Finite Group Databasebeta

← #38752897

Subgroups — #38752897

Order 25216.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161702C211yesmaximal, characteristic, complemented
2#27161554C211yesmaximal, characteristic, complemented
3#27161417C211yesmaximal, characteristic, complemented
4#27161404C211yesmaximal, characteristic, complemented
5#27161367C211yesmaximal, characteristic, complemented
6#27161357C211yesmaximal, characteristic, complemented
7#27161296C211yesmaximal, characteristic, complemented
8#27161296C211yesmaximal, characteristic, complemented
9#27161213C211yesmaximal, characteristic
10#27161188C211yesmaximal, characteristic, complemented
11#27161180C211yesmaximal, characteristic, complemented
12SD16×Dic197C211yesmaximal, characteristic, complemented
13#27161159C211yesmaximal, characteristic, complemented
14#27161137C211yesmaximal, characteristic, complemented
15#450×C197C211yesmaximal, characteristic, complemented
16C2×#15691343C2211yescharacteristic
17C2×#15691343C2211yescharacteristic
18#20263325C2211yescharacteristic, complemented
19#20263321—21no—
20#20263315C2211yescharacteristic, complemented
21#20263312—21no—
22#20263308—21no—
23#20263306—21no—
24#20263300C2211yescharacteristic
25Q8×Dic197C2211yescharacteristic, complemented
26D4×Dic197—21no—
27D4×Dic197—21no—
28D4×Dic197C2211yescharacteristic
29D4×Dic197C2211yescharacteristic
30D4×Dic197C2211yescharacteristic
31D4×Dic197C2211yescharacteristic
32#20263288C2211yescharacteristic, complemented
33#20263285C2211yescharacteristic
34#20263285C2211yescharacteristic
35C2×C22⋊Dic197—21no—
36C2×C22⋊Dic197—21no—
37#20263260C2211yescharacteristic, complemented
38#20263259C2211yescharacteristic, complemented
39#20263258C2211yescharacteristic, complemented
40#20263257C2211yescharacteristic, complemented
41#20263255C2211yescharacteristic, complemented
42#20263255C2211yescharacteristic, complemented
43#20263250C2211yescharacteristic, complemented
44#20263250C2211yescharacteristic, complemented
45#20263248C2211yescharacteristic, complemented
46#20263248C2211yescharacteristic, complemented
47#20263247C2211yescharacteristic
48#20263247C2211yescharacteristic
49C22⋊#13096802C2211yescharacteristic
50#20263236C2211yescharacteristic, complemented
51C8×Dic197C2211yescharacteristic, complemented
52C22×D4×C197C2211yescharacteristic
53C2×SD16×C197C2211yescharacteristic
54C2×SD16×C197C2211yescharacteristic
55C22⋊Q8×C197C2211yescharacteristic
56#102×C197C2211yescharacteristic
57#102×C197C2211yescharacteristic
58C22⋊C8×C197C2211yescharacteristic
59C2×Dic394—21no—
60C2×D788C2311yescharacteristic
61C2×C22⋊D197—21no—
62C22×Dic197—21no—
63C22×Dic197—21no—
64C22×Dic197—21no—
65C22×Dic197—21no—
66C2×C4×D197—21no—
67#15691343—22no—
68#15691343—22no—
69C2×#13096802C2311yescharacteristic
70C2×#13096802C2311yescharacteristic
71C4⋊Dic197—21no—
72C4⋊Dic197D411yescharacteristic
73C4⋊Dic197D411yescharacteristic
74C4⋊Dic197C2311yescharacteristic
75C4⋊Dic197C2311yescharacteristic
76C4⋊Dic197C2311yescharacteristic
77#15691337—21no—
78C22⋊Dic197—21no—
79C22⋊Dic197—21no—
80C22⋊Dic197—21no—
81C22⋊Dic197—21no—
82C22⋊Dic197—21no—
83C22⋊Dic197D411yescharacteristic
84C22⋊Dic197D411yescharacteristic
85#15691335—21no—
86#15691335—21no—
87#15691335—21no—
88#15691335—21no—
89C4×Dic197—21no—
90C4×Dic197—21no—
91C4×Dic197C2311yescharacteristic
92C4×Dic197C2311yescharacteristic
93C24×C197—21no—
94C2×Q8×C197C2311yescharacteristic
95C2×D4×C197—22no—
96C2×D4×C197—22no—
97C2×D4×C197—21no—
98C2×D4×C197C2311yescharacteristic
99C2×D4×C197C2311yescharacteristic
100C22×C4×C197C2311yescharacteristic
101SD16×C197—22no—
102SD16×C197—22no—
103C2×C8×C197C2311yescharacteristic
104C2×C8×C197C2311yescharacteristic
105C4⋊C4×C197—21no—
106C4⋊C4×C197C2311yescharacteristic
107C22⋊C4×C197—21no—
108C22×D197—21no—
109Dic394—81no—
110D788—21no—
111D788—21no—
112C22⋊D197—81no—
113C2×Dic197—21no—
114C2×Dic197—21no—
115C2×Dic197—21no—
116C2×Dic197—81no—
117C2×Dic197—81no—
118C2×Dic197—81no—
119C2×Dic197—81no—
120C2×Dic197C2×D411yescharacteristic
121C2×Dic197C2×D411yescharacteristic
122C2×Dic197C2×D411yescharacteristic
123C2×Dic197C2×D411yescharacteristic
124C4×D197—81no—
125#13096802—21no—
126#13096802—21no—
127C23×C197—24no—
128C23×C197—21no—
129C23×C197—21no—
130C23×C197—81no—
131C23×C197—81no—
132C23×C197C2×D411yescharacteristic
133Q8×C197—21no—
134Q8×C197—21no—
135D4×C197—24no—
136D4×C197—21no—
137D4×C197—21no—
138D4×C197#4112yes—
139D4×C197#4112yes—
140C2×C4×C197—21no—
141C2×C4×C197—21no—
142C2×C4×C197—21no—
143C2×C4×C197—21no—
144C2×C4×C197C2×D411yescharacteristic
145C2×C4×C197C2411yescharacteristic
146C8×C197—21no—
147C8×C197—21no—
148C22×C197—21no—
149C22×C197—22no—
150C22×C197—22no—
151C22×C197—21no—
152C22×C197—21no—
153C22×C197—41no—
154C22×C197—41no—
155C22×C197—41no—
156C22×C197—41no—
157C22×C197—84no—
158C22×C197C22×D411yescharacteristic
159C2×D197—41no—
160C2×D197—41no—
161C2×D197—41no—
162C4×C197—21no—
163C4×C197—41no—
164C4×C197—41no—
165C4×C197C2×#4111yescharacteristic
166C4×C197C2×#4111yescharacteristic
167Dic197—21no—
168Dic197—21no—
169Dic197—21no—
170Dic197—21no—
171Dic197—41no—
172Dic197—41no—
173Dic197—41no—
174C2×C197—41no—
175C2×C197—42no—
176C2×C197—42no—
177C2×C197#52511yescharacteristic
178C2×C197#57611yescharacteristic
179C2×C197#57711yescharacteristic
180D197—81no—
181C197#311811yescharacteristic, complemented, Sylow
182#3118—1971nomaximal, Sylow
183#542—1971no—
184#518—1971no—
185#485—1971no—
186#482—1971no—
187#476—1971no—
188#474—1971no—
189#463—1971no—
190#463—1971no—
191#450D19711yescharacteristic, complemented
192#447—1971no—
193#442—1971no—
194#440—1971no—
195C4×SD16—1971no—
196C4×D8—1971no—
197#433—1971no—
198C22×D4C2×D19711yescharacteristic
199C2×SD16C2×D19711yescharacteristic
200C2×SD16C2×D19711yescharacteristic
201C2×D8—1971no—
202C2×D8—1971no—
203C4⋊Q8—1971no—
204#126—3941no—
205#124—3941no—
206#124—1971no—
207#123—3941no—
208C22⋊Q8—3941no—
209C22⋊Q8C2×D19711yescharacteristic
210#121—1971no—
211C4×Q8—1971no—
212C4×D4—3941no—
213C4×D4—3941no—
214C4×D4—1971no—
215C4×D4—1971no—
216C4×D4—1971no—
217C4×D4—1971no—
218C4×D4—1971no—
219#117—1971no—
220#117—1971no—
221C2×C22⋊C4—3941no—
222C2×C22⋊C4—3941no—
223#107—1971no—
224#107—1971no—
225#106—1971no—
226#106—1971no—
227C4⋊C8—1971no—
228C4⋊C8—1971no—
229#103—1971no—
230#103—1971no—
231#102—1971no—
232#102—1971no—
233#102—1971no—
234#102—1971no—
235#102C2×D19711yescharacteristic
236#102C2×D19711yescharacteristic
237C22⋊C8—1971no—
238C22⋊C8C2×D19711yescharacteristic
239#97—1971no—
240C4×C8—1971no—
241C24—21no—
242C2×Q8—3941no—
243C2×Q8C22×D19711yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—3941no—
248C2×D4—1971no—
249C2×D4C22×D19711yescharacteristic
250C2×D4C22×D19711yescharacteristic
251C22×C4—3941no—
252C22×C4—3941no—
253C22×C4—3941no—
254C22×C4—3941no—
255C22×C4—3941no—
256C22×C4C22×D19711yescharacteristic
257SD16—22no—
258SD16—22no—
259D8—3942no—
260D8—3942no—
261C2×C8—1971no—
262C2×C8—1971no—
263C2×C8C22×D19711yescharacteristic
264C2×C8C22×D19711yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—3941no—
267C4⋊C4—3941no—
268C4⋊C4—1971no—
269C4⋊C4—1971no—
270C4⋊C4—1971no—
271C4⋊C4—1971no—
272C4⋊C4—1971no—
273C4⋊C4C22×D19711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—3941no—
276C22⋊C4—3941no—
277C22⋊C4—3941no—
278C22⋊C4—3941no—
279C22⋊C4—3941no—
280C22⋊C4—3941no—
281C22⋊C4—3941no—
282C22⋊C4—3941no—
283C22⋊C4—3941no—
284C22⋊C4—1971no—
285C22⋊C4—1971no—
286C42—3941no—
287C42—3941no—
288C42—1971no—
289C42—1971no—
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—3941no—
296C23D4×D19711yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—15761no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—15761no—
304D4—3941no—
305D4—3941no—
306D4#1569135712yes—
307D4#1569135712yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—15761no—
313C2×C4—15761no—
314C2×C4—15761no—
315C2×C4—15761no—
316C2×C4—15761no—
317C2×C4—3941no—
318C2×C4—3941no—
319C2×C4—3941no—
320C2×C4—1971no—
321C2×C4—1971no—
322C2×C4—1971no—
323C2×C4—1971no—
324C2×C4D4×D19711yescharacteristic
325C2×C4C23×D19711yescharacteristic
326C8—21no—
327C8—21no—
328C8—3941no—
329C8—3941no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—22no—
334C22—22no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—84no—
340C22—7881no—
341C22—7881no—
342C22—7881no—
343C22C2×D4×D19711yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—7881no—
348C4—7881no—
349C4—7881no—
350C4—3941no—
351C4—3941no—
352C4—3941no—
353C4—3941no—
354C4C2×#1569135711yescharacteristic
355C4C2×#1569135711yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—15761no—
360C2#2716158511yescharacteristic
361C2#2716188311yescharacteristic
362C2#2716188511yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#450×C197C2
socleC22×C197C22×D4
prime radical#27161213C2
hypercenter#450D197
hypocenterC197#3118
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #27161702C2semidirect
2. #27161554C2semidirect
3. #27161417C2semidirect
4. #27161404C2semidirect
5. #27161367C2semidirect
6. #27161357C2semidirect
7. #27161296C2semidirect
8. #27161296C2semidirect
10. #27161188C2semidirect
11. #27161180C2semidirect
12. SD16×Dic197C2semidirect
13. #27161159C2semidirect
14. #27161137C2semidirect
15. #450×C197C2semidirect
18. #20263325C22semidirect
20. #20263315C22semidirect
25. Q8×Dic197C22semidirect
32. #20263288C22semidirect
37. #20263260C22semidirect
38. #20263259C22semidirect
39. #20263258C22semidirect
40. #20263257C22semidirect
41. #20263255C22semidirect
42. #20263255C22semidirect
43. #20263250C22semidirect
44. #20263250C22semidirect
45. #20263248C22semidirect
46. #20263248C22semidirect
50. #20263236C22semidirect
51. C8×Dic197C22semidirect
181. C197#3118semidirect
191. #450D197semidirect

Maximal subgroups

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

subgroupindex
#271617022
#271615542
#271614172
#271614042
#271613672
#271613572
#271612962
#271612132
#271611882
#271611802
SD16×Dic1972
#271611592
#271611372
#450×C1972
#3118197