Finite Group Databasebeta

← #37583304

Subgroups — #37583304

Order 24448.

420 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 115 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#27133176C211yesmaximal, characteristic
2C2×#20259298C211yesmaximal, characteristic, complemented
3#27133003C212yesmaximal, complemented
4#27132991C211yesmaximal, characteristic, complemented
5#27132977C211yesmaximal, characteristic, complemented
6#27132901C212yesmaximal, complemented
7#27132895C212yesmaximal, complemented
8#27132824C212yesmaximal, complemented
9#27132802C211yesmaximal, characteristic, complemented
10C22⋊C8×D191C211yesmaximal, characteristic, complemented
11#451×C191C211yesmaximal, characteristic, complemented
12C2×#15690288C2211yescharacteristic
13C22×C4×D191C2211yescharacteristic
14C2×#15690274C2212yescomplemented
15C2×C8×D191C2212yes—
16#20259307—21no—
17#20259298—21no—
18#20259298—21no—
19#20259298C2212yes—
20#20259298C2212yes—
21#20259298C2211yescharacteristic
22#20259297C2211yescharacteristic
23#20259296C2211yescharacteristic
24#20259292—21no—
25#20259284—21no—
26(C22⋊D191)⋊C4—21no—
27#20259280C2211yescharacteristic
28C2×C4⋊Dic191C2211yescharacteristic
29C2×Dic191⋊C4—21no—
30C2×(C2×D191)⋊C4—21no—
31#20259256C2212yescomplemented
32#20259253C2212yescomplemented
33#20259246C2212yescomplemented
34#20259246C2212yescomplemented
35#20259244C2212yescomplemented
36C22⋊(C191⋊C8)C2211yescharacteristic
37(C2×D191)⋊C8C2212yes—
38C22×Q8×C191C2211yescharacteristic
39C2×Q16×C191C2212yescomplemented
40C22⋊Q8×C191C2211yescharacteristic
41#103×C191C2212yescomplemented
42C22⋊C8×C191C2211yescharacteristic
43C23×D191D411yescharacteristic
44#15690288—42no—
45C2×D764—21no—
46C2×D764—21no—
47C2×D764C2311yescharacteristic
48C2×C22⋊D191—21no—
49C22×Dic191—21no—
50C22×Dic191D411yescharacteristic
51C2×C4×D191—21no—
52C2×C4×D191—42no—
53C2×C4×D191D411yescharacteristic
54C2×C4×D191D411yescharacteristic
55C2×C4×D191C2311yescharacteristic
56C2×C4×D191C2311yescharacteristic
57#15690274—42no—
58C2×C191⋊C8C2312yes—
59C8×D191—42no—
60C4⋊Dic191—21no—
61C4⋊Dic191C2312yes—
62C4⋊Dic191C2311yescharacteristic
63Dic191⋊C4—22no—
64Dic191⋊C4—22no—
65Dic191⋊C4—21no—
66(C2×D191)⋊C4—22no—
67(C2×D191)⋊C4—22no—
68(C2×D191)⋊C4—21no—
69(C2×D191)⋊C4—21no—
70(C2×D191)⋊C4—21no—
71(C2×D191)⋊C4—21no—
72C4×Dic191—21no—
73C2×Q8×C191—21no—
74C2×Q8×C191—42no—
75C2×Q8×C191C2312yes—
76C2×Q8×C191C2311yescharacteristic
77C22×C4×C191—21no—
78C22×C4×C191C2311yescharacteristic
79Q16×C191—42no—
80C2×C8×C191C2312yes—
81C4⋊C4×C191—21no—
82C4⋊C4×C191C2311yescharacteristic
83C22⋊C4×C191—21no—
84C22×D191—22no—
85C22×D191—22no—
86C22×D191—21no—
87C22×D191—41no—
88C22×D191C2×D411yescharacteristic
89C22×D191C2×D411yescharacteristic
90D764—22no—
91D764—42no—
92D764—42no—
93C22⋊D191—42no—
94C2×Dic191—21no—
95C2×Dic191—22no—
96C2×Dic191—41no—
97C2×Dic191—42no—
98C2×Dic191C2×D411yescharacteristic
99C2×Dic191C2×D411yescharacteristic
100C4×D191—22no—
101C4×D191—22no—
102C4×D191—41no—
103C4×D191—41no—
104C4×D191—42no—
105C191⋊C8—22no—
106C23×C191C2×D411yescharacteristic
107Q8×C191—22no—
108Q8×C191—22no—
109Q8×C191—22no—
110Q8×C191—42no—
111C2×C4×C191—22no—
112C2×C4×C191—21no—
113C2×C4×C191—21no—
114C2×C4×C191—22no—
115C2×C4×C191—42no—
116C2×C4×C191C2×D411yescharacteristic
117C2×C4×C191C2411yescharacteristic
118C8×C191—22no—
119C22×C191—21no—
120C22×C191—21no—
121C22×C191#13512yes—
122C22×C191C22×D411yescharacteristic
123C2×D191—21no—
124C2×D191—21no—
125C2×D191—21no—
126C2×D191—21no—
127C2×D191—21no—
128C2×D191—44no—
129C2×D191—41no—
130C2×D191—41no—
131C2×D191—41no—
132C2×D191#13712yes—
133C4×C191—21no—
134C4×C191—42no—
135C4×C191—41no—
136C4×C191—41no—
137C4×C191H(5,𝔽2)12yes—
138Dic191—21no—
139Dic191—21no—
140Dic191—42no—
141Dic191—41no—
142C2×C191—22no—
143C2×C191#53411yescharacteristic
144C2×C191C2×#13511yescharacteristic
145C2×C191C2×#13711yescharacteristic
146D191—22no—
147D191—41no—
148D191—81no—
149C191#319011yescharacteristic, complemented, Sylow
150#3190—1911nomaximal, Sylow
151#525—1911no—
152C2×C22⋊Q8—1911no—
153#485—1912no—
154#483—1911no—
155#480—1911no—
156#467—1912no—
157#465—1912no—
158#452—1912no—
159#451D19111yescharacteristic, complemented
160#448—1911no—
161C2×C22⋊C8—1911no—
162C2×#41—1911no—
163C22×Q8C2×D19111yescharacteristic
164C23×C4—1911no—
165C2×Q16C2×D19112yescomplemented
166C2×SD16—1912no—
167C22×C8—1912no—
168#123—3821no—
169C22⋊Q8—3821no—
170C22⋊Q8—3821no—
171C22⋊Q8—1912no—
172C22⋊Q8—1912no—
173C22⋊Q8—1911no—
174C22⋊Q8C2×D19111yescharacteristic
175#121—1911no—
176#121—1911no—
177#120—3821no—
178C4×D4—3821no—
179C4×D4—3821no—
180#117—1911no—
181C2×C4⋊C4—3821no—
182C2×C4⋊C4—1911no—
183C2×C22⋊C4—3821no—
184#107—1912no—
185#106—1912no—
186#103—1912no—
187#103—1912no—
188#103C2×D19112yescomplemented
189#102—1912no—
190C22⋊C8—1911no—
191C22⋊C8—1912no—
192C22⋊C8C2×D19111yescharacteristic
193C24—1911no—
194#41—7642no—
195C2×Q8—21no—
196C2×Q8—42no—
197C2×Q8C22×D19111yescharacteristic
198C2×Q8C22×D19112yes—
199C2×D4—3821no—
200C2×D4—3821no—
201C2×D4—3821no—
202C2×D4—1911no—
203C22×C4—21no—
204C22×C4—3821no—
205C22×C4—3821no—
206C22×C4—1911no—
207C22×C4—1911no—
208C22×C4—1911no—
209C22×C4—1911no—
210C22×C4—1911no—
211C22×C4—7642no—
212C22×C4C22×D19111yescharacteristic
213Q16—42no—
214SD16—7642no—
215C2×C8—1912no—
216C2×C8—7642no—
217C2×C8C22×D19112yes—
218C4⋊C4—21no—
219C4⋊C4—3822no—
220C4⋊C4—3822no—
221C4⋊C4—3821no—
222C4⋊C4—3821no—
223C4⋊C4—1912no—
224C4⋊C4—1911no—
225C4⋊C4C22×D19111yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—3822no—
228C22⋊C4—3822no—
229C22⋊C4—3821no—
230C22⋊C4—3821no—
231C22⋊C4—3821no—
232C22⋊C4—3821no—
233C42—3821no—
234C23—3822no—
235C23—3822no—
236C23—3821no—
237C23—1911no—
238C23—1911no—
239C23—7641no—
240C23D4×D19111yescharacteristic
241Q8—22no—
242Q8—22no—
243Q8—22no—
244Q8—42no—
245D4—3822no—
246D4—7642no—
247D4—7642no—
248D4—7642no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—3822no—
255C2×C4—3822no—
256C2×C4—3821no—
257C2×C4—3822no—
258C2×C4—1911no—
259C2×C4—1911no—
260C2×C4—7641no—
261C2×C4—7641no—
262C2×C4—7642no—
263C2×C4—7641no—
264C2×C4—7642no—
265C2×C4D4×D19111yescharacteristic
266C2×C4C23×D19111yescharacteristic
267C8—22no—
268C8—3822no—
269C22—21no—
270C22—21no—
271C22—3821no—
272C22—3821no—
273C22—3821no—
274C22—3821no—
275C22—3821no—
276C22—1912no—
277C22—7644no—
278C22—7641no—
279C22—7641no—
280C22—7641no—
281C22#2025934512yes—
282C22C2×D4×D19111yescharacteristic
283C4—21no—
284C4—41no—
285C4—42no—
286C4—41no—
287C4—3821no—
288C4—3821no—
289C4—7642no—
290C4—7641no—
291C4#2025936712yes—
292C2—22no—
293C2—3822no—
294C2—15281no—
295C2—7641no—
296C2#2713322311yescharacteristic
297C2C2×#2025934511yescharacteristic
298C2#137×D19111yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#451×C191C2
socleC22×C191C22×D4
prime radical#27133176C2
hypercenter#451D191
hypocenterC191#3190
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×#20259298C2semidirect
3. #27133003C2semidirect
4. #27132991C2semidirect
5. #27132977C2semidirect
6. #27132901C2semidirect
7. #27132895C2semidirect
8. #27132824C2semidirect
9. #27132802C2semidirect
10. C22⋊C8×D191C2semidirect
11. #451×C191C2semidirect
14. C2×#15690274C22semidirect
31. #20259256C22semidirect
32. #20259253C22semidirect
33. #20259246C22semidirect
34. #20259246C22semidirect
35. #20259244C22semidirect
39. C2×Q16×C191C22semidirect
41. #103×C191C22semidirect
149. C191#3190semidirect
159. #451D191semidirect
165. C2×Q16C2×D191semidirect
188. #103C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271331762
C2×#202592982
#271330032
#271329912
#271329772
#271329012
#271328952
#271328242
#271328022
C22⋊C8×D1912
#451×C1912
#3190191