Finite Group Databasebeta

← #26031025

Subgroups — #26031025

Order 11392.

416 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 121 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×#15667988C211yesmaximal, characteristic
2C22⋊Q8×D89C211yesmaximal, characteristic
3#19151934C211yesmaximal, characteristic, complemented
4#19151932C212yesmaximal, complemented
5#19151931C211yesmaximal, characteristic, complemented
6#19151838C212yesmaximal, complemented
7#19151836C212yesmaximal, complemented
8#19151754C211yesmaximal, characteristic, complemented
9#19151752C212yesmaximal, complemented
10C22⋊C8×D89C211yesmaximal, characteristic
11#451×C89C211yesmaximal, characteristic, complemented
12C2×Q8×D89C2211yescharacteristic
13C22×C4×D89C2211yescharacteristic
14C2×#13083564C2212yescomplemented
15C2×C8×D89C2212yes—
16#15667992C2211yescharacteristic
17#15667991C2211yescharacteristic
18#15667990—21no—
19#15667989—21no—
20#15667988—21no—
21#15667988—21no—
22#15667988C2212yes—
23#15667988C2211yescharacteristic
24#15667988C2212yes—
25C2×C4⋊Dic89C2211yescharacteristic
26C2×#13083553—21no—
27C4⋊C4×D89—21no—
28C4⋊C4×D89C2211yescharacteristic
29C2×#13083551—21no—
30C22⋊C4×D89—21no—
31#15667946C2212yescomplemented
32#15667945C2212yescomplemented
33#15667937C2212yescomplemented
34#15667936C2212yescomplemented
35#15667936C2212yescomplemented
36C22⋊#10623494C2211yescharacteristic
37#15667924C2212yes—
38C22×Q8×C89C2211yescharacteristic
39C2×Q16×C89C2212yescomplemented
40C22⋊Q8×C89C2211yescharacteristic
41#103×C89C2212yescomplemented
42C22⋊C8×C89C2211yescharacteristic
43C23×D89D411yescharacteristic
44C2×Dic178—21no—
45C2×Dic178C2311yescharacteristic
46Q8×D89—42no—
47C22×Dic89—21no—
48C22×Dic89D411yescharacteristic
49C2×C4×D89—21no—
50C2×C4×D89—21no—
51C2×C4×D89—42no—
52C2×C4×D89D411yescharacteristic
53C2×C4×D89D411yescharacteristic
54C2×C4×D89C2311yescharacteristic
55C2×C4×D89C2311yescharacteristic
56#13083564—42no—
57C2×#10623494C2312yes—
58C8×D89—42no—
59C4⋊Dic89—21no—
60C4⋊Dic89—21no—
61C4⋊Dic89C2312yes—
62C4⋊Dic89C2311yescharacteristic
63#13083553—21no—
64#13083553—22no—
65#13083553—22no—
66#13083553—21no—
67#13083553—21no—
68C22⋊Dic89—21no—
69#13083551—22no—
70#13083551—22no—
71#13083551—21no—
72#13083551—21no—
73C2×Q8×C89—21no—
74C2×Q8×C89—42no—
75C2×Q8×C89C2312yes—
76C2×Q8×C89C2311yescharacteristic
77C22×C4×C89—21no—
78C22×C4×C89C2311yescharacteristic
79Q16×C89—42no—
80C2×C8×C89C2312yes—
81C4⋊C4×C89—21no—
82C4⋊C4×C89C2311yescharacteristic
83C22⋊C4×C89—21no—
84C23×C89C2×D411yescharacteristic
85C22×D89—41no—
86C22×D89—41no—
87C22×D89Q1614yescomplemented
88C22×D89C2×D411yescharacteristic
89C22×D89C2×D411yescharacteristic
90Q8×C89—22no—
91Q8×C89—22no—
92Q8×C89—22no—
93Q8×C89—42no—
94C2×C4×C89—22no—
95C2×C4×C89—21no—
96C2×C4×C89—21no—
97C2×C4×C89—22no—
98C2×C4×C89—42no—
99C2×C4×C89C2×D411yescharacteristic
100C2×C4×C89C2411yescharacteristic
101C2×Dic89—22no—
102C2×Dic89—21no—
103C2×Dic89—21no—
104C2×Dic89—41no—
105C2×Dic89—42no—
106C2×Dic89C2×D411yescharacteristic
107C2×Dic89C2×D411yescharacteristic
108C4×D89—22no—
109C4×D89—22no—
110C4×D89—41no—
111C4×D89—41no—
112C4×D89—42no—
113C4×D89—42no—
114Dic178—22no—
115Dic178—42no—
116C8×C89—22no—
117#10623494—22no—
118C22×C89—21no—
119C22×C89—21no—
120C22×C89C2×Q1612yes—
121C22×C89C22×D411yescharacteristic
122C2×D89—21no—
123C2×D89—21no—
124C2×D89—21no—
125C2×D89—21no—
126C2×D89—44no—
127C2×D89C2×Q1612yes—
128C2×D89C2×Q1612yes—
129C4×C89—21no—
130C4×C89—41no—
131C4×C89—42no—
132C4×C89—41no—
133C4×C89H(5,𝔽2)12yes—
134Dic89—21no—
135Dic89—21no—
136Dic89—41no—
137Dic89—41no—
138Dic89—42no—
139C2×C89—22no—
140C2×C89#53411yescharacteristic
141C2×C89C22×Q1611yescharacteristic
142C2×C89#57511yescharacteristic
143D89—22no—
144D89—22no—
145#3189—891nomaximal, Sylow
146C89#318911yescharacteristic, complemented, Sylow
147C2×C22⋊Q8—891no—
148C2×C22⋊Q8—891no—
149#484—891no—
150#484—892no—
151#484—891no—
152#467—892no—
153#467—892no—
154#451—891no—
155#451—892no—
156#451D8911yescharacteristic, complemented
157C2×C22⋊C8—891no—
158C22×Q8—891no—
159C22×Q8C2×D8911yescharacteristic
160C23×C4—891no—
161C2×Q16—892no—
162C2×Q16C2×D8912yescomplemented
163C22×C8—892no—
164C22⋊Q8—892no—
165C22⋊Q8—891no—
166C22⋊Q8—892no—
167C22⋊Q8—891no—
168C22⋊Q8—891no—
169C22⋊Q8—1781no—
170C22⋊Q8—1781no—
171C22⋊Q8—1781no—
172C22⋊Q8—1781no—
173C22⋊Q8C2×D8911yescharacteristic
174C2×C4⋊C4—891no—
175C2×C4⋊C4—891no—
176C2×C4⋊C4—1781no—
177C2×C4⋊C4—1781no—
178C2×C22⋊C4—1781no—
179C2×C22⋊C4—1781no—
180#107—892no—
181#107—892no—
182#103—892no—
183#103—892no—
184#103—892no—
185#103C2×D8912yescomplemented
186C22⋊C8—891no—
187C22⋊C8—892no—
188C22⋊C8C2×D8911yescharacteristic
189C24—891no—
190C2×Q8—21no—
191C2×Q8—42no—
192C2×Q8—891no—
193C2×Q8—3562no—
194C2×Q8—1781no—
195C2×Q8C22×D8911yescharacteristic
196C2×Q8C22×D8912yes—
197C22×C4—21no—
198C22×C4—891no—
199C22×C4—891no—
200C22×C4—891no—
201C22×C4—891no—
202C22×C4—891no—
203C22×C4—3562no—
204C22×C4—1781no—
205C22×C4—1781no—
206C22×C4—1781no—
207C22×C4C22×D8911yescharacteristic
208Q16—42no—
209Q16—3562no—
210C2×C8—892no—
211C2×C8—3562no—
212C2×C8C22×D8912yes—
213C4⋊C4—21no—
214C4⋊C4—891no—
215C4⋊C4—892no—
216C4⋊C4—1781no—
217C4⋊C4—1781no—
218C4⋊C4—1782no—
219C4⋊C4—1782no—
220C4⋊C4—1781no—
221C4⋊C4—1781no—
222C4⋊C4—1781no—
223C4⋊C4C22×D8911yescharacteristic
224C22⋊C4—21no—
225C22⋊C4—1781no—
226C22⋊C4—1781no—
227C22⋊C4—1782no—
228C22⋊C4—1782no—
229C22⋊C4—1781no—
230C23—894no—
231C23—891no—
232C23—891no—
233C23—3561no—
234C23—3561no—
235C23D4×D8911yescharacteristic
236Q8—22no—
237Q8—22no—
238Q8—22no—
239Q8—42no—
240Q8—3562no—
241Q8—1782no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—891no—
248C2×C4—891no—
249C2×C4—3561no—
250C2×C4—3561no—
251C2×C4—3562no—
252C2×C4—3562no—
253C2×C4—3562no—
254C2×C4—3561no—
255C2×C4—1782no—
256C2×C4—1781no—
257C2×C4—1781no—
258C2×C4—1782no—
259C2×C4—1782no—
260C2×C4D4×D8911yescharacteristic
261C2×C4C23×D8911yescharacteristic
262C8—22no—
263C8—1782no—
264C22—21no—
265C22—21no—
266C22—892no—
267C22—892no—
268C22—3564no—
269C22—1781no—
270C22—1781no—
271C22—1781no—
272C22—1781no—
273C22Q16×D8912yes—
274C22C2×D4×D8911yescharacteristic
275C4—21no—
276C4—41no—
277C4—42no—
278C4—41no—
279C4—3561no—
280C4—3562no—
281C4—3561no—
282C4—1781no—
283C4—1781no—
284C4#1566805712yes—
285C2—22no—
286C2—1782no—
287C2—1782no—
288C2#1915215811yescharacteristic
289C2C2×Q16×D8911yescharacteristic
290C2#1915239711yescharacteristic
subgroupG/H
centerC22C2×D4×D89
commutator, cosocleC2×C4×C89C24
FrattiniC2×C4C23×D89
Fitting#451×C89C2
socleC22×C89C22×D4
prime radicalC23×D89D4
hypercenter#451D89
hypocenterC89#3189
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #19151934C2semidirect
4. #19151932C2semidirect
5. #19151931C2semidirect
6. #19151838C2semidirect
7. #19151836C2semidirect
8. #19151754C2semidirect
9. #19151752C2semidirect
11. #451×C89C2semidirect
14. C2×#13083564C22semidirect
31. #15667946C22semidirect
32. #15667945C22semidirect
33. #15667937C22semidirect
34. #15667936C22semidirect
35. #15667936C22semidirect
39. C2×Q16×C89C22semidirect
41. #103×C89C22semidirect
87. C22×D89Q16semidirect
146. C89#3189semidirect
156. #451D89semidirect
162. C2×Q16C2×D89semidirect
185. #103C2×D89semidirect

Maximal subgroups

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

subgroupindex
C2×#156679882
C22⋊Q8×D892
#191519342
#191519322
#191519312
#191518382
#191518362
#191517542
#191517522
C22⋊C8×D892
#451×C892
#318989