Finite Group Databasebeta

← #38744357

Subgroups — #38744357

Order 25216.

280 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 55 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#27161640C211yesmaximal, characteristic
2#100×D197C211yesmaximal, characteristic, complemented
3#99×D197C211yesmaximal, characteristic, complemented
4#27160869C211yesmaximal, characteristic, complemented
5#27160867C211yesmaximal, characteristic, complemented
6#27160865C211yesmaximal, characteristic, complemented
7#352×C197C211yesmaximal, characteristic, complemented
8C22×D788C411yescharacteristic, complemented
9C2×D4×D197C2211yescharacteristic
10M16×D197—21no—
11#20263311—21no—
12#20263309C411yescharacteristic, complemented
13#20263301—21no—
14#20263299—21no—
15#20263296—21no—
16#20263296C411yescharacteristic, complemented
17#20263294C2211yescharacteristic
18C22⋊C4×D197—21no—
19C22⋊C4×D197C411yescharacteristic, complemented
20#20263243C2211yescharacteristic
21#20263242—21no—
22#20263241C2211yescharacteristic
23#20263240—21no—
24#123×C197C2211yescharacteristic
25#100×C197C2211yescharacteristic
26#99×C197C2211yescharacteristic
27C23×D197C2×C411yescharacteristic
28C23×D197C2×C411yescharacteristic, complemented
29C23×D197D411yescharacteristic
30C2×D788—21no—
31C2×D788—21no—
32C2×D788—21no—
33C2×D788—41no—
34C2×D788—41no—
35C2×D788D411yescharacteristic
36C2×C22⋊D197—21no—
37C2×C22⋊D197C2×C411yescharacteristic
38C2×C22⋊D197D411yescharacteristic
39D4×D197—41no—
40D4×D197—41no—
41C2×C4×D197—21no—
42C2×C4×D197—41no—
43C2×C4×D197D411yescharacteristic
44#15691342—21no—
45#15691341—41no—
46C8×D197—41no—
47#15691337—21no—
48C22⋊Dic197—21no—
49C22⋊Dic197C2×C411yescharacteristic, complemented
50#15691335—21no—
51#15691335—21no—
52#15691335—21no—
53#15691335—21no—
54#15691335—41no—
55C2×D4×C197C2311yescharacteristic
56C22×C4×C197C2×C411yescharacteristic, complemented
57M16×C197—21no—
58C4⋊C4×C197—21no—
59C22⋊C4×C197—21no—
60C22⋊C4×C197—21no—
61C22⋊C4×C197C2×C411yescharacteristic, complemented
62C22×D197—21no—
63C22×D197—21no—
64C22×D197—21no—
65C22×D197—21no—
66C22×D197—21no—
67C22×D197—41no—
68C22×D197—81no—
69C22×D197—81no—
70C22×D197—81no—
71C22×D197C22⋊C411yescharacteristic
72C22×D197C22⋊C411yescharacteristic
73C22×D197C22⋊C411yescharacteristic
74D788—41no—
75D788—42no—
76D788—41no—
77D788—41no—
78D788—81no—
79C22⋊D197—41no—
80C22⋊D197—41no—
81C22⋊D197—81no—
82C2×Dic197—21no—
83C2×Dic197—41no—
84C2×Dic197C22⋊C411yescharacteristic
85C4×D197—21no—
86C4×D197—21no—
87C4×D197—81no—
88C4×D197—82no—
89#13096802—41no—
90C23×C197C22×C411yescharacteristic
91C23×C197C2×D411yescharacteristic
92D4×C197—41no—
93C2×C4×C197—21no—
94C2×C4×C197—21no—
95C2×C4×C197—41no—
96C2×C4×C197—41no—
97C2×C4×C197C2×D411yescharacteristic
98C8×C197—41no—
99C22×C197—21no—
100C22×C197—21no—
101C22×C197—41no—
102C22×C197—41no—
103C22×C197C2×C22⋊C411yescharacteristic
104C2×D197—21no—
105C2×D197—21no—
106C2×D197—41no—
107C2×D197—42no—
108C2×D197—41no—
109C2×D197—41no—
110C2×D197—41no—
111C2×D197—41no—
112C2×D197—41no—
113C2×D197—81no—
114C2×D197—81no—
115C2×D197—81no—
116C2×D197—82no—
117C2×D197#9911yescharacteristic
118C2×D197#9911yescharacteristic
119C4×C197—21no—
120C4×C197—41no—
121C4×C197—41no—
122C4×C197—81no—
123Dic197—21no—
124Dic197—41no—
125Dic197—81no—
126C2×C197—21no—
127C2×C197—41no—
128C2×C197—41no—
129C2×C197C2×#9911yescharacteristic
130D197—21no—
131D197—21no—
132D197—41no—
133D197—41no—
134D197—81no—
135C197#212111yescharacteristic, complemented, Sylow
136#2121—1971nomaximal, Sylow
137#534—1971no—
138C2×#100—1971no—
139C2×#99—1971no—
140#352—1971no—
141#352D19711yescharacteristic, complemented
142AΣL(1,𝔽16)—1971no—
143AΣL(1,𝔽16)—1971no—
144C22×D4—1971no—
145C22×D4—1971no—
146C2×M16—3941no—
147#123—3941no—
148#123—1971no—
149#123C2×D19711yescharacteristic
150#121—3941no—
151#121—3941no—
152#120—3941no—
153#120—1971no—
154#120—1971no—
155C2×C22⋊C4—3941no—
156C2×C22⋊C4—1971no—
157#100—3941no—
158#100—1971no—
159#100C2×D19711yescharacteristic
160#99—3941no—
161#99—1971no—
162#99C2×D19711yescharacteristic
163C24—1971no—
164C24—1971no—
165C24—1971no—
166C2×D4—7881no—
167C2×D4—7881no—
168C2×D4—7881no—
169C2×D4—7881no—
170C2×D4—3941no—
171C2×D4—3941no—
172C2×D4—3941no—
173C2×D4—3941no—
174C2×D4—1971no—
175C2×D4—1971no—
176C2×D4—1971no—
177C2×D4C22×D19711yescharacteristic
178C22×C4—7881no—
179C22×C4—3941no—
180C22×C4—1971no—
181C22×C4C4×D19711yescharacteristic, complemented
182M16—21no—
183M16—7881no—
184M16—3941no—
185C2×C8—7881no—
186C4⋊C4—21no—
187C4⋊C4—3941no—
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—7881no—
191C22⋊C4—3941no—
192C22⋊C4—3941no—
193C22⋊C4—3941no—
194C22⋊C4—3941no—
195C22⋊C4—3941no—
196C22⋊C4—1971no—
197C22⋊C4C4×D19711yescharacteristic, complemented
198C23—7881no—
199C23—15761no—
200C23—15761no—
201C23—15761no—
202C23—3941no—
203C23—3941no—
204C23—3941no—
205C23—3941no—
206C23—3941no—
207C23—1971no—
208C23—1971no—
209C23—1971no—
210C23C2×C4×D19711yescharacteristic
211C23D4×D19711yescharacteristic
212D4—41no—
213D4—7881no—
214D4—7882no—
215D4—7881no—
216D4—7881no—
217D4—7881no—
218D4—7881no—
219D4—15761no—
220D4—15761no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—7881no—
226C2×C4—15761no—
227C2×C4—15762no—
228C2×C4—3941no—
229C2×C4—3941no—
230C2×C4—3941no—
231C2×C4—1971no—
232C2×C4D4×D19711yescharacteristic
233C8—41no—
234C8—7881no—
235C22—21no—
236C22—21no—
237C22—41no—
238C22—41no—
239C22—7881no—
240C22—7881no—
241C22—7881no—
242C22—7881no—
243C22—7881no—
244C22—7882no—
245C22—7881no—
246C22—15761no—
247C22—15761no—
248C22—15761no—
249C22—15762no—
250C22—3941no—
251C22—3941no—
252C22—1971no—
253C22—1971no—
254C22C22⋊C4×D19711yescharacteristic
255C4—21no—
256C4—41no—
257C4—41no—
258C4—81no—
259C4—7881no—
260C4—15761no—
261C4—3941no—
262C2—21no—
263C2—41no—
264C2—41no—
265C2—7881no—
266C2—7881no—
267C2—15761no—
268C2—3941no—
269C2—3941no—
270C2#99×D19711yescharacteristic
subgroupG/H
centerC2#99×D197
commutatorC23×C197C22×C4
FrattiniC2×D4C22×D197
Fitting#352×C197C2
socleC2×C197C2×#99
prime radical#27161640C2
cosocleC2×D4×C197C23
hypercenter#352D197
hypocenterC197#2121
LCSG ▷ C23×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
2. #100×D197C2semidirect
3. #99×D197C2semidirect
4. #27160869C2semidirect
5. #27160867C2semidirect
6. #27160865C2semidirect
7. #352×C197C2semidirect
8. C22×D788C4semidirect
12. #20263309C4semidirect
16. #20263296C4semidirect
19. C22⋊C4×D197C4semidirect
28. C23×D197C2×C4semidirect
49. C22⋊Dic197C2×C4semidirect
56. C22×C4×C197C2×C4semidirect
61. C22⋊C4×C197C2×C4semidirect
135. C197#2121semidirect
141. #352D197semidirect
181. C22×C4C4×D197semidirect
197. C22⋊C4C4×D197semidirect

Maximal subgroups

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

subgroupindex
#271616402
#100×D1972
#99×D1972
#271608692
#271608672
#271608652
#352×C1972
#2121197