Finite Group Databasebeta

← #38755259

Subgroups — #38755259

Order 25216.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161799C211yesmaximal, characteristic, complemented
2#27161775C211yesmaximal, characteristic
3#27161488C211yesmaximal, characteristic, complemented
4#27161421C211yesmaximal, characteristic, complemented
5#27161410C211yesmaximal, characteristic, complemented
6#27161404C211yesmaximal, characteristic, complemented
7#27161404C211yesmaximal, characteristic, complemented
8#27161387C211yesmaximal, characteristic, complemented
9#27161370C211yesmaximal, characteristic, complemented
10#27161325C211yesmaximal, characteristic, complemented
11#27161308C211yesmaximal, characteristic, complemented
12#27161181C211yesmaximal, characteristic, complemented
13#27161121C211yesmaximal, characteristic, complemented
14#27161117C211yesmaximal, characteristic, complemented
15#435×C197C211yesmaximal, characteristic, complemented
16C2×#15691347C2211yescharacteristic
17#20263326C2211yescharacteristic
18#20263321—21no—
19#20263319—21no—
20#20263319—21no—
21#20263318C2211yescharacteristic
22#20263313—21no—
23#20263308—21no—
24#20263305C2211yescharacteristic
25#20263305C2211yescharacteristic
26#20263300C2211yescharacteristic
27#20263300C2211yescharacteristic
28#20263298—21no—
29C4×Dic394C2211yescharacteristic
30#20263291—21no—
31C4×D788C2211yescharacteristic
32#20263286—21no—
33#20263285C2211yescharacteristic
34#20263285C2211yescharacteristic
35#20263285C2211yescharacteristic
36C2×C4⋊Dic197C2211yescharacteristic
37#20263260C2211yescharacteristic
38#20263260C2211yescharacteristic
39#20263260C2211yescharacteristic
40#20263260C2211yescharacteristic
41#20263258C2211yescharacteristic
42#20263258C2211yescharacteristic
43#20263258C2211yescharacteristic
44#20263258C2211yescharacteristic
45#20263258C2211yescharacteristic
46#20263251C2211yescharacteristic
47#20263251C2211yescharacteristic
48#20263251C2211yescharacteristic
49#20263248C2211yescharacteristic
50#20263248C2211yescharacteristic
51#20263248C2211yescharacteristic
52C2×M16×C197C2211yescharacteristic
53C22×C8×C197C2211yescharacteristic
54C4×D4×C197C2211yescharacteristic
55C4⋊C8×C197C2211yescharacteristic
56C22⋊C8×C197C2211yescharacteristic
57C22⋊C8×C197C2211yescharacteristic
58#97×C197C2211yescharacteristic
59C2×Dic394—21no—
60C2×Dic394C2311yescharacteristic
61C2×D788C2311yescharacteristic
62C2×C22⋊D197—21no—
63C2×C22⋊D197—21no—
64C22×Dic197—21no—
65C2×C4×D197—21no—
66#15691347—22no—
67C4⋊Dic197—21no—
68C4⋊Dic197—21no—
69C4⋊Dic197C2311yescharacteristic
70C4⋊Dic197C2311yescharacteristic
71C4⋊Dic197C2311yescharacteristic
72C4⋊Dic197C2311yescharacteristic
73C4⋊Dic197C2311yescharacteristic
74C4⋊Dic197C2311yescharacteristic
75#15691337—21no—
76#15691337—21no—
77#15691337—21no—
78#15691337—21no—
79#15691337—21no—
80C22⋊Dic197—21no—
81C22⋊Dic197—21no—
82C22⋊Dic197—21no—
83C22⋊Dic197—21no—
84C22⋊Dic197—21no—
85#15691335—21no—
86#15691335—21no—
87#15691335—21no—
88C4×Dic197—21no—
89C4×Dic197—21no—
90C4×Dic197—21no—
91C4×Dic197—21no—
92C2×D4×C197D411yescharacteristic
93C22×C4×C197C2311yescharacteristic
94C22×C4×C197C2311yescharacteristic
95M16×C197—22no—
96C2×C8×C197—22no—
97C2×C8×C197C2311yescharacteristic
98C2×C8×C197C2311yescharacteristic
99C2×C8×C197C2311yescharacteristic
100C2×C8×C197C2311yescharacteristic
101C4⋊C4×C197D411yescharacteristic
102C22⋊C4×C197D411yescharacteristic
103C22⋊C4×C197D411yescharacteristic
104C42×C197C2311yescharacteristic
105C22×D197—21no—
106Dic394—21no—
107Dic394—21no—
108Dic394—42no—
109D788—21no—
110D788—21no—
111C22⋊D197—42no—
112C22⋊D197—81no—
113C2×Dic197—21no—
114C2×Dic197—21no—
115C2×Dic197—21no—
116C2×Dic197—21no—
117C2×Dic197—21no—
118C2×Dic197—21no—
119C2×Dic197—21no—
120C2×Dic197—81no—
121C4×D197—81no—
122C23×C197C2×D411yescharacteristic
123C23×C197C2×D411yescharacteristic
124D4×C197—41no—
125C2×C4×C197—21no—
126C2×C4×C197—21no—
127C2×C4×C197—21no—
128C2×C4×C197—21no—
129C2×C4×C197C2×D411yescharacteristic
130C2×C4×C197C2×D411yescharacteristic
131C2×C4×C197C2×D411yescharacteristic
132C2×C4×C197C2×D411yescharacteristic
133C2×C4×C197C2411yescharacteristic
134C8×C197—21no—
135C8×C197—21no—
136C8×C197—21no—
137C8×C197#4112yes—
138C22×C197—21no—
139C22×C197—21no—
140C22×C197—21no—
141C22×C197—21no—
142C22×C197—21no—
143C22×C197—21no—
144C22×C197C22×D411yescharacteristic
145C2×D197—41no—
146C2×D197—41no—
147C2×D197—41no—
148C4×C197—21no—
149C4×C197—21no—
150C4×C197—21no—
151C4×C197—21no—
152C4×C197C2×#4111yescharacteristic
153C4×C1972−1+411yescharacteristic
154Dic197—41no—
155Dic197—41no—
156Dic197—41no—
157Dic197—41no—
158Dic197—41no—
159Dic197—41no—
160Dic197—41no—
161C2×C197—41no—
162C2×C197—41no—
163C2×C197#54711yescharacteristic
164C2×C197#57511yescharacteristic
165C2×C197#57711yescharacteristic
166D197—81no—
167C197#335011yescharacteristic, complemented, Sylow
168#3350—1971nomaximal, Sylow
169#556—1971no—
170#553—1971no—
171#501—1971no—
172#485—1971no—
173#483—1971no—
174#482—1971no—
175#482—1971no—
176#479—1971no—
177#476—1971no—
178#468—1971no—
179#465—1971no—
180#440—1971no—
181#435D19711yescharacteristic, complemented
182#428—1971no—
183#427—1971no—
184C2×SD16—1971no—
185C2×M16C2×D19711yescharacteristic
186C22×C8C2×D19711yescharacteristic
187C4⋊Q8—1971no—
188#126—3941no—
189#126—3941no—
190#126—3941no—
191#125—1971no—
192#124—3941no—
193#123—3941no—
194C22⋊Q8—1971no—
195C22⋊Q8—1971no—
196#121—3941no—
197#121—1971no—
198#121—1971no—
199C4×Q8—3941no—
200C4×Q8—1971no—
201C4×D4—3941no—
202C4×D4—1971no—
203C4×D4C2×D19711yescharacteristic
204#117—1971no—
205#117—1971no—
206#117—1971no—
207C2×C4⋊C4—1971no—
208#107—1971no—
209#107—1971no—
210#107—1971no—
211#107—1971no—
212#106—1971no—
213#106—1971no—
214#106—1971no—
215#106—1971no—
216#106—1971no—
217C4⋊C8C2×D19711yescharacteristic
218#103—1971no—
219#103—1971no—
220#103—1971no—
221#102—1971no—
222#102—1971no—
223#102—1971no—
224C22⋊C8C2×D19711yescharacteristic
225C22⋊C8C2×D19711yescharacteristic
226#97C2×D19711yescharacteristic
227C2×Q8—3941no—
228C2×Q8—1971no—
229C2×D4—3941no—
230C2×D4—3941no—
231C2×D4—1971no—
232C2×D4D78811yescharacteristic
233C22×C4—3941no—
234C22×C4—3941no—
235C22×C4C22×D19711yescharacteristic
236C22×C4C22×D19711yescharacteristic
237SD16—3942no—
238M16—22no—
239C2×C8—22no—
240C2×C8C22×D19711yescharacteristic
241C2×C8C22×D19711yescharacteristic
242C2×C8C22×D19711yescharacteristic
243C2×C8C22×D19711yescharacteristic
244C4⋊C4—3941no—
245C4⋊C4—3941no—
246C4⋊C4—3941no—
247C4⋊C4—3941no—
248C4⋊C4—3941no—
249C4⋊C4—3941no—
250C4⋊C4—3941no—
251C4⋊C4—1971no—
252C4⋊C4—1971no—
253C4⋊C4—1971no—
254C4⋊C4—1971no—
255C4⋊C4—1971no—
256C4⋊C4—1971no—
257C4⋊C4D78811yescharacteristic
258C22⋊C4—3941no—
259C22⋊C4—3941no—
260C22⋊C4—3941no—
261C22⋊C4—3941no—
262C22⋊C4—3941no—
263C22⋊C4—3941no—
264C22⋊C4—3941no—
265C22⋊C4—3941no—
266C22⋊C4D78811yescharacteristic
267C22⋊C4D78811yescharacteristic
268C42—3941no—
269C42—3941no—
270C42—3941no—
271C42—3941no—
272C42C22×D19711yescharacteristic
273C23—3941no—
274C23C2×D78811yescharacteristic
275C23C2×D78811yescharacteristic
276Q8—7882no—
277Q8—3941no—
278Q8—3941no—
279D4—41no—
280D4—7882no—
281D4—15761no—
282D4—3941no—
283D4—3941no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—15761no—
289C2×C4—15761no—
290C2×C4—3941no—
291C2×C4—3941no—
292C2×C4—3941no—
293C2×C4—3941no—
294C2×C4—3941no—
295C2×C4—3941no—
296C2×C4—3941no—
297C2×C4C2×D78811yescharacteristic
298C2×C4C2×D78811yescharacteristic
299C2×C4C2×D78811yescharacteristic
300C2×C4C2×D78811yescharacteristic
301C2×C4C23×D19711yescharacteristic
302C8—21no—
303C8—21no—
304C8—21no—
305C8#1569135712yes—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—7881no—
313C22—7881no—
314C22—7881no—
315C22C22×D78811yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—7881no—
321C4—7881no—
322C4—7881no—
323C4—7881no—
324C4—7881no—
325C4—7881no—
326C4—7881no—
327C4C2×#1569135711yescharacteristic
328C4#2026337311yescharacteristic
329C2—41no—
330C2—41no—
331C2—15761no—
332C2#2716173311yescharacteristic
333C2#2716187911yescharacteristic
334C2#2716189111yescharacteristic
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#435×C197C2
socleC22×C197C22×D4
prime radical#27161775C2
hypercenter#435D197
hypocenterC197#3350
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #27161799C2semidirect
3. #27161488C2semidirect
4. #27161421C2semidirect
5. #27161410C2semidirect
6. #27161404C2semidirect
7. #27161404C2semidirect
8. #27161387C2semidirect
9. #27161370C2semidirect
10. #27161325C2semidirect
11. #27161308C2semidirect
12. #27161181C2semidirect
13. #27161121C2semidirect
14. #27161117C2semidirect
15. #435×C197C2semidirect
167. C197#3350semidirect
181. #435D197semidirect

Maximal subgroups

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

subgroupindex
#271617992
#271617752
#271614882
#271614212
#271614102
#271614042
#271613872
#271613702
#271613252
#271613082
#271611812
#271611212
#271611172
#435×C1972
#3350197