Finite Group Databasebeta

← #38755312

Subgroups — #38755312

Order 25216.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161775C211yesmaximal, characteristic, complemented
2#27161478C211yesmaximal, characteristic, complemented
3#27161403C211yesmaximal, characteristic, complemented
4#27161400C211yesmaximal, characteristic, complemented
5#27161392C211yesmaximal, characteristic, complemented
6#27161390C211yesmaximal, characteristic, complemented
7#27161379C211yesmaximal, characteristic, complemented
8#27161376C211yesmaximal, characteristic, complemented
9#27161330C211yesmaximal, characteristic, complemented
10#27161330C211yesmaximal, characteristic, complemented
11C4×#15691343C211yesmaximal, characteristic, complemented
12#27161157C211yesmaximal, characteristic, complemented
13#27161122C211yesmaximal, characteristic, complemented
14#27161119C211yesmaximal, characteristic, complemented
15#553×C197C211yesmaximal, characteristic, complemented
16C2×#15691343C2211yescharacteristic
17C2×#15691342C2211yescharacteristic
18C2×#15691342C2211yescharacteristic
19#20263321—21no—
20#20263318C2211yescharacteristic
21#20263308—21no—
22#20263300C2211yescharacteristic
23#20263300C2211yescharacteristic
24#20263298—21no—
25C4×D788C2211yescharacteristic
26#20263286—21no—
27#20263285C2211yescharacteristic
28C2×C4⋊Dic197C2211yescharacteristic
29#20263259C2211yescharacteristic
30#20263259C2211yescharacteristic
31#20263259C2211yescharacteristic
32#20263259C2211yescharacteristic
33#20263259C2211yescharacteristic
34#20263257C2211yescharacteristic
35#20263257C2211yescharacteristic
36#20263257C2211yescharacteristic
37#20263257C2211yescharacteristic
38C4⋊#13096802C2211yescharacteristic
39#20263247C2211yescharacteristic
40#20263247C2211yescharacteristic
41#20263247C2211yescharacteristic
42#20263246C2211yescharacteristic
43#20263246C2211yescharacteristic
44#20263246C2211yescharacteristic
45C22⋊#13096802C2211yescharacteristic
46C22⋊#13096802C2211yescharacteristic
47C4×#13096802C2211yescharacteristic
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
59C2×D788C2311yescharacteristic
60C2×C22⋊D197—21no—
61C2×C22⋊D197—21no—
62C22×Dic197—21no—
63C2×C4×D197—21no—
64#15691343—22no—
65#15691342—22no—
66#15691342—22no—
67C2×#13096802C2311yescharacteristic
68C2×#13096802C2311yescharacteristic
69C2×#13096802C2311yescharacteristic
70C2×#13096802C2311yescharacteristic
71C4⋊Dic197—21no—
72C4⋊Dic197C2311yescharacteristic
73C4⋊Dic197C2311yescharacteristic
74C4⋊Dic197C2311yescharacteristic
75#15691337—21no—
76#15691337—21no—
77C22⋊Dic197—21no—
78#15691335—21no—
79#15691335—21no—
80#15691335—21no—
81C4×Dic197—21no—
82C2×D4×C197—21no—
83C2×D4×C197—21no—
84C2×D4×C197D411yescharacteristic
85C2×D4×C197C2311yescharacteristic
86C22×C4×C197—21no—
87C22×C4×C197—21no—
88C22×C4×C197C2311yescharacteristic
89C22×C4×C197C2311yescharacteristic
90C4⋊C4×C197—21no—
91C4⋊C4×C197—21no—
92C4⋊C4×C197—21no—
93C4⋊C4×C197D411yescharacteristic
94C4⋊C4×C197C2311yescharacteristic
95C4⋊C4×C197C2311yescharacteristic
96C4⋊C4×C197C2311yescharacteristic
97C22⋊C4×C197—21no—
98C22⋊C4×C197—21no—
99C22⋊C4×C197—21no—
100C22⋊C4×C197—21no—
101C22⋊C4×C197D411yescharacteristic
102C22⋊C4×C197D411yescharacteristic
103C42×C197—21no—
104C42×C197C2311yescharacteristic
105C22×D197—21no—
106D788—21no—
107D788—21no—
108C22⋊D197—42no—
109C22⋊D197—81no—
110C2×Dic197—21no—
111C2×Dic197—21no—
112C2×Dic197—21no—
113C2×Dic197—81no—
114C4×D197—81no—
115#13096802—21no—
116#13096802—21no—
117#13096802—21no—
118#13096802#4112yes—
119C23×C197—21no—
120C23×C197C2×D411yescharacteristic
121C23×C197C2×D411yescharacteristic
122D4×C197—21no—
123D4×C197—21no—
124D4×C197—41no—
125D4×C197—42no—
126D4×C197—81no—
127C2×C4×C197—21no—
128C2×C4×C197—21no—
129C2×C4×C197—21no—
130C2×C4×C197—21no—
131C2×C4×C197—21no—
132C2×C4×C197—21no—
133C2×C4×C197—21no—
134C2×C4×C197—81no—
135C2×C4×C197—81no—
136C2×C4×C197C2×D411yescharacteristic
137C2×C4×C197C2×D411yescharacteristic
138C2×C4×C197C2×D411yescharacteristic
139C2×C4×C197C2×D411yescharacteristic
140C2×C4×C197C2411yescharacteristic
141C22×C197—21no—
142C22×C197—21no—
143C22×C197—21no—
144C22×C197—21no—
145C22×C197—21no—
146C22×C197—21no—
147C22×C197—41no—
148C22×C197—41no—
149C22×C197—41no—
150C22×C197C22×D411yescharacteristic
151C2×D197—41no—
152C2×D197—41no—
153C2×D197—41no—
154C4×C197—21no—
155C4×C197—21no—
156C4×C197—21no—
157C4×C197—21no—
158C4×C197—41no—
159C4×C197—41no—
160C4×C197—41no—
161C4×C197C2×#4111yescharacteristic
162C4×C1972−1+411yescharacteristic
163Dic197—41no—
164Dic197—41no—
165Dic197—41no—
166C2×C197—41no—
167C2×C197—41no—
168C2×C197—81no—
169C2×C197#54711yescharacteristic
170C2×C197#57511yescharacteristic
171C2×C197#57611yescharacteristic
172D197—81no—
173C197#335511yescharacteristic, complemented, Sylow
174#3355—1971nomaximal, Sylow
175#553—1971no—
176#553D19711yescharacteristic, complemented
177#499—1971no—
178#482—1971no—
179#482—1971no—
180#480—1971no—
181#480—1971no—
182#478—1971no—
183#478—1971no—
184#469—1971no—
185#469—1971no—
186C4×D8—1971no—
187#436—1971no—
188#428—1971no—
189#428—1971no—
190C2×D8—1971no—
191C2×M16—1971no—
192C2×M16—1971no—
193#126—21no—
194#126—3941no—
195#125—1971no—
196#125C2×D19711yescharacteristic
197#123—21no—
198#123—3941no—
199#121—21no—
200#121—3941no—
201#121—1971no—
202#121—1971no—
203#121C2×D19711yescharacteristic
204#121C2×D19711yescharacteristic
205C4×D4—21no—
206C4×D4—3941no—
207C4×D4—1971no—
208C4×D4C2×D19711yescharacteristic
209C4×D4C2×D19711yescharacteristic
210#117—1971no—
211#117C2×D19711yescharacteristic
212C2×C4⋊C4—1971no—
213C2×C4⋊C4C2×D19711yescharacteristic
214#107—1971no—
215#107—1971no—
216#107—1971no—
217#107—1971no—
218#107—1971no—
219#106—1971no—
220#106—1971no—
221#106—1971no—
222#106—1971no—
223C4⋊C8—1971no—
224#102—1971no—
225#102—1971no—
226#102—1971no—
227#102—1971no—
228#102—1971no—
229#102—1971no—
230C22⋊C8—1971no—
231C22⋊C8—1971no—
232C4×C8—1971no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—3941no—
236C2×D4—3941no—
237C2×D4—1971no—
238C2×D4C22⋊D19711yescharacteristic
239C2×D4C22×D19711yescharacteristic
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—3941no—
243C22×C4—3941no—
244C22×C4C22×D19711yescharacteristic
245C22×C4C22×D19711yescharacteristic
246D8—3942no—
247M16—3942no—
248M16—3942no—
249C2×C8—1971no—
250C2×C8—1971no—
251C2×C8—1971no—
252C2×C8—1971no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—3941no—
257C4⋊C4—3941no—
258C4⋊C4—3941no—
259C4⋊C4—1971no—
260C4⋊C4—1971no—
261C4⋊C4—1971no—
262C4⋊C4C22⋊D19711yescharacteristic
263C4⋊C4C22×D19711yescharacteristic
264C4⋊C4C22×D19711yescharacteristic
265C4⋊C4C22×D19711yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—3941no—
271C22⋊C4—3941no—
272C22⋊C4—3941no—
273C22⋊C4—3941no—
274C22⋊C4C22⋊D19711yescharacteristic
275C22⋊C4C22⋊D19711yescharacteristic
276C42—21no—
277C42—3941no—
278C42C22×D19711yescharacteristic
279C23—21no—
280C23—3941no—
281C23C2×C22⋊D19711yescharacteristic
282C23C2×C22⋊D19711yescharacteristic
283D4—21no—
284D4—21no—
285D4—41no—
286D4—42no—
287D4—81no—
288D4—7882no—
289D4—15761no—
290D4—3941no—
291D4—3941no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—15761no—
302C2×C4—15761no—
303C2×C4—3941no—
304C2×C4—3941no—
305C2×C4—3941no—
306C2×C4C2×C22⋊D19711yescharacteristic
307C2×C4C2×C22⋊D19711yescharacteristic
308C2×C4C2×C22⋊D19711yescharacteristic
309C2×C4C2×C22⋊D19711yescharacteristic
310C2×C4C23×D19711yescharacteristic
311C8—3941no—
312C8—3941no—
313C8—3941no—
314C8—1972no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—7881no—
325C22—7881no—
326C22—7881no—
327C22C22×C22⋊D19711yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—7881no—
336C4—7881no—
337C4—7881no—
338C4#41×D19711yescharacteristic
339C4#2026337311yescharacteristic
340C2—41no—
341C2—41no—
342C2—81no—
343C2—15761no—
344C2#2716172811yescharacteristic
345C2#2716187711yescharacteristic
346C2#2716188211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#553×C197C2
socleC22×C197C22×D4
hypercenter#553D197
hypocenterC197#3355
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. #27161775C2semidirect
2. #27161478C2semidirect
3. #27161403C2semidirect
4. #27161400C2semidirect
5. #27161392C2semidirect
6. #27161390C2semidirect
7. #27161379C2semidirect
8. #27161376C2semidirect
9. #27161330C2semidirect
10. #27161330C2semidirect
11. C4×#15691343C2semidirect
12. #27161157C2semidirect
13. #27161122C2semidirect
14. #27161119C2semidirect
15. #553×C197C2semidirect
173. C197#3355semidirect
176. #553D197semidirect

Maximal subgroups

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

subgroupindex
#271617752
#271614782
#271614032
#271614002
#271613922
#271613902
#271613792
#271613762
#271613302
C4×#156913432
#271611572
#271611222
#271611192
#553×C1972
#3355197