Finite Group Databasebeta

← #37582128

Subgroups — #37582128

Order 24448.

500 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 133 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×#15690270C211yesmaximal, characteristic, complemented
3#27133175C211yesmaximal, characteristic, complemented
4#27133170C211yesmaximal, characteristic, complemented
5#27132920C212yesmaximal, complemented
6#27132915C212yesmaximal, complemented
7#27132912C212yesmaximal, complemented
8#27132908C212yesmaximal, complemented
9#27132699C211yesmaximal, characteristic, complemented
10#27132660C211yesmaximal, characteristic, complemented
11#416×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2211yescharacteristic
13C22×C4×D191C2211yescharacteristic
14#20259342—24no—
15C2×#15690277C2211yescharacteristic, complemented
16C2×#15690275C2211yescharacteristic, complemented
17C2×#15690274C2211yescharacteristic, complemented
18C2×#15690272C2211yescharacteristic, complemented
19C2×#15690270—24no—
20C2×#15690270C2212yes—
21C2×#15690270C2212yes—
22C22×C191⋊C8C2211yescharacteristic
23#20259306—21no—
24#20259303—21no—
25#20259302C2212yes—
26#20259298C2212yes—
27#20259297C2212yes—
28#20259293C2212yes—
29#20259292—21no—
30#20259290—21no—
31C4×D764—22no—
32C4×C22⋊D191—22no—
33#20259281C2211yescharacteristic
34#20259255C2212yescomplemented
35#20259253C2212yescomplemented
36#20259247C2212yescomplemented
37#20259244C2212yescomplemented
38C2×#41×C191C2211yescharacteristic
39C22×C8×C191C2211yescharacteristic
40#117×C191C2211yescharacteristic
41#103×C191C2212yescomplemented
42#102×C191C2212yescomplemented
43C23×D191D411yescharacteristic
44#15690287—22no—
45#15690287—22no—
46C2×Dic382C2311yescharacteristic
47C2×D764—22no—
48C2×D764C2311yescharacteristic
49C2×C22⋊D191—22no—
50C2×C22⋊D191—21no—
51C22×Dic191D411yescharacteristic
52C2×C4×D191—22no—
53C2×C4×D191—21no—
54C2×C4×D191—22no—
55C2×C4×D191D412yes—
56C2×C4×D191C2312yes—
57#15690277—22no—
58#15690275—22no—
59#15690274—22no—
60#15690272—22no—
61#15690270—24no—
62#15690270—24no—
63C2×C191⋊C8D414yescomplemented
64C2×C191⋊C8C2311yescharacteristic
65C2×C191⋊C8C2311yescharacteristic
66C4⋊Dic191—22no—
67C4⋊Dic191C2312yes—
68Dic191⋊C4—22no—
69C22⋊Dic191—22no—
70C22⋊Dic191—21no—
71(C2×D191)⋊C4—22no—
72(C2×D191)⋊C4—22no—
73(C2×D191)⋊C4—22no—
74C4×Dic191—21no—
75#41×C191—22no—
76#41×C191—22no—
77C2×Q8×C191C2311yescharacteristic
78C2×D4×C191—21no—
79C2×D4×C191C2311yescharacteristic
80C22×C4×C191—21no—
81C22×C4×C191C2311yescharacteristic
82C2×C8×C191—22no—
83C2×C8×C191C2312yes—
84C4⋊C4×C191C2312yes—
85C22⋊C4×C191—21no—
86C42×C191—21no—
87C22×D191—22no—
88C22×D191—21no—
89C22×D191—42no—
90C22×D191C2×D412yes—
91Dic382—21no—
92Dic382—21no—
93D764—21no—
94D764—21no—
95D764—44no—
96C22⋊D191—44no—
97C22⋊D191—42no—
98C2×Dic191—22no—
99C2×Dic191—22no—
100C2×Dic191—21no—
101C2×Dic191C2×D412yes—
102C4×D191—24no—
103C4×D191—22no—
104C4×D191—22no—
105C4×D191—42no—
106C191⋊C8C2×D412yes—
107C191⋊C8C2×D412yes—
108C23×C191—21no—
109C23×C191C2×D411yescharacteristic
110Q8×C191—21no—
111Q8×C191—21no—
112D4×C191—21no—
113D4×C191—21no—
114D4×C191—42no—
115C2×C4×C191—22no—
116C2×C4×C191—21no—
117C2×C4×C191—42no—
118C2×C4×C191C2×D412yes—
119C2×C4×C191C2×D411yescharacteristic
120C2×C4×C191#4112yes—
121C2×C4×C191C2411yescharacteristic
122C8×C191—22no—
123C22×C191—22no—
124C22×C191—41no—
125C22×C191—42no—
126C22×C191#12112yes—
127C22×C191C22×D411yescharacteristic
128C2×D191—22no—
129C2×D191—22no—
130C2×D191—22no—
131C2×D191—44no—
132C2×D191—41no—
133C2×D191—42no—
134C4×C191—41no—
135C4×C191—42no—
136C4×C191#12112yes—
137C4×C191C22×D411yescharacteristic
138C4×C191C2×#4111yescharacteristic
139Dic191—22no—
140Dic191—41no—
141Dic191—42no—
142C2×C191—22no—
143C2×C191—81no—
144C2×C191C2×#12111yescharacteristic
145C2×C191#57512yes—
146D191—42no—
147D191—81no—
148C191#305411yescharacteristic, complemented, Sylow
149#3054—1911nomaximal, Sylow
150C2×#135—1911no—
151C22×M16—1911no—
152#525—1911no—
153#525—1911no—
154#470—1912no—
155#469—1912no—
156#468—1912no—
157#468—1912no—
158#427—1911no—
159#416—1911no—
160#416D19111yescharacteristic, complemented
161C2×#41—1911no—
162C2×#41C2×D19111yescharacteristic
163C23×C4—1911no—
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×C8C2×D19111yescharacteristic
174#123—3821no—
175#123—3821no—
176C22⋊Q8—1912no—
177C22⋊Q8—1912no—
178#121—1912no—
179#121—1912no—
180#120—3821no—
181#120—3821no—
182C4×D4—3822no—
183C4×D4—3822no—
184#117—1911no—
185#117C2×D19111yescharacteristic
186#107—1912no—
187#106—1912no—
188#103—1912no—
189#103C2×D19112yescomplemented
190#102—1912no—
191#102C2×D19112yescomplemented
192C24—1911no—
193#41—22no—
194#41—22no—
195#41—3822no—
196#41—3822no—
197C2×Q8—1911no—
198C2×Q8C22×D19111yescharacteristic
199C2×D4—21no—
200C2×D4—3822no—
201C2×D4—3822no—
202C2×D4—3821no—
203C2×D4—1911no—
204C2×D4C22×D19111yescharacteristic
205C22×C4—21no—
206C22×C4—3822no—
207C22×C4—3822no—
208C22×C4—3821no—
209C22×C4—1911no—
210C22×C4—1912no—
211C22×C4—1912no—
212C22×C4C22×D19111yescharacteristic
213Q16—3822no—
214SD16—3822no—
215SD16—3822no—
216D8—3822no—
217M16—3824no—
218M16—3824no—
219C2×C8—22no—
220C2×C8—1914no—
221C2×C8—1911no—
222C2×C8—1911no—
223C2×C8C22×D19112yes—
224C4⋊C4—3822no—
225C4⋊C4—3822no—
226C4⋊C4—1912no—
227C4⋊C4C22×D19112yes—
228C22⋊C4—21no—
229C22⋊C4—3822no—
230C22⋊C4—3822no—
231C22⋊C4—3822no—
232C22⋊C4—3821no—
233C22⋊C4—3822no—
234C42—21no—
235C42—3821no—
236C23—21no—
237C23—3822no—
238C23—3821no—
239C23—1912no—
240C23—7642no—
241C23D4×D19111yescharacteristic
242Q8—21no—
243Q8—21no—
244Q8—3821no—
245Q8—3821no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—3821no—
250D4—3821no—
251D4—7644no—
252D4—7644no—
253D4—7642no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—3822no—
258C2×C4—3822no—
259C2×C4—3822no—
260C2×C4—3824no—
261C2×C4—3822no—
262C2×C4—3821no—
263C2×C4—1912no—
264C2×C4—7642no—
265C2×C4D4×D19112yes—
266C2×C4D4×D19111yescharacteristic
267C2×C4#1569028712yes—
268C2×C4C23×D19111yescharacteristic
269C8—22no—
270C8—1912no—
271C8—1912no—
272C22—22no—
273C22—42no—
274C22—41no—
275C22—3822no—
276C22—3822no—
277C22—3822no—
278C22—7644no—
279C22—7642no—
280C22—7641no—
281C22#2025929512yes—
282C22C2×D4×D19111yescharacteristic
283C4—42no—
284C4—41no—
285C4—3822no—
286C4—7642no—
287C4—7641no—
288C4#2025929512yes—
289C4C2×D4×D19111yescharacteristic
290C4C2×#1569028711yescharacteristic
291C2—22no—
292C2—81no—
293C2—15281no—
294C2—7642no—
295C2C2×#2025929511yescharacteristic
296C2#2713346012yes—
subgroupG/H
centerC2×C4D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#416×C191C2
socleC22×C191C22×D4
hypercenter#416D191
hypocenterC191#3054
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#20259342C2semidirect
2. C22×#15690270C2semidirect
3. #27133175C2semidirect
4. #27133170C2semidirect
5. #27132920C2semidirect
6. #27132915C2semidirect
7. #27132912C2semidirect
8. #27132908C2semidirect
9. #27132699C2semidirect
10. #27132660C2semidirect
11. #416×C191C2semidirect
15. C2×#15690277C22semidirect
16. C2×#15690275C22semidirect
17. C2×#15690274C22semidirect
18. C2×#15690272C22semidirect
34. #20259255C22semidirect
35. #20259253C22semidirect
36. #20259247C22semidirect
37. #20259244C22semidirect
41. #103×C191C22semidirect
42. #102×C191C22semidirect
63. C2×C191⋊C8D4semidirect
148. C191#3054semidirect
160. #416D191semidirect
189. #103C2×D191semidirect
191. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593422
C22×#156902702
#271331752
#271331702
#271329202
#271329152
#271329122
#271329082
#271326992
#271326602
#416×C1912
#3054191