Finite Group Databasebeta

← #38752666

Subgroups — #38752666

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#27161646C211yesmaximal, characteristic
2#27161560C211yesmaximal, characteristic, complemented
3#27161402C212yesmaximal, complemented
4#27161401C212yesmaximal, complemented
5#27161396C212yesmaximal, complemented
6#27161391C212yesmaximal, complemented
7#27161119C211yesmaximal, characteristic, complemented
8#27161117C211yesmaximal, characteristic, complemented
9#27161084C211yesmaximal, characteristic, complemented
10#27161045C211yesmaximal, characteristic, complemented
11C2×#102×C197C211yesmaximal, characteristic, complemented
12C2×#15691358C2211yescharacteristic
13C2×#15691342C2211yescharacteristic
14#20263322—21no—
15#20263313—21no—
16#20263297C2212yes—
17#20263297C2212yes—
18#20263295—22no—
19#20263291—21no—
20#20263288—21no—
21C4×C22⋊D197—22no—
22#20263285C2211yescharacteristic
23#20263285C2211yescharacteristic
24#20263284C2212yescomplemented
25C4⋊C4×D197C2212yescomplemented
26#20263260C2211yescharacteristic, complemented
27#20263260C2211yescharacteristic, complemented
28#20263259C2212yescomplemented
29#20263258C2211yescharacteristic, complemented
30#20263258C2211yescharacteristic, complemented
31#20263257C2212yescomplemented
32#20263247C2212yes—
33#20263247C2212yes—
34#20263238C2212yescomplemented
35#20263238C2212yescomplemented
36C22×D4×C197C2211yescharacteristic
37C22×C8×C197C2211yescharacteristic
38C2×C4⋊C4×C197C2211yescharacteristic
39#102×C197C2212yes—
40#102×C197C2212yes—
41#15691358—22no—
42#15691358—22no—
43C2×Dic394D411yescharacteristic
44C2×D788D411yescharacteristic
45C2×C22⋊D197—22no—
46C2×C22⋊D197—22no—
47C2×C22⋊D197D412yes—
48C2×C4×D197—22no—
49C2×C4×D197C2312yes—
50#15691342—22no—
51C2×#13096802C2312yes—
52C4⋊Dic197C2311yescharacteristic
53C4⋊Dic197C2311yescharacteristic
54C4⋊Dic197C2311yescharacteristic
55C4⋊Dic197C2311yescharacteristic
56#15691337—22no—
57C22⋊Dic197—22no—
58C22⋊Dic197—22no—
59C22⋊Dic197—21no—
60C22⋊Dic197—21no—
61#15691335—22no—
62C4×Dic197—22no—
63C4×Dic197—21no—
64C4×Dic197—21no—
65C24×C197—21no—
66C2×D4×C197—21no—
67C2×D4×C197—22no—
68C2×D4×C197—22no—
69C2×D4×C197C2312yes—
70C22×C4×C197—21no—
71C22×C4×C197C2311yescharacteristic
72C2×C8×C197—22no—
73C2×C8×C197C2311yescharacteristic
74C2×C8×C197C2311yescharacteristic
75C4⋊C4×C197—21no—
76C4⋊C4×C197C2312yes—
77C22×D197C2×D412yes—
78Dic394—22no—
79D788—22no—
80C22⋊D197—24no—
81C22⋊D197—82no—
82C22⋊D197—82no—
83C2×Dic197—21no—
84C2×Dic197—21no—
85C2×Dic197—21no—
86C2×Dic197—21no—
87C2×Dic197C2×D412yes—
88C4×D197—22no—
89C4×D197—22no—
90C4×D197—82no—
91#13096802—22no—
92C23×C197—22no—
93C23×C197—42no—
94C23×C197—81no—
95C23×C197—81no—
96C23×C197C2×D411yescharacteristic
97D4×C197—22no—
98D4×C197—22no—
99D4×C197—42no—
100C2×C4×C197—22no—
101C2×C4×C197—81no—
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—42no—
111C22×C197—42no—
112C22×C197—42no—
113C22×C197—84no—
114C22×C197#12312yes—
115C22×C197C22×D411yescharacteristic
116C2×D197—22no—
117C2×D197—22no—
118C2×D197—22no—
119C4×C197—42no—
120C4×C197#12312yes—
121C4×C197C2×#4111yescharacteristic
122C4×C197C2×#4111yescharacteristic
123Dic197—22no—
124Dic197—41no—
125Dic197—41no—
126Dic197—41no—
127Dic197—41no—
128C2×C197—22no—
129C2×C197—82no—
130C2×C197C2×#12311yescharacteristic
131C2×C197#57611yescharacteristic
132C2×C197#57711yescharacteristic
133D197—42no—
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#427—1971no—
144#418—1971no—
145C2×#102D19711yescharacteristic, complemented
146#408—1971no—
147C2×#41—1971no—
148C22×D4C2×D19711yescharacteristic
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—1971no—
160#117—1971no—
161#117—1972no—
162C2×C4⋊C4—1972no—
163C2×C4⋊C4C2×D19711yescharacteristic
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#102C2×D19712yes—
173#102C2×D19712yes—
174C22⋊C8—1972no—
175C22⋊C8—1972no—
176C24—21no—
177#41—3942no—
178#41—3942no—
179C2×Q8—1971no—
180C2×D4—21no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—3942no—
184C2×D4—3942no—
185C2×D4—1972no—
186C2×D4—1971no—
187C2×D4C22×D19712yes—
188C22×C4—21no—
189C22×C4—3942no—
190C22×C4—1972no—
191C22×C4C22×D19711yescharacteristic
192M16—3942no—
193C2×C8—22no—
194C2×C8—1972no—
195C2×C8C22×D19711yescharacteristic
196C2×C8C22×D19711yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—3942no—
199C4⋊C4—1971no—
200C4⋊C4—1971no—
201C4⋊C4—1971no—
202C4⋊C4—1971no—
203C4⋊C4C22×D19712yes—
204C22⋊C4—3942no—
205C22⋊C4—3941no—
206C22⋊C4—3941no—
207C22⋊C4—3942no—
208C22⋊C4—3942no—
209C42—3942no—
210C42—3941no—
211C42—3941no—
212C23—22no—
213C23—42no—
214C23—81no—
215C23—81no—
216C23—1972no—
217C23D4×D19711yescharacteristic
218Q8—3942no—
219D4—22no—
220D4—22no—
221D4—42no—
222D4—15762no—
223D4—15762no—
224D4—3944no—
225D4—3942no—
226C2×C4—22no—
227C2×C4—81no—
228C2×C4—15762no—
229C2×C4—3941no—
230C2×C4—3941no—
231C2×C4—3941no—
232C2×C4—3941no—
233C2×C4—3942no—
234C2×C4—3942no—
235C2×C4—1972no—
236C2×C4D4×D19711yescharacteristic
237C2×C4#1569135712yes—
238C2×C4#1569135812yes—
239C2×C4C23×D19711yescharacteristic
240C8—21no—
241C8—21no—
242C8—3942no—
243C22—21no—
244C22—21no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—84no—
249C22—3942no—
250C22—3942no—
251C22—3942no—
252C22#2026330812yes—
253C22C2×D4×D19711yescharacteristic
254C4—42no—
255C4—7881no—
256C4—7881no—
257C4—7881no—
258C4—7881no—
259C4—3942no—
260C4#2026330812yes—
261C4C2×#1569135711yescharacteristic
262C4C2×#1569135811yescharacteristic
263C2—22no—
264C2—82no—
265C2—7882no—
266C2C2×#2026330811yescharacteristic
267C2#2716188011yescharacteristic
268C2#2716188511yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#102×C197C2
socleC22×C197C22×D4
prime radical#27161646C2
hypercenterC2×#102D197
hypocenterC197#3096
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. #27161560C2semidirect
3. #27161402C2semidirect
4. #27161401C2semidirect
5. #27161396C2semidirect
6. #27161391C2semidirect
7. #27161119C2semidirect
8. #27161117C2semidirect
9. #27161084C2semidirect
10. #27161045C2semidirect
11. C2×#102×C197C2semidirect
24. #20263284C22semidirect
25. C4⋊C4×D197C22semidirect
26. #20263260C22semidirect
27. #20263260C22semidirect
28. #20263259C22semidirect
29. #20263258C22semidirect
30. #20263258C22semidirect
31. #20263257C22semidirect
34. #20263238C22semidirect
35. #20263238C22semidirect
134. C197#3096semidirect
145. C2×#102D197semidirect

Maximal subgroups

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

subgroupindex
#271616462
#271615602
#271614022
#271614012
#271613962
#271613912
#271611192
#271611172
#271610842
#271610452
C2×#102×C1972
#3096197