Finite Group Databasebeta

← #38754568

Subgroups — #38754568

Order 25216.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161781C211yesmaximal, characteristic, complemented
2#27161720C211yesmaximal, characteristic, complemented
3#27161435C211yesmaximal, characteristic, complemented
4#27161412C211yesmaximal, characteristic, complemented
5#27161405C211yesmaximal, characteristic, complemented
6#27161378C211yesmaximal, characteristic, complemented
7#27161314C211yesmaximal, characteristic, complemented
8#27161296C211yesmaximal, characteristic, complemented
9#27161223C211yesmaximal, characteristic, complemented
10#27161212C211yesmaximal, characteristic, complemented
11#27161159C211yesmaximal, characteristic, complemented
12C8×C22⋊D197C211yesmaximal, characteristic, complemented
13#27161074C211yesmaximal, characteristic, complemented
14C2×#20263248C211yesmaximal, characteristic, complemented
15#465×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic
17C22×D788C2211yescharacteristic
18C2×#15691343C2211yescharacteristic
19C2×#15691343C2211yescharacteristic
20C2×C8×D197C2211yescharacteristic, complemented
21#20263319—21no—
22#20263317C2211yescharacteristic, complemented
23#20263315C2211yescharacteristic
24#20263311—21no—
25#20263310—21no—
26#20263301—21no—
27#20263301C2211yescharacteristic
28#20263298—21no—
29#20263297C2211yescharacteristic, complemented
30#20263296—21no—
31D4×Dic197—21no—
32D4×Dic197C2211yescharacteristic, complemented
33#20263288C2211yescharacteristic
34C4×C22⋊D197C2211yescharacteristic
35#20263284C2211yescharacteristic, complemented
36C2×C4⋊Dic197C2211yescharacteristic
37C22⋊C4×D197—21no—
38#20263259C2211yescharacteristic, complemented
39#20263259C2211yescharacteristic, 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
51C8×Dic197C2211yescharacteristic, 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—
62C2×D788—22no—
63C2×D788—21no—
64C2×D788—21no—
65C2×D788—21no—
66C2×D788—41no—
67C2×D788C2311yescharacteristic
68C2×D788C2311yescharacteristic
69C2×C22⋊D197—21no—
70C2×C22⋊D197D411yescharacteristic
71C22×Dic197—21no—
72C22×Dic197—21no—
73C2×C4×D197—21no—
74C2×C4×D197—21no—
75C2×C4×D197C2311yescharacteristic
76#15691343—22no—
77#15691343—41no—
78C2×#13096802C2311yescharacteristic
79C2×#13096802C2311yescharacteristic
80C8×D197—41no—
81C4⋊Dic197—21no—
82C4⋊Dic197—21no—
83C4⋊Dic197C2311yescharacteristic
84C4⋊Dic197C2311yescharacteristic
85#15691337—21no—
86#15691337D411yescharacteristic
87C22⋊Dic197—21no—
88C22⋊Dic197—21no—
89C22⋊Dic197D411yescharacteristic
90#15691335—21no—
91#15691335—21no—
92#15691335—21no—
93#15691335—21no—
94#15691335—21no—
95#15691335D411yescharacteristic
96C4×Dic197—21no—
97C4×Dic197C2311yescharacteristic
98C2×Q8×C197C2311yescharacteristic
99C2×D4×C197—21no—
100C2×D4×C197C2311yescharacteristic
101C22×C4×C197C2311yescharacteristic
102SD16×C197—41no—
103C2×C8×C197—41no—
104C2×C8×C197C2311yescharacteristic
105C2×C8×C197C2311yescharacteristic
106C4⋊C4×C197—21no—
107C4⋊C4×C197C2311yescharacteristic
108C4⋊C4×C197C2311yescharacteristic
109C22⋊C4×C197—21no—
110C22⋊C4×C197—21no—
111C22×D197—21no—
112C22×D197—21no—
113C22×D197—42no—
114C22×D197—42no—
115C22×D197—81no—
116C22×D197C2×D411yescharacteristic
117D788—21no—
118D788—21no—
119D788—21no—
120D788—42no—
121D788—42no—
122D788—81no—
123D788#4112yes—
124C22⋊D197—22no—
125C22⋊D197—81no—
126C2×Dic197—21no—
127C2×Dic197—21no—
128C2×Dic197—81no—
129C2×Dic197—81no—
130C2×Dic197C2×D411yescharacteristic
131C2×Dic197C2×D411yescharacteristic
132C2×Dic197C2×D411yescharacteristic
133C4×D197—21no—
134C4×D197—21no—
135C4×D197—42no—
136C4×D197—81no—
137#13096802—21no—
138#13096802—21no—
139C23×C197—21no—
140C23×C197C2×D411yescharacteristic
141Q8×C197—21no—
142Q8×C197—21no—
143D4×C197—21no—
144D4×C197—21no—
145D4×C197—42no—
146C2×C4×C197—21no—
147C2×C4×C197—21no—
148C2×C4×C197—21no—
149C2×C4×C197—21no—
150C2×C4×C197—21no—
151C2×C4×C197C2×D411yescharacteristic
152C2×C4×C197C2411yescharacteristic
153C8×C197—21no—
154C8×C197—21no—
155C22×C197—21no—
156C22×C197—21no—
157C22×C197—41no—
158C22×C197—41no—
159C22×C197—41no—
160C22×C197#13512yes—
161C22×C197C22×D411yescharacteristic
162C2×D197—21no—
163C2×D197—22no—
164C2×D197—21no—
165C2×D197—21no—
166C2×D197—41no—
167C2×D197—41no—
168C2×D197—41no—
169C2×D197—41no—
170C2×D197—41no—
171C2×D197—84no—
172C4×C197—21no—
173C4×C197—41no—
174C4×C197—41no—
175C4×C197—41no—
176C4×C197C2×#4111yescharacteristic
177C4×C197H(5,𝔽2)11yescharacteristic
178Dic197—21no—
179Dic197—21no—
180Dic197—21no—
181Dic197—41no—
182Dic197—41no—
183C2×C197—22no—
184C2×C197—81no—
185C2×C197#54611yescharacteristic
186C2×C197C2×#13511yescharacteristic
187C2×C197#57611yescharacteristic
188D197—41no—
189D197—42no—
190D197—81no—
191C197#329711yescharacteristic, complemented, Sylow
192#3297—1971nomaximal, Sylow
193#553—1971no—
194#546—1971no—
195#488—1971no—
196#484—1971no—
197#482—1971no—
198#478—1971no—
199#466—1971no—
200#465D19711yescharacteristic, complemented
201#463—1971no—
202#449—1971no—
203#447—1971no—
204C4×D8—1971no—
205C8×D4—1971no—
206#416—1971no—
207C2×#102—1971no—
208C2×#41—1971no—
209C22×D4—1971no—
210C2×SD16C2×D19711yescharacteristic, complemented
211C2×D8—1971no—
212C2×D8—1971no—
213C22×C8—1971no—
214C22×C8C2×D19711yescharacteristic
215#126—3941no—
216#125—1971no—
217#124—1971no—
218#123—3941no—
219#123—3941no—
220C22⋊Q8C2×D19711yescharacteristic
221#121—3941no—
222#121—3941no—
223#121—1971no—
224#121—1971no—
225#121C2×D19711yescharacteristic
226#120—3941no—
227C4×D4—3941no—
228C4×D4—1971no—
229C4×D4—1971no—
230C4×D4—1971no—
231#117—1971no—
232C2×C4⋊C4—1971no—
233C2×C22⋊C4—3941no—
234#107—1971no—
235#107—1971no—
236#106C2×D19711yescharacteristic, complemented
237C4⋊C8—1971no—
238#103—1971no—
239#103—1971no—
240#103C2×D19711yescharacteristic, complemented
241#102—1971no—
242#102—1971no—
243#102—1971no—
244#102—1971no—
245#102—1971no—
246#102—1971no—
247#102C2×D19711yescharacteristic, complemented
248C22⋊C8—1971no—
249C22⋊C8—1971no—
250C4×C8—1971no—
251C24—3941no—
252#41—7881no—
253#41—3942no—
254C2×Q8C22×D19711yescharacteristic
255C2×D4—21no—
256C2×D4—7881no—
257C2×D4—3941no—
258C2×D4—3942no—
259C2×D4—3941no—
260C2×D4—3941no—
261C2×D4—3941no—
262C2×D4—1971no—
263C2×D4—1971no—
264C2×D4—1971no—
265C2×D4C22×D19711yescharacteristic
266C22×C4—3941no—
267C22×C4—3941no—
268C22×C4—3941no—
269C22×C4—3941no—
270C22×C4—1971no—
271C22×C4C22×D19711yescharacteristic
272SD16—41no—
273D8—7881no—
274D8—3942no—
275C2×C8—41no—
276C2×C8—7881no—
277C2×C8—1971no—
278C2×C8—1971no—
279C2×C8C22×D19711yescharacteristic
280C2×C8C22×D19711yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—3941no—
283C4⋊C4—3941no—
284C4⋊C4—3941no—
285C4⋊C4—1971no—
286C4⋊C4—1971no—
287C4⋊C4—1971no—
288C4⋊C4C22×D19711yescharacteristic
289C4⋊C4C22×D19711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—3941no—
293C22⋊C4—3941no—
294C22⋊C4—3941no—
295C22⋊C4—3941no—
296C22⋊C4—3941no—
297C22⋊C4—3941no—
298C22⋊C4—3941no—
299C22⋊C4—1971no—
300C22⋊C4—1971no—
301C42—3941no—
302C42—1971no—
303C23—21no—
304C23—7882no—
305C23—7882no—
306C23—15761no—
307C23—3941no—
308C23—3941no—
309C23—1971no—
310C23D4×D19711yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—42no—
316D4—7882no—
317D4—7882no—
318D4—15761no—
319D4—15761no—
320D4—3942no—
321D4—3941no—
322D4—3941no—
323D4—3941no—
324D4—1972no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—7882no—
331C2×C4—15761no—
332C2×C4—15761no—
333C2×C4—15761no—
334C2×C4—3941no—
335C2×C4—3941no—
336C2×C4—3941no—
337C2×C4—3941no—
338C2×C4—1971no—
339C2×C4—1971no—
340C2×C4—1971no—
341C2×C4D4×D19711yescharacteristic
342C2×C4C23×D19711yescharacteristic
343C8—21no—
344C8—21no—
345C8—3941no—
346C8—3941no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—7881no—
353C22—7881no—
354C22—7881no—
355C22—7881no—
356C22—7881no—
357C22—15764no—
358C22—3941no—
359C22—3941no—
360C22—3942no—
361C22—3941no—
362C22#2026334712yes—
363C22C2×D4×D19711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—7881no—
369C4—7881no—
370C4—3941no—
371C4—3941no—
372C4—3941no—
373C4#41×D19711yescharacteristic
374C4#2026337111yescharacteristic
375C2—22no—
376C2—81no—
377C2—7882no—
378C2—7881no—
379C2—15761no—
380C2#2716171811yescharacteristic
381C2C2×#2026334711yescharacteristic
382C2#2716188111yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#465×C197C2
socleC22×C197C22×D4
hypercenter#465D197
hypocenterC197#3297
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. #27161781C2semidirect
2. #27161720C2semidirect
3. #27161435C2semidirect
4. #27161412C2semidirect
5. #27161405C2semidirect
6. #27161378C2semidirect
7. #27161314C2semidirect
8. #27161296C2semidirect
9. #27161223C2semidirect
10. #27161212C2semidirect
11. #27161159C2semidirect
12. C8×C22⋊D197C2semidirect
13. #27161074C2semidirect
14. C2×#20263248C2semidirect
15. #465×C197C2semidirect
20. C2×C8×D197C22semidirect
22. #20263317C22semidirect
29. #20263297C22semidirect
32. D4×Dic197C22semidirect
35. #20263284C22semidirect
38. #20263259C22semidirect
39. #20263259C22semidirect
40. #20263255C22semidirect
41. #20263250C22semidirect
42. #20263250C22semidirect
50. #20263238C22semidirect
51. C8×Dic197C22semidirect
52. C2×SD16×C197C22semidirect
56. #106×C197C22semidirect
57. #103×C197C22semidirect
58. #102×C197C22semidirect
191. C197#3297semidirect
200. #465D197semidirect
210. C2×SD16C2×D197semidirect
236. #106C2×D197semidirect
240. #103C2×D197semidirect
247. #102C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617812
#271617202
#271614352
#271614122
#271614052
#271613782
#271613142
#271612962
#271612232
#271612122
#271611592
C8×C22⋊D1972
#271610742
C2×#202632482
#465×C1972
#3297197