Finite Group Databasebeta

← #38754712

Subgroups — #38754712

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#27161750C211yesmaximal, characteristic, complemented
2#27161445C211yesmaximal, characteristic, complemented
3#27161403C211yesmaximal, characteristic, complemented
4#27161392C211yesmaximal, characteristic, complemented
5#27161383C211yesmaximal, characteristic, complemented
6#27161330C211yesmaximal, characteristic, complemented
7#27161313C211yesmaximal, characteristic, complemented
8#27161274C211yesmaximal, characteristic, complemented
9#27161222C211yesmaximal, characteristic, complemented
10#27161222C211yesmaximal, characteristic, complemented
11#27161179C211yesmaximal, characteristic, complemented
12#27161151C211yesmaximal, characteristic, complemented
13#27161078C211yesmaximal, characteristic, complemented
14#27161073C211yesmaximal, characteristic, complemented
15#553×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic
17C2×#15691358C2211yescharacteristic
18C2×#15691346C2211yescharacteristic
19C2×#15691343C2211yescharacteristic
20C2×#15691343C2211yescharacteristic
21C2×#15691342C2211yescharacteristic
22C22×#13096802C2211yescharacteristic
23#20263323C2211yescharacteristic
24#20263322—21no—
25#20263300C2211yescharacteristic
26#20263300C2211yescharacteristic
27#20263299—21no—
28#20263299—21no—
29C4×Dic394C2211yescharacteristic
30#20263288—21no—
31#20263259C2211yescharacteristic
32#20263259C2211yescharacteristic
33#20263257C2211yescharacteristic
34C4⋊#13096802C2211yescharacteristic
35#20263249C2211yescharacteristic
36#20263249C2211yescharacteristic
37#20263249C2211yescharacteristic
38#20263247C2211yescharacteristic
39#20263246C2211yescharacteristic
40#20263246C2211yescharacteristic
41#20263246C2211yescharacteristic
42#20263246C2211yescharacteristic
43#20263246C2211yescharacteristic
44#20263246C2211yescharacteristic
45C22⋊#13096802C2211yescharacteristic
46C22⋊#13096802C2211yescharacteristic
47#20263237C2211yescharacteristic
48#126×C197—21no—
49#125×C197C2211yescharacteristic
50#123×C197—21no—
51#121×C197—21no—
52#121×C197C2211yescharacteristic
53#121×C197C2211yescharacteristic
54C4×D4×C197—21no—
55C4×D4×C197C2211yescharacteristic
56C4×D4×C197C2211yescharacteristic
57#117×C197C2211yescharacteristic
58C2×C4⋊C4×C197C2211yescharacteristic
59#15691358—22no—
60#15691358—22no—
61#15691358—41no—
62#15691358—41no—
63C2×Dic394C2311yescharacteristic
64C2×D788—21no—
65C2×D788—21no—
66C2×D788C2311yescharacteristic
67C2×D788C2311yescharacteristic
68C2×C22⋊D197—21no—
69C2×C22⋊D197—21no—
70C2×C22⋊D197—21no—
71C2×C22⋊D197—21no—
72C2×C4×D197—21no—
73C2×C4×D197—21no—
74#15691346—22no—
75#15691343—41no—
76#15691343—41no—
77#15691342—41no—
78C2×#13096802—41no—
79C2×#13096802C2311yescharacteristic
80C2×#13096802C2311yescharacteristic
81C2×#13096802C2311yescharacteristic
82C2×#13096802C2311yescharacteristic
83C4⋊Dic197C2311yescharacteristic
84#15691337—21no—
85#15691335—21no—
86#15691335—21no—
87C4×Dic197—21no—
88C2×D4×C197—21no—
89C2×D4×C197—21no—
90C2×D4×C197D411yescharacteristic
91C2×D4×C197C2311yescharacteristic
92C22×C4×C197—21no—
93C22×C4×C197—21no—
94C22×C4×C197C2311yescharacteristic
95C22×C4×C197C2311yescharacteristic
96C4⋊C4×C197—21no—
97C4⋊C4×C197—21no—
98C4⋊C4×C197—21no—
99C4⋊C4×C197D411yescharacteristic
100C4⋊C4×C197C2311yescharacteristic
101C4⋊C4×C197C2311yescharacteristic
102C4⋊C4×C197C2311yescharacteristic
103C22⋊C4×C197—21no—
104C22⋊C4×C197—21no—
105C22⋊C4×C197—21no—
106C22⋊C4×C197—21no—
107C22⋊C4×C197D411yescharacteristic
108C22⋊C4×C197D411yescharacteristic
109C42×C197—21no—
110C42×C197C2311yescharacteristic
111C22×D197—21no—
112C22×D197—21no—
113Dic394—21no—
114Dic394#4112yes—
115D788—21no—
116D788—21no—
117D788—21no—
118D788—21no—
119D788—42no—
120D788—81no—
121C22⋊D197—42no—
122C22⋊D197—42no—
123C22⋊D197—81no—
124C22⋊D197—81no—
125C2×Dic197—21no—
126C2×Dic197—21no—
127C4×D197—42no—
128C4×D197—42no—
129#13096802—21no—
130#13096802—21no—
131#13096802—21no—
132#13096802—21no—
133C23×C197—21no—
134C23×C197C2×D411yescharacteristic
135C23×C197C2×D411yescharacteristic
136D4×C197—21no—
137D4×C197—21no—
138D4×C197—41no—
139D4×C197—42no—
140D4×C197—81no—
141C2×C4×C197—21no—
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—81no—
149C2×C4×C197—81no—
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×C197C22×D411yescharacteristic
165C2×D197—41no—
166C2×D197—41no—
167C2×D197—41no—
168C2×D197—41no—
169C2×D197—41no—
170C2×D197—41no—
171C4×C197—21no—
172C4×C197—21no—
173C4×C197—21no—
174C4×C197—21no—
175C4×C197—41no—
176C4×C197—41no—
177C4×C197—41no—
178C4×C197C2×#4111yescharacteristic
179C4×C197H(5,𝔽2)11yescharacteristic
180Dic197—22no—
181Dic197—41no—
182C2×C197—41no—
183C2×C197—41no—
184C2×C197—81no—
185C2×C197#54611yescharacteristic
186C2×C197#57511yescharacteristic
187C2×C197#57611yescharacteristic
188D197—81no—
189D197—81no—
190C197#330911yescharacteristic, complemented, Sylow
191#3309—1971nomaximal, Sylow
192#553D19711yescharacteristic, complemented
193#550—1971no—
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×#41—1971no—
208C2×#41—1971no—
209C2×SD16—1971no—
210C2×D8—1971no—
211C2×D8—1971no—
212C2×M16—1971no—
213C22×C8—1971no—
214#127—3941no—
215#127—1971no—
216#126—21no—
217#125C2×D19711yescharacteristic
218#123—21no—
219#121—21no—
220#121—3941no—
221#121—3941no—
222#121—1971no—
223#121—1971no—
224#121C2×D19711yescharacteristic
225#121C2×D19711yescharacteristic
226C4×Q8—1971no—
227C4×D4—21no—
228C4×D4—3941no—
229C4×D4C2×D19711yescharacteristic
230C4×D4C2×D19711yescharacteristic
231#117C2×D19711yescharacteristic
232C2×C4⋊C4C2×D19711yescharacteristic
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—7881no—
251#41—7881no—
252#41—3942no—
253#41—3942no—
254C2×Q8—1971no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—3941no—
258C2×D4—3941no—
259C2×D4—3941no—
260C2×D4—3941no—
261C2×D4—3941no—
262C2×D4—3941no—
263C2×D4—1971no—
264C2×D4—1971no—
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—21no—
284C4⋊C4—21no—
285C4⋊C4—3941no—
286C4⋊C4—1971no—
287C4⋊C4C22⋊D19711yescharacteristic
288C4⋊C4C22×D19711yescharacteristic
289C4⋊C4C22×D19711yescharacteristic
290C4⋊C4C22×D19711yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—3941no—
296C22⋊C4—3941no—
297C22⋊C4C22⋊D19711yescharacteristic
298C22⋊C4C22⋊D19711yescharacteristic
299C42—21no—
300C42—3941no—
301C42C22×D19711yescharacteristic
302C23—21no—
303C23—3941no—
304C23—3941no—
305C23C2×C22⋊D19711yescharacteristic
306C23C2×C22⋊D19711yescharacteristic
307Q8—3941no—
308Q8—1972no—
309D4—21no—
310D4—21no—
311D4—41no—
312D4—42no—
313D4—81no—
314D4—7882no—
315D4—7882no—
316D4—7882no—
317D4—15761no—
318D4—15761no—
319D4—15761no—
320D4—3941no—
321D4—3941no—
322D4—3941no—
323D4—3941no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—7882no—
334C2×C4—7882no—
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—7881no—
356C22—7881no—
357C22—7881no—
358C22—7881no—
359C22—7881no—
360C22—7881no—
361C22C22×C22⋊D19711yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—7881no—
370C4—3942no—
371C4#41×D19711yescharacteristic
372C4#2026337211yescharacteristic
373C2—41no—
374C2—41no—
375C2—81no—
376C2—15761no—
377C2—15761no—
378C2#2716171711yescharacteristic
379C2#2716187711yescharacteristic
380C2#2716188211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#553×C197C2
socleC22×C197C22×D4
hypercenter#553D197
hypocenterC197#3309
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #27161750C2semidirect
2. #27161445C2semidirect
3. #27161403C2semidirect
4. #27161392C2semidirect
5. #27161383C2semidirect
6. #27161330C2semidirect
7. #27161313C2semidirect
8. #27161274C2semidirect
9. #27161222C2semidirect
10. #27161222C2semidirect
11. #27161179C2semidirect
12. #27161151C2semidirect
13. #27161078C2semidirect
14. #27161073C2semidirect
15. #553×C197C2semidirect
190. C197#3309semidirect
192. #553D197semidirect

Maximal subgroups

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

subgroupindex
#271617502
#271614452
#271614032
#271613922
#271613832
#271613302
#271613132
#271612742
#271612222
#271611792
#271611512
#271610782
#271610732
#553×C1972
#3309197