Finite Group Databasebeta

← #38754679

Subgroups — #38754679

Order 25216.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161729C211yesmaximal, characteristic, complemented
2#27161720C211yesmaximal, characteristic, complemented
3#27161440C211yesmaximal, characteristic, complemented
4#27161407C211yesmaximal, characteristic, complemented
5#27161399C211yesmaximal, characteristic, complemented
6#27161368C211yesmaximal, characteristic, complemented
7#27161331C211yesmaximal, characteristic, complemented
8#27161296C211yesmaximal, characteristic, complemented
9#27161223C211yesmaximal, characteristic, complemented
10#27161212C211yesmaximal, characteristic, complemented
11#27161188C211yesmaximal, characteristic, complemented
12#27161150C211yesmaximal, characteristic, complemented
13#27161079C211yesmaximal, characteristic, complemented
14C2×#20263248C211yesmaximal, characteristic, complemented
15#465×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic
17C2×#15691357—21no—
18C22×D788C2211yescharacteristic
19C2×#15691343C2211yescharacteristic
20C2×#15691343C2211yescharacteristic
21C2×#15691341C2211yescharacteristic, complemented
22#20263325C2211yescharacteristic, complemented
23#20263315C2211yescharacteristic
24#20263311—21no—
25#20263308—21no—
26#20263307—21no—
27#20263303—21no—
28#20263301—21no—
29#20263301C2211yescharacteristic
30#20263297C2211yescharacteristic, complemented
31#20263296—21no—
32D4×Dic197C2211yescharacteristic, complemented
33#20263288C2211yescharacteristic
34C4×C22⋊D197C2211yescharacteristic
35C2×C4⋊Dic197C2211yescharacteristic
36C4⋊C4×D197C2211yescharacteristic, complemented
37C22⋊C4×D197—21no—
38#20263257C2211yescharacteristic, complemented
39#20263257C2211yescharacteristic, complemented
40#20263255C2211yescharacteristic, complemented
41#20263250C2211yescharacteristic, complemented
42#20263250C2211yescharacteristic, complemented
43#20263248C2211yescharacteristic
44#20263248C2211yescharacteristic
45#20263248C2211yescharacteristic
46#20263248C2211yescharacteristic
47#20263246C2211yescharacteristic
48#20263246C2211yescharacteristic
49C22⋊#13096802C2211yescharacteristic
50#20263238C2211yescharacteristic, complemented
51#20263236C2211yescharacteristic, complemented
52C2×SD16×C197C2211yescharacteristic, complemented
53C22×C8×C197C2211yescharacteristic
54C22⋊Q8×C197C2211yescharacteristic
55#121×C197C2211yescharacteristic
56#106×C197C2211yescharacteristic, complemented
57#103×C197C2211yescharacteristic, complemented
58#102×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60#15691359—22no—
61#15691359—41no—
62#15691357—44no—
63C2×Dic394—21no—
64C2×D788—22no—
65C2×D788—21no—
66C2×D788—21no—
67C2×D788—21no—
68C2×D788—41no—
69C2×D788C2311yescharacteristic
70C2×D788C2311yescharacteristic
71C2×C22⋊D197—21no—
72C2×C22⋊D197D411yescharacteristic
73C22×Dic197—21no—
74C22×Dic197—21no—
75C2×C4×D197—21no—
76C2×C4×D197—21no—
77C2×C4×D197—21no—
78C2×C4×D197C2311yescharacteristic
79#15691343—22no—
80#15691343—41no—
81#15691341—41no—
82C2×#13096802C2311yescharacteristic
83C2×#13096802C2311yescharacteristic
84C4⋊Dic197—21no—
85C4⋊Dic197C2311yescharacteristic
86C4⋊Dic197C2311yescharacteristic
87#15691337—21no—
88#15691337—21no—
89#15691337D411yescharacteristic
90C22⋊Dic197—21no—
91C22⋊Dic197—21no—
92C22⋊Dic197D411yescharacteristic
93#15691335—21no—
94#15691335—21no—
95#15691335—21no—
96#15691335—21no—
97#15691335D411yescharacteristic
98C4×Dic197C2311yescharacteristic
99C2×Q8×C197C2311yescharacteristic
100C2×D4×C197—21no—
101C2×D4×C197C2311yescharacteristic
102C22×C4×C197C2311yescharacteristic
103SD16×C197—41no—
104C2×C8×C197—41no—
105C2×C8×C197C2311yescharacteristic
106C2×C8×C197C2311yescharacteristic
107C4⋊C4×C197—21no—
108C4⋊C4×C197C2311yescharacteristic
109C4⋊C4×C197C2311yescharacteristic
110C22⋊C4×C197—21no—
111C22⋊C4×C197—21no—
112C22×D197—21no—
113C22×D197—21no—
114C22×D197—42no—
115C22×D197—42no—
116C22×D197—81no—
117C22×D197C2×D411yescharacteristic
118Dic394—42no—
119D788—21no—
120D788—21no—
121D788—21no—
122D788—42no—
123D788—42no—
124D788—81no—
125D788#4112yescomplemented
126C22⋊D197—22no—
127C22⋊D197—42no—
128C2×Dic197—21no—
129C2×Dic197—21no—
130C2×Dic197—42no—
131C2×Dic197—42no—
132C2×Dic197C2×D411yescharacteristic
133C2×Dic197C2×D411yescharacteristic
134C2×Dic197C2×D411yescharacteristic
135C4×D197—21no—
136C4×D197—21no—
137C4×D197—42no—
138C4×D197—42no—
139C4×D197—81no—
140#13096802—21no—
141#13096802—21no—
142C23×C197—21no—
143C23×C197C2×D411yescharacteristic
144Q8×C197—21no—
145Q8×C197—21no—
146D4×C197—21no—
147D4×C197—21no—
148D4×C197—42no—
149C2×C4×C197—21no—
150C2×C4×C197—21no—
151C2×C4×C197—21no—
152C2×C4×C197—21no—
153C2×C4×C197—21no—
154C2×C4×C197C2×D411yescharacteristic
155C2×C4×C197C2411yescharacteristic
156C8×C197—21no—
157C8×C197—21no—
158C22×C197—21no—
159C22×C197—21no—
160C22×C197—41no—
161C22×C197—41no—
162C22×C197—41no—
163C22×C197AGL(1,ℤ/8)12yes—
164C22×C197C22×D411yescharacteristic
165C2×D197—21no—
166C2×D197—22no—
167C2×D197—21no—
168C2×D197—21no—
169C2×D197—41no—
170C2×D197—41no—
171C2×D197—41no—
172C2×D197—41no—
173C2×D197—41no—
174C2×D197—84no—
175C4×C197—21no—
176C4×C197—41no—
177C4×C197—41no—
178C4×C197—41no—
179C4×C197C2×#4111yescharacteristic
180C4×C197H(5,𝔽2)11yescharacteristic
181Dic197—21no—
182Dic197—21no—
183Dic197—21no—
184Dic197—41no—
185Dic197—41no—
186C2×C197—22no—
187C2×C197—81no—
188C2×C197#54611yescharacteristic
189C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C197#57711yescharacteristic
191D197—41no—
192D197—42no—
193D197—81no—
194C197#330611yescharacteristic, complemented, Sylow
195#3306—1971nomaximal, Sylow
196#547—1971no—
197#546—1971no—
198#489—1971no—
199#483—1971no—
200#481—1971no—
201#476—1971no—
202#469—1971no—
203#465D19711yescharacteristic, complemented
204#463—1971no—
205#449—1971no—
206#447—1971no—
207#442—1971no—
208#435—1971no—
209#417—1971no—
210C2×#102—1971no—
211C2×#41—3941no—
212C2×#41—1971no—
213C22×D4—1971no—
214C2×SD16C2×D19711yescharacteristic, complemented
215C2×D8—1971no—
216C2×D8—1971no—
217C2×M16—1971no—
218C22×C8C2×D19711yescharacteristic
219C4⋊Q8—1971no—
220#124—1971no—
221#123—3941no—
222#123—3941no—
223#123—3941no—
224C22⋊Q8—3941no—
225C22⋊Q8C2×D19711yescharacteristic
226#121—3941no—
227#121—1971no—
228#121—1971no—
229#121C2×D19711yescharacteristic
230#120—3941no—
231C4×D4—1971no—
232C4×D4—1971no—
233C4×D4—1971no—
234C2×C4⋊C4—1971no—
235C2×C4⋊C4—1971no—
236C2×C22⋊C4—3941no—
237#106—1971no—
238#106—1971no—
239#106C2×D19711yescharacteristic, complemented
240C4⋊C8—1971no—
241#103—1971no—
242#103—1971no—
243#103C2×D19711yescharacteristic, complemented
244#102—1971no—
245#102—1971no—
246#102—1971no—
247#102—1971no—
248#102—1971no—
249#102—1971no—
250#102C2×D19711yescharacteristic, complemented
251C22⋊C8—1971no—
252C22⋊C8—1971no—
253#97—1971no—
254C24—3941no—
255#41—7884no—
256#41—7881no—
257#41—3942no—
258C2×Q8—3941no—
259C2×Q8C22×D19711yescharacteristic
260C2×D4—21no—
261C2×D4—7881no—
262C2×D4—3941no—
263C2×D4—3942no—
264C2×D4—3941no—
265C2×D4—3941no—
266C2×D4—3941no—
267C2×D4—1971no—
268C2×D4—1971no—
269C2×D4—1971no—
270C2×D4C22×D19711yescharacteristic
271C22×C4—3941no—
272C22×C4—3941no—
273C22×C4—3941no—
274C22×C4—3941no—
275C22×C4—3941no—
276C22×C4—1971no—
277C22×C4C22×D19711yescharacteristic
278SD16—41no—
279D8—7881no—
280D8—3942no—
281M16—7881no—
282C2×C8—41no—
283C2×C8—1971no—
284C2×C8—1971no—
285C2×C8C22×D19711yescharacteristic
286C2×C8C22×D19711yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—3941no—
289C4⋊C4—3941no—
290C4⋊C4—3941no—
291C4⋊C4—1971no—
292C4⋊C4—1971no—
293C4⋊C4—1971no—
294C4⋊C4C22×D19711yescharacteristic
295C4⋊C4C22×D19711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—3941no—
299C22⋊C4—3941no—
300C22⋊C4—3941no—
301C22⋊C4—3941no—
302C22⋊C4—3941no—
303C22⋊C4—3941no—
304C22⋊C4—1971no—
305C22⋊C4—1971no—
306C42—1971no—
307C23—21no—
308C23—7882no—
309C23—7882no—
310C23—15761no—
311C23—3941no—
312C23—3941no—
313C23—1971no—
314C23D4×D19711yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—7882no—
318D4—21no—
319D4—21no—
320D4—42no—
321D4—7882no—
322D4—7882no—
323D4—7882no—
324D4—15761no—
325D4—3942no—
326D4—3941no—
327D4—3941no—
328D4—3941no—
329D4—1972no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—7882no—
336C2×C4—7882no—
337C2×C4—7882no—
338C2×C4—7882no—
339C2×C4—15761no—
340C2×C4—3941no—
341C2×C4—3941no—
342C2×C4—3941no—
343C2×C4—3941no—
344C2×C4—1971no—
345C2×C4—1971no—
346C2×C4—1971no—
347C2×C4D4×D19711yescharacteristic
348C2×C4C23×D19711yescharacteristic
349C8—21no—
350C8—21no—
351C8—3941no—
352C8—3941no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—7881no—
359C22—7881no—
360C22—7881no—
361C22—7881no—
362C22—7881no—
363C22—15764no—
364C22—3941no—
365C22—3941no—
366C22—3942no—
367C22—3941no—
368C22#2026335312yes—
369C22C2×D4×D19711yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—7881no—
375C4—7881no—
376C4—3941no—
377C4—3941no—
378C4—3941no—
379C4#41×D19711yescharacteristic
380C4#2026337111yescharacteristic
381C2—22no—
382C2—81no—
383C2—7882no—
384C2—7881no—
385C2—15761no—
386C2#2716171811yescharacteristic
387C2C2×#2026335311yescharacteristic
388C2#2716188611yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#465×C197C2
socleC22×C197C22×D4
hypercenter#465D197
hypocenterC197#3306
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #27161729C2semidirect
2. #27161720C2semidirect
3. #27161440C2semidirect
4. #27161407C2semidirect
5. #27161399C2semidirect
6. #27161368C2semidirect
7. #27161331C2semidirect
8. #27161296C2semidirect
9. #27161223C2semidirect
10. #27161212C2semidirect
11. #27161188C2semidirect
12. #27161150C2semidirect
13. #27161079C2semidirect
14. C2×#20263248C2semidirect
15. #465×C197C2semidirect
21. C2×#15691341C22semidirect
22. #20263325C22semidirect
30. #20263297C22semidirect
32. D4×Dic197C22semidirect
36. C4⋊C4×D197C22semidirect
38. #20263257C22semidirect
39. #20263257C22semidirect
40. #20263255C22semidirect
41. #20263250C22semidirect
42. #20263250C22semidirect
50. #20263238C22semidirect
51. #20263236C22semidirect
52. C2×SD16×C197C22semidirect
56. #106×C197C22semidirect
57. #103×C197C22semidirect
58. #102×C197C22semidirect
125. D788#41semidirect
194. C197#3306semidirect
203. #465D197semidirect
214. C2×SD16C2×D197semidirect
239. #106C2×D197semidirect
243. #103C2×D197semidirect
250. #102C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617292
#271617202
#271614402
#271614072
#271613992
#271613682
#271613312
#271612962
#271612232
#271612122
#271611882
#271611502
#271610792
C2×#202632482
#465×C1972
#3306197