Finite Group Databasebeta

← #37582433

Subgroups — #37582433

Order 24448.

420 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 137 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#27133174C211yesmaximal, characteristic
2#27133004C212yesmaximal, complemented
3#27132998C211yesmaximal, characteristic, complemented
4#27132993C211yesmaximal, characteristic, complemented
5#27132987C212yesmaximal, complemented
6#27132981C211yesmaximal, characteristic, complemented
7#27132976C211yesmaximal, characteristic, complemented
8#27132699C212yesmaximal, complemented
9#27132657C212yesmaximal, complemented
10C2×C22⋊(C191⋊C8)C211yesmaximal, characteristic, complemented
11C2×#117×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2211yescharacteristic
13C22×C191⋊C8C2212yes—
14#20259305—21no—
15#20259301C2211yescharacteristic
16#20259301C2211yescharacteristic
17#20259296C2211yescharacteristic
18#20259296C2211yescharacteristic
19#20259291—21no—
20C4×C22⋊D191—21no—
21C4×C22⋊D191—21no—
22#20259281C2211yescharacteristic
23#20259255C2212yescomplemented
24#20259255C2212yescomplemented
25#20259253C2212yescomplemented
26#20259253C2212yescomplemented
27#20259245C2212yescomplemented
28#20259245C2212yescomplemented
29#20259242C2212yescomplemented
30#20259242C2212yescomplemented
31C22⋊(C191⋊C8)C2211yescharacteristic
32C22⋊(C191⋊C8)C2211yescharacteristic
33C22⋊(C191⋊C8)C2212yes—
34C23×C4×C191C2211yescharacteristic
35#117×C191—21no—
36#117×C191—21no—
37#117×C191C2212yes—
38#117×C191C2212yes—
39C2×C4⋊C4×C191C2211yescharacteristic
40C2×C4⋊C4×C191C2211yescharacteristic
41C2×C22⋊C4×C191—21no—
42C2×C42×C191—21no—
43#15690287—24no—
44C2×Dic382C2311yescharacteristic
45C2×D764C2311yescharacteristic
46C2×C22⋊D191—21no—
47C2×C22⋊D191—21no—
48C2×C22⋊D191—21no—
49C2×C4×D191—21no—
50C2×C191⋊C8—24no—
51C2×C191⋊C8C2312yes—
52C2×C191⋊C8C2312yes—
53C4⋊Dic191C2311yescharacteristic
54C4⋊Dic191C2311yescharacteristic
55Dic191⋊C4—21no—
56Dic191⋊C4—21no—
57C22⋊Dic191—21no—
58C22⋊Dic191—21no—
59C22⋊Dic191—21no—
60(C2×D191)⋊C4—21no—
61(C2×D191)⋊C4—21no—
62C4×Dic191—21no—
63C24×C191D411yescharacteristic
64C22×C4×C191—24no—
65C22×C4×C191—21no—
66C22×C4×C191—21no—
67C22×C4×C191D411yescharacteristic
68C22×C4×C191D411yescharacteristic
69C22×C4×C191D411yescharacteristic
70C22×C4×C191C2311yescharacteristic
71C22×C4×C191C2311yescharacteristic
72C22×C4×C191C2311yescharacteristic
73C4⋊C4×C191—21no—
74C4⋊C4×C191—21no—
75C4⋊C4×C191C2312yes—
76C4⋊C4×C191C2312yes—
77C22⋊C4×C191—22no—
78C22⋊C4×C191—22no—
79C42×C191—22no—
80C42×C191—22no—
81C22×D191—21no—
82Dic382—22no—
83D764—22no—
84C22⋊D191—42no—
85C22⋊D191—42no—
86C22⋊D191—42no—
87C2×Dic191—21no—
88C2×Dic191—21no—
89C2×Dic191—21no—
90C4×D191—42no—
91C191⋊C8—22no—
92C191⋊C8—22no—
93C23×C191—22no—
94C23×C191—22no—
95C23×C191—41no—
96C23×C191C2×D411yescharacteristic
97C23×C191C2×D411yescharacteristic
98C23×C191C2×D411yescharacteristic
99C2×C4×C191—22no—
100C2×C4×C191—22no—
101C2×C4×C191—22no—
102C2×C4×C191—22no—
103C2×C4×C191—22no—
104C2×C4×C191—22no—
105C2×C4×C191—22no—
106C2×C4×C191—42no—
107C2×C4×C191—42no—
108C2×C4×C191C2×D411yescharacteristic
109C2×C4×C191C2×D411yescharacteristic
110C2×C4×C191C2×D411yescharacteristic
111C2×C4×C191#4114yes—
112C2×C4×C191C2411yescharacteristic
113C22×C191—21no—
114C22×C191—21no—
115C22×C191—21no—
116C22×C191—21no—
117C22×C191—21no—
118C22×C191—21no—
119C22×C191—21no—
120C22×C191—44no—
121C22×C191#12312yes—
122C22×C191#13512yes—
123C22×C191C22×D411yescharacteristic
124C2×D191—41no—
125C2×D191—41no—
126C2×D191—41no—
127C4×C191—21no—
128C4×C191—21no—
129C4×C191—42no—
130C4×C191—42no—
131C4×C191#12312yes—
132C4×C191C2×#4112yes—
133Dic191—41no—
134Dic191—41no—
135Dic191—41no—
136C2×C191—22no—
137C2×C191—22no—
138C2×C191—41no—
139C2×C191C2×#12311yescharacteristic
140C2×C191C2×#13511yescharacteristic
141C2×C191#57511yescharacteristic
142D191—81no—
143C191#309111yescharacteristic, complemented, Sylow
144#3091—1911nomaximal, Sylow
145#525—1911no—
146C2×#117D19111yescharacteristic, complemented
147#485—1912no—
148#484—1911no—
149#483—1911no—
150#482—1912no—
151#481—1911no—
152#480—1911no—
153#427—1912no—
154#416—1912no—
155C2×C22⋊C8—1911no—
156C2×#41—1911no—
157C23×C4C2×D19111yescharacteristic
158C22×C8—1912no—
159#123—3821no—
160C22⋊Q8—1911no—
161C22⋊Q8—1911no—
162#121—1911no—
163#121—1911no—
164#120—3821no—
165C4×D4—3821no—
166C4×D4—3821no—
167#117—21no—
168#117—21no—
169#117—1911no—
170#117C2×D19112yes—
171#117C2×D19112yes—
172C2×C4⋊C4C2×D19111yescharacteristic
173C2×C4⋊C4C2×D19111yescharacteristic
174C2×C22⋊C4—21no—
175C2×C42—21no—
176#107—1912no—
177#107—1912no—
178#106—1912no—
179#106—1912no—
180#103—1912no—
181#103—1912no—
182#102—1912no—
183#102—1912no—
184C22⋊C8—1911no—
185C22⋊C8—1911no—
186C22⋊C8—1912no—
187C24C22⋊D19111yescharacteristic
188#41—3824no—
189C2×Q8—1911no—
190C2×D4—3821no—
191C2×D4—3821no—
192C2×D4—3821no—
193C2×D4—1911no—
194C22×C4—24no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—3821no—
198C22×C4C22⋊D19111yescharacteristic
199C22×C4C22⋊D19111yescharacteristic
200C22×C4C22⋊D19111yescharacteristic
201C22×C4C22×D19111yescharacteristic
202C22×C4C22×D19111yescharacteristic
203C22×C4C22×D19111yescharacteristic
204C2×C8—3824no—
205C2×C8—1912no—
206C2×C8—1912no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—3821no—
210C4⋊C4—3821no—
211C4⋊C4—1911no—
212C4⋊C4—1911no—
213C4⋊C4C22×D19112yes—
214C4⋊C4C22×D19112yes—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—3821no—
218C22⋊C4—3821no—
219C22⋊C4—3821no—
220C22⋊C4—3821no—
221C22⋊C4—3821no—
222C42—22no—
223C42—22no—
224C42—3821no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—3821no—
229C23C2×C22⋊D19111yescharacteristic
230C23C2×C22⋊D19111yescharacteristic
231C23C2×C22⋊D19111yescharacteristic
232Q8—3822no—
233D4—3822no—
234D4—7642no—
235D4—7642no—
236D4—7642no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—3821no—
247C2×C4—3821no—
248C2×C4—3821no—
249C2×C4—7642no—
250C2×C4C2×C22⋊D19111yescharacteristic
251C2×C4C2×C22⋊D19111yescharacteristic
252C2×C4C2×C22⋊D19111yescharacteristic
253C2×C4#1569028714yes—
254C2×C4C23×D19111yescharacteristic
255C8—3822no—
256C8—3822no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—7641no—
266C22—7641no—
267C22—7641no—
268C22#2025930512yes—
269C22#2025934212yes—
270C22C22×C22⋊D19111yescharacteristic
271C4—21no—
272C4—21no—
273C4—42no—
274C4—42no—
275C4—7641no—
276C4—7641no—
277C4—7641no—
278C4#2025930512yes—
279C4C2×#1569028712yes—
280C2—22no—
281C2—22no—
282C2—41no—
283C2—15281no—
284C2C2×#2025930511yescharacteristic
285C2C2×#2025934211yescharacteristic
286C2#2713346111yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC2×#117×C191C2
socleC22×C191C22×D4
prime radical#27133174C2
hypercenterC2×#117D191
hypocenterC191#3091
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #27133004C2semidirect
3. #27132998C2semidirect
4. #27132993C2semidirect
5. #27132987C2semidirect
6. #27132981C2semidirect
7. #27132976C2semidirect
8. #27132699C2semidirect
9. #27132657C2semidirect
10. C2×C22⋊(C191⋊C8)C2semidirect
11. C2×#117×C191C2semidirect
23. #20259255C22semidirect
24. #20259255C22semidirect
25. #20259253C22semidirect
26. #20259253C22semidirect
27. #20259245C22semidirect
28. #20259245C22semidirect
29. #20259242C22semidirect
30. #20259242C22semidirect
143. C191#3091semidirect
146. C2×#117D191semidirect

Maximal subgroups

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

subgroupindex
#271331742
#271330042
#271329982
#271329932
#271329872
#271329812
#271329762
#271326992
#271326572
C2×C22⋊(C191⋊C8)2
C2×#117×C1912
#3091191