Finite Group Databasebeta

← #38751584

Subgroups — #38751584

Order 25216.

484 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
1C2×#20263347C211yesmaximal, characteristic
2#27161561C211yesmaximal, characteristic, complemented
3#27161558C211yesmaximal, characteristic, complemented
4#27161181C211yesmaximal, characteristic, complemented
5#27161180C211yesmaximal, characteristic, complemented
6#27161178C211yesmaximal, characteristic, complemented
7C4×#15691348C211yesmaximal, characteristic, complemented
8#27161171C211yesmaximal, characteristic, complemented
9C4×#15691343C211yesmaximal, characteristic, complemented
10#27161159C211yesmaximal, characteristic, complemented
11#107×D197C211yesmaximal, characteristic, complemented
12#27161079C211yesmaximal, characteristic, complemented
13#27161077C211yesmaximal, characteristic, complemented
14#27161036C211yesmaximal, characteristic, complemented
15#440×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic
17C2×#15691357C2211yescharacteristic
18#20263347C418yescomplemented
19C2×#15691348C2211yescharacteristic
20C2×#15691347C2211yescharacteristic
21C2×#15691343C2211yescharacteristic
22C2×#15691341C2211yescharacteristic
23C2×C8×D197C2211yescharacteristic
24Q8×Dic197C2211yescharacteristic
25C4×Dic394C2211yescharacteristic
26#20263291C2211yescharacteristic
27C4×D788—21no—
28C4×D788C2211yescharacteristic
29D4×Dic197C2211yescharacteristic
30#20263288—21no—
31#20263288C2211yescharacteristic
32C4×C22⋊D197—21no—
33#20263286—21no—
34#20263283—21no—
35#20263283C2211yescharacteristic
36#20263282—21no—
37C2×#15691337—21no—
38C4⋊C4×D197—21no—
39C4⋊C4×D197C2211yescharacteristic
40C4⋊C4×D197C2211yescharacteristic
41#20263260C2211yescharacteristic
42#20263259C2211yescharacteristic
43#20263259C2211yescharacteristic
44#20263251C2211yescharacteristic
45#20263250C2211yescharacteristic
46#20263249C2211yescharacteristic
47#20263248C2211yescharacteristic
48#20263247C2211yescharacteristic
49#20263246C2211yescharacteristic
50#20263236C2211yescharacteristic
51C4×#13096802C2211yescharacteristic
52C8×Dic197C2211yescharacteristic
53C2×SD16×C197C2211yescharacteristic
54C4×Q8×C197C2211yescharacteristic
55C4×D4×C197C2211yescharacteristic
56#107×C197C2211yescharacteristic
57#103×C197C2211yescharacteristic
58#102×C197C2211yescharacteristic
59#97×C197C2211yescharacteristic
60#15691359—22no—
61#15691359C2×C414yescomplemented
62#15691357—22no—
63#15691357C2×C414yescomplemented
64C2×Dic394C2311yescharacteristic
65C2×D788—21no—
66C2×D788C2311yescharacteristic
67C2×C22⋊D197—21no—
68C22×Dic197—21no—
69C22×Dic197—21no—
70C2×C4×D197—21no—
71C2×C4×D197—21no—
72C2×C4×D197—21no—
73C2×C4×D197—21no—
74C2×C4×D197—21no—
75C2×C4×D197C2311yescharacteristic
76#15691348C2×C414yescomplemented
77#15691347C2×C414yescomplemented
78#15691343C2×C414yescomplemented
79#15691341—22no—
80C2×#13096802C2311yescharacteristic
81C2×#13096802C2311yescharacteristic
82C8×D197C2×C414yescomplemented
83C4⋊Dic197—21no—
84C4⋊Dic197C2311yescharacteristic
85C4⋊Dic197C2311yescharacteristic
86#15691337—21no—
87#15691337—21no—
88#15691337—21no—
89#15691337—21no—
90#15691337D411yescharacteristic
91#15691337D411yescharacteristic
92C22⋊Dic197—21no—
93#15691335—21no—
94#15691335—21no—
95#15691335D411yescharacteristic
96#15691335D411yescharacteristic
97C4×Dic197—21no—
98C4×Dic197—21no—
99C4×Dic197—21no—
100C4×Dic197C2311yescharacteristic
101C2×Q8×C197C2311yescharacteristic
102C2×D4×C197C2311yescharacteristic
103C22×C4×C197—21no—
104SD16×C197C2×C414yescomplemented
105C2×C8×C197C2311yescharacteristic
106C2×C8×C197C2311yescharacteristic
107C4⋊C4×C197—21no—
108C4⋊C4×C197C2311yescharacteristic
109C4⋊C4×C197C2311yescharacteristic
110C22⋊C4×C197—21no—
111C42×C197—21no—
112C42×C197C2311yescharacteristic
113C22×D197—21no—
114C22×D197C2×D411yescharacteristic
115Dic394—21no—
116Dic394C22×C412yes—
117D788—24no—
118D788—21no—
119D788C22×C412yes—
120C22⋊D197—24no—
121C2×Dic197—21no—
122C2×Dic197—24no—
123C2×Dic197—21no—
124C2×Dic197—21no—
125C2×Dic197—81no—
126C2×Dic197C2×D411yescharacteristic
127C2×Dic197C2×D411yescharacteristic
128C2×Dic197C2×D411yescharacteristic
129C4×D197—24no—
130C4×D197—81no—
131C4×D197—81no—
132C4×D197—81no—
133C4×D197—81no—
134C4×D197C22×C412yes—
135C4×D197#4112yes—
136#13096802—21no—
137#13096802C22×C412yes—
138C23×C197—21no—
139Q8×C197—21no—
140Q8×C197C22×C412yes—
141D4×C197—21no—
142D4×C197C22×C412yes—
143C2×C4×C197—21no—
144C2×C4×C197—21no—
145C2×C4×C197—21no—
146C2×C4×C197—81no—
147C2×C4×C197C2×D411yescharacteristic
148C2×C4×C197C2×D411yescharacteristic
149C2×C4×C197C2411yescharacteristic
150C8×C197—21no—
151C8×C197C22×C412yes—
152C22×C197—22no—
153C22×C197—41no—
154C22×C197—41no—
155C22×C197C22×D411yescharacteristic
156C2×D197—21no—
157C2×D197—21no—
158C2×D197—22no—
159C2×D197—41no—
160C2×D197—41no—
161C2×D197C4×D412yes—
162C4×C197—22no—
163C4×C197—21no—
164C4×C197—21no—
165C4×C197—41no—
166C4×C197—41no—
167C4×C197C23×C411yescharacteristic
168C4×C197C2×#4111yescharacteristic
169Dic197—22no—
170Dic197—21no—
171Dic197—21no—
172Dic197—41no—
173Dic197—41no—
174Dic197C4×D412yes—
175C2×C197—42no—
176C2×C197C2×C4×D411yescharacteristic
177C2×C197#57611yescharacteristic
178C2×C197#57811yescharacteristic
179D197—22no—
180D197—42no—
181C197#295111yescharacteristic, complemented, Sylow
182#2951—1971nomaximal, Sylow
183C2×#135—1971no—
184#520—1971no—
185#520—1971no—
186#440—1971no—
187#440—1971no—
188#440—1971no—
189#440D19711yescharacteristic, complemented
190C4×Q16—1971no—
191C4×Q16—1971no—
192C4×D8—1971no—
193C4×D8—1971no—
194C2×#107—1971no—
195#417—1971no—
196#417—1971no—
197#405—1971no—
198C2×#41—1971no—
199C2×#41—1971no—
200#135—1978no—
201C2×Q16—1971no—
202C2×SD16—1971no—
203C2×SD16C2×D19711yescharacteristic
204C2×D8—1971no—
205C2×M16—1971no—
206C22×C8—1971no—
207C4×Q8—1971no—
208C4×Q8—1971no—
209C4×Q8—1971no—
210C4×Q8C2×D19711yescharacteristic
211C4×D4—3941no—
212C4×D4—3941no—
213C4×D4—3941no—
214C4×D4—3941no—
215C4×D4—1971no—
216C4×D4—1971no—
217C4×D4—1971no—
218C4×D4C2×D19711yescharacteristic
219#117—3941no—
220#117—3941no—
221#117—1971no—
222C2×C4⋊C4—3941no—
223C2×C4⋊C4—3941no—
224C2×C4⋊C4—1971no—
225C2×C4⋊C4—1971no—
226#107—1971no—
227#107—1971no—
228#107—1971no—
229#107C2×D19711yescharacteristic
230#103—1971no—
231#103—1971no—
232#103—1971no—
233#103C2×D19711yescharacteristic
234#102—1971no—
235#102—1971no—
236#102—1971no—
237#102C2×D19711yescharacteristic
238#97—1971no—
239#97C2×D19711yescharacteristic
240C4×C8—1971no—
241C4×C8—1971no—
242#41—3942no—
243#41—3942no—
244#41—1974no—
245#41—1974no—
246C2×Q8—1971no—
247C2×Q8C22×D19711yescharacteristic
248C2×D4—3941no—
249C2×D4—3941no—
250C2×D4—1971no—
251C2×D4C22×D19711yescharacteristic
252C22×C4—21no—
253C22×C4—3941no—
254C22×C4—3941no—
255C22×C4—3941no—
256C22×C4—3941no—
257C22×C4—3941no—
258C22×C4—3941no—
259C22×C4—3941no—
260C22×C4—1971no—
261Q16—1974no—
262SD16—1974no—
263SD16C4×D19714yescomplemented
264D8—1974no—
265M16—3942no—
266C2×C8—1971no—
267C2×C8—1971no—
268C2×C8—1974no—
269C2×C8C22×D19711yescharacteristic
270C2×C8C22×D19711yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—3941no—
273C4⋊C4—3941no—
274C4⋊C4—3941no—
275C4⋊C4—3941no—
276C4⋊C4—3941no—
277C4⋊C4—1971no—
278C4⋊C4—1971no—
279C4⋊C4—1971no—
280C4⋊C4—1971no—
281C4⋊C4C22×D19711yescharacteristic
282C4⋊C4C22×D19711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—3941no—
285C22⋊C4—3941no—
286C22⋊C4—3941no—
287C22⋊C4—1971no—
288C22⋊C4—1971no—
289C42—21no—
290C42—3941no—
291C42—3941no—
292C42—3941no—
293C42—1971no—
294C42C22×D19711yescharacteristic
295C23—21no—
296C23—3941no—
297C23—1971no—
298Q8—21no—
299Q8—3941no—
300Q8—1972no—
301Q8C2×C4×D19712yes—
302D4—21no—
303D4—3944no—
304D4—3944no—
305D4—3941no—
306D4—1972no—
307D4C2×C4×D19712yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—81no—
312C2×C4—15761no—
313C2×C4—15761no—
314C2×C4—15761no—
315C2×C4—15761no—
316C2×C4—15761no—
317C2×C4—3944no—
318C2×C4—3941no—
319C2×C4—3941no—
320C2×C4—3941no—
321C2×C4—3944no—
322C2×C4—1971no—
323C2×C4—1971no—
324C2×C4—1971no—
325C2×C4—1972no—
326C2×C4—1972no—
327C2×C4D4×D19711yescharacteristic
328C2×C4D4×D19711yescharacteristic
329C2×C4C23×D19711yescharacteristic
330C8—21no—
331C8—3941no—
332C8—1972no—
333C8C2×C4×D19712yes—
334C22—22no—
335C22—41no—
336C22—41no—
337C22—7881no—
338C22—7881no—
339C22—3942no—
340C22—3941no—
341C22—3941no—
342C22—1972no—
343C22C2×D4×D19711yescharacteristic
344C4—21no—
345C4—21no—
346C4—22no—
347C4—41no—
348C4—41no—
349C4—7881no—
350C4—7881no—
351C4—3941no—
352C4—3941no—
353C4—3942no—
354C4—1972no—
355C4C22×C4×D19711yescharacteristic
356C4#41×D19711yescharacteristic
357C2—42no—
358C2—7882no—
359C2—3942no—
360C2C4×D4×D19711yescharacteristic
361C2#2716188311yescharacteristic
362C2#2716189511yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutatorC4×C197C23×C4
FrattiniC2×C4C23×D197
Fitting#440×C197C2
socleC22×C197C22×D4
prime radicalC2×#20263347C2
cosocleC2×C4×C197C24
hypercenter#440D197
hypocenterC197#2951
LCSG ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. #27161561C2semidirect
3. #27161558C2semidirect
4. #27161181C2semidirect
5. #27161180C2semidirect
6. #27161178C2semidirect
7. C4×#15691348C2semidirect
8. #27161171C2semidirect
9. C4×#15691343C2semidirect
10. #27161159C2semidirect
11. #107×D197C2semidirect
12. #27161079C2semidirect
13. #27161077C2semidirect
14. #27161036C2semidirect
15. #440×C197C2semidirect
18. #20263347C4semidirect
61. #15691359C2×C4semidirect
63. #15691357C2×C4semidirect
76. #15691348C2×C4semidirect
77. #15691347C2×C4semidirect
78. #15691343C2×C4semidirect
82. C8×D197C2×C4semidirect
104. SD16×C197C2×C4semidirect
181. C197#2951semidirect
189. #440D197semidirect
263. SD16C4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633472
#271615612
#271615582
#271611812
#271611802
#271611782
C4×#156913482
#271611712
C4×#156913432
#271611592
#107×D1972
#271610792
#271610772
#271610362
#440×C1972
#2951197