Finite Group Databasebeta

← #37582782

Subgroups — #37582782

Order 24448.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27133284C211yesmaximal, characteristic, complemented
2#27132998C211yesmaximal, characteristic, complemented
3#27132976C211yesmaximal, characteristic, complemented
4#27132958C211yesmaximal, characteristic, complemented
5#27132943C211yesmaximal, characteristic, complemented
6#27132885C211yesmaximal, characteristic, complemented
7#27132878C211yesmaximal, characteristic, complemented
8#27132825C211yesmaximal, characteristic, complemented
9#27132808C211yesmaximal, characteristic
10#27132763C211yesmaximal, characteristic, complemented
11#27132761C211yesmaximal, characteristic, complemented
12C4×#15690277C211yesmaximal, characteristic, complemented
13C4×#15690272C211yesmaximal, characteristic, complemented
14#27132724C211yesmaximal, characteristic, complemented
15#520×C191C211yesmaximal, characteristic, complemented
16C2×#15690277C2211yescharacteristic
17C2×#15690275C2211yescharacteristic
18C2×#15690274C2211yescharacteristic
19C2×#15690272C2211yescharacteristic
20#20259322C2211yescharacteristic
21#20259317—21no—
22#20259310C2211yescharacteristic
23#20259308—21no—
24#20259304—21no—
25#20259301C2211yescharacteristic
26#20259300—21no—
27#20259296C2211yescharacteristic
28C4×Dic382C2211yescharacteristic
29C4×D764C2211yescharacteristic
30#20259255C2211yescharacteristic
31#20259255C2211yescharacteristic
32#20259255C2211yescharacteristic
33#20259255C2211yescharacteristic
34C4⋊(C191⋊C8)C2211yescharacteristic
35C4⋊(C191⋊C8)C2211yescharacteristic
36#20259246C2211yescharacteristic
37#20259246C2211yescharacteristic
38#20259245C2211yescharacteristic
39#20259245C2211yescharacteristic
40#20259243C2211yescharacteristic
41#20259243C2211yescharacteristic
42#20259242C2211yescharacteristic
43#20259242C2211yescharacteristic
44C22⋊(C191⋊C8)C2211yescharacteristic
45C22⋊(C191⋊C8)C2211yescharacteristic
46#20259233C2211yescharacteristic
47C4×C191⋊C8C2211yescharacteristic
48C2×#41×C191C2211yescharacteristic
49C4×Q8×C191C2211yescharacteristic
50C4×Q8×C191C2211yescharacteristic
51C4×D4×C191—21no—
52C4×D4×C191—21no—
53C4×D4×C191C2211yescharacteristic
54C4×D4×C191C2211yescharacteristic
55#117×C191—21no—
56#117×C191C2211yescharacteristic
57C2×C4⋊C4×C191—21no—
58C2×C4⋊C4×C191C2211yescharacteristic
59C2×Dic382—21no—
60C2×Dic382C2311yescharacteristic
61C2×D764C2311yescharacteristic
62C2×C22⋊D191—21no—
63C2×C4×D191—21no—
64#15690277—22no—
65#15690275—22no—
66#15690274—22no—
67#15690272—22no—
68C2×C191⋊C8C2311yescharacteristic
69C2×C191⋊C8C2311yescharacteristic
70C2×C191⋊C8C2311yescharacteristic
71C2×C191⋊C8C2311yescharacteristic
72C4⋊Dic191—21no—
73C4⋊Dic191C2311yescharacteristic
74C4⋊Dic191C2311yescharacteristic
75Dic191⋊C4—21no—
76Dic191⋊C4—21no—
77C22⋊Dic191—21no—
78(C2×D191)⋊C4—21no—
79(C2×D191)⋊C4—21no—
80(C2×D191)⋊C4—21no—
81(C2×D191)⋊C4—21no—
82C4×Dic191—21no—
83#41×C191—22no—
84#41×C191—22no—
85C2×Q8×C191C2311yescharacteristic
86C2×D4×C191—21no—
87C2×D4×C191C2311yescharacteristic
88C22×C4×C191—21no—
89C22×C4×C191—21no—
90C22×C4×C191—21no—
91C22×C4×C191—21no—
92C22×C4×C191C2311yescharacteristic
93C4⋊C4×C191—21no—
94C4⋊C4×C191—21no—
95C4⋊C4×C191—21no—
96C4⋊C4×C191D411yescharacteristic
97C4⋊C4×C191D411yescharacteristic
98C4⋊C4×C191C2311yescharacteristic
99C4⋊C4×C191C2311yescharacteristic
100C22⋊C4×C191—21no—
101C22⋊C4×C191—21no—
102C22⋊C4×C191D411yescharacteristic
103C22⋊C4×C191D411yescharacteristic
104C42×C191—21no—
105C42×C191—21no—
106C42×C191C2311yescharacteristic
107C42×C191C2311yescharacteristic
108C22×D191—21no—
109Dic382—21no—
110Dic382—21no—
111Dic382—81no—
112D764—21no—
113D764—21no—
114C22⋊D191—81no—
115C2×Dic191—21no—
116C2×Dic191—21no—
117C2×Dic191—21no—
118C4×D191—81no—
119C191⋊C8—21no—
120C191⋊C8—21no—
121C191⋊C8—21no—
122C191⋊C8—21no—
123C23×C191—21no—
124C23×C191C2×D411yescharacteristic
125Q8×C191—21no—
126Q8×C191#4112yes—
127D4×C191—24no—
128D4×C191—21no—
129D4×C191#4112yes—
130C2×C4×C191—24no—
131C2×C4×C191—21no—
132C2×C4×C191—21no—
133C2×C4×C191—21no—
134C2×C4×C191—21no—
135C2×C4×C191—21no—
136C2×C4×C191—81no—
137C2×C4×C191—81no—
138C2×C4×C191—81no—
139C2×C4×C191C2×D411yescharacteristic
140C2×C4×C191C2×D411yescharacteristic
141C2×C4×C191C2×D411yescharacteristic
142C2×C4×C191C2×D411yescharacteristic
143C2×C4×C191C2×D411yescharacteristic
144C2×C4×C191C2411yescharacteristic
145C22×C191—21no—
146C22×C191—22no—
147C22×C191—21no—
148C22×C191—21no—
149C22×C191—41no—
150C22×C191—41no—
151C22×C191C22×D411yescharacteristic
152C2×D191—41no—
153C2×D191—41no—
154C2×D191—41no—
155C4×C191—21no—
156C4×C191—22no—
157C4×C191—21no—
158C4×C191—21no—
159C4×C191—21no—
160C4×C191—21no—
161C4×C191—41no—
162C4×C191—41no—
163C4×C191C2×#4111yescharacteristic
164C4×C191C2×#4111yescharacteristic
165Dic191—41no—
166Dic191—41no—
167Dic191—41no—
168C2×C191—41no—
169C2×C191—42no—
170C2×C191#52511yescharacteristic
171C2×C191#57611yescharacteristic
172C2×C191#57811yescharacteristic
173D191—81no—
174C191#312411yescharacteristic, complemented, Sylow
175#3124—1911nomaximal, Sylow
176#542—1911no—
177#520D19111yescharacteristic, complemented
178#484—1911no—
179#480—1911no—
180#477—1911no—
181#474—1911no—
182#464—1911no—
183#463—1911no—
184#452—1911no—
185#449—1911no—
186#440—1911no—
187#440—1911no—
188C4×Q16—1911no—
189C4×D8—1911no—
190#433—1911no—
191C2×#41C2×D19111yescharacteristic
192C2×Q16—1911no—
193C2×SD16—1911no—
194C2×SD16—1911no—
195C2×D8—1911no—
196C4⋊Q8—1911no—
197#126—3821no—
198#124—3821no—
199#124—1911no—
200#123—3821no—
201C22⋊Q8—3821no—
202C22⋊Q8—1911no—
203#121—1911no—
204C4×Q8—1911no—
205C4×Q8C2×D19111yescharacteristic
206C4×Q8C2×D19111yescharacteristic
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—1911no—
210C4×D4C2×D19111yescharacteristic
211C4×D4C2×D19111yescharacteristic
212#117—21no—
213#117C2×D19111yescharacteristic
214C2×C4⋊C4—21no—
215C2×C4⋊C4C2×D19111yescharacteristic
216#107—1911no—
217#107—1911no—
218#107—1911no—
219#107—1911no—
220C4⋊C8—1911no—
221C4⋊C8—1911no—
222#103—1911no—
223#103—1911no—
224#103—1911no—
225#103—1911no—
226#102—1911no—
227#102—1911no—
228#102—1911no—
229#102—1911no—
230C22⋊C8—1911no—
231C22⋊C8—1911no—
232#97—1911no—
233C4×C8—1911no—
234#41—22no—
235#41—22no—
236C2×Q8—3821no—
237C2×Q8—1911no—
238C2×Q8C22×D19111yescharacteristic
239C2×D4—21no—
240C2×D4—3821no—
241C2×D4—1911no—
242C2×D4C22×D19111yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—3821no—
248C22×C4C22×D19111yescharacteristic
249Q16—3822no—
250SD16—3822no—
251SD16—3822no—
252D8—3822no—
253C2×C8—1911no—
254C2×C8—1911no—
255C2×C8—1911no—
256C2×C8—1911no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—3821no—
261C4⋊C4—3821no—
262C4⋊C4—3821no—
263C4⋊C4—1911no—
264C4⋊C4—1911no—
265C4⋊C4C22⋊D19111yescharacteristic
266C4⋊C4C22⋊D19111yescharacteristic
267C4⋊C4C22×D19111yescharacteristic
268C4⋊C4C22×D19111yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—3821no—
272C22⋊C4—3821no—
273C22⋊C4—3821no—
274C22⋊C4—3821no—
275C22⋊C4—3821no—
276C22⋊C4C22⋊D19111yescharacteristic
277C22⋊C4C22⋊D19111yescharacteristic
278C42—21no—
279C42—21no—
280C42—3821no—
281C42C22×D19111yescharacteristic
282C42C22×D19111yescharacteristic
283C23—21no—
284C23—3821no—
285C23C2×C22⋊D19111yescharacteristic
286Q8—21no—
287Q8—3821no—
288Q8—3821no—
289Q8—15281no—
290Q8#1569028712yes—
291D4—24no—
292D4—21no—
293D4—3821no—
294D4—3821no—
295D4—15281no—
296D4#1569028712yes—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—3821no—
307C2×C4—3821no—
308C2×C4—3821no—
309C2×C4—15281no—
310C2×C4C2×C22⋊D19111yescharacteristic
311C2×C4C2×C22⋊D19111yescharacteristic
312C2×C4C2×C22⋊D19111yescharacteristic
313C2×C4C2×C22⋊D19111yescharacteristic
314C2×C4C2×C22⋊D19111yescharacteristic
315C2×C4C23×D19111yescharacteristic
316C8—3821no—
317C8—3821no—
318C8—3821no—
319C8—3821no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—7641no—
327C22—7641no—
328C22—7641no—
329C22C22×C22⋊D19111yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—7641no—
339C4—7641no—
340C4—7641no—
341C4C2×#1569028711yescharacteristic
342C4C2×#1569028711yescharacteristic
343C2—41no—
344C2—42no—
345C2—15281no—
346C2#2713317411yescharacteristic
347C2#2713346611yescharacteristic
348C2#2713347811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#520×C191C2
socleC22×C191C22×D4
prime radical#27132808C2
hypercenter#520D191
hypocenterC191#3124
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #27133284C2semidirect
2. #27132998C2semidirect
3. #27132976C2semidirect
4. #27132958C2semidirect
5. #27132943C2semidirect
6. #27132885C2semidirect
7. #27132878C2semidirect
8. #27132825C2semidirect
10. #27132763C2semidirect
11. #27132761C2semidirect
12. C4×#15690277C2semidirect
13. C4×#15690272C2semidirect
14. #27132724C2semidirect
15. #520×C191C2semidirect
174. C191#3124semidirect
177. #520D191semidirect

Maximal subgroups

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

subgroupindex
#271332842
#271329982
#271329762
#271329582
#271329432
#271328852
#271328782
#271328252
#271328082
#271327632
#271327612
C4×#156902772
C4×#156902722
#271327242
#520×C1912
#3124191