Finite Group Databasebeta

← #37581804

Subgroups — #37581804

Order 24448.

732 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 149 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×#20259369C211yesmaximal, characteristic, complemented
2C2×#20259348C211yesmaximal, characteristic, complemented
3C22×#15690275C211yesmaximal, characteristic, complemented
4#27132806C214yesmaximal, complemented
5#27132801C214yesmaximal, complemented
6#27132671C211yesmaximal, characteristic, complemented
7#27132657C211yesmaximal, characteristic, complemented
8#27132630C211yesmaximal, characteristic, complemented
9#535×C191C211yesmaximal, characteristic, complemented
10#20259369—24no—
11#20259369—24no—
12C2×#15690287C2211yescharacteristic
13C2×#15690287C2212yescomplemented
14C2×#15690286—22no—
15C2×#15690286—21no—
16C22×Dic382C2211yescharacteristic
17C22×Dic382C2212yescomplemented
18C2×Q8×D191—21no—
19#20259348—24no—
20C2×#15690275—24no—
21C2×#15690275C2212yes—
22C2×#15690275C2214yes—
23C2×#15690272C2212yes—
24C2×#15690271C2211yescharacteristic
25C22×C191⋊C8C2211yescharacteristic
26#20259245C2212yescomplemented
27#20259245C2214yescomplemented
28#20259242C2212yescomplemented
29C22⋊(C191⋊C8)C2214yescomplemented
30C2×#41×C191C2211yescharacteristic
31C22×D4×C191C2211yescharacteristic
32#127×C191—21no—
33#124×C191—21no—
34#121×C191C2214yes—
35#120×C191—22no—
36#117×C191C2211yescharacteristic
37#15690287—24no—
38#15690287—24no—
39#15690287—22no—
40#15690287D414yescomplemented
41#15690286—216no—
42#15690286—28no—
43C2×Dic382—24no—
44C2×Dic382—24no—
45C2×Dic382—21no—
46C2×Dic382—22no—
47C2×Dic382—22no—
48C2×Dic382D414yescomplemented
49C2×Dic382C2312yes—
50C2×Dic382C2311yescharacteristic
51Q8×D191—28no—
52C2×D764C2311yescharacteristic
53C2×C22⋊D191—21no—
54C2×C22⋊D191—22no—
55C22×Dic191—21no—
56C22×Dic191—22no—
57C2×C4×D191—21no—
58C2×C4×D191—22no—
59#15690275—28no—
60#15690275—24no—
61#15690272—24no—
62#15690271—22no—
63C2×C191⋊C8—22no—
64C2×C191⋊C8C2312yes—
65C2×C191⋊C8C2312yes—
66C24×C191—21no—
67#41×C191—22no—
68#41×C191—22no—
69C2×Q8×C191D411yescharacteristic
70C2×D4×C191—24no—
71C2×D4×C191—22no—
72C2×D4×C191—22no—
73C2×D4×C191—21no—
74C2×D4×C191D412yes—
75C2×D4×C191D411yescharacteristic
76C2×D4×C191C2312yes—
77C22×C4×C191C2312yes—
78C22×C4×C191C2311yescharacteristic
79C4⋊C4×C191C2312yes—
80C22⋊C4×C191—21no—
81C22⋊C4×C191—24no—
82C42×C191—21no—
83C22×D191—21no—
84Dic382—28no—
85Dic382—24no—
86Dic382—21no—
87Dic382—22no—
88Dic382C2×D412yes—
89Dic382C2×D414yes—
90D764—21no—
91D764C2×D412yes—
92C22⋊D191—28no—
93C22⋊D191—24no—
94C2×Dic191—28no—
95C2×Dic191—24no—
96C2×Dic191—21no—
97C2×Dic191—22no—
98C4×D191—28no—
99C4×D191—24no—
100C191⋊C8—22no—
101C191⋊C8—22no—
102C23×C191—22no—
103C23×C191—42no—
104C23×C191—82no—
105C23×C191C2×D412yes—
106C23×C191C2×D411yescharacteristic
107Q8×C191—22no—
108D4×C191—24no—
109D4×C191—22no—
110D4×C191—22no—
111D4×C191—22no—
112D4×C191—42no—
113D4×C191—84no—
114C2×C4×C191—22no—
115C2×C4×C191—22no—
116C2×C4×C191—22no—
117C2×C4×C191C2×D412yes—
118C2×C4×C191C2×D412yes—
119C2×C4×C191C2×D412yes—
120C2×C4×C191C2×D411yescharacteristic
121C2×C4×C191C2411yescharacteristic
122C22×C191—22no—
123C22×C191—22no—
124C22×C191—24no—
125C22×C191—42no—
126C22×C191—44no—
127C22×C191—84no—
128C22×C191#12012yes—
129C22×C191C22×D411yescharacteristic
130C2×D191—22no—
131C2×D191—42no—
132C4×C191—22no—
133C4×C191—42no—
134C4×C191#12012yes—
135C4×C191C22×D411yescharacteristic
136C4×C191C22×D411yescharacteristic
137Dic191—24no—
138Dic191—22no—
139C2×C191—22no—
140C2×C191—42no—
141C2×C191—82no—
142C2×C191C2×#12011yescharacteristic
143C2×C191#57712yes—
144D191—42no—
145C191#301011yescharacteristic, complemented, Sylow
146#3010—1911nomaximal, Sylow
147C2×2−1+4—1911no—
148C2×AGL(1,ℤ/8)—1911no—
149C22×SD16—1911no—
150#535D19111yescharacteristic, complemented
151#449—1914no—
152#448—1914no—
153#419—1911no—
154#416—1911no—
155#408—1911no—
1562−1+4—3824no—
1572−1+4—3824no—
158C2×#41—3822no—
159C2×#41—3821no—
160C2×#41—1912no—
161C2×#41—1911no—
162C2×#41C2×D19111yescharacteristic
163C22×Q8—3821no—
164C22×Q8—1912no—
165C22×Q8—1911no—
166C22×D4C2×D19111yescharacteristic
167AGL(1,ℤ/8)—3824no—
168C2×SD16—3824no—
169C2×SD16—1914no—
170C2×SD16—1912no—
171C2×D8—1912no—
172C2×M16—1911no—
173C22×C8—1911no—
174#127—21no—
175#124—21no—
176#121C2×D19114yes—
177#120—22no—
178#117C2×D19111yescharacteristic
179#103—1912no—
180#103—1914no—
181#102—1912no—
182C22⋊C8—1914no—
183C24—21no—
184#41—22no—
185#41—22no—
186#41—3824no—
187#41—3824no—
188#41—38216no—
189#41—3828no—
190#41—3822no—
191#41—1914no—
192C2×Q8—3824no—
193C2×Q8—3824no—
194C2×Q8—3828no—
195C2×Q8—3822no—
196C2×Q8—3822no—
197C2×Q8—3821no—
198C2×Q8—1914no—
199C2×Q8—1911no—
200C2×Q8—1912no—
201C2×Q8C22⋊D19111yescharacteristic
202C2×D4—24no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—3822no—
207C2×D4—3821no—
208C2×D4—1911no—
209C2×D4C22⋊D19112yes—
210C2×D4C22⋊D19111yescharacteristic
211C2×D4C22×D19112yes—
212C22×C4—3822no—
213C22×C4—3821no—
214C22×C4—3822no—
215C22×C4—3821no—
216C22×C4C22×D19112yes—
217C22×C4C22×D19111yescharacteristic
218SD16—3828no—
219SD16—3824no—
220D8—3824no—
221M16—3822no—
222C2×C8—3822no—
223C2×C8—1912no—
224C2×C8—1912no—
225C4⋊C4C22×D19112yes—
226C22⋊C4—21no—
227C22⋊C4—24no—
228C42—21no—
229C23—22no—
230C23—42no—
231C23—82no—
232C23—3821no—
233C23C2×C22⋊D19112yes—
234C23C2×C22⋊D19111yescharacteristic
235Q8—22no—
236Q8—3828no—
237Q8—3821no—
238Q8—3822no—
239Q8—3824no—
240Q8—1912no—
241Q8—1914no—
242D4—24no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—42no—
247D4—84no—
248D4—3828no—
249D4—3824no—
250D4—3821no—
251D4—1912no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—3828no—
256C2×C4—3828no—
257C2×C4—3822no—
258C2×C4—3821no—
259C2×C4—3824no—
260C2×C4—3824no—
261C2×C4D4×D19112yes—
262C2×C4D4×D19112yes—
263C2×C4C2×C22⋊D19112yes—
264C2×C4C2×C22⋊D19111yescharacteristic
265C2×C4C23×D19111yescharacteristic
266C8—3822no—
267C8—3822no—
268C22—22no—
269C22—24no—
270C22—22no—
271C22—44no—
272C22—42no—
273C22—84no—
274C22—3822no—
275C22—7642no—
276C22#2025929012yes—
277C22C22×C22⋊D19111yescharacteristic
278C4—22no—
279C4—42no—
280C4—3824no—
281C4—3822no—
282C4#2025929012yes—
283C4C2×D4×D19111yescharacteristic
284C4C2×D4×D19111yescharacteristic
285C2—22no—
286C2—42no—
287C2—82no—
288C2—7642no—
289C2C2×#2025929011yescharacteristic
290C2#2713347412yes—
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#535×C191C2
socleC22×C191C22×D4
hypercenter#535D191
hypocenterC191#3010
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#20259369C2semidirect
2. C2×#20259348C2semidirect
3. C22×#15690275C2semidirect
4. #27132806C2semidirect
5. #27132801C2semidirect
6. #27132671C2semidirect
7. #27132657C2semidirect
8. #27132630C2semidirect
9. #535×C191C2semidirect
13. C2×#15690287C22semidirect
17. C22×Dic382C22semidirect
26. #20259245C22semidirect
27. #20259245C22semidirect
28. #20259242C22semidirect
29. C22⋊(C191⋊C8)C22semidirect
40. #15690287D4semidirect
48. C2×Dic382D4semidirect
145. C191#3010semidirect
150. #535D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593692
C2×#202593482
C22×#156902752
#271328062
#271328012
#271326712
#271326572
#271326302
#535×C1912
#3010191