes—
64C4⋊C4×C89—24no—
65C4⋊C4×C89—41no—
66C22⋊C4×C89—24no—
67C22⋊C4×C89—41no—
68C23×C89—22no—
69C23×C89—22no—
70C23×C89—21no—
71C23×C89—21no—
72C23×C89C2×Q812yes—
73C23×C89C2411yescharacteristic
74C22×D89—22no—
75C22×D89—22no—
76C22×D89—22no—
77C22×D89—22no—
78C22×D89—216no—
79C22×D89—44no—
80C22×D89—44no—
81C22×D89—48no—
82C22×D89—44no—
83C22×D89C2×D414yes—
84C22×D89C2×Q814yes—
85C22×D89#4114yes—
86C2×C4×C89—22no—
87C2×C4×C89—22no—
88C2×C4×C89—22no—
89C2×C4×C89—22no—
90C2×C4×C89—22no—
91C2×C4×C89—22no—
92C2×C4×C89—21no—
93C2×C4×C89—21no—
94C2×C4×C89—22no—
95C2×C4×C89C2×D414yes—
96C2×Dic89—22no—
97C2×Dic89—22no—
98C2×Dic89—22no—
99C2×Dic89—22no—
100C2×Dic89—22no—
101C2×Dic89—22no—
102C2×Dic89—22no—
103C2×Dic89—22no—
104C2×Dic89C2×D414yes—
105C4×D89—48no—
106C4×D89—48no—
107Dic178—44no—
108C22×C89—24no—
109C22×C89—22no—
110C22×C89—41no—
111C22×C89—42no—
112C22×C89—41no—
113C22×C89C22×D412yes—
114C22×C89C22×Q811yescharacteristic
115C22×C89C2×#4112yes—
116C22×C89H(5,𝔽2)11yescharacteristic
117C22×C89H(5,𝔽2)11yescharacteristic
118C2×D89—24no—
119C2×D89—24no—
120C2×D89—24no—
121C2×D89—24no—
122C2×D89—24no—
123C2×D89—24no—
124C2×D89—416no—
125C2×D89C22⋊Q818yescomplemented
126C4×C89—24no—
127C4×C89—41no—
128C4×C89—42no—
129C4×C89—41no—
130Dic89—24no—
131Dic89—42no—
132Dic89—42no—
133C2×C89—42no—
134C2×C89C2×C22⋊Q812yes—
135C2×C89#54311yescharacteristic
136C2×C89D4211yescharacteristic
137C2×C89#54612yes—
138C2×C89#55111yescharacteristic
139D89—28no—
140#2437—891nomaximal, Sylow
141C89#243711yescharacteristic, complemented, Sylow
142C2×C22⋊Q8—892no—
143C22×C22⋊C4—892no—
144#399D8911yescharacteristic, complemented
145#396—892no—
146#393—891no—
147#385—892no—
148#385—892no—
149#380—892no—
150#379—891no—
151C25—891no—
152C22×Q8—891no—
153C23×C4—892no—
154C23×C4—892no—
155C22⋊Q8—1788no—
156C2×C4⋊C4—892no—
157C2×C4⋊C4—892no—
158C2×C4⋊C4—892no—
159C2×C4⋊C4C2×D8912yescomplemented
160C2×C4⋊C4C2×D8911yescharacteristic, complemented
161C2×C22⋊C4—892no—
162C2×C22⋊C4—891no—
163C2×C22⋊C4—892no—
164C2×C22⋊C4—892no—
165C2×C22⋊C4—892no—
166C2×C22⋊C4—892no—
167C2×C22⋊C4—1788no—
168C2×C22⋊C4C2×D8912yes—
169C2×C22⋊C4C2×D8911yescharacteristic
170#95—891no—
171#95—892no—
172#95—892no—
173#95C2×D8911yescharacteristic, complemented
174C24—892no—
175C24—894no—
176C24—3561no—
177C24—3561no—
178C24—1784no—
179C24—1784no—
180C24C22×D8911yescharacteristic
181C2×Q8—3561no—
182C2×Q8—1784no—
183C22×C4—892no—
184C22×C4—892no—
185C22×C4—892no—
186C22×C4—3562no—
187C22×C4—3562no—
188C22×C4—1784no—
189C22×C4—1784no—
190C22×C4—1784no—
191C22×C4—1784no—
192C22×C4C22×D8911yescharacteristic
193C22×C4C22×D8911yescharacteristic
194C22×C4C22×D8912yes—
195C22×C4C22×D8912yes—
196C4⋊C4—24no—
197C4⋊C4—41no—
198C4⋊C4—3562no—
199C4⋊C4—1784no—
200C4⋊C4—1784no—
201C22⋊C4—24no—
202C22⋊C4—41no—
203C22⋊C4—898no—
204C22⋊C4—3561no—
205C22⋊C4—1784no—
206C22⋊C4—1784no—
207C22⋊C4—1784no—
208C22⋊C4—1784no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—21no—
213C23—894no—
214C23—894no—
215C23—894no—
216C23—3564no—
217C23—3564no—
218C23—3568no—
219C23—3564no—
220C23—17816no—
221C23—1782no—
222C23—1782no—
223C23—1782no—
224C23—1782no—
225C23Q8×D8912yes—
226C23C23×D8911yescharacteristic
227Q8—3564no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—894no—
238C2×C4—3568no—
239C2×C4—3568no—
240C2×C4—1782no—
241C2×C4—1782no—
242C2×C4—1782no—
243C2×C4—1782no—
244C2×C4—1782no—
245C2×C4—1782no—
246C2×C4—1782no—
247C2×C4—1782no—
248C2×C4D4×D8914yes—
249C22—22no—
250C22—24no—
251C22—41no—
252C22—42no—
253C22—41no—
254C22—898no—
255C22—35616no—
256C22—1784no—
257C22—1784no—
258C22—1784no—
259C22—1784no—
260C22—1784no—
261C22—1784no—
262C22C2×D4×D8912yes—
263C22C2×Q8×D8911yescharacteristic
264C22#41×D8912yes—
265C22#1566805711yescharacteristic
266C22#1566805811yescharacteristic
267C4—24no—
268C4—42no—
269C4—41no—
270C4—41no—
271C4—3562no—
272C4—3562no—
273C4—1784no—
274C2—42no—
275C2—1788no—
276C2C22⋊Q8×D8912yes—
277C2#1915222311yescharacteristic
278C2#1915223311yescharacteristic
279C2#1915224012yes—
280C2#1915227611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D89
commutator, socle, cosocleC23×C89C24
Fitting#399×C89C2
prime radicalC24×D89C22
hypercenter#399D89
hypocenterC89#2437
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#15667990C2semidirect
3. #19151521C2semidirect
4. #19151510C2semidirect
5. #19151472C2semidirect
6. #19151469C2semidirect
7. #19151455C2semidirect
9. #399×C89C2semidirect
11. C22×Dic178C22semidirect
15. C2×C4⋊Dic89C22semidirect
16. C2×#13083553C22semidirect
17. C2×#13083553C22semidirect
25. #15667919C22semidirect
26. #15667919C22semidirect
27. #15667919C22semidirect
28. C2×C4⋊C4×C89C22semidirect
29. C2×C4⋊C4×C89C22semidirect
32. #95×C89C22semidirect
37. C23×D89Q8semidirect
125. C2×D89C22⋊Q8semidirect
141. C89#2437semidirect
144. #399D89semidirect
159. C2×C4⋊C4C2×D89semidirect
160. C2×C4⋊C4C2×D89semidirect
173. #95C2×D89semidirect

Maximal subgroups

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

subgroupindex
C2×#156679902
C2×C22⋊C4×D892
#191515212
#191515102
#191514722
#191514692
#191514552
#191514532
#399×C892
#243789