Finite Group Databasebeta

← #38752644

Subgroups — #38752644

Order 25216.

414 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161592C211yesmaximal, characteristic
2#117×D197C211yesmaximal, characteristic, complemented
3#27161419C211yesmaximal, characteristic, complemented
4#27161418C211yesmaximal, characteristic, complemented
5#27161417C211yesmaximal, characteristic, complemented
6#27161406C211yesmaximal, characteristic, complemented
7#27161405C211yesmaximal, characteristic, complemented
8#27161404C211yesmaximal, characteristic, complemented
9#27161403C211yesmaximal, characteristic, complemented
10#27161120C211yesmaximal, characteristic, complemented
11#27161118C211yesmaximal, characteristic, complemented
12#27161079C211yesmaximal, characteristic, complemented
13#27161079C211yesmaximal, characteristic, complemented
14#27161041C211yesmaximal, characteristic, complemented
15#485×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic
17C22×C4×D197C2211yescharacteristic
18C2×#15691341C2211yescharacteristic
19C2×#15691341C2211yescharacteristic
20#20263311—21no—
21#20263302C2211yescharacteristic
22#20263301C2211yescharacteristic
23#20263300C2211yescharacteristic
24#20263296—21no—
25#20263288—21no—
26#20263286—21no—
27#20263285C2211yescharacteristic
28#20263284C2211yescharacteristic
29#20263284C2211yescharacteristic
30#20263284C2211yescharacteristic
31#20263283—21no—
32#20263282—21no—
33C4⋊C4×D197C2211yescharacteristic
34C4⋊C4×D197C2211yescharacteristic
35C22⋊C4×D197—21no—
36C42×D197—21no—
37#20263260C2211yescharacteristic, complemented
38#20263259C2211yescharacteristic, complemented
39#20263259C2211yescharacteristic, complemented
40#20263258C2211yescharacteristic, complemented
41#20263257C2211yescharacteristic, complemented
42#20263257C2211yescharacteristic, complemented
43#20263250C2211yescharacteristic, complemented
44#20263250C2211yescharacteristic, complemented
45#20263248C2211yescharacteristic, complemented
46#20263248C2211yescharacteristic, complemented
47#20263246C2211yescharacteristic, complemented
48#20263246C2211yescharacteristic, complemented
49C22⋊#13096802C2211yescharacteristic
50#20263238C2211yescharacteristic
51#20263238C2211yescharacteristic
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×D197D411yescharacteristic
60#15691359—22no—
61#15691359—22no—
62C2×D788—21no—
63C2×D788—21no—
64C2×D788C2311yescharacteristic
65C2×C22⋊D197—21no—
66C22×Dic197D411yescharacteristic
67C2×C4×D197—22no—
68C2×C4×D197—22no—
69C2×C4×D197—21no—
70C2×C4×D197—21no—
71C2×C4×D197—21no—
72C2×C4×D197D411yescharacteristic
73C2×C4×D197D411yescharacteristic
74C2×C4×D197C2311yescharacteristic
75C2×C4×D197C2311yescharacteristic
76#15691341—22no—
77#15691341—22no—
78C2×#13096802C2311yescharacteristic
79C2×#13096802C2311yescharacteristic
80C4⋊Dic197C2311yescharacteristic
81C4⋊Dic197C2311yescharacteristic
82C4⋊Dic197C2311yescharacteristic
83#15691337—21no—
84#15691337—21no—
85#15691337—21no—
86C22⋊Dic197—21no—
87#15691335—21no—
88#15691335—21no—
89#15691335—21no—
90#15691335—21no—
91#15691335—21no—
92#15691335—21no—
93C4×Dic197—21no—
94C4×Dic197—21no—
95C4×Dic197—21no—
96C4×Dic197—21no—
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—22no—
109C22×D197—21no—
110C22×D197—41no—
111C22×D197—41no—
112C22×D197C2×D411yescharacteristic
113C22×D197C2×D411yescharacteristic
114D788—21no—
115D788—21no—
116D788—42no—
117D788—42no—
118C22⋊D197—42no—
119C2×Dic197—22no—
120C2×Dic197—21no—
121C2×Dic197—21no—
122C2×Dic197—21no—
123C2×Dic197C2×D411yescharacteristic
124C2×Dic197C2×D411yescharacteristic
125C4×D197—22no—
126C4×D197—21no—
127C4×D197—21no—
128C4×D197—22no—
129C4×D197—42no—
130C4×D197—42no—
131C4×D197—42no—
132C4×D197#4112yes—
133C4×D197#4112yes—
134#13096802—21no—
135#13096802—21no—
136C23×C197C2×D411yescharacteristic
137Q8×C197—21no—
138Q8×C197—21no—
139C2×C4×C197—21no—
140C2×C4×C197—21no—
141C2×C4×C197—21no—
142C2×C4×C197—21no—
143C2×C4×C197—21no—
144C2×C4×C197—21no—
145C2×C4×C197C2×D411yescharacteristic
146C2×C4×C197C2411yescharacteristic
147C8×C197—21no—
148C8×C197—21no—
149C22×C197—21no—
150C22×C197—21no—
151C22×C197—21no—
152C22×C197C22×D411yescharacteristic
153C2×D197—21no—
154C2×D197—21no—
155C2×D197—21no—
156C2×D197—21no—
157C2×D197—21no—
158C2×D197—44no—
159C2×D197—41no—
160C2×D197—41no—
161C2×D197—41no—
162C2×D197#12312yes—
163C4×C197—21no—
164C4×C197—41no—
165C4×C197—41no—
166C4×C197—41no—
167C4×C197—41no—
168C4×C197C2×#4111yescharacteristic
169C4×C197C2×#4111yescharacteristic
170Dic197—21no—
171Dic197—41no—
172Dic197—41no—
173Dic197—41no—
174Dic197#12312yes—
175C2×C197—41no—
176C2×C197C2×#12311yescharacteristic
177C2×C197#57511yescharacteristic
178C2×C197#57511yescharacteristic
179D197—22no—
180D197—41no—
181D197—81no—
182C197#309311yescharacteristic, complemented, Sylow
183#3093—1971nomaximal, Sylow
184#525—1971no—
185C2×#117—1971no—
186#485—1971no—
187#485—1971no—
188#485—1971no—
189#485D19711yescharacteristic, complemented
190#482—1971no—
191#482—1971no—
192#482—1971no—
193#482—1971no—
194#428—1971no—
195#428—1971no—
196#417—1971no—
197#417—1971no—
198#407—1971no—
199C2×#41—1971no—
200C23×C4—1971no—
201C2×M16—1971no—
202C2×M16—1971no—
203#123—3941no—
204C22⋊Q8—1971no—
205C22⋊Q8C2×D19711yescharacteristic
206#121—1971no—
207#121—1971no—
208#120—3941no—
209C4×D4—3941no—
210C4×D4—3941no—
211#117—3941no—
212#117—3941no—
213#117—1971no—
214#117—1971no—
215#117—1971no—
216#117—1971no—
217#117C2×D19711yescharacteristic
218C2×C4⋊C4—1971no—
219C2×C4⋊C4—1971no—
220C2×C22⋊C4—3941no—
221C2×C42—3941no—
222#107—1971no—
223#107—1971no—
224#107—1971no—
225#107C2×D19711yescharacteristic, complemented
226#106—1971no—
227#106—1971no—
228#106—1971no—
229#106C2×D19711yescharacteristic, complemented
230#103—1971no—
231#103—1971no—
232#103C2×D19711yescharacteristic, complemented
233#103C2×D19711yescharacteristic, complemented
234#102—1971no—
235#102—1971no—
236#102—1971no—
237#102—1971no—
238C22⋊C8—1971no—
239C22⋊C8—1971no—
240C22⋊C8—1971no—
241C22⋊C8C2×D19711yescharacteristic
242C24—1971no—
243#41—3942no—
244#41—3942no—
245C2×Q8C22×D19711yescharacteristic
246C2×D4—3941no—
247C2×D4—3941no—
248C2×D4—3941no—
249C2×D4—1971no—
250C22×C4—3941no—
251C22×C4—3941no—
252C22×C4—3941no—
253C22×C4—3942no—
254C22×C4—3942no—
255C22×C4—1971no—
256C22×C4—1971no—
257C22×C4—1971no—
258C22×C4—1971no—
259C22×C4—1971no—
260C22×C4C22×D19711yescharacteristic
261M16—3942no—
262M16—3942no—
263C2×C8—1971no—
264C2×C8—1971no—
265C2×C8C22×D19711yescharacteristic
266C2×C8C22×D19711yescharacteristic
267C4⋊C4—21no—
268C4⋊C4—3941no—
269C4⋊C4—3941no—
270C4⋊C4—3941no—
271C4⋊C4—1971no—
272C4⋊C4—1971no—
273C4⋊C4—1971no—
274C4⋊C4C22×D19711yescharacteristic
275C4⋊C4C22×D19711yescharacteristic
276C4⋊C4C22×D19711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—3941no—
280C22⋊C4—3941no—
281C22⋊C4—3941no—
282C22⋊C4—3941no—
283C22⋊C4—3941no—
284C22⋊C4—3941no—
285C22⋊C4—3941no—
286C42—21no—
287C42—3941no—
288C42—3941no—
289C42—3941no—
290C42—3941no—
291C23—7881no—
292C23—7881no—
293C23—3942no—
294C23—3941no—
295C23—1971no—
296C23—1971no—
297C23D4×D19711yescharacteristic
298Q8—21no—
299Q8—21no—
300D4—7882no—
301D4—7882no—
302D4—7882no—
303D4—3941no—
304D4—3941no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—7882no—
312C2×C4—7882no—
313C2×C4—7882no—
314C2×C4—3942no—
315C2×C4—3942no—
316C2×C4—3941no—
317C2×C4—3941no—
318C2×C4—3941no—
319C2×C4—3942no—
320C2×C4—3941no—
321C2×C4—3941no—
322C2×C4—1971no—
323C2×C4—1971no—
324C2×C4—1972no—
325C2×C4—1972no—
326C2×C4D4×D19711yescharacteristic
327C2×C4C23×D19711yescharacteristic
328C8—21no—
329C8—21no—
330C8—3941no—
331C8—3941no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—7884no—
336C22—7881no—
337C22—7881no—
338C22—7881no—
339C22—3941no—
340C22—3941no—
341C22—3941no—
342C22—3941no—
343C22—3941no—
344C22—1972no—
345C22C2×D4×D19711yescharacteristic
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—7881no—
352C4—7881no—
353C4—7881no—
354C4—3941no—
355C4—1972no—
356C4#41×D19711yescharacteristic
357C4#41×D19711yescharacteristic
358C2—41no—
359C2—7881no—
360C2—15761no—
361C2—3942no—
362C2#123×D19711yescharacteristic
363C2#2716187611yescharacteristic
364C2#2716187811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#485×C197C2
socleC22×C197C22×D4
prime radical#27161592C2
hypercenter#485D197
hypocenterC197#3093
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
2. #117×D197C2semidirect
3. #27161419C2semidirect
4. #27161418C2semidirect
5. #27161417C2semidirect
6. #27161406C2semidirect
7. #27161405C2semidirect
8. #27161404C2semidirect
9. #27161403C2semidirect
10. #27161120C2semidirect
11. #27161118C2semidirect
12. #27161079C2semidirect
13. #27161079C2semidirect
14. #27161041C2semidirect
15. #485×C197C2semidirect
37. #20263260C22semidirect
38. #20263259C22semidirect
39. #20263259C22semidirect
40. #20263258C22semidirect
41. #20263257C22semidirect
42. #20263257C22semidirect
43. #20263250C22semidirect
44. #20263250C22semidirect
45. #20263248C22semidirect
46. #20263248C22semidirect
47. #20263246C22semidirect
48. #20263246C22semidirect
54. #107×C197C22semidirect
55. #106×C197C22semidirect
56. #103×C197C22semidirect
57. #103×C197C22semidirect
182. C197#3093semidirect
189. #485D197semidirect
225. #107C2×D197semidirect
229. #106C2×D197semidirect
232. #103C2×D197semidirect
233. #103C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271615922
#117×D1972
#271614192
#271614182
#271614172
#271614062
#271614052
#271614042
#271614032
#271611202
#271611182
#271610792
#271610412
#485×C1972
#3093197