Finite Group Databasebeta

← #38752961

Subgroups — #38752961

Order 25216.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161687C211yesmaximal, characteristic, complemented
2#27161551C211yesmaximal, characteristic
3#27161423C211yesmaximal, characteristic, complemented
4#27161418C211yesmaximal, characteristic, complemented
5#27161381C211yesmaximal, characteristic, complemented
6#27161378C211yesmaximal, characteristic, complemented
7#27161283C211yesmaximal, characteristic, complemented
8#27161282C211yesmaximal, characteristic, complemented
9#27161231C211yesmaximal, characteristic
10#27161228C211yesmaximal, characteristic
11#27161179C211yesmaximal, characteristic, complemented
12#27161177C211yesmaximal, characteristic, complemented
13C4×#15691347C211yesmaximal, characteristic, complemented
14#27161138C211yesmaximal, characteristic, complemented
15C4×SD16×C197C211yesmaximal, characteristic, complemented
16C2×D4×D197C2211yescharacteristic
17C2×#15691347C2211yescharacteristic
18C2×#15691346C2211yescharacteristic
19C2×#15691345C2211yescharacteristic
20#20263324C2211yescharacteristic, complemented
21#20263321—21no—
22#20263320—21no—
23#20263318—21no—
24#20263317C2211yescharacteristic, complemented
25#20263316—21no—
26#20263306C2211yescharacteristic
27#20263302C2211yescharacteristic
28C4×Dic394C2211yescharacteristic, complemented
29C4×D788C2211yescharacteristic
30D4×Dic197C2211yescharacteristic
31#20263288C2211yescharacteristic
32C4×C22⋊D197—21no—
33#20263286—21no—
34#20263284C2211yescharacteristic
35#20263283C2211yescharacteristic
36C2×#15691335—21no—
37C22⋊C4×D197—21no—
38#20263259C2211yescharacteristic, complemented
39#20263259C2211yescharacteristic, complemented
40#20263258C2211yescharacteristic, complemented
41#20263255C2211yescharacteristic, complemented
42#20263255C2211yescharacteristic, complemented
43#20263251C2211yescharacteristic, complemented
44#20263250C2211yescharacteristic, complemented
45#20263249C2211yescharacteristic, complemented
46#20263248C2211yescharacteristic
47#20263247C2211yescharacteristic
48#20263246C2211yescharacteristic
49#20263238C2211yescharacteristic
50#20263238C2211yescharacteristic
51#20263237C2211yescharacteristic, complemented
52C2×SD16×C197C2211yescharacteristic
53C4×Q8×C197C2211yescharacteristic, complemented
54C4×D4×C197C2211yescharacteristic
55#106×C197C2211yescharacteristic, complemented
56#103×C197C2211yescharacteristic, complemented
57#102×C197C2211yescharacteristic
58C4×C8×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60C2×Dic394C2311yescharacteristic
61C2×D788C2311yescharacteristic
62C2×C22⋊D197—21no—
63D4×D197—22no—
64D4×D197—22no—
65C22×Dic197—21no—
66C2×C4×D197—21no—
67C2×C4×D197—21no—
68C2×C4×D197C2311yescharacteristic
69#15691347—22no—
70#15691346—22no—
71#15691345—22no—
72C2×#13096802C2311yescharacteristic
73C2×#13096802C2311yescharacteristic
74C4⋊Dic197—21no—
75C4⋊Dic197—21no—
76C4⋊Dic197C2311yescharacteristic
77C4⋊Dic197C2311yescharacteristic
78#15691337—21no—
79#15691337—21no—
80#15691337—21no—
81#15691337—21no—
82#15691337—21no—
83#15691337D411yescharacteristic
84#15691337D411yescharacteristic
85C22⋊Dic197—21no—
86#15691335—21no—
87#15691335—21no—
88#15691335—21no—
89#15691335—21no—
90#15691335—21no—
91#15691335D411yescharacteristic
92#15691335D411yescharacteristic
93C4×Dic197—21no—
94C4×Dic197—21no—
95C4×Dic197C2311yescharacteristic
96C2×Q8×C197C2311yescharacteristic
97C2×D4×C197C2311yescharacteristic
98C22×C4×C197—21no—
99SD16×C197—22no—
100C2×C8×C197C2311yescharacteristic
101C2×C8×C197C2311yescharacteristic
102C4⋊C4×C197—21no—
103C4⋊C4×C197C2311yescharacteristic
104C4⋊C4×C197C2311yescharacteristic
105C22⋊C4×C197—21no—
106C42×C197—21no—
107C42×C197C2311yescharacteristic
108C22×D197—21no—
109C22×D197—24no—
110C22×D197—81no—
111C22×D197—81no—
112C22×D197C2×D411yescharacteristic
113Dic394—21no—
114Dic394—21no—
115D788—21no—
116D788#4112yes—
117C22⋊D197—24no—
118C2×Dic197—21no—
119C2×Dic197—21no—
120C2×Dic197—21no—
121C2×Dic197—81no—
122C2×Dic197C2×D411yescharacteristic
123C2×Dic197C2×D411yescharacteristic
124C2×Dic197C2×D411yescharacteristic
125C4×D197—21no—
126C4×D197—21no—
127C4×D197—81no—
128C4×D197—81no—
129#13096802—21no—
130#13096802—21no—
131C23×C197—21no—
132Q8×C197—21no—
133Q8×C197—21no—
134D4×C197—21no—
135D4×C197#4112yes—
136C2×C4×C197—21no—
137C2×C4×C197—21no—
138C2×C4×C197—21no—
139C2×C4×C197—81no—
140C2×C4×C197C2×D411yescharacteristic
141C2×C4×C197C2×D411yescharacteristic
142C2×C4×C197C2411yescharacteristic
143C8×C197—21no—
144C8×C197—21no—
145C22×C197—22no—
146C22×C197—41no—
147C22×C197—41no—
148C22×C197C22×D411yescharacteristic
149C2×D197—21no—
150C2×D197—22no—
151C2×D197—21no—
152C2×D197—21no—
153C2×D197—41no—
154C2×D197—41no—
155C2×D197—84no—
156C4×C197—21no—
157C4×C197—21no—
158C4×C197—41no—
159C4×C197—41no—
160C4×C197—41no—
161C4×C197C2×#4111yescharacteristic
162C4×C197C2×#4111yescharacteristic
163Dic197—21no—
164Dic197—21no—
165Dic197—21no—
166Dic197—41no—
167Dic197—41no—
168Dic197—41no—
169C2×C197—42no—
170C2×C197#52511yescharacteristic
171C2×C197#57711yescharacteristic
172C2×C197#57711yescharacteristic
173D197—41no—
174D197—42no—
175C197#312311yescharacteristic, complemented, Sylow
176#3123—1971nomaximal, Sylow
177#541—1971no—
178#518—1971no—
179#485—1971no—
180#485—1971no—
181#478—1971no—
182#478—1971no—
183#461—1971no—
184#461—1971no—
185#450—1971no—
186#450—1971no—
187#440—1971no—
188#440—1971no—
189C4×SD16—1971no—
190C4×SD16D19711yescharacteristic, complemented
191#433—1971no—
192C22×D4—1971no—
193C2×SD16—1971no—
194C2×SD16—1971no—
195C2×SD16—1971no—
196C2×SD16C2×D19711yescharacteristic
197C4⋊Q8—1971no—
198#126—3941no—
199#126—3941no—
200#125—3941no—
201#125—3941no—
202#125—1971no—
203C22⋊Q8—1971no—
204C22⋊Q8—1971no—
205C4×Q8—1971no—
206C4×Q8C2×D19711yescharacteristic, complemented
207C4×D4—3941no—
208C4×D4—3941no—
209C4×D4—1971no—
210C4×D4—1971no—
211C4×D4—1971no—
212C4×D4C2×D19711yescharacteristic
213#117—1971no—
214#117—1971no—
215C2×C22⋊C4—3941no—
216C2×C22⋊C4—3941no—
217#107—1971no—
218#107—1971no—
219#106—1971no—
220#106C2×D19711yescharacteristic, complemented
221C4⋊C8—1971no—
222C4⋊C8—1971no—
223#103—1971no—
224#103—1971no—
225#103—1971no—
226#103C2×D19711yescharacteristic, complemented
227#102—1971no—
228#102—1971no—
229#102—1971no—
230#102C2×D19711yescharacteristic
231C22⋊C8—1971no—
232C22⋊C8—1971no—
233#97—1971no—
234C4×C8C2×D19711yescharacteristic, complemented
235C24—3941no—
236C2×Q8—1971no—
237C2×Q8C22×D19711yescharacteristic
238C2×D4—3942no—
239C2×D4—3942no—
240C2×D4—3941no—
241C2×D4—1971no—
242C2×D4C22×D19711yescharacteristic
243C22×C4—21no—
244C22×C4—3941no—
245C22×C4—3941no—
246C22×C4—3941no—
247C22×C4—1971no—
248SD16—22no—
249SD16—3942no—
250SD16—3942no—
251SD16—3942no—
252C2×C8—1971no—
253C2×C8—1971no—
254C2×C8C22×D19711yescharacteristic
255C2×C8C22×D19711yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—3941no—
258C4⋊C4—3941no—
259C4⋊C4—3941no—
260C4⋊C4—3941no—
261C4⋊C4—3941no—
262C4⋊C4—3941no—
263C4⋊C4—3941no—
264C4⋊C4—1971no—
265C4⋊C4—1971no—
266C4⋊C4—1971no—
267C4⋊C4—1971no—
268C4⋊C4C22×D19711yescharacteristic
269C4⋊C4C22×D19711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—3941no—
272C22⋊C4—3941no—
273C22⋊C4—3941no—
274C22⋊C4—3941no—
275C22⋊C4—3941no—
276C22⋊C4—3941no—
277C22⋊C4—1971no—
278C22⋊C4—1971no—
279C42—21no—
280C42—3941no—
281C42—3941no—
282C42—1971no—
283C42C22×D19711yescharacteristic
284C23—21no—
285C23—15761no—
286C23—15761no—
287C23—3944no—
288C23—3941no—
289C23—1971no—
290Q8—21no—
291Q8—21no—
292Q8—3941no—
293Q8—3941no—
294D4—21no—
295D4—3944no—
296D4—3941no—
297D4—1972no—
298D4#1569135812yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—15761no—
304C2×C4—15761no—
305C2×C4—15761no—
306C2×C4—3941no—
307C2×C4—3941no—
308C2×C4—3941no—
309C2×C4—3941no—
310C2×C4—3941no—
311C2×C4—1971no—
312C2×C4—1971no—
313C2×C4—1971no—
314C2×C4D4×D19711yescharacteristic
315C2×C4D4×D19711yescharacteristic
316C2×C4C23×D19711yescharacteristic
317C8—21no—
318C8—21no—
319C8—3941no—
320C8—3941no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—7881no—
325C22—7881no—
326C22—15764no—
327C22—3941no—
328C22—3941no—
329C22—3941no—
330C22—3942no—
331C22C2×D4×D19711yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—7881no—
338C4—7881no—
339C4—7881no—
340C4—3941no—
341C4—3941no—
342C4—3941no—
343C4#41×D19711yescharacteristic
344C4C2×#1569135811yescharacteristic
345C2—42no—
346C2—7882no—
347C2—7881no—
348C2#2716158811yescharacteristic
349C2#2716188511yescharacteristic
350C2#2716188611yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC4×SD16×C197C2
socleC22×C197C22×D4
prime radicalC2×D4×D197C22
hypercenterC4×SD16D197
hypocenterC197#3123
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #27161687C2semidirect
3. #27161423C2semidirect
4. #27161418C2semidirect
5. #27161381C2semidirect
6. #27161378C2semidirect
7. #27161283C2semidirect
8. #27161282C2semidirect
11. #27161179C2semidirect
12. #27161177C2semidirect
13. C4×#15691347C2semidirect
14. #27161138C2semidirect
15. C4×SD16×C197C2semidirect
20. #20263324C22semidirect
24. #20263317C22semidirect
28. C4×Dic394C22semidirect
38. #20263259C22semidirect
39. #20263259C22semidirect
40. #20263258C22semidirect
41. #20263255C22semidirect
42. #20263255C22semidirect
43. #20263251C22semidirect
44. #20263250C22semidirect
45. #20263249C22semidirect
51. #20263237C22semidirect
53. C4×Q8×C197C22semidirect
55. #106×C197C22semidirect
56. #103×C197C22semidirect
58. C4×C8×C197C22semidirect
175. C197#3123semidirect
190. C4×SD16D197semidirect
206. C4×Q8C2×D197semidirect
220. #106C2×D197semidirect
226. #103C2×D197semidirect
234. C4×C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271616872
#271615512
#271614232
#271614182
#271613812
#271613782
#271612832
#271612822
#271612312
#271612282
#271611792
#271611772
C4×#156913472
#271611382
C4×SD16×C1972
#3123197