Finite Group Databasebeta

← #37582655

Subgroups — #37582655

Order 24448.

450 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#27133194C211yesmaximal, characteristic, complemented
2C2×#20259284C211yesmaximal, characteristic, complemented
3#27132993C211yesmaximal, characteristic, complemented
4#27132966C212yesmaximal, complemented
5#27132877C212yesmaximal, complemented
6#27132816C211yesmaximal, characteristic, complemented
7#27132796C211yesmaximal, characteristic
8#27132772C212yesmaximal, complemented
9#27132762C212yesmaximal, complemented
10#27132717C211yesmaximal, characteristic, complemented
11#480×C191C211yesmaximal, characteristic, complemented
12C22×D764C2211yescharacteristic
13C22×C4×D191—21no—
14C2×#15690276C2212yes—
15C2×#15690272C2212yes—
16#20259315—21no—
17#20259313C2211yescharacteristic, complemented
18#20259309C2211yescharacteristic, complemented
19#20259303—21no—
20#20259301C2211yescharacteristic
21#20259287C2211yescharacteristic, complemented
22D4×Dic191—21no—
23D4×Dic191C2211yescharacteristic, complemented
24#20259284—21no—
25#20259284—21no—
26#20259284C2212yes—
27#20259284C2212yes—
28#20259278—21no—
29C2×(C2×D191)⋊C4—21no—
30C2×C4×Dic191C2211yescharacteristic
31#20259253C2212yescomplemented
32Dic191⋊C8C2212yescomplemented
33#20259245C2212yescomplemented
34#20259244C2212yes—
35#20259242C2212yes—
36C22⋊(C191⋊C8)C2211yescharacteristic
37#20259232C2212yescomplemented
38#121×C191C2211yescharacteristic
39C2×C4⋊C4×C191C2211yescharacteristic
40#107×C191C2212yescomplemented
41#102×C191C2212yescomplemented
42C22⋊C8×C191C2211yescharacteristic
43C23×D191—21no—
44C2×Dic382C2311yescharacteristic
45C2×D764—24no—
46C2×D764—21no—
47C2×D764C2312yes—
48C22×Dic191—21no—
49C22×Dic191D411yescharacteristic
50C22×Dic191D411yescharacteristic
51C2×C4×D191—22no—
52C2×C4×D191—22no—
53C2×C4×D191—44no—
54#15690276—24no—
55#15690272—24no—
56C2×C191⋊C8C2312yes—
57C4⋊Dic191—21no—
58C4⋊Dic191C2311yescharacteristic
59Dic191⋊C4—21no—
60Dic191⋊C4—21no—
61Dic191⋊C4—21no—
62C22⋊Dic191—21no—
63C22⋊Dic191—21no—
64C22⋊Dic191—21no—
65C22⋊Dic191—21no—
66(C2×D191)⋊C4—22no—
67(C2×D191)⋊C4—22no—
68C4×Dic191—21no—
69C4×Dic191D411yescharacteristic
70C4×Dic191D411yescharacteristic
71C4×Dic191C2311yescharacteristic
72C4×Dic191C2311yescharacteristic
73C2×D4×C191—21no—
74C2×D4×C191C2311yescharacteristic
75C22×C4×C191—21no—
76C22×C4×C191C2311yescharacteristic
77C2×C8×C191C2312yes—
78C4⋊C4×C191—21no—
79C4⋊C4×C191C2312yes—
80C4⋊C4×C191C2311yescharacteristic
81C22⋊C4×C191—21no—
82C22×D191—24no—
83C22×D191—22no—
84C22×D191—42no—
85C22×D191—42no—
86Dic382—22no—
87D764—24no—
88D764—22no—
89D764#4114yes—
90C2×Dic191—22no—
91C2×Dic191—21no—
92C2×Dic191—21no—
93C2×Dic191—41no—
94C2×Dic191—42no—
95C2×Dic191C2×D411yescharacteristic
96C2×Dic191C2×D411yescharacteristic
97C2×Dic191C2×D411yescharacteristic
98C2×Dic191C2×D411yescharacteristic
99C4×D191—44no—
100C4×D191—44no—
101C191⋊C8—22no—
102C23×C191—21no—
103C23×C191C2×D411yescharacteristic
104D4×C191—22no—
105D4×C191—42no—
106C2×C4×C191—22no—
107C2×C4×C191—21no—
108C2×C4×C191—22no—
109C2×C4×C191—42no—
110C2×C4×C191C2×D411yescharacteristic
111C2×C4×C191C2411yescharacteristic
112C8×C191—22no—
113C22×C191—21no—
114C22×C191—21no—
115C22×C191—41no—
116C22×C191—41no—
117C22×C191—41no—
118C22×C191AGL(1,ℤ/8)12yes—
119C22×C191C22×D411yescharacteristic
120C2×D191—24no—
121C2×D191—48no—
122C2×D191—42no—
123C2×D191—42no—
124C4×C191—21no—
125C4×C191—42no—
126C4×C191—41no—
127C4×C191C2×#4112yes—
128Dic191—21no—
129Dic191—21no—
130Dic191—21no—
131Dic191—21no—
132Dic191—41no—
133Dic191—41no—
134C2×C191—22no—
135C2×C191—81no—
136C2×C191#52511yescharacteristic
137C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C191#57511yescharacteristic
139D191—44no—
140C191#311411yescharacteristic, complemented, Sylow
141#3114—1911nomaximal, Sylow
142#529—1911no—
143C2×C4×D4—1911no—
144#483—1911no—
145#480D19111yescharacteristic, complemented
146#478—1912no—
147#463—1912no—
148#450—1911no—
149#447—1911no—
150#442—1912no—
151#440—1912no—
152#432—1911no—
153C22×D4—1911no—
154C23×C4—3821no—
155C2×SD16—1912no—
156C2×D8—1912no—
157#126—3821no—
158#125—1911no—
159#124—1911no—
160#123—3821no—
161C22⋊Q8—1911no—
162#121C2×D19111yescharacteristic
163C4×Q8—1911no—
164C4×D4—3821no—
165C4×D4—3821no—
166C4×D4—3821no—
167C4×D4—1911no—
168C4×D4—1912no—
169C4×D4—1912no—
170#117—3821no—
171C2×C4⋊C4C2×D19111yescharacteristic
172C2×C22⋊C4—3821no—
173C2×C42—1911no—
174#107C2×D19112yescomplemented
175#106—1912no—
176C4⋊C8—1912no—
177#103—1912no—
178#102—1912no—
179#102—1912no—
180#102C2×D19112yescomplemented
181C22⋊C8—1911no—
182C22⋊C8C2×D19111yescharacteristic
183#97—1912no—
184C24—3821no—
185C2×Q8—1911no—
186C2×D4—21no—
187C2×D4—3824no—
188C2×D4—3821no—
189C2×D4—1912no—
190C2×D4C22×D19111yescharacteristic
191C22×C4—21no—
192C22×C4—3822no—
193C22×C4—3822no—
194C22×C4—3821no—
195C22×C4—1911no—
196C22×C4—1911no—
197C22×C4—7644no—
198C22×C4C22×D19111yescharacteristic
199SD16—3824no—
200D8—3824no—
201C2×C8—1912no—
202C2×C8C22×D19112yes—
203C4⋊C4—21no—
204C4⋊C4—3821no—
205C4⋊C4—3821no—
206C4⋊C4—3821no—
207C4⋊C4—3821no—
208C4⋊C4—1911no—
209C4⋊C4C22×D19111yescharacteristic
210C4⋊C4C22×D19112yes—
211C22⋊C4—21no—
212C22⋊C4—3822no—
213C22⋊C4—3822no—
214C22⋊C4—3821no—
215C22⋊C4—3821no—
216C22⋊C4—3821no—
217C22⋊C4—3821no—
218C42—3821no—
219C42—1911no—
220C42—1911no—
221C42—1911no—
222C42—1911no—
223C23—21no—
224C23—3824no—
225C23—3822no—
226C23—7642no—
227C23—7642no—
228C23D4×D19111yescharacteristic
229Q8—3822no—
230D4—22no—
231D4—42no—
232D4—3824no—
233D4—3822no—
234D4—1914no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—3822no—
240C2×C4—3821no—
241C2×C4—3821no—
242C2×C4—1911no—
243C2×C4—1911no—
244C2×C4—1911no—
245C2×C4—1911no—
246C2×C4—7644no—
247C2×C4—7644no—
248C2×C4—7641no—
249C2×C4—7642no—
250C2×C4D4×D19111yescharacteristic
251C2×C4C23×D19111yescharacteristic
252C8—22no—
253C8—3822no—
254C22—21no—
255C22—21no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—3824no—
260C22—7648no—
261C22—7642no—
262C22—7642no—
263C22#2025934612yes—
264C22C2×D4×D19111yescharacteristic
265C4—21no—
266C4—42no—
267C4—41no—
268C4—3821no—
269C4—3821no—
270C4—3821no—
271C4—3821no—
272C4—7641no—
273C4—7641no—
274C4#41×D19112yes—
275C2—22no—
276C2—81no—
277C2—7644no—
278C2#2713317111yescharacteristic
279C2C2×#2025934611yescharacteristic
280C2#2713346211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#480×C191C2
socleC22×C191C22×D4
prime radical#27132796C2
hypercenter#480D191
hypocenterC191#3114
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #27133194C2semidirect
2. C2×#20259284C2semidirect
3. #27132993C2semidirect
4. #27132966C2semidirect
5. #27132877C2semidirect
6. #27132816C2semidirect
8. #27132772C2semidirect
9. #27132762C2semidirect
10. #27132717C2semidirect
11. #480×C191C2semidirect
17. #20259313C22semidirect
18. #20259309C22semidirect
21. #20259287C22semidirect
23. D4×Dic191C22semidirect
31. #20259253C22semidirect
32. Dic191⋊C8C22semidirect
33. #20259245C22semidirect
37. #20259232C22semidirect
40. #107×C191C22semidirect
41. #102×C191C22semidirect
140. C191#3114semidirect
145. #480D191semidirect
174. #107C2×D191semidirect
180. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271331942
C2×#202592842
#271329932
#271329662
#271328772
#271328162
#271327962
#271327722
#271327622
#271327172
#480×C1912
#3114191