Finite Group Databasebeta

← #38754718

Subgroups — #38754718

Order 25216.

404 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161775C211yesmaximal, characteristic, complemented
2#27161443C211yesmaximal, characteristic, complemented
3#27161400C211yesmaximal, characteristic, complemented
4#27161390C211yesmaximal, characteristic, complemented
5#27161386C211yesmaximal, characteristic, complemented
6#27161330C211yesmaximal, characteristic, complemented
7#27161313C211yesmaximal, characteristic, complemented
8#27161271C211yesmaximal, characteristic, complemented
9#27161224C211yesmaximal, characteristic, complemented
10#27161224C211yesmaximal, characteristic, complemented
11#27161177C211yesmaximal, characteristic, complemented
12#27161151C211yesmaximal, characteristic, complemented
13#27161080C211yesmaximal, characteristic, complemented
14#27161075C211yesmaximal, characteristic, complemented
15#550×C197C211yesmaximal, characteristic, complemented
16C2×#15691345C2211yescharacteristic
17C2×#15691343C2211yescharacteristic
18C2×#15691343C2211yescharacteristic
19C2×#15691342C2211yescharacteristic
20C22×#13096802C2211yescharacteristic
21#20263321—21no—
22#20263318C2211yescharacteristic
23#20263308—21no—
24#20263300C2211yescharacteristic
25#20263300C2211yescharacteristic
26#20263298—21no—
27C4×D788C2211yescharacteristic
28#20263286—21no—
29#20263285C2211yescharacteristic
30C2×C4⋊Dic197C2211yescharacteristic
31#20263259C2211yescharacteristic
32#20263259C2211yescharacteristic
33#20263257C2211yescharacteristic
34C4⋊#13096802C2211yescharacteristic
35#20263250C2211yescharacteristic
36#20263250C2211yescharacteristic
37#20263250C2211yescharacteristic
38#20263247C2211yescharacteristic
39#20263247C2211yescharacteristic
40#20263247C2211yescharacteristic
41#20263247C2211yescharacteristic
42#20263247C2211yescharacteristic
43#20263247C2211yescharacteristic
44#20263246C2211yescharacteristic
45C22⋊#13096802C2211yescharacteristic
46C22⋊#13096802C2211yescharacteristic
47#20263237C2211yescharacteristic
48C2×#41×C197C2211yescharacteristic
49C2×#41×C197C2211yescharacteristic
50#127×C197—21no—
51#127×C197C2211yescharacteristic
52#121×C197—21no—
53#121×C197—21no—
54#121×C197C2211yescharacteristic
55#121×C197C2211yescharacteristic
56C4×Q8×C197C2211yescharacteristic
57C4×D4×C197—21no—
58C4×D4×C197C2211yescharacteristic
59C2×D788C2311yescharacteristic
60C2×C22⋊D197—21no—
61C2×C22⋊D197—21no—
62C22×Dic197—21no—
63C2×C4×D197—21no—
64#15691345—22no—
65#15691343—41no—
66#15691343—41no—
67#15691342—41no—
68C2×#13096802—41no—
69C2×#13096802C2311yescharacteristic
70C2×#13096802C2311yescharacteristic
71C2×#13096802C2311yescharacteristic
72C2×#13096802C2311yescharacteristic
73C4⋊Dic197—21no—
74C4⋊Dic197C2311yescharacteristic
75C4⋊Dic197C2311yescharacteristic
76C4⋊Dic197C2311yescharacteristic
77#15691337—21no—
78#15691337—21no—
79C22⋊Dic197—21no—
80#15691335—21no—
81#15691335—21no—
82#15691335—21no—
83C4×Dic197—21no—
84#41×C197—22no—
85#41×C197—22no—
86#41×C197—41no—
87#41×C197—41no—
88C2×Q8×C197C2311yescharacteristic
89C2×D4×C197—21no—
90C2×D4×C197—21no—
91C2×D4×C197—21no—
92C2×D4×C197—21no—
93C2×D4×C197—21no—
94C2×D4×C197—21no—
95C2×D4×C197D411yescharacteristic
96C2×D4×C197C2311yescharacteristic
97C2×D4×C197C2311yescharacteristic
98C22×C4×C197—21no—
99C22×C4×C197—21no—
100C22×C4×C197C2311yescharacteristic
101C22×C4×C197C2311yescharacteristic
102C4⋊C4×C197—21no—
103C4⋊C4×C197D411yescharacteristic
104C4⋊C4×C197C2311yescharacteristic
105C22⋊C4×C197—21no—
106C22⋊C4×C197—21no—
107C22⋊C4×C197D411yescharacteristic
108C22⋊C4×C197D411yescharacteristic
109C42×C197—21no—
110C42×C197C2311yescharacteristic
111C22×D197—21no—
112D788—21no—
113D788—21no—
114C22⋊D197—42no—
115C22⋊D197—81no—
116C2×Dic197—21no—
117C2×Dic197—21no—
118C2×Dic197—21no—
119C2×Dic197—81no—
120C4×D197—81no—
121#13096802—21no—
122#13096802—21no—
123#13096802—21no—
124#13096802—21no—
125C23×C197—21no—
126C23×C197—21no—
127C23×C197C2×D411yescharacteristic
128C23×C197C2×D411yescharacteristic
129Q8×C197—21no—
130Q8×C197#4112yes—
131D4×C197—21no—
132D4×C197—21no—
133D4×C197—21no—
134D4×C197—21no—
135D4×C197—41no—
136D4×C197—42no—
137D4×C197—42no—
138D4×C197—42no—
139D4×C197—81no—
140D4×C197—81no—
141D4×C197—81no—
142C2×C4×C197—21no—
143C2×C4×C197—21no—
144C2×C4×C197—21no—
145C2×C4×C197—21no—
146C2×C4×C197—21no—
147C2×C4×C197—21no—
148C2×C4×C197—42no—
149C2×C4×C197—42no—
150C2×C4×C197C2×D411yescharacteristic
151C2×C4×C197C2×D411yescharacteristic
152C2×C4×C197C2×D411yescharacteristic
153C2×C4×C197C2×D411yescharacteristic
154C2×C4×C197C2411yescharacteristic
155C22×C197—21no—
156C22×C197—21no—
157C22×C197—21no—
158C22×C197—21no—
159C22×C197—21no—
160C22×C197—21no—
161C22×C197—41no—
162C22×C197—41no—
163C22×C197—41no—
164C22×C197—41no—
165C22×C197—41no—
166C22×C197—41no—
167C22×C197C22×D411yescharacteristic
168C2×D197—41no—
169C2×D197—41no—
170C2×D197—41no—
171C4×C197—21no—
172C4×C197—21no—
173C4×C197—22no—
174C4×C197—21no—
175C4×C197—21no—
176C4×C197—41no—
177C4×C197C2×#4111yescharacteristic
178C4×C197H(5,𝔽2)11yescharacteristic
179Dic197—41no—
180Dic197—41no—
181Dic197—41no—
182C2×C197—41no—
183C2×C197—41no—
184C2×C197—81no—
185C2×C197—81no—
186C2×C197#54611yescharacteristic
187C2×C197#57511yescharacteristic
188C2×C197#57611yescharacteristic
189D197—81no—
190C197#330911yescharacteristic, complemented, Sylow
191#3309—1971nomaximal, Sylow
192#553—1971no—
193#550D19711yescharacteristic, complemented
194#490—1971no—
195#482—1971no—
196#480—1971no—
197#479—1971no—
198#469—1971no—
199#466—1971no—
200#459—1971no—
201#449—1971no—
202#449—1971no—
203#440—1971no—
204#435—1971no—
205#417—1971no—
206#416—1971no—
207C2×#41C2×D19711yescharacteristic
208C2×#41C2×D19711yescharacteristic
209C2×SD16—1971no—
210C2×D8—1971no—
211C2×D8—1971no—
212C2×M16—1971no—
213C22×C8—1971no—
214#127—21no—
215#127C2×D19711yescharacteristic
216#126—3941no—
217#125—1971no—
218#123—3941no—
219#121—21no—
220#121—21no—
221#121—3941no—
222#121—1971no—
223#121—1971no—
224#121C2×D19711yescharacteristic
225#121C2×D19711yescharacteristic
226C4×Q8C2×D19711yescharacteristic
227C4×D4—21no—
228C4×D4—3941no—
229C4×D4—1971no—
230C4×D4C2×D19711yescharacteristic
231#117—1971no—
232C2×C4⋊C4—1971no—
233#107—1971no—
234#107—1971no—
235#106—1971no—
236C4⋊C8—1971no—
237#103—1971no—
238#103—1971no—
239#103—1971no—
240#102—1971no—
241#102—1971no—
242#102—1971no—
243#102—1971no—
244#102—1971no—
245#102—1971no—
246#102—1971no—
247C22⋊C8—1971no—
248C22⋊C8—1971no—
249#97—1971no—
250#41—22no—
251#41—22no—
252#41—41no—
253#41—41no—
254C2×Q8C22×D19711yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—3941no—
262C2×D4—3941no—
263C2×D4—1971no—
264C2×D4C22⋊D19711yescharacteristic
265C2×D4C22×D19711yescharacteristic
266C2×D4C22×D19711yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—3941no—
270C22×C4—3941no—
271C22×C4C22×D19711yescharacteristic
272C22×C4C22×D19711yescharacteristic
273SD16—3942no—
274D8—7881no—
275D8—7881no—
276M16—7881no—
277C2×C8—7881no—
278C2×C8—1971no—
279C2×C8—1971no—
280C2×C8—1971no—
281C2×C8—1971no—
282C4⋊C4—21no—
283C4⋊C4—3941no—
284C4⋊C4—3941no—
285C4⋊C4—3941no—
286C4⋊C4—1971no—
287C4⋊C4—1971no—
288C4⋊C4—1971no—
289C4⋊C4C22⋊D19711yescharacteristic
290C4⋊C4C22×D19711yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—3941no—
294C22⋊C4—3941no—
295C22⋊C4—3941no—
296C22⋊C4—3941no—
297C22⋊C4C22⋊D19711yescharacteristic
298C22⋊C4C22⋊D19711yescharacteristic
299C42—21no—
300C42—3941no—
301C42C22×D19711yescharacteristic
302C23—21no—
303C23—21no—
304C23—3941no—
305C23C2×C22⋊D19711yescharacteristic
306C23C2×C22⋊D19711yescharacteristic
307Q8—21no—
308Q8#1569135712yes—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—41no—
314D4—42no—
315D4—42no—
316D4—42no—
317D4—81no—
318D4—81no—
319D4—81no—
320D4—7882no—
321D4—15761no—
322D4—3941no—
323D4—3941no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—15761no—
333C2×C4—15761no—
334C2×C4—3941no—
335C2×C4—3941no—
336C2×C4—3941no—
337C2×C4C2×C22⋊D19711yescharacteristic
338C2×C4C2×C22⋊D19711yescharacteristic
339C2×C4C2×C22⋊D19711yescharacteristic
340C2×C4C2×C22⋊D19711yescharacteristic
341C2×C4C23×D19711yescharacteristic
342C8—3941no—
343C8—3941no—
344C8—3941no—
345C8—3941no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—7881no—
359C22—7881no—
360C22—7881no—
361C22C22×C22⋊D19711yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—22no—
367C4—41no—
368C4—7881no—
369C4—7881no—
370C4—7881no—
371C4C2×#1569135711yescharacteristic
372C4#2026337111yescharacteristic
373C2—41no—
374C2—41no—
375C2—81no—
376C2—81no—
377C2—15761no—
378C2#2716172311yescharacteristic
379C2#2716187711yescharacteristic
380C2#2716188211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#550×C197C2
socleC22×C197C22×D4
hypercenter#550D197
hypocenterC197#3309
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #27161775C2semidirect
2. #27161443C2semidirect
3. #27161400C2semidirect
4. #27161390C2semidirect
5. #27161386C2semidirect
6. #27161330C2semidirect
7. #27161313C2semidirect
8. #27161271C2semidirect
9. #27161224C2semidirect
10. #27161224C2semidirect
11. #27161177C2semidirect
12. #27161151C2semidirect
13. #27161080C2semidirect
14. #27161075C2semidirect
15. #550×C197C2semidirect
190. C197#3309semidirect
193. #550D197semidirect

Maximal subgroups

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

subgroupindex
#271617752
#271614432
#271614002
#271613902
#271613862
#271613302
#271613132
#271612712
#271612242
#271611772
#271611512
#271610802
#271610752
#550×C1972
#3309197