Finite Group Databasebeta

← #38752638

Subgroups — #38752638

Order 25216.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161591C211yesmaximal, characteristic
2C2×C4⋊C4×D197C211yesmaximal, characteristic, complemented
3#27161416C212yesmaximal, complemented
4#27161411C212yesmaximal, complemented
5#27161399C212yesmaximal, complemented
6#27161394C212yesmaximal, complemented
7#27161122C211yesmaximal, characteristic, complemented
8#27161081C211yesmaximal, characteristic, complemented
9#27161078C211yesmaximal, characteristic, complemented
10#27161042C211yesmaximal, characteristic, complemented
11#428×C197C211yesmaximal, characteristic, complemented
12C2×#15691358C2211yescharacteristic
13C22×C4×D197—21no—
14C22×C4×D197C2211yescharacteristic
15C2×#15691342C2211yescharacteristic
16#20263310—21no—
17#20263306C2211yescharacteristic
18#20263306C2211yescharacteristic
19#20263301C2211yescharacteristic
20#20263301C2211yescharacteristic
21#20263296—21no—
22C4×D788—21no—
23C4×D788—21no—
24C2×C4⋊Dic197C2211yescharacteristic
25C2×#15691337—21no—
26C4⋊C4×D197—21no—
27C4⋊C4×D197—21no—
28C4⋊C4×D197C2212yes—
29C4⋊C4×D197C2212yes—
30#20263259C2212yescomplemented
31#20263257C2212yescomplemented
32#20263251C2212yescomplemented
33#20263249C2212yescomplemented
34#20263248C2212yescomplemented
35#20263246C2212yescomplemented
36#20263238C2212yes—
37#20263238C2212yes—
38C2×M16×C197C2211yescharacteristic
39#117×C197C2211yescharacteristic
40C2×C4⋊C4×C197C2211yescharacteristic
41#107×C197C2212yescomplemented
42#106×C197C2212yescomplemented
43C23×D197D411yescharacteristic
44#15691358—22no—
45#15691358—22no—
46C2×Dic394C2311yescharacteristic
47C2×D788—21no—
48C2×D788—21no—
49C2×D788C2311yescharacteristic
50C2×C22⋊D197—21no—
51C22×Dic197—21no—
52C22×Dic197D411yescharacteristic
53C2×C4×D197—22no—
54C2×C4×D197—22no—
55C2×C4×D197—21no—
56C2×C4×D197—22no—
57C2×C4×D197—22no—
58C2×C4×D197—44no—
59C2×C4×D197D411yescharacteristic
60C2×C4×D197D411yescharacteristic
61C2×C4×D197C2311yescharacteristic
62C2×C4×D197C2311yescharacteristic
63#15691342—22no—
64C2×#13096802C2312yes—
65C4⋊Dic197—21no—
66C4⋊Dic197—21no—
67C4⋊Dic197—21no—
68C4⋊Dic197C2312yes—
69#15691337—22no—
70#15691337—22no—
71#15691335—21no—
72#15691335—21no—
73#15691335—21no—
74#15691335—21no—
75C22×C4×C197—21no—
76C22×C4×C197C2311yescharacteristic
77M16×C197—22no—
78C2×C8×C197C2312yes—
79C4⋊C4×C197—21no—
80C4⋊C4×C197C2311yescharacteristic
81C4⋊C4×C197C2312yes—
82C4⋊C4×C197C2311yescharacteristic
83C22⋊C4×C197—21no—
84C42×C197—21no—
85C22×D197—22no—
86C22×D197—22no—
87C22×D197—22no—
88C22×D197—21no—
89C22×D197C2×D411yescharacteristic
90C22×D197C2×D411yescharacteristic
91Dic394—21no—
92Dic394—21no—
93D788—21no—
94D788—21no—
95D788—42no—
96D788—42no—
97C22⋊D197—42no—
98C2×Dic197—22no—
99C2×Dic197—21no—
100C2×Dic197—22no—
101C2×Dic197—42no—
102C2×Dic197C2×D411yescharacteristic
103C2×Dic197C2×D411yescharacteristic
104C4×D197—22no—
105C4×D197—22no—
106C4×D197—21no—
107C4×D197—21no—
108C4×D197—21no—
109C4×D197—21no—
110C4×D197—44no—
111C4×D197—44no—
112C4×D197—42no—
113#13096802—22no—
114C23×C197C2×D411yescharacteristic
115C2×C4×C197—21no—
116C2×C4×C197—21no—
117C2×C4×C197—22no—
118C2×C4×C197—42no—
119C2×C4×C197C2×D411yescharacteristic
120C2×C4×C197#4112yes—
121C2×C4×C197#4112yes—
122C2×C4×C197C2411yescharacteristic
123C8×C197—22no—
124C22×C197—21no—
125C22×C197—21no—
126C22×C197#12312yes—
127C22×C197C22×D411yescharacteristic
128C2×D197—28no—
129C2×D197—21no—
130C2×D197—21no—
131C2×D197—21no—
132C2×D197—21no—
133C2×D197—41no—
134C2×D197—41no—
135C2×D197—41no—
136C2×D197AGL(1,ℤ/8)12yes—
137C2×D197#13712yes—
138C4×C197—42no—
139C4×C197—41no—
140C4×C197—41no—
141C4×C197#12312yes—
142C4×C197C2×#4111yescharacteristic
143C4×C197C2×#4111yescharacteristic
144Dic197—21no—
145Dic197—21no—
146Dic197—41no—
147Dic197—42no—
148C2×C197—22no—
149C2×C197C2×#12311yescharacteristic
150C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C197C2×#13711yescharacteristic
152D197—22no—
153D197—22no—
154D197—81no—
155C197#309211yescharacteristic, complemented, Sylow
156#3092—1971nomaximal, Sylow
157#525—1971no—
158C22×C4⋊C4—1971no—
159#484—1972no—
160#483—1972no—
161#481—1972no—
162#480—1972no—
163#428—1971no—
164#428D19711yescharacteristic, complemented
165#417—1971no—
166#417—1971no—
167#407—1971no—
168C2×#41—1971no—
169C23×C4—3941no—
170C23×C4—1971no—
171C2×M16—1971no—
172C2×M16C2×D19711yescharacteristic
173#123—3941no—
174C22⋊Q8—1971no—
175C22⋊Q8—1971no—
176#121—1971no—
177#121—1971no—
178#120—3941no—
179C4×D4—3941no—
180C4×D4—3941no—
181#117C2×D19711yescharacteristic
182C2×C4⋊C4—3941no—
183C2×C4⋊C4—3941no—
184C2×C4⋊C4—3941no—
185C2×C4⋊C4—1971no—
186C2×C4⋊C4—1972no—
187C2×C4⋊C4—1972no—
188C2×C4⋊C4C2×D19711yescharacteristic
189#107—1972no—
190#107C2×D19712yescomplemented
191#106—1972no—
192#106C2×D19712yescomplemented
193#103—1972no—
194#103—1972no—
195#102—1972no—
196#102—1972no—
197C22⋊C8—1972no—
198C22⋊C8—1972no—
199C24—1971no—
200#41—3942no—
201#41—3942no—
202C2×Q8—1971no—
203C2×D4—3941no—
204C2×D4—3941no—
205C2×D4—3941no—
206C2×D4—1971no—
207C22×C4—21no—
208C22×C4—7884no—
209C22×C4—3942no—
210C22×C4—3942no—
211C22×C4—3941no—
212C22×C4—3941no—
213C22×C4—3942no—
214C22×C4—3942no—
215C22×C4—1971no—
216C22×C4—1971no—
217C22×C4—1971no—
218C22×C4—1971no—
219C22×C4—1971no—
220C22×C4C22×D19711yescharacteristic
221M16—22no—
222M16—3942no—
223C2×C8—1972no—
224C2×C8C22×D19712yes—
225C4⋊C4—21no—
226C4⋊C4—3942no—
227C4⋊C4—3942no—
228C4⋊C4—3941no—
229C4⋊C4—3941no—
230C4⋊C4—3941no—
231C4⋊C4—1972no—
232C4⋊C4C22×D19712yes—
233C4⋊C4C22×D19711yescharacteristic
234C4⋊C4C22×D19711yescharacteristic
235C22⋊C4—21no—
236C22⋊C4—3941no—
237C22⋊C4—3941no—
238C22⋊C4—3941no—
239C22⋊C4—3941no—
240C42—21no—
241C23—3942no—
242C23—3942no—
243C23—3941no—
244C23—3942no—
245C23—1971no—
246C23—1971no—
247C23D4×D19711yescharacteristic
248Q8—3941no—
249Q8—3941no—
250D4—7882no—
251D4—7882no—
252D4—7882no—
253D4—3941no—
254D4—3941no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—7884no—
260C2×C4—7884no—
261C2×C4—7882no—
262C2×C4—7882no—
263C2×C4—3941no—
264C2×C4—3942no—
265C2×C4—3942no—
266C2×C4—3942no—
267C2×C4—3942no—
268C2×C4—3941no—
269C2×C4—3941no—
270C2×C4—3941no—
271C2×C4—3941no—
272C2×C4—1971no—
273C2×C4—1971no—
274C2×C4D4×D19711yescharacteristic
275C2×C4#1569135812yes—
276C2×C4#1569135912yes—
277C2×C4C23×D19711yescharacteristic
278C8—22no—
279C8—3942no—
280C22—21no—
281C22—21no—
282C22—7881no—
283C22—7881no—
284C22—7881no—
285C22—3948no—
286C22—3941no—
287C22—3941no—
288C22—3941no—
289C22—3941no—
290C22—1972no—
291C22—1972no—
292C22#2026331112yes—
293C22C2×D4×D19711yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—7882no—
298C4—7881no—
299C4—3941no—
300C4—3941no—
301C4#2026331112yes—
302C4C2×#1569135811yescharacteristic
303C4C2×#1569135911yescharacteristic
304C2—22no—
305C2—15761no—
306C2—3942no—
307C2—3942no—
308C2C2×#2026331111yescharacteristic
309C2AGL(1,ℤ/8)×D19711yescharacteristic
310C2#137×D19711yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#428×C197C2
socleC22×C197C22×D4
prime radical#27161591C2
hypercenter#428D197
hypocenterC197#3092
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D197C2semidirect
3. #27161416C2semidirect
4. #27161411C2semidirect
5. #27161399C2semidirect
6. #27161394C2semidirect
7. #27161122C2semidirect
8. #27161081C2semidirect
9. #27161078C2semidirect
10. #27161042C2semidirect
11. #428×C197C2semidirect
30. #20263259C22semidirect
31. #20263257C22semidirect
32. #20263251C22semidirect
33. #20263249C22semidirect
34. #20263248C22semidirect
35. #20263246C22semidirect
41. #107×C197C22semidirect
42. #106×C197C22semidirect
155. C197#3092semidirect
164. #428D197semidirect
190. #107C2×D197semidirect
192. #106C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271615912
C2×C4⋊C4×D1972
#271614162
#271614112
#271613992
#271613942
#271611222
#271610812
#271610782
#271610422
#428×C1972
#3092197