Finite Group Databasebeta

← #38752623

Subgroups — #38752623

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#27161586C211yesmaximal, characteristic
2#117×D197C211yesmaximal, characteristic, complemented
3#27161419C211yesmaximal, characteristic, complemented
4#27161418C211yesmaximal, characteristic, complemented
5#27161413C211yesmaximal, characteristic, complemented
6#27161408C211yesmaximal, characteristic, complemented
7#27161402C211yesmaximal, characteristic, complemented
8#27161401C211yesmaximal, characteristic, complemented
9#27161396C211yesmaximal, characteristic, complemented
10#27161391C211yesmaximal, characteristic, complemented
11#27161117C211yesmaximal, characteristic, complemented
12#27161115C211yesmaximal, characteristic, complemented
13#27161075C211yesmaximal, characteristic, complemented
14C2×#20263238C211yesmaximal, characteristic, complemented
15#416×C197C211yesmaximal, characteristic, complemented
16C22×C4×D197C2211yescharacteristic
17C22×#13096802C2211yescharacteristic
18#20263307—21no—
19#20263302C2211yescharacteristic
20#20263302C2211yescharacteristic
21#20263297C2211yescharacteristic
22#20263297C2211yescharacteristic
23#20263294—21no—
24C4×C22⋊D197—21no—
25C4×C22⋊D197—21no—
26#20263285C2211yescharacteristic
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#20263260C2211yescharacteristic, complemented
37#20263260C2211yescharacteristic, complemented
38#20263259C2211yescharacteristic, complemented
39#20263259C2211yescharacteristic, complemented
40#20263258C2211yescharacteristic, complemented
41#20263258C2211yescharacteristic, complemented
42#20263257C2211yescharacteristic, complemented
43#20263257C2211yescharacteristic, complemented
44#20263250C2211yescharacteristic, complemented
45#20263250C2211yescharacteristic, complemented
46#20263247C2211yescharacteristic, complemented
47#20263247C2211yescharacteristic, complemented
48#20263238C2211yescharacteristic
49#20263238C2211yescharacteristic
50#20263238C2211yescharacteristic
51#20263238C2211yescharacteristic
52C2×#41×C197C2211yescharacteristic
53C22×C8×C197C2211yescharacteristic
54#117×C197C2211yescharacteristic
55#103×C197C2211yescharacteristic, complemented
56#103×C197C2211yescharacteristic, complemented
57#102×C197C2211yescharacteristic, complemented
58#102×C197C2211yescharacteristic, complemented
59C23×D197D411yescharacteristic
60C2×C22⋊D197—21no—
61C2×C22⋊D197—21no—
62C22×Dic197D411yescharacteristic
63C2×C4×D197—21no—
64C2×C4×D197—21no—
65C2×C4×D197—22no—
66C2×C4×D197—22no—
67C2×C4×D197D411yescharacteristic
68C2×C4×D197D411yescharacteristic
69C2×C4×D197C2311yescharacteristic
70C2×C4×D197C2311yescharacteristic
71C2×#13096802—22no—
72C2×#13096802C2311yescharacteristic
73C2×#13096802C2311yescharacteristic
74C4⋊Dic197C2311yescharacteristic
75C4⋊Dic197C2311yescharacteristic
76C4⋊Dic197C2311yescharacteristic
77C4⋊Dic197C2311yescharacteristic
78#15691337—21no—
79#15691337—21no—
80#15691337—21no—
81#15691337—21no—
82C22⋊Dic197—21no—
83C22⋊Dic197—21no—
84C22⋊Dic197—21no—
85C22⋊Dic197—21no—
86#15691335—21no—
87#15691335—21no—
88#15691335—21no—
89#15691335—21no—
90C4×Dic197—21no—
91C4×Dic197—21no—
92C4×Dic197—21no—
93C4×Dic197—21no—
94#41×C197—22no—
95#41×C197—22no—
96C2×Q8×C197C2311yescharacteristic
97C2×D4×C197—21no—
98C2×D4×C197C2311yescharacteristic
99C22×C4×C197—21no—
100C22×C4×C197C2311yescharacteristic
101C2×C8×C197—22no—
102C2×C8×C197C2311yescharacteristic
103C2×C8×C197C2311yescharacteristic
104C4⋊C4×C197C2311yescharacteristic
105C4⋊C4×C197C2311yescharacteristic
106C22⋊C4×C197—21no—
107C42×C197—21no—
108C22×D197—22no—
109C22×D197—22no—
110C22×D197—41no—
111C22×D197C2×D411yescharacteristic
112C22×D197C2×D411yescharacteristic
113C22⋊D197—42no—
114C22⋊D197—42no—
115C2×Dic197—22no—
116C2×Dic197—21no—
117C2×Dic197—21no—
118C2×Dic197—21no—
119C2×Dic197—21no—
120C2×Dic197C2×D411yescharacteristic
121C2×Dic197C2×D411yescharacteristic
122C4×D197—22no—
123C4×D197—22no—
124C4×D197—21no—
125C4×D197—21no—
126C4×D197—21no—
127C4×D197—21no—
128C4×D197—42no—
129C4×D197—42no—
130#13096802—21no—
131#13096802—21no—
132C23×C197—21no—
133C23×C197C2×D411yescharacteristic
134Q8×C197—21no—
135Q8×C197—21no—
136D4×C197—21no—
137D4×C197—21no—
138D4×C197—42no—
139C2×C4×C197—21no—
140C2×C4×C197—21no—
141C2×C4×C197—21no—
142C2×C4×C197—42no—
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—41no—
152C22×C197—41no—
153C22×C197—41no—
154C22×C197#12312yes—
155C22×C197C22×D411yescharacteristic
156C2×D197—21no—
157C2×D197—21no—
158C2×D197—21no—
159C2×D197—21no—
160C2×D197—21no—
161C2×D197—44no—
162C2×D197#13512yes—
163C4×C197—41no—
164C4×C197—41no—
165C4×C197—41no—
166C4×C197#12312yes—
167C4×C197C2×#4111yescharacteristic
168C4×C197C2×#4111yescharacteristic
169Dic197—21no—
170Dic197—21no—
171Dic197—41no—
172Dic197—41no—
173Dic197—41no—
174Dic197—41no—
175C2×C197—22no—
176C2×C197—81no—
177C2×C197C2×#12311yescharacteristic
178C2×C197C2×#13511yescharacteristic
179C2×C197#57511yescharacteristic
180D197—22no—
181D197—41no—
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#427—1971no—
196#416—1971no—
197#416D19711yescharacteristic, complemented
198C2×C22⋊C8—1971no—
199C2×#41C2×D19711yescharacteristic
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#117—1971no—
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#107—1971no—
225#107—1971no—
226#106—1971no—
227#106—1971no—
228#106—1971no—
229#106—1971no—
230#103—1971no—
231#103—1971no—
232#103C2×D19711yescharacteristic, complemented
233#103C2×D19711yescharacteristic, complemented
234#102—1971no—
235#102—1971no—
236#102C2×D19711yescharacteristic, complemented
237#102C2×D19711yescharacteristic, complemented
238C22⋊C8—1971no—
239C22⋊C8—1971no—
240C22⋊C8—1971no—
241C22⋊C8—1971no—
242C24—1971no—
243#41—22no—
244#41—22no—
245C2×Q8C22×D19711yescharacteristic
246C2×D4—21no—
247C2×D4—3941no—
248C2×D4—3941no—
249C2×D4C22×D19711yescharacteristic
250C22×C4—21no—
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⋊C4—1971no—
274C4⋊C4—1971no—
275C4⋊C4C22×D19711yescharacteristic
276C4⋊C4C22×D19711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—3941no—
279C22⋊C4—3941no—
280C22⋊C4—3941no—
281C22⋊C4—3941no—
282C22⋊C4—3941no—
283C22⋊C4—3941no—
284C22⋊C4—3941no—
285C22⋊C4—3941no—
286C42—21no—
287C42—3941no—
288C42—3941no—
289C42—3941no—
290C42—3941no—
291C23—21no—
292C23—7881no—
293C23—3942no—
294C23—3942no—
295C23—1971no—
296C23—1971no—
297C23D4×D19711yescharacteristic
298Q8—21no—
299Q8—21no—
300D4—21no—
301D4—21no—
302D4—42no—
303D4—7882no—
304D4—7882no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—7882no—
310C2×C4—7882no—
311C2×C4—3941no—
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#1569135712yes—
326C2×C4#1569135812yes—
327C2×C4C23×D19711yescharacteristic
328C8—21no—
329C8—21no—
330C8—3941no—
331C8—3941no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—7884no—
338C22—3941no—
339C22—3941no—
340C22—3941no—
341C22—3941no—
342C22—3941no—
343C22—1972no—
344C22#2026330812yes—
345C22C2×D4×D19711yescharacteristic
346C4—41no—
347C4—41no—
348C4—41no—
349C4—7881no—
350C4—7881no—
351C4—7881no—
352C4—7881no—
353C4—3941no—
354C4—3941no—
355C4#2026330812yes—
356C4C2×#1569135711yescharacteristic
357C4C2×#1569135811yescharacteristic
358C2—22no—
359C2—81no—
360C2—7881no—
361C2—3942no—
362C2C2×#2026330811yescharacteristic
363C2#135×D19711yescharacteristic
364C2#2716187611yescharacteristic
subgroupG/H
centerC2×C4D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#416×C197C2
socleC22×C197C22×D4
prime radical#27161586C2
hypercenter#416D197
hypocenterC197#3091
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. #117×D197C2semidirect
3. #27161419C2semidirect
4. #27161418C2semidirect
5. #27161413C2semidirect
6. #27161408C2semidirect
7. #27161402C2semidirect
8. #27161401C2semidirect
9. #27161396C2semidirect
10. #27161391C2semidirect
11. #27161117C2semidirect
12. #27161115C2semidirect
13. #27161075C2semidirect
14. C2×#20263238C2semidirect
15. #416×C197C2semidirect
36. #20263260C22semidirect
37. #20263260C22semidirect
38. #20263259C22semidirect
39. #20263259C22semidirect
40. #20263258C22semidirect
41. #20263258C22semidirect
42. #20263257C22semidirect
43. #20263257C22semidirect
44. #20263250C22semidirect
45. #20263250C22semidirect
46. #20263247C22semidirect
47. #20263247C22semidirect
55. #103×C197C22semidirect
56. #103×C197C22semidirect
57. #102×C197C22semidirect
58. #102×C197C22semidirect
182. C197#3091semidirect
197. #416D197semidirect
232. #103C2×D197semidirect
233. #103C2×D197semidirect
236. #102C2×D197semidirect
237. #102C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271615862
#117×D1972
#271614192
#271614182
#271614132
#271614082
#271614022
#271614012
#271613962
#271613912
#271611172
#271611152
#271610752
C2×#202632382
#416×C1972
#3091197