Finite Group Databasebeta

← #38752670

Subgroups — #38752670

Order 25216.

418 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 133 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#27161643C211yesmaximal, characteristic
2#27161403C212yesmaximal, complemented
3#27161403C212yesmaximal, complemented
4#27161397C212yesmaximal, complemented
5#27161392C212yesmaximal, complemented
6#27161119C211yesmaximal, characteristic, complemented
7#27161115C211yesmaximal, characteristic, complemented
8#27161083C211yesmaximal, characteristic, complemented
9C2×#20263246C211yesmaximal, characteristic, complemented
10#27161046C211yesmaximal, characteristic, complemented
11#520×C197C211yesmaximal, characteristic, complemented
12C22×D788C2211yescharacteristic
13C2×#15691342C2211yescharacteristic
14C22×#13096802C2211yescharacteristic
15#20263322—21no—
16#20263315—21no—
17#20263300C2212yes—
18#20263300C2212yes—
19#20263294—22no—
20#20263285C2211yescharacteristic
21#20263259C2211yescharacteristic, complemented
22#20263259C2211yescharacteristic, complemented
23#20263259C2212yescomplemented
24#20263257C2211yescharacteristic, complemented
25#20263257C2211yescharacteristic, complemented
26#20263257C2212yescomplemented
27#20263246C2212yes—
28#20263246C2212yes—
29#20263246C2212yes—
30#20263246C2212yes—
31C22⋊#13096802C2212yescomplemented
32C22⋊#13096802C2212yescomplemented
33C2×#41×C197C2211yescharacteristic
34C4×Q8×C197—21no—
35C4×D4×C197—21no—
36C4×D4×C197—22no—
37#117×C197C2211yescharacteristic
38#117×C197C2212yescomplemented
39C2×C4⋊C4×C197C2211yescharacteristic
40C2×C4⋊C4×C197C2212yescomplemented
41C23×D197—21no—
42C2×D788—21no—
43C2×D788—22no—
44C2×D788—22no—
45C2×D788C2312yes—
46C2×C22⋊D197—22no—
47C2×C22⋊D197—22no—
48#15691342—22no—
49C2×#13096802—22no—
50C2×#13096802C2311yescharacteristic
51C2×#13096802C2312yes—
52C2×#13096802C2311yescharacteristic
53C4⋊Dic197C2311yescharacteristic
54C4⋊Dic197C2311yescharacteristic
55C22⋊Dic197—21no—
56#15691335—22no—
57#15691335—22no—
58C4×Dic197—21no—
59#41×C197—22no—
60#41×C197—22no—
61C2×Q8×C197D411yescharacteristic
62C2×D4×C197D412yes—
63C2×D4×C197D411yescharacteristic
64C22×C4×C197—21no—
65C22×C4×C197—22no—
66C22×C4×C197C2312yes—
67C22×C4×C197C2311yescharacteristic
68C4⋊C4×C197—21no—
69C4⋊C4×C197—22no—
70C4⋊C4×C197C2312yes—
71C4⋊C4×C197C2311yescharacteristic
72C4⋊C4×C197C2311yescharacteristic
73C22⋊C4×C197—21no—
74C22⋊C4×C197—22no—
75C42×C197—21no—
76C42×C197—22no—
77C22×D197—22no—
78C22×D197—42no—
79C22×D197—81no—
80C22×D197—81no—
81D788—22no—
82D788—22no—
83D788—42no—
84C22⋊D197—82no—
85C22⋊D197—82no—
86C2×Dic197—21no—
87C2×Dic197—21no—
88#13096802—21no—
89#13096802—21no—
90#13096802—22no—
91C23×C197C2×D412yes—
92C23×C197C2×D411yescharacteristic
93Q8×C197—22no—
94D4×C197—24no—
95D4×C197—22no—
96C2×C4×C197—22no—
97C2×C4×C197—22no—
98C2×C4×C197—21no—
99C2×C4×C197—21no—
100C2×C4×C197—22no—
101C2×C4×C197—81no—
102C2×C4×C197—82no—
103C2×C4×C197C2×D412yes—
104C2×C4×C197C2×D411yescharacteristic
105C2×C4×C197#4112yes—
106C2×C4×C197#4112yes—
107C2×C4×C197C2411yescharacteristic
108C22×C197—21no—
109C22×C197—21no—
110C22×C197—22no—
111C22×C197—22no—
112C22×C197—22no—
113C22×C197#12312yes—
114C22×C197C22×D411yescharacteristic
115C2×D197—42no—
116C2×D197—42no—
117C2×D197—42no—
118C2×D197—84no—
119C4×C197—22no—
120C4×C197—41no—
121C4×C197—42no—
122C4×C197—41no—
123C4×C197#12312yes—
124C4×C197C2×#4111yescharacteristic
125C4×C197C2×#4111yescharacteristic
126Dic197—41no—
127Dic197—41no—
128C2×C197—22no—
129C2×C197—42no—
130C2×C197C2×#12311yescharacteristic
131C2×C197#57611yescharacteristic
132C2×C197#57711yescharacteristic
133D197—82no—
134C197#309611yescharacteristic, complemented, Sylow
135#3096—1971nomaximal, Sylow
136#535—1971no—
137#520D19711yescharacteristic, complemented
138#482—1972no—
139#482—1972no—
140#481—1972no—
141#480—1972no—
142#428—1971no—
143#427—1971no—
144#418—1971no—
145C2×#102—1971no—
146#408—1971no—
147C2×#41C2×D19711yescharacteristic
148C22×D4—1971no—
149C2×M16—1971no—
150C22×C8—1971no—
151#127—3941no—
152#124—3941no—
153#121—1972no—
154#121—1972no—
155#120—3942no—
156C4×Q8—21no—
157C4×D4—21no—
158C4×D4—22no—
159#117—1971no—
160#117C2×D19711yescharacteristic
161#117C2×D19712yescomplemented
162C2×C4⋊C4C2×D19711yescharacteristic
163C2×C4⋊C4C2×D19712yescomplemented
164#107—1971no—
165#107—1971no—
166#107—1972no—
167#106—1971no—
168#106—1971no—
169#106—1972no—
170#102—1972no—
171#102—1972no—
172#102—1972no—
173#102—1972no—
174C22⋊C8—1972no—
175C22⋊C8—1972no—
176C24—3941no—
177#41—22no—
178#41—22no—
179C2×Q8C22⋊D19711yescharacteristic
180C2×D4—3942no—
181C2×D4—3942no—
182C2×D4—3941no—
183C2×D4—3942no—
184C2×D4—3942no—
185C2×D4—1972no—
186C2×D4C22⋊D19712yes—
187C2×D4C22⋊D19711yescharacteristic
188C22×C4—21no—
189C22×C4—22no—
190C22×C4C22×D19712yes—
191C22×C4C22×D19711yescharacteristic
192M16—3942no—
193C2×C8—3942no—
194C2×C8—1971no—
195C2×C8—1972no—
196C2×C8—1971no—
197C4⋊C4—21no—
198C4⋊C4—22no—
199C4⋊C4—1971no—
200C4⋊C4—1971no—
201C4⋊C4C22×D19712yes—
202C4⋊C4C22×D19711yescharacteristic
203C4⋊C4C22×D19711yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—22no—
206C22⋊C4—3941no—
207C22⋊C4—3942no—
208C22⋊C4—3942no—
209C42—21no—
210C42—22no—
211C42—3941no—
212C23—7882no—
213C23—15761no—
214C23—15761no—
215C23—3942no—
216C23C2×C22⋊D19712yes—
217C23C2×C22⋊D19711yescharacteristic
218Q8—22no—
219D4—24no—
220D4—22no—
221D4—7882no—
222D4—15762no—
223D4—15762no—
224D4—3942no—
225D4—3942no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—81no—
232C2×C4—82no—
233C2×C4—3941no—
234C2×C4—3941no—
235C2×C4C2×C22⋊D19712yes—
236C2×C4C2×C22⋊D19711yescharacteristic
237C2×C4#1569135812yes—
238C2×C4#1569135812yes—
239C2×C4C23×D19711yescharacteristic
240C8—3941no—
241C8—3941no—
242C8—3942no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—7882no—
249C22—7882no—
250C22—7882no—
251C22—15764no—
252C22#2026330912yes—
253C22C22×C22⋊D19711yescharacteristic
254C4—22no—
255C4—41no—
256C4—42no—
257C4—41no—
258C4—7881no—
259C4—7881no—
260C4#2026330912yes—
261C4C2×#1569135811yescharacteristic
262C4C2×#1569135811yescharacteristic
263C2—22no—
264C2—42no—
265C2—15762no—
266C2C2×#2026330911yescharacteristic
267C2#2716188211yescharacteristic
268C2#2716188711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#520×C197C2
socleC22×C197C22×D4
prime radical#27161643C2
hypercenter#520D197
hypocenterC197#3096
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C2×#41 ◁ #520

Complements

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

NH ≅ G/Ntype
2. #27161403C2semidirect
3. #27161403C2semidirect
4. #27161397C2semidirect
5. #27161392C2semidirect
6. #27161119C2semidirect
7. #27161115C2semidirect
8. #27161083C2semidirect
9. C2×#20263246C2semidirect
10. #27161046C2semidirect
11. #520×C197C2semidirect
21. #20263259C22semidirect
22. #20263259C22semidirect
23. #20263259C22semidirect
24. #20263257C22semidirect
25. #20263257C22semidirect
26. #20263257C22semidirect
31. C22⋊#13096802C22semidirect
32. C22⋊#13096802C22semidirect
38. #117×C197C22semidirect
40. C2×C4⋊C4×C197C22semidirect
134. C197#3096semidirect
137. #520D197semidirect
161. #117C2×D197semidirect
163. C2×C4⋊C4C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271616432
#271614032
#271613972
#271613922
#271611192
#271611152
#271610832
C2×#202632462
#271610462
#520×C1972
#3096197