Finite Group Databasebeta

← #38752673

Subgroups — #38752673

Order 25216.

418 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 125 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#27161648C211yesmaximal, characteristic
2#27161559C211yesmaximal, characteristic, complemented
3#27161406C212yesmaximal, complemented
4#27161405C212yesmaximal, complemented
5#27161399C212yesmaximal, complemented
6#27161394C212yesmaximal, complemented
7#27161122C211yesmaximal, characteristic, complemented
8#27161083C211yesmaximal, characteristic, complemented
9C2×#20263248C211yesmaximal, characteristic, complemented
10#27161045C211yesmaximal, characteristic, complemented
11#427×C197C211yesmaximal, characteristic, complemented
12C2×#15691358C2211yescharacteristic
13C22×D788C2211yescharacteristic
14C2×#15691342C2211yescharacteristic
15#20263323—21no—
16#20263314—21no—
17#20263301C2212yes—
18#20263301C2212yes—
19#20263296—22no—
20C4×Dic394—21no—
21C4×D788—21no—
22#20263286—22no—
23#20263284C2212yescomplemented
24C2×C4⋊Dic197C2211yescharacteristic
25C4⋊C4×D197C2212yescomplemented
26#20263259C2212yescomplemented
27#20263257C2212yescomplemented
28#20263248C2212yes—
29#20263248C2212yes—
30#20263246C2212yes—
31#20263246C2212yes—
32#20263238C2212yescomplemented
33#20263238C2212yescomplemented
34C22×C8×C197C2211yescharacteristic
35#117×C197C2211yescharacteristic
36#117×C197C2211yescharacteristic
37#107×C197C2211yescharacteristic, complemented
38#107×C197C2211yescharacteristic, complemented
39#106×C197C2211yescharacteristic, complemented
40#106×C197C2211yescharacteristic, complemented
41C23×D197—21no—
42#15691358—22no—
43#15691358—22no—
44C2×Dic394D411yescharacteristic
45C2×D788—22no—
46C2×D788—22no—
47C2×D788—21no—
48C2×D788—22no—
49C2×D788—22no—
50C2×D788D411yescharacteristic
51C2×D788C2312yes—
52C2×C22⋊D197D412yes—
53C22×Dic197—21no—
54C2×C4×D197—22no—
55C2×C4×D197C2312yes—
56#15691342—22no—
57C2×#13096802C2312yes—
58C4⋊Dic197—21no—
59C4⋊Dic197C2312yes—
60#15691337—22no—
61#15691335—22no—
62#15691335—22no—
63#15691335—22no—
64C4×Dic197—22no—
65C22×C4×C197C2311yescharacteristic
66C2×C8×C197—22no—
67C2×C8×C197C2311yescharacteristic
68C2×C8×C197C2311yescharacteristic
69C4⋊C4×C197C2311yescharacteristic
70C4⋊C4×C197C2311yescharacteristic
71C4⋊C4×C197C2311yescharacteristic
72C4⋊C4×C197C2311yescharacteristic
73C22⋊C4×C197—21no—
74C22⋊C4×C197—21no—
75C42×C197—21no—
76C42×C197—21no—
77C22×D197—22no—
78C22×D197—42no—
79C22×D197—81no—
80C22×D197—81no—
81C22×D197C2×D412yes—
82Dic394—22no—
83D788—22no—
84D788—22no—
85D788—22no—
86D788—42no—
87D788—82no—
88D788—82no—
89C22⋊D197—24no—
90C2×Dic197—22no—
91C2×Dic197—81no—
92C2×Dic197C2×D412yes—
93C4×D197—22no—
94C4×D197—22no—
95C4×D197—82no—
96#13096802—22no—
97C23×C197C2×D411yescharacteristic
98C2×C4×C197—21no—
99C2×C4×C197—21no—
100C2×C4×C197—21no—
101C2×C4×C197—21no—
102C2×C4×C197C2×D411yescharacteristic
103C2×C4×C197#4112yes—
104C2×C4×C197#4112yes—
105C2×C4×C197C2411yescharacteristic
106C8×C197—21no—
107C8×C197—21no—
108C22×C197—21no—
109C22×C197—21no—
110C22×C197#12312yes—
111C22×C197C22×D411yescharacteristic
112C2×D197—22no—
113C2×D197—22no—
114C2×D197—22no—
115C2×D197—42no—
116C2×D197—42no—
117C2×D197—42no—
118C2×D197—84no—
119C4×C197—41no—
120C4×C197—41no—
121C4×C197—41no—
122C4×C197—41no—
123C4×C197#12312yes—
124C4×C197C2×#4111yescharacteristic
125C4×C197C2×#4111yescharacteristic
126Dic197—22no—
127Dic197—42no—
128C2×C197—22no—
129C2×C197C2×#12311yescharacteristic
130C2×C197#57611yescharacteristic
131C2×C197#57711yescharacteristic
132D197—42no—
133D197—82no—
134C197#309611yescharacteristic, complemented, Sylow
135#3096—1971nomaximal, Sylow
136#535—1971no—
137#520—1971no—
138#482—1972no—
139#482—1972no—
140#481—1972no—
141#480—1972no—
142#428—1971no—
143#427D19711yescharacteristic, complemented
144#418—1971no—
145C2×#102—1971no—
146#408—1971no—
147C2×#41—1971no—
148C22×D4—1971no—
149C2×M16—1971no—
150C22×C8C2×D19711yescharacteristic
151#127—3941no—
152#124—3941no—
153#121—1972no—
154#121—1972no—
155#120—3942no—
156C4×Q8—3941no—
157C4×D4—3941no—
158C4×D4—3942no—
159#117—1972no—
160#117C2×D19711yescharacteristic
161#117C2×D19711yescharacteristic
162C2×C4⋊C4—1971no—
163C2×C4⋊C4—1972no—
164#107—1972no—
165#107C2×D19711yescharacteristic, complemented
166#107C2×D19711yescharacteristic, complemented
167#106—1972no—
168#106C2×D19711yescharacteristic, complemented
169#106C2×D19711yescharacteristic, complemented
170#102—1972no—
171#102—1972no—
172#102—1972no—
173#102—1972no—
174C22⋊C8—1972no—
175C22⋊C8—1972no—
176C24—3941no—
177#41—3942no—
178#41—3942no—
179C2×Q8—1971no—
180C2×D4—3942no—
181C2×D4—3942no—
182C2×D4—3941no—
183C2×D4—3942no—
184C2×D4—3942no—
185C2×D4—1972no—
186C2×D4—1971no—
187C2×D4—1972no—
188C22×C4—3942no—
189C22×C4—3941no—
190C22×C4—1972no—
191C22×C4C22×D19711yescharacteristic
192M16—3942no—
193C2×C8—22no—
194C2×C8—1972no—
195C2×C8C22×D19711yescharacteristic
196C2×C8C22×D19711yescharacteristic
197C4⋊C4—3942no—
198C4⋊C4—3941no—
199C4⋊C4—1972no—
200C4⋊C4C22×D19711yescharacteristic
201C4⋊C4C22×D19711yescharacteristic
202C4⋊C4C22×D19711yescharacteristic
203C4⋊C4C22×D19711yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—3942no—
207C22⋊C4—3942no—
208C22⋊C4—3942no—
209C42—21no—
210C42—21no—
211C42—3942no—
212C23—7882no—
213C23—15761no—
214C23—15761no—
215C23—3942no—
216C23—1972no—
217C23D4×D19711yescharacteristic
218Q8—3942no—
219D4—7882no—
220D4—15762no—
221D4—15762no—
222D4—3944no—
223D4—3942no—
224D4—3942no—
225D4—3942no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—15762no—
231C2×C4—15761no—
232C2×C4—3942no—
233C2×C4—3942no—
234C2×C4—3942no—
235C2×C4—1972no—
236C2×C4D4×D19711yescharacteristic
237C2×C4#1569135812yes—
238C2×C4#1569135912yes—
239C2×C4C23×D19711yescharacteristic
240C8—21no—
241C8—21no—
242C8—3942no—
243C22—21no—
244C22—21no—
245C22—7882no—
246C22—7882no—
247C22—7882no—
248C22—15764no—
249C22—3942no—
250C22—3942no—
251C22—3942no—
252C22#2026331112yes—
253C22C2×D4×D19711yescharacteristic
254C4—41no—
255C4—41no—
256C4—41no—
257C4—41no—
258C4—7882no—
259C4—3942no—
260C4#2026331112yes—
261C4C2×#1569135811yescharacteristic
262C4C2×#1569135911yescharacteristic
263C2—22no—
264C2—7882no—
265C2—15762no—
266C2C2×#2026331111yescharacteristic
267C2#2716188111yescharacteristic
268C2#2716188611yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#427×C197C2
socleC22×C197C22×D4
prime radical#27161648C2
hypercenter#427D197
hypocenterC197#3096
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
2. #27161559C2semidirect
3. #27161406C2semidirect
4. #27161405C2semidirect
5. #27161399C2semidirect
6. #27161394C2semidirect
7. #27161122C2semidirect
8. #27161083C2semidirect
9. C2×#20263248C2semidirect
10. #27161045C2semidirect
11. #427×C197C2semidirect
23. #20263284C22semidirect
25. C4⋊C4×D197C22semidirect
26. #20263259C22semidirect
27. #20263257C22semidirect
32. #20263238C22semidirect
33. #20263238C22semidirect
37. #107×C197C22semidirect
38. #107×C197C22semidirect
39. #106×C197C22semidirect
40. #106×C197C22semidirect
134. C197#3096semidirect
143. #427D197semidirect
165. #107C2×D197semidirect
166. #107C2×D197semidirect
168. #106C2×D197semidirect
169. #106C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271616482
#271615592
#271614062
#271614052
#271613992
#271613942
#271611222
#271610832
C2×#202632482
#271610452
#427×C1972
#3096197