Finite Group Databasebeta

← #37583426

Subgroups — #37583426

Order 24448.

402 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27133241C211yesmaximal, characteristic, complemented
2#27132993C211yesmaximal, characteristic, complemented
3#27132991C211yesmaximal, characteristic, complemented
4#27132981C211yesmaximal, characteristic, complemented
5#27132979C211yesmaximal, characteristic, complemented
6#27132919C211yesmaximal, characteristic, complemented
7#27132914C211yesmaximal, 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
14#27132630C211yesmaximal, characteristic, complemented
15#537×C191C211yesmaximal, characteristic, complemented
16C2×#15690287C2211yescharacteristic
17C2×#15690277C2211yescharacteristic
18C2×#15690275C2211yescharacteristic
19C2×#15690274C2211yescharacteristic
20C2×#15690272C2211yescharacteristic
21C2×#15690271C2211yescharacteristic
22C22×C191⋊C8C2211yescharacteristic
23#20259301C2211yescharacteristic
24#20259301C2211yescharacteristic
25#20259298—21no—
26#20259296C2211yescharacteristic
27#20259296C2211yescharacteristic
28#20259294—21no—
29#20259294—21no—
30#20259293—21no—
31C2×C4⋊Dic191C2211yescharacteristic
32#20259253C2211yescharacteristic
33#20259253C2211yescharacteristic
34#20259253C2211yescharacteristic
35#20259253C2211yescharacteristic
36#20259246C2211yescharacteristic
37#20259246C2211yescharacteristic
38#20259245C2211yescharacteristic
39#20259245C2211yescharacteristic
40#20259243C2211yescharacteristic
41#20259243C2211yescharacteristic
42#20259242C2211yescharacteristic
43#20259242C2211yescharacteristic
44C22⋊(C191⋊C8)C2211yescharacteristic
45C22⋊(C191⋊C8)C2211yescharacteristic
46C22⋊(C191⋊C8)C2211yescharacteristic
47C22⋊(C191⋊C8)C2211yescharacteristic
48C2×#41×C191C2211yescharacteristic
49C2×#41×C191C2211yescharacteristic
50C22⋊Q8×C191—21no—
51C22⋊Q8×C191C2211yescharacteristic
52C22⋊Q8×C191C2211yescharacteristic
53#121×C191—21no—
54#121×C191—21no—
55#121×C191—21no—
56#121×C191C2211yescharacteristic
57#121×C191C2211yescharacteristic
58C2×C4⋊C4×C191C2211yescharacteristic
59#15690287—41no—
60#15690287—41no—
61C2×Dic382C2311yescharacteristic
62C2×D764C2311yescharacteristic
63C2×C22⋊D191—21no—
64C2×C22⋊D191—21no—
65C2×C22⋊D191—21no—
66C22×Dic191—21no—
67C2×C4×D191—21no—
68#15690277—41no—
69#15690275—41no—
70#15690274—41no—
71#15690272—41no—
72#15690271—41no—
73C2×C191⋊C8—41no—
74C2×C191⋊C8C2311yescharacteristic
75C2×C191⋊C8C2311yescharacteristic
76C2×C191⋊C8C2311yescharacteristic
77C2×C191⋊C8C2311yescharacteristic
78C4⋊Dic191—21no—
79C4⋊Dic191C2311yescharacteristic
80C4⋊Dic191C2311yescharacteristic
81Dic191⋊C4—21no—
82Dic191⋊C4—21no—
83C22⋊Dic191—21no—
84C22⋊Dic191—21no—
85(C2×D191)⋊C4—21no—
86(C2×D191)⋊C4—21no—
87#41×C191—41no—
88#41×C191—41no—
89#41×C191—41no—
90#41×C191—41no—
91C2×Q8×C191D411yescharacteristic
92C2×Q8×C191C2311yescharacteristic
93C2×D4×C191—21no—
94C2×D4×C191—21no—
95C2×D4×C191—21no—
96C2×D4×C191D411yescharacteristic
97C2×D4×C191D411yescharacteristic
98C2×D4×C191D411yescharacteristic
99C2×D4×C191C2311yescharacteristic
100C22×C4×C191—21no—
101C22×C4×C191—21no—
102C22×C4×C191C2311yescharacteristic
103C22×C4×C191C2311yescharacteristic
104C22×C4×C191C2311yescharacteristic
105C4⋊C4×C191—21no—
106C4⋊C4×C191—21no—
107C4⋊C4×C191—21no—
108C4⋊C4×C191C2311yescharacteristic
109C4⋊C4×C191C2311yescharacteristic
110C22⋊C4×C191—21no—
111C22⋊C4×C191—21no—
112C22⋊C4×C191—21no—
113C22⋊C4×C191—21no—
114C22×D191—21no—
115Dic382—21no—
116Dic382—21no—
117D764—21no—
118D764—21no—
119C22⋊D191—42no—
120C22⋊D191—81no—
121C22⋊D191—81no—
122C2×Dic191—21no—
123C2×Dic191—21no—
124C2×Dic191—21no—
125C2×Dic191—81no—
126C4×D191—42no—
127C191⋊C8—21no—
128C191⋊C8—21no—
129C191⋊C8—21no—
130C191⋊C8—21no—
131C23×C191—21no—
132C23×C191C2×D411yescharacteristic
133C23×C191C2×D411yescharacteristic
134C23×C191C2×D411yescharacteristic
135Q8×C191—21no—
136Q8×C191—21no—
137Q8×C191—41no—
138D4×C191—21no—
139D4×C191—21no—
140D4×C191—41no—
141D4×C191—41no—
142D4×C191—41no—
143D4×C191—42no—
144D4×C191—81no—
145D4×C191—81no—
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—21no—
154C2×C4×C191—21no—
155C2×C4×C191—42no—
156C2×C4×C191—81no—
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—21no—
170C22×C191—41no—
171C22×C191—41no—
172C22×C191—41no—
173C22×C191C22×D411yescharacteristic
174C2×D191—41no—
175C2×D191—41no—
176C2×D191—41no—
177C4×C191—21no—
178C4×C191—21no—
179C4×C191—21no—
180C4×C191—41no—
181C4×C191—41no—
182C4×C191—41no—
183C4×C191H(5,𝔽2)11yescharacteristic
184C4×C191H(5,𝔽2)11yescharacteristic
185Dic191—41no—
186Dic191—41no—
187Dic191—41no—
188C2×C191—41no—
189C2×C191—41no—
190C2×C191—41no—
191C2×C191—81no—
192C2×C191#53411yescharacteristic
193C2×C191#57711yescharacteristic
194C2×C191#57711yescharacteristic
195D191—81no—
196C191#320411yescharacteristic, complemented, Sylow
197#3204—1911nomaximal, Sylow
198#537—1911no—
199#537D19111yescharacteristic, complemented
200#483—1911no—
201#483—1911no—
202#481—1911no—
203#481—1911no—
204#470—1911no—
205#469—1911no—
206#465—1911no—
207#465—1911no—
208#452—1911no—
209#452—1911no—
210#449—1911no—
211#449—1911no—
212#408—1911no—
213C2×#41—1911no—
214C2×#41C2×D19111yescharacteristic
215C2×#41C2×D19111yescharacteristic
216C2×Q16—1911no—
217C2×SD16—1911no—
218C2×SD16—1911no—
219C2×D8—1911no—
220C2×M16—1911no—
221C22×C8—1911no—
222C22⋊Q8—21no—
223C22⋊Q8—3821no—
224C22⋊Q8—1911no—
225C22⋊Q8—1911no—
226C22⋊Q8C2×D19111yescharacteristic
227C22⋊Q8C2×D19111yescharacteristic
228#121—21no—
229#121—21no—
230#121—21no—
231#121—3821no—
232#121—3821no—
233#121—3821no—
234#121—1911no—
235#121—1911no—
236#121C2×D19111yescharacteristic
237#121C2×D19111yescharacteristic
238C2×C4⋊C4—1911no—
239C2×C4⋊C4C2×D19111yescharacteristic
240#106—1911no—
241#106—1911no—
242#106—1911no—
243#106—1911no—
244#103—1911no—
245#103—1911no—
246#103—1911no—
247#103—1911no—
248#102—1911no—
249#102—1911no—
250#102—1911no—
251#102—1911no—
252C22⋊C8—1911no—
253C22⋊C8—1911no—
254C22⋊C8—1911no—
255C22⋊C8—1911no—
256#41—41no—
257#41—41no—
258#41—41no—
259#41—41no—
260#41—7641no—
261#41—7641no—
262C2×Q8—1911no—
263C2×Q8C22⋊D19111yescharacteristic
264C2×Q8C22×D19111yescharacteristic
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—3821no—
269C2×D4—3821no—
270C2×D4—3821no—
271C2×D4—1911no—
272C2×D4C22⋊D19111yescharacteristic
273C2×D4C22⋊D19111yescharacteristic
274C2×D4C22⋊D19111yescharacteristic
275C2×D4C22×D19111yescharacteristic
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—3821no—
279C22×C4—3821no—
280C22×C4C22×D19111yescharacteristic
281C22×C4C22×D19111yescharacteristic
282C22×C4C22×D19111yescharacteristic
283Q16—7641no—
284SD16—7641no—
285SD16—7641no—
286D8—7641no—
287M16—7641no—
288C2×C8—1911no—
289C2×C8—1911no—
290C2×C8—1911no—
291C2×C8—1911no—
292C2×C8—7641no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—3821no—
297C4⋊C4—3821no—
298C4⋊C4—3821no—
299C4⋊C4—1911no—
300C4⋊C4—1911no—
301C4⋊C4C22×D19111yescharacteristic
302C4⋊C4C22×D19111yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—3821no—
308C22⋊C4—3821no—
309C22⋊C4—3821no—
310C22⋊C4—3821no—
311C23—21no—
312C23—3821no—
313C23C2×C22⋊D19111yescharacteristic
314C23C2×C22⋊D19111yescharacteristic
315C23C2×C22⋊D19111yescharacteristic
316Q8—21no—
317Q8—21no—
318Q8—41no—
319Q8—3821no—
320Q8—3821no—
321D4—21no—
322D4—21no—
323D4—41no—
324D4—41no—
325D4—41no—
326D4—42no—
327D4—81no—
328D4—81no—
329D4—3821no—
330D4—3821no—
331D4—15281no—
332D4—15281no—
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—42no—
344C2×C4—81no—
345C2×C4—3821no—
346C2×C4—3821no—
347C2×C4—3821no—
348C2×C4—15281no—
349C2×C4—7642no—
350C2×C4C2×C22⋊D19111yescharacteristic
351C2×C4C2×C22⋊D19111yescharacteristic
352C2×C4C2×C22⋊D19111yescharacteristic
353C2×C4C23×D19111yescharacteristic
354C8—3821no—
355C8—3821no—
356C8—3821no—
357C8—3821no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—7641no—
371C22—7641no—
372C22—7641no—
373C22C22×C22⋊D19111yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—7641no—
381C4—7641no—
382C4—7641no—
383C4#2025936711yescharacteristic
384C4#2025936711yescharacteristic
385C2—41no—
386C2—41no—
387C2—41no—
388C2—81no—
389C2—15281no—
390C2#2713322211yescharacteristic
391C2#2713347111yescharacteristic
392C2#2713347411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#537×C191C2
socleC22×C191C22×D4
hypercenter#537D191
hypocenterC191#3204
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #27133241C2semidirect
2. #27132993C2semidirect
3. #27132991C2semidirect
4. #27132981C2semidirect
5. #27132979C2semidirect
6. #27132919C2semidirect
7. #27132914C2semidirect
8. #27132894C2semidirect
9. #27132890C2semidirect
10. #27132825C2semidirect
11. #27132823C2semidirect
12. #27132808C2semidirect
13. #27132806C2semidirect
14. #27132630C2semidirect
15. #537×C191C2semidirect
196. C191#3204semidirect
199. #537D191semidirect

Maximal subgroups

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

subgroupindex
#271332412
#271329932
#271329912
#271329812
#271329792
#271329192
#271329142
#271328942
#271328902
#271328252
#271328232
#271328082
#271328062
#271326302
#537×C1912
#3204191