Finite Group Databasebeta

← #38752960

Subgroups — #38752960

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#27161689C211yesmaximal, characteristic, complemented
2#27161553C211yesmaximal, characteristic
3#27161420C211yesmaximal, characteristic, complemented
4#27161382C211yesmaximal, characteristic, complemented
5#27161378C211yesmaximal, characteristic, complemented
6#27161283C211yesmaximal, characteristic, complemented
7#27161283C211yesmaximal, characteristic, complemented
8#27161232C211yesmaximal, characteristic
9#27161229C211yesmaximal, characteristic
10#27161182C211yesmaximal, characteristic, complemented
11#27161178C211yesmaximal, characteristic, complemented
12#27161166C211yesmaximal, characteristic, complemented
13#27161164C211yesmaximal, characteristic, complemented
14#27161137C211yesmaximal, characteristic, complemented
15#485×C197C211yesmaximal, characteristic, complemented
16C22×D788C2211yescharacteristic
17C2×#15691347C2211yescharacteristic
18C2×#15691347C2211yescharacteristic
19C2×#15691345C2211yescharacteristic
20C2×#15691345C2211yescharacteristic
21#20263324C2211yescharacteristic, complemented
22#20263319—21no—
23#20263319—21no—
24#20263317—21no—
25#20263317C2211yescharacteristic, complemented
26#20263316—21no—
27#20263305C2211yescharacteristic
28Q8×Dic197C2211yescharacteristic, complemented
29#20263291C2211yescharacteristic, complemented
30C4×D788—21no—
31C4×D788—21no—
32#20263288C2211yescharacteristic
33#20263288C2211yescharacteristic
34#20263288C2211yescharacteristic
35#20263288C2211yescharacteristic
36#20263285C2211yescharacteristic
37C22⋊C4×D197—21no—
38C22⋊C4×D197—21no—
39#20263259C2211yescharacteristic, complemented
40#20263257C2211yescharacteristic, complemented
41#20263255C2211yescharacteristic, complemented
42#20263255C2211yescharacteristic, complemented
43#20263249C2211yescharacteristic, complemented
44#20263249C2211yescharacteristic, complemented
45#20263248C2211yescharacteristic
46#20263248C2211yescharacteristic
47#20263246C2211yescharacteristic
48#20263246C2211yescharacteristic
49C22⋊#13096802C2211yescharacteristic
50#20263236C2211yescharacteristic, complemented
51C8×Dic197C2211yescharacteristic, complemented
52C22⋊Q8×C197C2211yescharacteristic
53#117×C197C2211yescharacteristic
54#107×C197C2211yescharacteristic, complemented
55#106×C197C2211yescharacteristic, complemented
56#103×C197C2211yescharacteristic, complemented
57#103×C197C2211yescharacteristic, complemented
58C22⋊C8×C197C2211yescharacteristic
59C23×D197—21no—
60C2×Dic394C2311yescharacteristic
61C2×D788—22no—
62C2×D788—22no—
63C2×D788—21no—
64C2×D788C2311yescharacteristic
65C2×D788C2311yescharacteristic
66C2×C4×D197—21no—
67C2×C4×D197—21no—
68C2×C4×D197—21no—
69C2×C4×D197—21no—
70#15691347—22no—
71#15691347—22no—
72#15691345—22no—
73#15691345—22no—
74C2×#13096802C2311yescharacteristic
75C2×#13096802C2311yescharacteristic
76C4⋊Dic197—21no—
77C4⋊Dic197—21no—
78C4⋊Dic197D411yescharacteristic
79C4⋊Dic197D411yescharacteristic
80C4⋊Dic197C2311yescharacteristic
81#15691337—21no—
82#15691337—21no—
83#15691337—21no—
84#15691337—21no—
85#15691337—21no—
86C22⋊Dic197—21no—
87C22⋊Dic197D411yescharacteristic
88C22⋊Dic197D411yescharacteristic
89#15691335—21no—
90#15691335—21no—
91#15691335—21no—
92#15691335—21no—
93C4×Dic197—21no—
94C4×Dic197—21no—
95C4×Dic197C2311yescharacteristic
96C4×Dic197C2311yescharacteristic
97C2×Q8×C197C2311yescharacteristic
98C22×C4×C197C2311yescharacteristic
99C2×C8×C197C2311yescharacteristic
100C2×C8×C197C2311yescharacteristic
101C4⋊C4×C197—21no—
102C4⋊C4×C197C2311yescharacteristic
103C4⋊C4×C197C2311yescharacteristic
104C4⋊C4×C197C2311yescharacteristic
105C22⋊C4×C197—21no—
106C22⋊C4×C197—21no—
107C42×C197—21no—
108C22×D197—21no—
109C22×D197—21no—
110C22×D197—24no—
111C22×D197—81no—
112C22×D197—81no—
113Dic394—21no—
114Dic394—21no—
115D788—24no—
116D788—21no—
117D788—21no—
118D788#4112yes—
119D788#4112yes—
120C2×Dic197—21no—
121C2×Dic197—21no—
122C2×Dic197C2×D411yescharacteristic
123C2×Dic197C2×D411yescharacteristic
124C2×Dic197C2×D411yescharacteristic
125C2×Dic197C2×D411yescharacteristic
126C4×D197—81no—
127C4×D197—81no—
128C4×D197—81no—
129C4×D197—81no—
130#13096802—21no—
131#13096802—21no—
132C23×C197C2×D411yescharacteristic
133Q8×C197—21no—
134Q8×C197—21no—
135C2×C4×C197—21no—
136C2×C4×C197—21no—
137C2×C4×C197—21no—
138C2×C4×C197—21no—
139C2×C4×C197—21no—
140C2×C4×C197—21no—
141C2×C4×C197C2×D411yescharacteristic
142C2×C4×C197C2411yescharacteristic
143C8×C197—21no—
144C8×C197—21no—
145C22×C197—21no—
146C22×C197—21no—
147C22×C197—21no—
148C22×C197C22×D411yescharacteristic
149C2×D197—22no—
150C2×D197—22no—
151C2×D197—41no—
152C2×D197—41no—
153C2×D197—41no—
154C2×D197—41no—
155C2×D197—84no—
156C4×C197—21no—
157C4×C197—41no—
158C4×C197—41no—
159C4×C197—41no—
160C4×C197—41no—
161C4×C197C2×#4111yescharacteristic
162C4×C197C2×#4111yescharacteristic
163Dic197—21no—
164Dic197—21no—
165Dic197—21no—
166Dic197—21no—
167Dic197—41no—
168Dic197—41no—
169C2×C197—41no—
170C2×C197#52511yescharacteristic
171C2×C197#57711yescharacteristic
172C2×C197#57711yescharacteristic
173D197—42no—
174D197—42no—
175C197#312311yescharacteristic, complemented, Sylow
176#3123—1971nomaximal, Sylow
177#541—1971no—
178#518—1971no—
179#485—1971no—
180#485D19711yescharacteristic, complemented
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×SD16—1971no—
191#433—1971no—
192C22×D4—1971no—
193C2×SD16—1971no—
194C2×SD16—1971no—
195C2×SD16—1971no—
196C2×SD16—1971no—
197C4⋊Q8—1971no—
198#126—3941no—
199#126—3941no—
200#125—3941no—
201#125—3941no—
202#125—1971no—
203C22⋊Q8—1971no—
204C22⋊Q8C2×D19711yescharacteristic
205C4×Q8—1971no—
206C4×Q8—1971no—
207C4×D4—3941no—
208C4×D4—3941no—
209C4×D4—1971no—
210C4×D4—1971no—
211C4×D4—1971no—
212C4×D4—1971no—
213#117—1971no—
214#117C2×D19711yescharacteristic
215C2×C22⋊C4—3941no—
216C2×C22⋊C4—3941no—
217#107—1971no—
218#107C2×D19711yescharacteristic, complemented
219#106—1971no—
220#106C2×D19711yescharacteristic, complemented
221C4⋊C8—1971no—
222C4⋊C8—1971no—
223#103—1971no—
224#103—1971no—
225#103C2×D19711yescharacteristic, complemented
226#103C2×D19711yescharacteristic, complemented
227#102—1971no—
228#102—1971no—
229#102—1971no—
230#102—1971no—
231C22⋊C8—1971no—
232C22⋊C8C2×D19711yescharacteristic
233#97—1971no—
234C4×C8—1971no—
235C24—3941no—
236C2×Q8—1971no—
237C2×Q8C22×D19711yescharacteristic
238C2×D4—3942no—
239C2×D4—3942no—
240C2×D4—3941no—
241C2×D4—1971no—
242C2×D4—1971no—
243C22×C4—3941no—
244C22×C4—3941no—
245C22×C4—3941no—
246C22×C4—3941no—
247C22×C4C22×D19711yescharacteristic
248SD16—3942no—
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⋊C4C22×D19711yescharacteristic
268C4⋊C4C22×D19711yescharacteristic
269C4⋊C4C22×D19711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—21no—
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—
283C42—1971no—
284C23—15761no—
285C23—15761no—
286C23—3944no—
287C23—3941no—
288C23—3941no—
289C23D4×D19711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—3941no—
293Q8—3941no—
294D4—3944no—
295D4—3941no—
296D4—3941no—
297D4—1972no—
298D4—1972no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—15761no—
306C2×C4—15761no—
307C2×C4—15761no—
308C2×C4—15761no—
309C2×C4—3941no—
310C2×C4—3941no—
311C2×C4—1971no—
312C2×C4—1971no—
313C2×C4—1971no—
314C2×C4—1971no—
315C2×C4D4×D19711yescharacteristic
316C2×C4C23×D19711yescharacteristic
317C8—21no—
318C8—21no—
319C8—3941no—
320C8—3941no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—7881no—
325C22—7881no—
326C22—7881no—
327C22—7881no—
328C22—15764no—
329C22—3942no—
330C22—3942no—
331C22C2×D4×D19711yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—7881no—
338C4—7881no—
339C4—3941no—
340C4—3941no—
341C4—3941no—
342C4—3941no—
343C4#41×D19711yescharacteristic
344C4#41×D19711yescharacteristic
345C2—41no—
346C2—7882no—
347C2—7882no—
348C2#2716158711yescharacteristic
349C2#2716188611yescharacteristic
350C2#2716188811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#485×C197C2
socleC22×C197C22×D4
prime radicalC22×D788C22
hypercenter#485D197
hypocenterC197#3123
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #27161689C2semidirect
3. #27161420C2semidirect
4. #27161382C2semidirect
5. #27161378C2semidirect
6. #27161283C2semidirect
7. #27161283C2semidirect
10. #27161182C2semidirect
11. #27161178C2semidirect
12. #27161166C2semidirect
13. #27161164C2semidirect
14. #27161137C2semidirect
15. #485×C197C2semidirect
21. #20263324C22semidirect
25. #20263317C22semidirect
28. Q8×Dic197C22semidirect
29. #20263291C22semidirect
39. #20263259C22semidirect
40. #20263257C22semidirect
41. #20263255C22semidirect
42. #20263255C22semidirect
43. #20263249C22semidirect
44. #20263249C22semidirect
50. #20263236C22semidirect
51. C8×Dic197C22semidirect
54. #107×C197C22semidirect
55. #106×C197C22semidirect
56. #103×C197C22semidirect
57. #103×C197C22semidirect
175. C197#3123semidirect
180. #485D197semidirect
218. #107C2×D197semidirect
220. #106C2×D197semidirect
225. #103C2×D197semidirect
226. #103C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271616892
#271615532
#271614202
#271613822
#271613782
#271612832
#271612322
#271612292
#271611822
#271611782
#271611662
#271611642
#271611372
#485×C1972
#3123197