Finite Group Databasebeta

← #38752629

Subgroups — #38752629

Order 25216.

420 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161591C211yesmaximal, characteristic
2#117×D197C211yesmaximal, characteristic, complemented
3#27161423C211yesmaximal, characteristic, complemented
4#27161422C211yesmaximal, characteristic, complemented
5#27161416C211yesmaximal, characteristic, complemented
6#27161411C211yesmaximal, characteristic, complemented
7#27161406C211yesmaximal, characteristic, complemented
8#27161405C211yesmaximal, characteristic, complemented
9#27161399C211yesmaximal, characteristic, complemented
10#27161394C211yesmaximal, characteristic, complemented
11#27161115C211yesmaximal, characteristic, complemented
12#27161076C211yesmaximal, characteristic, complemented
13#27161073C211yesmaximal, characteristic, complemented
14C2×#20263238C211yesmaximal, characteristic, complemented
15#427×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic
17C22×C4×D197C2211yescharacteristic
18C22×#13096802C2211yescharacteristic
19#20263310—21no—
20#20263306C2211yescharacteristic
21#20263306C2211yescharacteristic
22#20263301C2211yescharacteristic
23#20263301C2211yescharacteristic
24#20263296—21no—
25C4×D788—21no—
26C4×D788—21no—
27#20263285C2211yescharacteristic
28#20263284C2211yescharacteristic
29#20263284C2211yescharacteristic
30#20263283—21no—
31#20263282—21no—
32C4⋊C4×D197C2211yescharacteristic
33C4⋊C4×D197C2211yescharacteristic
34C22⋊C4×D197—21no—
35C42×D197—21no—
36#20263259C2211yescharacteristic, complemented
37#20263259C2211yescharacteristic, complemented
38#20263257C2211yescharacteristic, complemented
39#20263257C2211yescharacteristic, complemented
40#20263251C2211yescharacteristic, complemented
41#20263251C2211yescharacteristic, complemented
42#20263249C2211yescharacteristic, complemented
43#20263249C2211yescharacteristic, complemented
44#20263248C2211yescharacteristic, complemented
45#20263248C2211yescharacteristic, complemented
46#20263246C2211yescharacteristic, complemented
47#20263246C2211yescharacteristic, complemented
48#20263238C2211yescharacteristic
49#20263238C2211yescharacteristic
50#20263238C2211yescharacteristic
51#20263238C2211yescharacteristic
52C22×C8×C197C2211yescharacteristic
53#117×C197C2211yescharacteristic
54#117×C197C2211yescharacteristic
55#107×C197C2211yescharacteristic, complemented
56#107×C197C2211yescharacteristic, complemented
57#106×C197C2211yescharacteristic, complemented
58#106×C197C2211yescharacteristic, complemented
59C23×D197D411yescharacteristic
60#15691358—22no—
61#15691358—22no—
62C2×Dic394C2311yescharacteristic
63C2×D788—21no—
64C2×D788—21no—
65C2×D788C2311yescharacteristic
66C2×C22⋊D197—21no—
67C22×Dic197D411yescharacteristic
68C2×C4×D197—21no—
69C2×C4×D197—21no—
70C2×C4×D197—21no—
71C2×C4×D197—22no—
72C2×C4×D197—22no—
73C2×C4×D197D411yescharacteristic
74C2×C4×D197D411yescharacteristic
75C2×C4×D197C2311yescharacteristic
76C2×C4×D197C2311yescharacteristic
77C2×#13096802—22no—
78C2×#13096802C2311yescharacteristic
79C2×#13096802C2311yescharacteristic
80C4⋊Dic197—21no—
81C4⋊Dic197—21no—
82C4⋊Dic197C2311yescharacteristic
83C4⋊Dic197C2311yescharacteristic
84#15691337—21no—
85#15691337—21no—
86C22⋊Dic197—21no—
87#15691335—21no—
88#15691335—21no—
89#15691335—21no—
90#15691335—21no—
91#15691335—21no—
92#15691335—21no—
93C4×Dic197—21no—
94C4×Dic197—21no—
95C4×Dic197—21no—
96C22×C4×C197C2311yescharacteristic
97C2×C8×C197—22no—
98C2×C8×C197C2311yescharacteristic
99C2×C8×C197C2311yescharacteristic
100C4⋊C4×C197C2311yescharacteristic
101C4⋊C4×C197C2311yescharacteristic
102C4⋊C4×C197C2311yescharacteristic
103C4⋊C4×C197C2311yescharacteristic
104C22⋊C4×C197—21no—
105C22⋊C4×C197—21no—
106C42×C197—21no—
107C42×C197—21no—
108C22×D197—22no—
109C22×D197—22no—
110C22×D197—21no—
111C22×D197—41no—
112C22×D197C2×D411yescharacteristic
113C22×D197C2×D411yescharacteristic
114Dic394—21no—
115Dic394—21no—
116D788—21no—
117D788—21no—
118D788—42no—
119D788—42no—
120C22⋊D197—42no—
121C2×Dic197—22no—
122C2×Dic197—21no—
123C2×Dic197—21no—
124C2×Dic197—21no—
125C2×Dic197C2×D411yescharacteristic
126C2×Dic197C2×D411yescharacteristic
127C4×D197—22no—
128C4×D197—22no—
129C4×D197—21no—
130C4×D197—21no—
131C4×D197—21no—
132C4×D197—21no—
133C4×D197—42no—
134C4×D197—42no—
135C4×D197—42no—
136#13096802—21no—
137#13096802—21no—
138C23×C197C2×D411yescharacteristic
139C2×C4×C197—21no—
140C2×C4×C197—21no—
141C2×C4×C197—21no—
142C2×C4×C197—21no—
143C2×C4×C197C2×D411yescharacteristic
144C2×C4×C197#4112yes—
145C2×C4×C197#4112yes—
146C2×C4×C197C2411yescharacteristic
147C8×C197—21no—
148C8×C197—21no—
149C22×C197—21no—
150C22×C197—21no—
151C22×C197#12312yes—
152C22×C197C22×D411yescharacteristic
153C2×D197—21no—
154C2×D197—21no—
155C2×D197—21no—
156C2×D197—21no—
157C2×D197—21no—
158C2×D197—44no—
159C2×D197—41no—
160C2×D197—41no—
161C2×D197—41no—
162C2×D197#13512yes—
163C4×C197—41no—
164C4×C197—41no—
165C4×C197—41no—
166C4×C197—41no—
167C4×C197#12312yes—
168C4×C197C2×#4111yescharacteristic
169C4×C197C2×#4111yescharacteristic
170Dic197—21no—
171Dic197—21no—
172Dic197—41no—
173Dic197—41no—
174Dic197—41no—
175C2×C197—22no—
176C2×C197C2×#12311yescharacteristic
177C2×C197C2×#13511yescharacteristic
178C2×C197#57511yescharacteristic
179D197—22no—
180D197—41no—
181D197—81no—
182C197#309111yescharacteristic, complemented, Sylow
183#3091—1971nomaximal, Sylow
184#525—1971no—
185C2×#117—1971no—
186#485—1971no—
187#485—1971no—
188#484—1971no—
189#483—1971no—
190#482—1971no—
191#482—1971no—
192#481—1971no—
193#480—1971no—
194#427—1971no—
195#427D19711yescharacteristic, complemented
196#416—1971no—
197#416—1971no—
198C2×C22⋊C8—1971no—
199C2×#41—1971no—
200C23×C4—1971no—
201C22×C8—1971no—
202C22×C8C2×D19711yescharacteristic
203#123—3941no—
204C22⋊Q8—1971no—
205C22⋊Q8—1971no—
206#121—1971no—
207#121—1971no—
208#120—3941no—
209C4×D4—3941no—
210C4×D4—3941no—
211#117—3941no—
212#117—3941no—
213#117—1971no—
214#117—1971no—
215#117—1971no—
216#117C2×D19711yescharacteristic
217#117C2×D19711yescharacteristic
218C2×C4⋊C4—1971no—
219C2×C4⋊C4—1971no—
220C2×C22⋊C4—3941no—
221C2×C42—3941no—
222#107—1971no—
223#107—1971no—
224#107C2×D19711yescharacteristic, complemented
225#107C2×D19711yescharacteristic, complemented
226#106—1971no—
227#106—1971no—
228#106C2×D19711yescharacteristic, complemented
229#106C2×D19711yescharacteristic, complemented
230#103—1971no—
231#103—1971no—
232#103—1971no—
233#103—1971no—
234#102—1971no—
235#102—1971no—
236#102—1971no—
237#102—1971no—
238C22⋊C8—1971no—
239C22⋊C8—1971no—
240C22⋊C8—1971no—
241C22⋊C8—1971no—
242C24—1971no—
243#41—3942no—
244#41—3942no—
245C2×Q8—1971no—
246C2×D4—3941no—
247C2×D4—3941no—
248C2×D4—3941no—
249C2×D4—1971no—
250C22×C4—3941no—
251C22×C4—3941no—
252C22×C4—3941no—
253C22×C4—3942no—
254C22×C4—3942no—
255C22×C4—1971no—
256C22×C4—1971no—
257C22×C4—1971no—
258C22×C4—1971no—
259C22×C4—1971no—
260C22×C4C22×D19711yescharacteristic
261C2×C8—22no—
262C2×C8—3942no—
263C2×C8—1971no—
264C2×C8—1971no—
265C2×C8C22×D19711yescharacteristic
266C2×C8C22×D19711yescharacteristic
267C4⋊C4—3941no—
268C4⋊C4—3941no—
269C4⋊C4—3941no—
270C4⋊C4—3941no—
271C4⋊C4—1971no—
272C4⋊C4—1971no—
273C4⋊C4C22×D19711yescharacteristic
274C4⋊C4C22×D19711yescharacteristic
275C4⋊C4C22×D19711yescharacteristic
276C4⋊C4C22×D19711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—3941no—
280C22⋊C4—3941no—
281C22⋊C4—3941no—
282C22⋊C4—3941no—
283C22⋊C4—3941no—
284C22⋊C4—3941no—
285C22⋊C4—3941no—
286C42—21no—
287C42—21no—
288C42—3941no—
289C42—3941no—
290C42—3941no—
291C23—7881no—
292C23—3942no—
293C23—3941no—
294C23—3942no—
295C23—1971no—
296C23—1971no—
297C23D4×D19711yescharacteristic
298Q8—3941no—
299Q8—3941no—
300D4—7882no—
301D4—7882no—
302D4—7882no—
303D4—3941no—
304D4—3941no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—7882no—
310C2×C4—7882no—
311C2×C4—7882no—
312C2×C4—3941no—
313C2×C4—3941no—
314C2×C4—3941no—
315C2×C4—3942no—
316C2×C4—3942no—
317C2×C4—3942no—
318C2×C4—3941no—
319C2×C4—3941no—
320C2×C4—3941no—
321C2×C4—3941no—
322C2×C4—1971no—
323C2×C4—1971no—
324C2×C4D4×D19711yescharacteristic
325C2×C4#1569135812yes—
326C2×C4#1569135912yes—
327C2×C4C23×D19711yescharacteristic
328C8—21no—
329C8—21no—
330C8—3941no—
331C8—3941no—
332C22—21no—
333C22—21no—
334C22—7884no—
335C22—7881no—
336C22—7881no—
337C22—7881no—
338C22—3941no—
339C22—3941no—
340C22—3941no—
341C22—3941no—
342C22—3941no—
343C22—1972no—
344C22#2026331112yes—
345C22C2×D4×D19711yescharacteristic
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—7881no—
351C4—7881no—
352C4—7881no—
353C4—3941no—
354C4—3941no—
355C4#2026331112yes—
356C4C2×#1569135811yescharacteristic
357C4C2×#1569135911yescharacteristic
358C2—22no—
359C2—7881no—
360C2—15761no—
361C2—3942no—
362C2C2×#2026331111yescharacteristic
363C2#135×D19711yescharacteristic
364C2#2716187611yescharacteristic
subgroupG/H
centerC2×C4D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#427×C197C2
socleC22×C197C22×D4
prime radical#27161591C2
hypercenter#427D197
hypocenterC197#3091
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
2. #117×D197C2semidirect
3. #27161423C2semidirect
4. #27161422C2semidirect
5. #27161416C2semidirect
6. #27161411C2semidirect
7. #27161406C2semidirect
8. #27161405C2semidirect
9. #27161399C2semidirect
10. #27161394C2semidirect
11. #27161115C2semidirect
12. #27161076C2semidirect
13. #27161073C2semidirect
14. C2×#20263238C2semidirect
15. #427×C197C2semidirect
36. #20263259C22semidirect
37. #20263259C22semidirect
38. #20263257C22semidirect
39. #20263257C22semidirect
40. #20263251C22semidirect
41. #20263251C22semidirect
42. #20263249C22semidirect
43. #20263249C22semidirect
44. #20263248C22semidirect
45. #20263248C22semidirect
46. #20263246C22semidirect
47. #20263246C22semidirect
55. #107×C197C22semidirect
56. #107×C197C22semidirect
57. #106×C197C22semidirect
58. #106×C197C22semidirect
182. C197#3091semidirect
195. #427D197semidirect
224. #107C2×D197semidirect
225. #107C2×D197semidirect
228. #106C2×D197semidirect
229. #106C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271615912
#117×D1972
#271614232
#271614222
#271614162
#271614112
#271614062
#271614052
#271613992
#271613942
#271611152
#271610762
#271610732
C2×#202632382
#427×C1972
#3091197