Finite Group Databasebeta

← #37582470

Subgroups — #37582470

Order 24448.

410 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 127 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×#20259302C211yesmaximal, characteristic
2#117×D191C211yesmaximal, characteristic
3#27133007C212yesmaximal, complemented
4#27133006C212yesmaximal, complemented
5#27133000C212yesmaximal, complemented
6#27132995C212yesmaximal, complemented
7#27132703C211yesmaximal, characteristic, complemented
8#27132673C211yesmaximal, characteristic, complemented
9#27132671C211yesmaximal, characteristic, complemented
10#27132626C211yesmaximal, characteristic
11#428×C191C211yesmaximal, characteristic, complemented
12C22×Dic382C2211yescharacteristic
13C22×C4×D191C2211yescharacteristic
14C2×#15690271C2211yescharacteristic
15#20259302—21no—
16#20259302—21no—
17#20259302C2212yes—
18#20259302C2212yes—
19#20259281C2211yescharacteristic
20#20259280C2211yescharacteristic
21#20259280C2211yescharacteristic
22#20259279—21no—
23#20259278—21no—
24C2×C4⋊Dic191—21no—
25C4⋊C4×D191C2211yescharacteristic
26C4⋊C4×D191C2211yescharacteristic
27C2×(C2×D191)⋊C4—21no—
28C22⋊C4×D191—21no—
29C42×D191—21no—
30#20259255C2212yescomplemented
31#20259253C2212yescomplemented
32#20259247C2212yescomplemented
33#20259247C2212yescomplemented
34#20259245C2212yescomplemented
35#20259245C2212yescomplemented
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
44C2×Dic382—21no—
45C2×Dic382—22no—
46C2×Dic382—22no—
47C2×Dic382C2312yes—
48C22×Dic191—21no—
49C22×Dic191D411yescharacteristic
50C2×C4×D191—21no—
51C2×C4×D191—21no—
52C2×C4×D191—22no—
53C2×C4×D191—22no—
54C2×C4×D191D411yescharacteristic
55C2×C4×D191D411yescharacteristic
56C2×C4×D191C2311yescharacteristic
57C2×C4×D191C2311yescharacteristic
58#15690271—22no—
59C2×C191⋊C8C2312yes—
60C4⋊Dic191—22no—
61C4⋊Dic191—22no—
62C4⋊Dic191C2311yescharacteristic
63C4⋊Dic191C2311yescharacteristic
64Dic191⋊C4—21no—
65Dic191⋊C4—21no—
66C22⋊Dic191—21no—
67(C2×D191)⋊C4—21no—
68(C2×D191)⋊C4—21no—
69(C2×D191)⋊C4—22no—
70(C2×D191)⋊C4—22no—
71C4×Dic191—21no—
72C4×Dic191—21no—
73C4×Dic191—21no—
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—22no—
86C22×D191—41no—
87C22×D191C2×D411yescharacteristic
88C22×D191C2×D411yescharacteristic
89Dic382—22no—
90Dic382—22no—
91Dic382—42no—
92C2×Dic191—22no—
93C2×Dic191—21no—
94C2×Dic191—21no—
95C2×Dic191—22no—
96C2×Dic191—42no—
97C2×Dic191C2×D411yescharacteristic
98C2×Dic191C2×D411yescharacteristic
99C4×D191—22no—
100C4×D191—22no—
101C4×D191—21no—
102C4×D191—21no—
103C4×D191—21no—
104C4×D191—21no—
105C4×D191—42no—
106C4×D191—42no—
107C191⋊C8—22no—
108C23×C191C2×D411yescharacteristic
109C2×C4×C191—21no—
110C2×C4×C191—21no—
111C2×C4×C191—22no—
112C2×C4×C191—42no—
113C2×C4×C191C2×D411yescharacteristic
114C2×C4×C191#4112yes—
115C2×C4×C191#4112yes—
116C2×C4×C191C2411yescharacteristic
117C8×C191—22no—
118C22×C191—21no—
119C22×C191—21no—
120C22×C191#12312yes—
121C22×C191C22×D411yescharacteristic
122C2×D191—21no—
123C2×D191—21no—
124C2×D191—21no—
125C2×D191—21no—
126C2×D191—21no—
127C2×D191—44no—
128C2×D191#13712yes—
129C4×C191—41no—
130C4×C191—42no—
131C4×C191—41no—
132C4×C191#12312yes—
133C4×C191C2×#4111yescharacteristic
134C4×C191C2×#4111yescharacteristic
135Dic191—21no—
136Dic191—21no—
137Dic191—42no—
138Dic191—41no—
139Dic191—41no—
140C2×C191—22no—
141C2×C191C2×#12311yescharacteristic
142C2×C191C2×#13711yescharacteristic
143C2×C191#57511yescharacteristic
144D191—22no—
145D191—41no—
146C191#309511yescharacteristic, complemented, Sylow
147#3095—1911nomaximal, Sylow
148C2×C22⋊Q8—1911no—
149C2×#117—1911no—
150#485—1912no—
151#485—1912no—
152#484—1912no—
153#483—1912no—
154#428—1911no—
155#428D19111yescharacteristic, complemented
156#419—1911no—
157#419—1911no—
158#407—1911no—
159C22×Q8—1911no—
160C23×C4—1911no—
161C2×M16—1911no—
162C2×M16C2×D19111yescharacteristic
163C22⋊Q8—3821no—
164C22⋊Q8—3821no—
165C22⋊Q8—1912no—
166C22⋊Q8—1912no—
167#117—3821no—
168#117—3821no—
169#117—1911no—
170#117—1911no—
171#117—1911no—
172#117C2×D19111yescharacteristic
173C2×C4⋊C4—3821no—
174C2×C4⋊C4—1911no—
175C2×C4⋊C4—1911no—
176C2×C4⋊C4C2×D19111yescharacteristic
177C2×C22⋊C4—3821no—
178C2×C22⋊C4—3821no—
179C2×C42—3821no—
180#107—1912no—
181#107C2×D19112yescomplemented
182#106—1912no—
183#106C2×D19112yescomplemented
184#103—1912no—
185#103—1912no—
186#103—1912no—
187#103—1912no—
188C22⋊C8—1912no—
189C22⋊C8—1912no—
190C24—1911no—
191C2×Q8—3821no—
192C2×Q8—3822no—
193C2×Q8—3822no—
194C2×Q8—1912no—
195C22×C4—21no—
196C22×C4—3821no—
197C22×C4—3821no—
198C22×C4—3822no—
199C22×C4—3822no—
200C22×C4—3821no—
201C22×C4—1911no—
202C22×C4—1911no—
203C22×C4—1911no—
204C22×C4—1911no—
205C22×C4—1911no—
206C22×C4C22×D19111yescharacteristic
207M16—22no—
208M16—3822no—
209C2×C8—1912no—
210C2×C8C22×D19112yes—
211C4⋊C4—21no—
212C4⋊C4—3821no—
213C4⋊C4—3821no—
214C4⋊C4—3822no—
215C4⋊C4—3822no—
216C4⋊C4—1911no—
217C4⋊C4—1911no—
218C4⋊C4C22×D19111yescharacteristic
219C4⋊C4C22×D19111yescharacteristic
220C4⋊C4C22×D19112yes—
221C22⋊C4—21no—
222C22⋊C4—3821no—
223C22⋊C4—3821no—
224C22⋊C4—3822no—
225C22⋊C4—3822no—
226C22⋊C4—3821no—
227C42—21no—
228C42—3821no—
229C42—3821no—
230C42—3821no—
231C23—3822no—
232C23—3822no—
233C23—1911no—
234C23—1911no—
235C23—7641no—
236C23D4×D19111yescharacteristic
237Q8—3822no—
238Q8—3822no—
239Q8—7642no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—3822no—
245C2×C4—3821no—
246C2×C4—3821no—
247C2×C4—3821no—
248C2×C4—3821no—
249C2×C4—3822no—
250C2×C4—3822no—
251C2×C4—3821no—
252C2×C4—3821no—
253C2×C4—3822no—
254C2×C4—1911no—
255C2×C4—1911no—
256C2×C4—7642no—
257C2×C4—7642no—
258C2×C4—7642no—
259C2×C4D4×D19111yescharacteristic
260C2×C4#1569028712yes—
261C2×C4#1569028812yes—
262C2×C4C23×D19111yescharacteristic
263C8—22no—
264C8—3822no—
265C22—21no—
266C22—21no—
267C22—3821no—
268C22—3821no—
269C22—3821no—
270C22—3821no—
271C22—3821no—
272C22—1912no—
273C22—7644no—
274C22#2025930712yes—
275C22C2×D4×D19111yescharacteristic
276C4—41no—
277C4—42no—
278C4—41no—
279C4—3821no—
280C4—3821no—
281C4—7641no—
282C4—7641no—
283C4—7642no—
284C4#2025930712yes—
285C4C2×#1569028711yescharacteristic
286C4C2×#1569028811yescharacteristic
287C2—22no—
288C2—3822no—
289C2—7641no—
290C2C2×#2025930711yescharacteristic
291C2#137×D19111yescharacteristic
292C2#2713345911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#428×C191C2
socleC22×C191C22×D4
prime radicalC23×D191D4
hypercenter#428D191
hypocenterC191#3095
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
3. #27133007C2semidirect
4. #27133006C2semidirect
5. #27133000C2semidirect
6. #27132995C2semidirect
7. #27132703C2semidirect
8. #27132673C2semidirect
9. #27132671C2semidirect
11. #428×C191C2semidirect
30. #20259255C22semidirect
31. #20259253C22semidirect
32. #20259247C22semidirect
33. #20259247C22semidirect
34. #20259245C22semidirect
35. #20259245C22semidirect
41. #107×C191C22semidirect
42. #106×C191C22semidirect
146. C191#3095semidirect
155. #428D191semidirect
181. #107C2×D191semidirect
183. #106C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593022
#117×D1912
#271330072
#271330062
#271330002
#271329952
#271327032
#271326732
#271326712
#271326262
#428×C1912
#3095191