Finite Group Databasebeta

← #37582454

Subgroups — #37582454

Order 24448.

414 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 125 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#27133175C211yesmaximal, characteristic
2C2×#20259280C211yesmaximal, characteristic, complemented
3#27133007C212yesmaximal, complemented
4#27133006C212yesmaximal, complemented
5#27132990C212yesmaximal, complemented
6#27132989C212yesmaximal, complemented
7#27132706C211yesmaximal, characteristic, complemented
8#27132665C211yesmaximal, characteristic, complemented
9#27132662C211yesmaximal, characteristic, complemented
10#27132626C211yesmaximal, characteristic, complemented
11#428×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2211yescharacteristic
13C22×C4×D191C2211yescharacteristic
14C2×#15690271C2211yescharacteristic
15#20259306—21no—
16#20259302C2211yescharacteristic
17#20259302C2211yescharacteristic
18#20259297C2211yescharacteristic
19#20259297C2211yescharacteristic
20#20259292—21no—
21C4×D764—21no—
22C4×D764—21no—
23#20259280—21no—
24#20259280—21no—
25#20259280C2212yes—
26#20259280C2212yes—
27C2×C4⋊Dic191C2211yescharacteristic
28C2×(C2×D191)⋊C4—21no—
29C2×C4×Dic191—21no—
30#20259255C2212yescomplemented
31#20259253C2212yescomplemented
32#20259247C2212yescomplemented
33#20259245C2212yescomplemented
34#20259244C2212yescomplemented
35#20259242C2212yescomplemented
36(C2×D191)⋊C8C2212yes—
37(C2×D191)⋊C8C2212yes—
38C2×M16×C191C2211yescharacteristic
39#117×C191C2211yescharacteristic
40C2×C4⋊C4×C191C2211yescharacteristic
41#107×C191C2212yescomplemented
42#106×C191C2212yescomplemented
43C23×D191D411yescharacteristic
44#15690287—22no—
45#15690287—22no—
46C2×Dic382C2311yescharacteristic
47C2×D764—21no—
48C2×D764—21no—
49C2×D764C2311yescharacteristic
50C2×C22⋊D191—21no—
51C22×Dic191—21no—
52C22×Dic191D411yescharacteristic
53C2×C4×D191—21no—
54C2×C4×D191—22no—
55C2×C4×D191—22no—
56C2×C4×D191D411yescharacteristic
57C2×C4×D191D411yescharacteristic
58C2×C4×D191C2311yescharacteristic
59C2×C4×D191C2311yescharacteristic
60#15690271—22no—
61C2×C191⋊C8C2312yes—
62C4⋊Dic191—21no—
63C4⋊Dic191—21no—
64C4⋊Dic191—21no—
65C4⋊Dic191C2312yes—
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—22no—
73C4×Dic191—22no—
74C22×C4×C191—21no—
75C22×C4×C191C2311yescharacteristic
76M16×C191—22no—
77C2×C8×C191C2312yes—
78C4⋊C4×C191—21no—
79C4⋊C4×C191C2311yescharacteristic
80C4⋊C4×C191C2312yes—
81C4⋊C4×C191C2311yescharacteristic
82C22⋊C4×C191—21no—
83C42×C191—21no—
84C22×D191—22no—
85C22×D191—21no—
86C22×D191—41no—
87C22×D191—41no—
88C22×D191C2×D411yescharacteristic
89C22×D191C2×D411yescharacteristic
90Dic382—21no—
91Dic382—21no—
92D764—21no—
93D764—21no—
94D764—42no—
95D764—42no—
96C22⋊D191—42no—
97C2×Dic191—22no—
98C2×Dic191—22no—
99C2×Dic191—21no—
100C2×Dic191—42no—
101C2×Dic191C2×D411yescharacteristic
102C2×Dic191C2×D411yescharacteristic
103C4×D191—22no—
104C4×D191—22no—
105C4×D191—21no—
106C4×D191—21no—
107C4×D191—21no—
108C4×D191—21no—
109C4×D191—42no—
110C191⋊C8—22no—
111C23×C191C2×D411yescharacteristic
112C2×C4×C191—21no—
113C2×C4×C191—21no—
114C2×C4×C191—22no—
115C2×C4×C191—42no—
116C2×C4×C191C2×D411yescharacteristic
117C2×C4×C191#4112yes—
118C2×C4×C191#4112yes—
119C2×C4×C191C2411yescharacteristic
120C8×C191—22no—
121C22×C191—21no—
122C22×C191—21no—
123C22×C191#12312yes—
124C22×C191C22×D411yescharacteristic
125C2×D191—21no—
126C2×D191—21no—
127C2×D191—21no—
128C2×D191—21no—
129C2×D191—21no—
130C2×D191—21no—
131C2×D191—44no—
132C2×D191—41no—
133C2×D191—41no—
134C2×D191—41no—
135C4×C191—42no—
136C4×C191—41no—
137C4×C191—41no—
138C4×C191#12312yes—
139C4×C191C2×#4111yescharacteristic
140C4×C191C2×#4111yescharacteristic
141Dic191—21no—
142Dic191—21no—
143Dic191—41no—
144Dic191—42no—
145C2×C191—22no—
146C2×C191C2×#12311yescharacteristic
147C2×C191#57511yescharacteristic
148C2×C191#57511yescharacteristic
149D191—41no—
150D191—41no—
151D191—81no—
152C191#309311yescharacteristic, complemented, Sylow
153#3093—1911nomaximal, Sylow
154#525—1911no—
155C2×#117—1911no—
156#485—1912no—
157#485—1912no—
158#482—1912no—
159#482—1912no—
160#428—1911no—
161#428D19111yescharacteristic, complemented
162#417—1911no—
163#417—1911no—
164#407—1911no—
165C2×#41—1911no—
166C23×C4—1911no—
167C2×M16—1911no—
168C2×M16C2×D19111yescharacteristic
169#123—3821no—
170C22⋊Q8—1911no—
171C22⋊Q8—1911no—
172#121—1911no—
173#121—1911no—
174#120—3821no—
175C4×D4—3821no—
176C4×D4—3821no—
177#117—3821no—
178#117—3821no—
179#117—1912no—
180#117—1912no—
181#117C2×D19111yescharacteristic
182C2×C4⋊C4—1911no—
183C2×C4⋊C4C2×D19111yescharacteristic
184C2×C22⋊C4—3821no—
185C2×C42—3821no—
186#107—1912no—
187#107C2×D19112yescomplemented
188#106—1912no—
189#106C2×D19112yescomplemented
190#103—1912no—
191#103—1912no—
192#102—1912no—
193#102—1912no—
194C22⋊C8—1912no—
195C22⋊C8—1912no—
196C24—1911no—
197#41—3822no—
198#41—3822no—
199C2×Q8—1911no—
200C2×D4—3821no—
201C2×D4—3821no—
202C2×D4—3821no—
203C2×D4—1911no—
204C22×C4—21no—
205C22×C4—3822no—
206C22×C4—3822no—
207C22×C4—3821no—
208C22×C4—3821no—
209C22×C4—1911no—
210C22×C4—1911no—
211C22×C4—1911no—
212C22×C4—1911no—
213C22×C4—1911no—
214C22×C4C22×D19111yescharacteristic
215M16—22no—
216M16—3822no—
217C2×C8—1912no—
218C2×C8C22×D19112yes—
219C4⋊C4—21no—
220C4⋊C4—3821no—
221C4⋊C4—3821no—
222C4⋊C4—3821no—
223C4⋊C4—1912no—
224C4⋊C4C22×D19112yes—
225C4⋊C4C22×D19111yescharacteristic
226C4⋊C4C22×D19111yescharacteristic
227C22⋊C4—21no—
228C22⋊C4—3822no—
229C22⋊C4—3822no—
230C22⋊C4—3821no—
231C22⋊C4—3821no—
232C22⋊C4—3821no—
233C22⋊C4—3821no—
234C42—21no—
235C42—3822no—
236C42—3822no—
237C23—3822no—
238C23—3821no—
239C23—1911no—
240C23—1911no—
241C23—7641no—
242C23—7641no—
243C23D4×D19111yescharacteristic
244Q8—3821no—
245Q8—3821no—
246D4—3821no—
247D4—3821no—
248D4—7642no—
249D4—7642no—
250D4—7642no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—42no—
255C2×C4—3822no—
256C2×C4—3821no—
257C2×C4—3821no—
258C2×C4—3821no—
259C2×C4—3821no—
260C2×C4—3822no—
261C2×C4—3822no—
262C2×C4—3822no—
263C2×C4—3821no—
264C2×C4—1911no—
265C2×C4—1911no—
266C2×C4—7642no—
267C2×C4—7642no—
268C2×C4D4×D19111yescharacteristic
269C2×C4#1569028712yes—
270C2×C4#1569028812yes—
271C2×C4C23×D19111yescharacteristic
272C8—22no—
273C8—3822no—
274C22—21no—
275C22—21no—
276C22—3821no—
277C22—3821no—
278C22—3821no—
279C22—3821no—
280C22—3821no—
281C22—3821no—
282C22—7644no—
283C22—7641no—
284C22—7641no—
285C22—7641no—
286C22#2025930712yes—
287C22C2×D4×D19111yescharacteristic
288C4—42no—
289C4—41no—
290C4—41no—
291C4—3821no—
292C4—3821no—
293C4—7642no—
294C4—7641no—
295C4#2025930712yes—
296C4C2×#1569028711yescharacteristic
297C4C2×#1569028811yescharacteristic
298C2—22no—
299C2—15281no—
300C2—7641no—
301C2—7641no—
302C2C2×#2025930711yescharacteristic
303C2#2713345911yescharacteristic
304C2#2713346211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#428×C191C2
socleC22×C191C22×D4
prime radical#27133175C2
hypercenter#428D191
hypocenterC191#3093
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. C2×#20259280C2semidirect
3. #27133007C2semidirect
4. #27133006C2semidirect
5. #27132990C2semidirect
6. #27132989C2semidirect
7. #27132706C2semidirect
8. #27132665C2semidirect
9. #27132662C2semidirect
10. #27132626C2semidirect
11. #428×C191C2semidirect
30. #20259255C22semidirect
31. #20259253C22semidirect
32. #20259247C22semidirect
33. #20259245C22semidirect
34. #20259244C22semidirect
35. #20259242C22semidirect
41. #107×C191C22semidirect
42. #106×C191C22semidirect
152. C191#3093semidirect
161. #428D191semidirect
187. #107C2×D191semidirect
189. #106C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271331752
C2×#202592802
#271330072
#271330062
#271329902
#271329892
#271327062
#271326652
#271326622
#271326262
#428×C1912
#3093191