Finite Group Databasebeta

← #37583308

Subgroups — #37583308

Order 24448.

424 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27133174C211yesmaximal, characteristic, complemented
2#27133004C211yesmaximal, characteristic, complemented
3#27133001C211yesmaximal, characteristic, complemented
4#27132987C211yesmaximal, characteristic, complemented
5#27132984C211yesmaximal, characteristic, complemented
6#27132902C211yesmaximal, characteristic, complemented
7#27132897C211yesmaximal, characteristic, complemented
8#27132894C211yesmaximal, characteristic, complemented
9#27132890C211yesmaximal, characteristic, complemented
10#27132825C211yesmaximal, characteristic, complemented
11#27132823C211yesmaximal, characteristic, complemented
12#27132808C211yesmaximal, characteristic, complemented
13#27132806C211yesmaximal, characteristic, complemented
14C2×C22⋊(C191⋊C8)C211yesmaximal, characteristic, complemented
15#525×C191C211yesmaximal, characteristic, complemented
16C2×#15690287C2211yescharacteristic
17C2×#15690277C2211yescharacteristic, complemented
18C2×#15690275C2211yescharacteristic, complemented
19C2×#15690274C2211yescharacteristic, complemented
20C2×#15690272C2211yescharacteristic, complemented
21C22×C191⋊C8C2211yescharacteristic
22C22×C191⋊C8C2211yescharacteristic
23#20259305—21no—
24#20259301C2211yescharacteristic
25#20259301C2211yescharacteristic
26#20259296C2211yescharacteristic
27#20259296C2211yescharacteristic
28#20259291—21no—
29C4×C22⋊D191—21no—
30C4×C22⋊D191—21no—
31#20259281C2211yescharacteristic
32#20259255C2211yescharacteristic, complemented
33#20259255C2211yescharacteristic, complemented
34#20259253C2211yescharacteristic, complemented
35#20259253C2211yescharacteristic, complemented
36#20259246C2211yescharacteristic, complemented
37#20259246C2211yescharacteristic, complemented
38#20259245C2211yescharacteristic, complemented
39#20259245C2211yescharacteristic, complemented
40#20259243C2211yescharacteristic, complemented
41#20259243C2211yescharacteristic, complemented
42#20259242C2211yescharacteristic, complemented
43#20259242C2211yescharacteristic, complemented
44C22⋊(C191⋊C8)C2211yescharacteristic
45C22⋊(C191⋊C8)C2211yescharacteristic
46C22⋊(C191⋊C8)C2211yescharacteristic
47C22⋊(C191⋊C8)C2211yescharacteristic
48C2×#41×C191C2211yescharacteristic
49C23×C4×C191C2211yescharacteristic
50#123×C191—21no—
51C22⋊Q8×C191C2211yescharacteristic
52C22⋊Q8×C191C2211yescharacteristic
53#121×C191C2211yescharacteristic
54#121×C191C2211yescharacteristic
55#120×C191—21no—
56C4×D4×C191—21no—
57C4×D4×C191—21no—
58#117×C191C2211yescharacteristic
59#15690287—41no—
60#15690287—41no—
61C2×Dic382C2311yescharacteristic
62C2×D764C2311yescharacteristic
63C2×C22⋊D191—21no—
64C2×C22⋊D191—21no—
65C2×C22⋊D191—21no—
66C2×C4×D191—21no—
67#15690277—41no—
68#15690275—41no—
69#15690274—41no—
70#15690272—41no—
71C2×C191⋊C8—41no—
72C2×C191⋊C8—41no—
73C2×C191⋊C8C2311yescharacteristic
74C2×C191⋊C8C2311yescharacteristic
75C2×C191⋊C8C2311yescharacteristic
76C2×C191⋊C8C2311yescharacteristic
77C4⋊Dic191C2311yescharacteristic
78C4⋊Dic191C2311yescharacteristic
79Dic191⋊C4—21no—
80Dic191⋊C4—21no—
81C22⋊Dic191—21no—
82C22⋊Dic191—21no—
83C22⋊Dic191—21no—
84(C2×D191)⋊C4—21no—
85(C2×D191)⋊C4—21no—
86C4×Dic191—21no—
87C24×C191D411yescharacteristic
88#41×C191—41no—
89#41×C191—41no—
90C2×Q8×C191C2311yescharacteristic
91C2×D4×C191—21no—
92C2×D4×C191—21no—
93C2×D4×C191—21no—
94C2×D4×C191C2311yescharacteristic
95C22×C4×C191—21no—
96C22×C4×C191—41no—
97C22×C4×C191—41no—
98C22×C4×C191D411yescharacteristic
99C22×C4×C191D411yescharacteristic
100C22×C4×C191D411yescharacteristic
101C22×C4×C191C2311yescharacteristic
102C22×C4×C191C2311yescharacteristic
103C22×C4×C191C2311yescharacteristic
104C4⋊C4×C191—21no—
105C4⋊C4×C191—21no—
106C4⋊C4×C191C2311yescharacteristic
107C4⋊C4×C191C2311yescharacteristic
108C22⋊C4×C191—21no—
109C22⋊C4×C191—21no—
110C22⋊C4×C191—21no—
111C22⋊C4×C191—21no—
112C22⋊C4×C191—21no—
113C42×C191—21no—
114C22×D191—21no—
115Dic382—21no—
116Dic382—21no—
117D764—21no—
118D764—21no—
119C22⋊D191—42no—
120C22⋊D191—42no—
121C22⋊D191—42no—
122C2×Dic191—21no—
123C2×Dic191—21no—
124C2×Dic191—21no—
125C4×D191—42no—
126C191⋊C8—21no—
127C191⋊C8—21no—
128C191⋊C8—21no—
129C191⋊C8—21no—
130C23×C191—22no—
131C23×C191—21no—
132C23×C191—41no—
133C23×C191—41no—
134C23×C191C2×D411yescharacteristic
135C23×C191C2×D411yescharacteristic
136C23×C191C2×D411yescharacteristic
137Q8×C191—21no—
138Q8×C191—21no—
139D4×C191—21no—
140D4×C191—21no—
141D4×C191—42no—
142D4×C191—42no—
143D4×C191—42no—
144C2×C4×C191—21no—
145C2×C4×C191—21no—
146C2×C4×C191—21no—
147C2×C4×C191—21no—
148C2×C4×C191—21no—
149C2×C4×C191—21no—
150C2×C4×C191—21no—
151C2×C4×C191—21no—
152C2×C4×C191—21no—
153C2×C4×C191—41no—
154C2×C4×C191—41no—
155C2×C4×C191—41no—
156C2×C4×C191—42no—
157C2×C4×C191C2×D411yescharacteristic
158C2×C4×C191C2×D411yescharacteristic
159C2×C4×C191C2×D411yescharacteristic
160C2×C4×C191C2411yescharacteristic
161C22×C191—21no—
162C22×C191—21no—
163C22×C191—21no—
164C22×C191—21no—
165C22×C191—21no—
166C22×C191—21no—
167C22×C191—21no—
168C22×C191—21no—
169C22×C191—44no—
170C22×C191—41no—
171C22×C191—41no—
172C22×C191—41no—
173C22×C191#13512yes—
174C22×C191C22×D411yescharacteristic
175C2×D191—41no—
176C2×D191—41no—
177C2×D191—41no—
178C4×C191—21no—
179C4×C191—21no—
180C4×C191—21no—
181C4×C191—41no—
182C4×C191—41no—
183C4×C191—41no—
184C4×C191H(5,𝔽2)11yescharacteristic
185C4×C191H(5,𝔽2)11yescharacteristic
186Dic191—41no—
187Dic191—41no—
188Dic191—41no—
189C2×C191—22no—
190C2×C191—41no—
191C2×C191—41no—
192C2×C191—81no—
193C2×C191#53411yescharacteristic
194C2×C191C2×#13511yescharacteristic
195C2×C191#57511yescharacteristic
196D191—81no—
197C191#319111yescharacteristic, complemented, Sylow
198#3191—1911nomaximal, Sylow
199#525—1911no—
200#525D19111yescharacteristic, complemented
201#485—1911no—
202#485—1911no—
203#482—1911no—
204#482—1911no—
205#467—1911no—
206#466—1911no—
207#465—1911no—
208#465—1911no—
209#452—1911no—
210#452—1911no—
211#449—1911no—
212#449—1911no—
213C2×C22⋊C8—1911no—
214C2×#41—1911no—
215C2×#41C2×D19111yescharacteristic
216C23×C4C2×D19111yescharacteristic
217C2×Q16—1911no—
218C2×SD16—1911no—
219C2×SD16—1911no—
220C2×D8—1911no—
221C22×C8—1911no—
222C22×C8—1911no—
223#123—21no—
224#123—3821no—
225C22⋊Q8—1911no—
226C22⋊Q8—1911no—
227C22⋊Q8C2×D19111yescharacteristic
228C22⋊Q8C2×D19111yescharacteristic
229#121—1911no—
230#121—1911no—
231#121C2×D19111yescharacteristic
232#121C2×D19111yescharacteristic
233#120—21no—
234#120—3821no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—3821no—
238C4×D4—3821no—
239#117—1911no—
240#117C2×D19111yescharacteristic
241#107—1911no—
242#107—1911no—
243#106—1911no—
244#106—1911no—
245#103—1911no—
246#103—1911no—
247#103—1911no—
248#103—1911no—
249#102—1911no—
250#102—1911no—
251#102—1911no—
252#102—1911no—
253C22⋊C8—1911no—
254C22⋊C8—1911no—
255C22⋊C8—1911no—
256C22⋊C8—1911no—
257C24C22⋊D19111yescharacteristic
258#41—41no—
259#41—41no—
260#41—7641no—
261#41—7641no—
262C2×Q8—1911no—
263C2×Q8C22×D19111yescharacteristic
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—3821no—
268C2×D4—3821no—
269C2×D4—3821no—
270C2×D4—1911no—
271C2×D4C22×D19111yescharacteristic
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—3821no—
276C22×C4C22⋊D19111yescharacteristic
277C22×C4C22⋊D19111yescharacteristic
278C22×C4C22⋊D19111yescharacteristic
279C22×C4C22×D19111yescharacteristic
280C22×C4C22×D19111yescharacteristic
281C22×C4C22×D19111yescharacteristic
282Q16—7641no—
283SD16—7641no—
284SD16—7641no—
285D8—7641no—
286C2×C8—1911no—
287C2×C8—1911no—
288C2×C8—1911no—
289C2×C8—1911no—
290C2×C8—7641no—
291C2×C8—7641no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—3821no—
295C4⋊C4—3821no—
296C4⋊C4—1911no—
297C4⋊C4—1911no—
298C4⋊C4C22×D19111yescharacteristic
299C4⋊C4C22×D19111yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—3821no—
306C22⋊C4—3821no—
307C22⋊C4—3821no—
308C22⋊C4—3821no—
309C22⋊C4—3821no—
310C42—21no—
311C42—3821no—
312C23—22no—
313C23—21no—
314C23—41no—
315C23—41no—
316C23—3821no—
317C23C2×C22⋊D19111yescharacteristic
318C23C2×C22⋊D19111yescharacteristic
319C23C2×C22⋊D19111yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—3821no—
323Q8—3821no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—42no—
328D4—42no—
329D4—3821no—
330D4—3821no—
331D4—7642no—
332D4—7642no—
333D4—7642no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—3821no—
348C2×C4—3821no—
349C2×C4—3821no—
350C2×C4—7642no—
351C2×C4C2×C22⋊D19111yescharacteristic
352C2×C4C2×C22⋊D19111yescharacteristic
353C2×C4C2×C22⋊D19111yescharacteristic
354C2×C4C23×D19111yescharacteristic
355C8—3821no—
356C8—3821no—
357C8—3821no—
358C8—3821no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—44no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—7641no—
372C22—7641no—
373C22—7641no—
374C22#2025934212yes—
375C22C22×C22⋊D19111yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—7641no—
383C4—7641no—
384C4—7641no—
385C4#2025936711yescharacteristic
386C4#2025936711yescharacteristic
387C2—22no—
388C2—41no—
389C2—41no—
390C2—81no—
391C2—15281no—
392C2#2713322211yescharacteristic
393C2C2×#2025934211yescharacteristic
394C2#2713346111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#525×C191C2
socleC22×C191C22×D4
hypercenter#525D191
hypocenterC191#3191
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #27133174C2semidirect
2. #27133004C2semidirect
3. #27133001C2semidirect
4. #27132987C2semidirect
5. #27132984C2semidirect
6. #27132902C2semidirect
7. #27132897C2semidirect
8. #27132894C2semidirect
9. #27132890C2semidirect
10. #27132825C2semidirect
11. #27132823C2semidirect
12. #27132808C2semidirect
13. #27132806C2semidirect
14. C2×C22⋊(C191⋊C8)C2semidirect
15. #525×C191C2semidirect
17. C2×#15690277C22semidirect
18. C2×#15690275C22semidirect
19. C2×#15690274C22semidirect
20. C2×#15690272C22semidirect
32. #20259255C22semidirect
33. #20259255C22semidirect
34. #20259253C22semidirect
35. #20259253C22semidirect
36. #20259246C22semidirect
37. #20259246C22semidirect
38. #20259245C22semidirect
39. #20259245C22semidirect
40. #20259243C22semidirect
41. #20259243C22semidirect
42. #20259242C22semidirect
43. #20259242C22semidirect
197. C191#3191semidirect
200. #525D191semidirect

Maximal subgroups

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

subgroupindex
#271331742
#271330042
#271330012
#271329872
#271329842
#271329022
#271328972
#271328942
#271328902
#271328252
#271328232
#271328082
#271328062
C2×C22⋊(C191⋊C8)2
#525×C1912
#3191191