Finite Group Databasebeta

← #37582126

Subgroups — #37582126

Order 24448.

500 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 141 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×#20259342C211yesmaximal, characteristic, complemented
2C22×#15690271C211yesmaximal, characteristic, complemented
3#27133174C211yesmaximal, characteristic, complemented
4#27132919C212yesmaximal, complemented
5#27132914C212yesmaximal, complemented
6#27132911C212yesmaximal, complemented
7#27132907C212yesmaximal, complemented
8#27132699C211yesmaximal, characteristic, complemented
9#27132659C211yesmaximal, characteristic, complemented
10#27132657C211yesmaximal, characteristic, complemented
11#525×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2211yescharacteristic
13#20259342—24no—
14C2×#15690277C2211yescharacteristic, complemented
15C2×#15690275C2211yescharacteristic, complemented
16C2×#15690274C2211yescharacteristic, complemented
17C2×#15690272C2211yescharacteristic, complemented
18C2×#15690271—24no—
19C2×#15690271C2212yes—
20C2×#15690271C2212yes—
21C22×C191⋊C8C2211yescharacteristic
22C22×C191⋊C8C2211yescharacteristic
23#20259305—21no—
24#20259301C2212yes—
25#20259296C2212yes—
26#20259291—21no—
27C4×C22⋊D191—22no—
28#20259281C2211yescharacteristic
29#20259255C2212yescomplemented
30#20259253C2212yescomplemented
31#20259246C2212yescomplemented
32#20259245C2212yescomplemented
33#20259243C2212yescomplemented
34#20259242C2212yescomplemented
35C2×#41×C191C2211yescharacteristic
36C23×C4×C191C2211yescharacteristic
37#123×C191—21no—
38C22⋊Q8×C191C2212yes—
39#121×C191C2212yes—
40#120×C191—21no—
41C4×D4×C191—22no—
42#117×C191C2211yescharacteristic
43#15690287—22no—
44#15690287—22no—
45C2×Dic382C2311yescharacteristic
46C2×D764C2311yescharacteristic
47C2×C22⋊D191—21no—
48C2×C22⋊D191—22no—
49C2×C4×D191—21no—
50#15690277—22no—
51#15690275—22no—
52#15690274—22no—
53#15690272—22no—
54#15690271—24no—
55#15690271—24no—
56C2×C191⋊C8—22no—
57C2×C191⋊C8D414yescomplemented
58C2×C191⋊C8C2312yes—
59C2×C191⋊C8C2311yescharacteristic
60C2×C191⋊C8C2311yescharacteristic
61C4⋊Dic191C2312yes—
62Dic191⋊C4—22no—
63C22⋊Dic191—21no—
64C22⋊Dic191—22no—
65(C2×D191)⋊C4—22no—
66C4×Dic191—21no—
67C24×C191D411yescharacteristic
68#41×C191—22no—
69#41×C191—22no—
70C2×Q8×C191C2311yescharacteristic
71C2×D4×C191—22no—
72C2×D4×C191—21no—
73C2×D4×C191C2311yescharacteristic
74C22×C4×C191—22no—
75C22×C4×C191—22no—
76C22×C4×C191—21no—
77C22×C4×C191D412yes—
78C22×C4×C191D411yescharacteristic
79C22×C4×C191C2312yes—
80C22×C4×C191C2311yescharacteristic
81C4⋊C4×C191—22no—
82C4⋊C4×C191C2312yes—
83C22⋊C4×C191—21no—
84C22⋊C4×C191—22no—
85C22⋊C4×C191—22no—
86C42×C191—21no—
87C22×D191—21no—
88Dic382—21no—
89Dic382—21no—
90D764—21no—
91D764—21no—
92C22⋊D191—44no—
93C22⋊D191—42no—
94C2×Dic191—22no—
95C2×Dic191—21no—
96C4×D191—42no—
97C191⋊C8—22no—
98C191⋊C8C2×D412yes—
99C191⋊C8C2×D412yes—
100C23×C191—22no—
101C23×C191—21no—
102C23×C191—42no—
103C23×C191C2×D412yes—
104C23×C191C2×D411yescharacteristic
105Q8×C191—21no—
106Q8×C191—21no—
107D4×C191—21no—
108D4×C191—21no—
109D4×C191—44no—
110D4×C191—42no—
111C2×C4×C191—22no—
112C2×C4×C191—24no—
113C2×C4×C191—22no—
114C2×C4×C191—22no—
115C2×C4×C191—22no—
116C2×C4×C191—21no—
117C2×C4×C191—42no—
118C2×C4×C191C2×D412yes—
119C2×C4×C191C2×D412yes—
120C2×C4×C191C2×D411yescharacteristic
121C2×C4×C191#4112yes—
122C2×C4×C191C2411yescharacteristic
123C22×C191—22no—
124C22×C191—22no—
125C22×C191—22no—
126C22×C191—22no—
127C22×C191—44no—
128C22×C191—41no—
129C22×C191—42no—
130C22×C191#12112yes—
131C22×C191C22×D411yescharacteristic
132C2×D191—41no—
133C2×D191—42no—
134C4×C191—22no—
135C4×C191—41no—
136C4×C191—42no—
137C4×C191#12112yes—
138C4×C191C22×D411yescharacteristic
139C4×C191C2×#4111yescharacteristic
140Dic191—41no—
141Dic191—42no—
142C2×C191—22no—
143C2×C191—42no—
144C2×C191—81no—
145C2×C191C2×#12111yescharacteristic
146C2×C191#57512yes—
147D191—81no—
148C191#305411yescharacteristic, complemented, Sylow
149#3054—1911nomaximal, Sylow
150C2×#135—1911no—
151C22×M16—1911no—
152#525—1911no—
153#525D19111yescharacteristic, complemented
154#470—1912no—
155#469—1912no—
156#468—1912no—
157#468—1912no—
158#427—1911no—
159#416—1911no—
160#416—1911no—
161C2×#41—1911no—
162C2×#41C2×D19111yescharacteristic
163C23×C4C2×D19111yescharacteristic
164#135—3824no—
165C2×Q16—1911no—
166C2×SD16—1911no—
167C2×SD16—1911no—
168C2×D8—1911no—
169C2×M16—3824no—
170C2×M16—1912no—
171C2×M16—1912no—
172C22×C8—1911no—
173C22×C8—1911no—
174#123—21no—
175#123—3821no—
176C22⋊Q8—1912no—
177C22⋊Q8C2×D19112yes—
178#121—1912no—
179#121C2×D19112yes—
180#120—21no—
181#120—3821no—
182C4×D4—22no—
183C4×D4—3822no—
184#117—1911no—
185#117C2×D19111yescharacteristic
186#107—1912no—
187#106—1912no—
188#103—1912no—
189#103—1912no—
190#102—1912no—
191#102—1912no—
192C24C22⋊D19111yescharacteristic
193#41—22no—
194#41—22no—
195#41—3822no—
196#41—3822no—
197C2×Q8—1911no—
198C2×Q8C22×D19111yescharacteristic
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—3822no—
202C2×D4—3821no—
203C2×D4—1911no—
204C2×D4C22×D19111yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—3821no—
209C22×C4C22⋊D19111yescharacteristic
210C22×C4C22⋊D19112yes—
211C22×C4C22×D19112yes—
212C22×C4C22×D19111yescharacteristic
213Q16—3822no—
214SD16—3822no—
215SD16—3822no—
216D8—3822no—
217M16—3824no—
218M16—3824no—
219C2×C8—3822no—
220C2×C8—1912no—
221C2×C8—1914no—
222C2×C8—1911no—
223C2×C8—1911no—
224C4⋊C4—22no—
225C4⋊C4—3822no—
226C4⋊C4—1912no—
227C4⋊C4C22×D19112yes—
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—3821no—
232C22⋊C4—3822no—
233C22⋊C4—3822no—
234C42—21no—
235C42—3821no—
236C23—22no—
237C23—21no—
238C23—42no—
239C23—3821no—
240C23C2×C22⋊D19112yes—
241C23C2×C22⋊D19111yescharacteristic
242Q8—21no—
243Q8—21no—
244Q8—3821no—
245Q8—3821no—
246D4—21no—
247D4—21no—
248D4—44no—
249D4—42no—
250D4—3821no—
251D4—3821no—
252D4—7644no—
253D4—7642no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—24no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—3822no—
262C2×C4—3821no—
263C2×C4—7642no—
264C2×C4D4×D19112yes—
265C2×C4C2×C22⋊D19112yes—
266C2×C4C2×C22⋊D19111yescharacteristic
267C2×C4#1569028612yes—
268C2×C4C23×D19111yescharacteristic
269C8—3822no—
270C8—1912no—
271C8—1912no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—44no—
277C22—42no—
278C22—41no—
279C22—7642no—
280C22—7641no—
281C22#2025929412yes—
282C22C22×C22⋊D19111yescharacteristic
283C4—22no—
284C4—42no—
285C4—41no—
286C4—7642no—
287C4—7641no—
288C4#2025929412yes—
289C4C2×D4×D19111yescharacteristic
290C4C2×#1569028611yescharacteristic
291C2—22no—
292C2—42no—
293C2—81no—
294C2—15281no—
295C2C2×#2025929411yescharacteristic
296C2#2713346112yes—
subgroupG/H
centerC2×C4C2×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#525×C191C2
socleC22×C191C22×D4
hypercenter#525D191
hypocenterC191#3054
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#20259342C2semidirect
2. C22×#15690271C2semidirect
3. #27133174C2semidirect
4. #27132919C2semidirect
5. #27132914C2semidirect
6. #27132911C2semidirect
7. #27132907C2semidirect
8. #27132699C2semidirect
9. #27132659C2semidirect
10. #27132657C2semidirect
11. #525×C191C2semidirect
14. C2×#15690277C22semidirect
15. C2×#15690275C22semidirect
16. C2×#15690274C22semidirect
17. C2×#15690272C22semidirect
29. #20259255C22semidirect
30. #20259253C22semidirect
31. #20259246C22semidirect
32. #20259245C22semidirect
33. #20259243C22semidirect
34. #20259242C22semidirect
57. C2×C191⋊C8D4semidirect
148. C191#3054semidirect
153. #525D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593422
C22×#156902712
#271331742
#271329192
#271329142
#271329112
#271329072
#271326992
#271326592
#271326572
#525×C1912
#3054191