Finite Group Databasebeta

← #38752106

Subgroups — #38752106

Order 25216.

766 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 145 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#27161915C211yesmaximal, characteristic, complemented
2#27161591C212yesmaximal, complemented
3#27161586C211yesmaximal, characteristic, complemented
4#27161270C214yesmaximal, complemented
5#27161267C214yesmaximal, complemented
6#27161052C212yesmaximal, complemented
7#410×C197C211yesmaximal, characteristic, complemented
8#20263374—21no—
9#20263373—22no—
10#20263372—22no—
11#20263371—21no—
12#20263371C2214yes—
13C2×#15691358—22no—
14C2×#15691358C2212yes—
15#41×D197—24no—
16C22×C4×D197C2211yescharacteristic, complemented
17#20263310C2214yescomplemented
18#20263307C2212yescomplemented
19#20263306—22no—
20#20263302—21no—
21#20263301—22no—
22#20263297—21no—
23#20263296C2214yescomplemented
24#20263294C2212yescomplemented
25C4×D788—24no—
26C4×C22⋊D197—22no—
27#20263285C2211yescharacteristic, complemented
28#20263240C2218yescomplemented
29C2×#41×C197C2211yescharacteristic
30#117×C197C2212yescomplemented
31#99×C197C2214yescomplemented
32C23×D197C2311yescharacteristic
33#15691359—24no—
34#15691358—24no—
35#15691358—24no—
36#15691358—22no—
37#15691358—22no—
38#15691358—24no—
39#15691358—22no—
40#15691358—22no—
41#15691357—24no—
42#15691357—28no—
43C2×Dic394—22no—
44C2×Dic394D412yescomplemented
45Q8×D197—24no—
46C2×D788—22no—
47C2×D788—24no—
48C2×D788D412yescomplemented
49C2×C22⋊D197—24no—
50C2×C22⋊D197—22no—
51C2×C22⋊D197C2314yes—
52D4×D197—24no—
53D4×D197—28no—
54C22×Dic197C2311yescharacteristic
55C2×C4×D197—24no—
56C2×C4×D197—21no—
57C2×C4×D197—22no—
58C2×C4×D197—21no—
59C2×C4×D197—22no—
60C2×C4×D197D414yescomplemented
61C4⋊Dic197—24no—
62C4⋊Dic197—21no—
63#15691337—22no—
64C22⋊Dic197—22no—
65C22⋊Dic197C2312yes—
66#15691335—24no—
67#15691335—24no—
68#15691335—22no—
69C4×Dic197—21no—
70#41×C197—22no—
71#41×C197—22no—
72C2×Q8×C197D411yescharacteristic, complemented
73C2×D4×C197D411yescharacteristic, complemented
74C2×D4×C197C2312yes—
75C22×C4×C197D412yescomplemented
76C22×C4×C197C2311yescharacteristic
77C4⋊C4×C197—22no—
78C22⋊C4×C197C2314yes—
79C42×C197—22no—
80C22×D197—21no—
81C22×D197—22no—
82C22×D197—24no—
83C22×D197—42no—
84C22×D197C2×D414yes—
85Dic394—24no—
86Dic394—22no—
87Dic394—22no—
88D788—24no—
89D788—22no—
90D788—22no—
91D788—48no—
92C22⋊D197—28no—
93C22⋊D197—24no—
94C22⋊D197—24no—
95C22⋊D197—44no—
96C2×Dic197—24no—
97C2×Dic197—22no—
98C2×Dic197—22no—
99C2×Dic197—21no—
100C2×Dic197C2×D414yes—
101C4×D197—28no—
102C4×D197—24no—
103C4×D197—24no—
104C4×D197—41no—
105C4×D197—42no—
106C4×D197—41no—
107C23×C197C2×D412yes—
108C23×C197C2411yescharacteristic
109Q8×C197—22no—
110D4×C197—24no—
111D4×C197—22no—
112C2×C4×C197—22no—
113C2×C4×C197—22no—
114C2×C4×C197—24no—
115C2×C4×C197C2×D412yes—
116C2×C4×C197C2×D412yes—
117C2×C4×C197C2×D412yes—
118C2×C4×C197C2×D411yescharacteristic
119C2×C4×C197C2×D411yescharacteristic
120C22×C197—22no—
121C22×C197—41no—
122C22×C197—42no—
123C22×C197C22×D412yes—
124C22×C197C22×D411yescharacteristic
125C2×D197—22no—
126C2×D197—24no—
127C2×D197—24no—
128C2×D197—41no—
129C2×D197—42no—
130C2×D197—41no—
131C2×D197—44no—
132C2×D197—84no—
133C4×C197—22no—
134C4×C197—44no—
135C4×C197#12011yescharacteristic
136C4×C197#12012yes—
137C4×C197#12011yescharacteristic
138Dic197—24no—
139Dic197—42no—
140Dic197—41no—
141C2×C197—21no—
142C2×C197—22no—
143C2×C197—42no—
144C2×C197C2×#12011yescharacteristic
145D197—42no—
146D197—44no—
147C197#302511yescharacteristic, complemented, Sylow
148#3025—1971nomaximal, Sylow
149#585—1971no—
150#525—1971no—
151#525—1972no—
152#458—1974no—
153B(4,𝔽2)—1974no—
154#410—1972no—
155#410D19711yescharacteristic, complemented
1562−1+4—3942no—
1572−1+4—3941no—
158H(5,𝔽2)—3942no—
159H(5,𝔽2)—3941no—
160H(5,𝔽2)—1974no—
161C2×#41—3944no—
162C2×#41—3942no—
163C2×#41—1972no—
164C2×#41C2×D19711yescharacteristic
165C23×C4—1971no—
166#123—1972no—
167#123—1974no—
168C22⋊Q8—3942no—
169C22⋊Q8—3941no—
170#121—3942no—
171#121—3941no—
172#120—1972no—
173#120—1974no—
174C4×D4—3944no—
175C4×D4—3942no—
176#117—1971no—
177#117C2×D19712yescomplemented
178#99—1978no—
179#99C2×D19714yescomplemented
180C24—1971no—
181#41—22no—
182#41—22no—
183#41—3944no—
184#41—3944no—
185#41—3944no—
186#41—3944no—
187#41—3948no—
188#41—3942no—
189#41—3942no—
190#41—3942no—
191#41—3944no—
192#41—3942no—
193C2×Q8—3942no—
194C2×Q8—3944no—
195C2×Q8—1972no—
196C2×Q8D78811yescharacteristic, complemented
197C2×D4—3944no—
198C2×D4—3942no—
199C2×D4—3944no—
200C2×D4—3948no—
201C2×D4—3942no—
202C2×D4—3944no—
203C2×D4—1974no—
204C2×D4—1972no—
205C2×D4D78811yescharacteristic, complemented
206C2×D4C22×D19712yes—
207C22×C4—3944no—
208C22×C4—3942no—
209C22×C4—3942no—
210C22×C4—3941no—
211C22×C4—3941no—
212C22×C4—1971no—
213C22×C4—1974no—
214C22×C4D78812yescomplemented
215C22×C4C22×D19711yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—3944no—
218C4⋊C4—3942no—
219C4⋊C4—3941no—
220C22⋊C4—3944no—
221C22⋊C4—3944no—
222C22⋊C4—3942no—
223C22⋊C4—3942no—
224C22⋊C4—1972no—
225C22⋊C4C22×D19714yes—
226C42—22no—
227C42—3941no—
228C23—7882no—
229C23—3944no—
230C23—3942no—
231C23—3941no—
232C23—1974no—
233C23C2×D78812yes—
234C23C23×D19711yescharacteristic
235Q8—22no—
236Q8—3944no—
237Q8—3942no—
238Q8—3942no—
239D4—24no—
240D4—22no—
241D4—7888no—
242D4—7884no—
243D4—3944no—
244D4—3948no—
245D4—3944no—
246D4—3944no—
247D4—3942no—
248D4—3942no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—7881no—
253C2×C4—7882no—
254C2×C4—7881no—
255C2×C4—3948no—
256C2×C4—3942no—
257C2×C4—3942no—
258C2×C4—3941no—
259C2×C4—3944no—
260C2×C4—3944no—
261C2×C4—3944no—
262C2×C4—1974no—
263C2×C4D4×D19712yes—
264C2×C4D4×D19712yes—
265C2×C4C2×D78812yes—
266C2×C4C2×D78811yescharacteristic
267C2×C4C2×D78811yescharacteristic
268C22—22no—
269C22—41no—
270C22—42no—
271C22—7884no—
272C22—7882no—
273C22—7881no—
274C22—7881no—
275C22—15764no—
276C22—3944no—
277C22—3942no—
278C22—3944no—
279C22C2×D4×D19712yes—
280C22C22×D78811yescharacteristic
281C4—22no—
282C4—44no—
283C4—7882no—
284C4—7881no—
285C4—3944no—
286C4#2026329611yescharacteristic
287C4#2026329612yes—
288C4#2026329611yescharacteristic
289C2—22no—
290C2—21no—
291C2—42no—
292C2—7882no—
293C2—7884no—
294C2C2×#2026329611yescharacteristic
subgroupG/H
centerC4#20263296
commutator, cosocleC23×C197C24
FrattiniC23C23×D197
Fitting#410×C197C2
socleC2×C197C2×#120
hypercenter#410D197
hypocenterC197#3025
LCSG ▷ C23×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #27161915C2semidirect
2. #27161591C2semidirect
3. #27161586C2semidirect
4. #27161270C2semidirect
5. #27161267C2semidirect
6. #27161052C2semidirect
7. #410×C197C2semidirect
16. C22×C4×D197C22semidirect
17. #20263310C22semidirect
18. #20263307C22semidirect
23. #20263296C22semidirect
24. #20263294C22semidirect
27. #20263285C22semidirect
28. #20263240C22semidirect
30. #117×C197C22semidirect
31. #99×C197C22semidirect
44. C2×Dic394D4semidirect
48. C2×D788D4semidirect
60. C2×C4×D197D4semidirect
72. C2×Q8×C197D4semidirect
73. C2×D4×C197D4semidirect
75. C22×C4×C197D4semidirect
147. C197#3025semidirect
155. #410D197semidirect
177. #117C2×D197semidirect
179. #99C2×D197semidirect
196. C2×Q8D788semidirect
205. C2×D4D788semidirect
214. C22×C4D788semidirect

Maximal subgroups

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

subgroupindex
#271619152
#271615912
#271615862
#271612702
#271612672
#271610522
#410×C1972
#3025197