Finite Group Databasebeta

← #38744355

Subgroups — #38744355

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#27161639C211yesmaximal, characteristic
2#100×D197C211yesmaximal, characteristic, complemented
3#99×D197C211yesmaximal, characteristic, complemented
4#27160870C211yesmaximal, characteristic, complemented
5#27160868C211yesmaximal, characteristic, complemented
6#27160866C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C197C211yesmaximal, characteristic, complemented
8C22×C22⋊D197C411yescharacteristic, complemented
9C2×D4×D197C2211yescharacteristic
10M16×D197—21no—
11#20263310C411yescharacteristic, complemented
12#20263308—21no—
13#20263307C2211yescharacteristic
14#20263299—21no—
15#20263297—21no—
16#20263295C411yescharacteristic, complemented
17#20263294—21no—
18C22⋊C4×D197—21no—
19C22⋊C4×D197C411yescharacteristic, complemented
20#20263243C2211yescharacteristic
21#20263242—21no—
22#20263241C2211yescharacteristic
23#20263240—21no—
24#120×C197C2211yescharacteristic
25#100×C197C2211yescharacteristic
26#99×C197C2211yescharacteristic
27C23×D197C2×C411yescharacteristic
28C23×D197D411yescharacteristic
29C2×D788D411yescharacteristic
30C2×C22⋊D197—21no—
31C2×C22⋊D197—21no—
32C2×C22⋊D197—21no—
33C2×C22⋊D197—41no—
34C2×C22⋊D197—41no—
35C2×C22⋊D197C2×C411yescharacteristic
36C2×C22⋊D197D411yescharacteristic
37D4×D197—41no—
38D4×D197—41no—
39C22×Dic197C2×C411yescharacteristic, complemented
40C2×C4×D197—21no—
41C2×C4×D197—41no—
42C2×C4×D197D411yescharacteristic
43#15691342—21no—
44#15691341—41no—
45C8×D197—41no—
46C4⋊Dic197—21no—
47#15691337—21no—
48C22⋊Dic197—21no—
49C22⋊Dic197—21no—
50C22⋊Dic197—21no—
51C22⋊Dic197C2×C411yescharacteristic, complemented
52#15691335—21no—
53#15691335—21no—
54#15691335—41no—
55C24×C197C2×C411yescharacteristic, complemented
56C2×D4×C197—21no—
57C2×D4×C197C2311yescharacteristic
58M16×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—81no—
67C22×D197—81no—
68C22×D197C22⋊C411yescharacteristic
69C22×D197C22⋊C411yescharacteristic
70C22×D197C22⋊C411yescharacteristic
71D788—41no—
72C22⋊D197—41no—
73C22⋊D197—41no—
74C22⋊D197—42no—
75C22⋊D197—41no—
76C22⋊D197—41no—
77C22⋊D197—81no—
78C2×Dic197—21no—
79C2×Dic197—21no—
80C2×Dic197—41no—
81C2×Dic197—41no—
82C2×Dic197C22⋊C411yescharacteristic
83C4×D197—21no—
84C4×D197—21no—
85C4×D197—81no—
86C4×D197—82no—
87#13096802—41no—
88C23×C197—21no—
89C23×C197—41no—
90C23×C197—81no—
91C23×C197C22×C411yescharacteristic
92C23×C197C2×D411yescharacteristic
93D4×C197—41no—
94D4×C197—81no—
95C2×C4×C197—21no—
96C2×C4×C197—41no—
97C2×C4×C197C2×D411yescharacteristic
98C8×C197—41no—
99C22×C197—21no—
100C22×C197—21no—
101C22×C197—41no—
102C22×C197—41no—
103C22×C197—41no—
104C22×C197—41no—
105C22×C197—41no—
106C22×C197—82no—
107C22×C197C2×C22⋊C411yescharacteristic
108C2×D197—21no—
109C2×D197—21no—
110C2×D197—41no—
111C2×D197—42no—
112C2×D197—41no—
113C2×D197—41no—
114C2×D197—81no—
115C2×D197—81no—
116C2×D197—81no—
117C2×D197#9911yescharacteristic
118C2×D197#9911yescharacteristic
119C4×C197—21no—
120C4×C197—41no—
121C4×C197—81no—
122Dic197—21no—
123Dic197—41no—
124Dic197—41no—
125Dic197—81no—
126C2×C197—21no—
127C2×C197—41no—
128C2×C197—41no—
129C2×C197—81no—
130C2×C197C2×#9911yescharacteristic
131D197—21no—
132D197—21no—
133D197—41no—
134D197—41no—
135C197#212111yescharacteristic, complemented, Sylow
136#2121—1971nomaximal, Sylow
137#534—1971no—
138C2×#100—1971no—
139C2×#99—1971no—
140#352—1971no—
141#352—1971no—
142AΣL(1,𝔽16)—1971no—
143AΣL(1,𝔽16)D19711yescharacteristic, complemented
144C22×D4—1971no—
145C22×D4—1971no—
146C2×M16—3941no—
147#123—3941no—
148#123—1971no—
149#123—1971no—
150#121—3941no—
151#121—3941no—
152#120—3941no—
153#120—1971no—
154#120C2×D19711yescharacteristic
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—
165C24C4×D19711yescharacteristic, complemented
166C2×D4—21no—
167C2×D4—7881no—
168C2×D4—7881no—
169C2×D4—7881no—
170C2×D4—7881no—
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×C4—1971no—
182M16—21no—
183M16—7881no—
184M16—3941no—
185C2×C8—7881no—
186C4⋊C4—3941no—
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—21no—
199C23—41no—
200C23—81no—
201C23—15761no—
202C23—15761no—
203C23—3941no—
204C23—3941no—
205C23—3941no—
206C23—3941no—
207C23—1971no—
208C23—1971no—
209C23—1971no—
210C23C2×C4×D19711yescharacteristic
211C23D4×D19711yescharacteristic
212D4—41no—
213D4—81no—
214D4—7881no—
215D4—7882no—
216D4—7881no—
217D4—7881no—
218D4—7881no—
219D4—7881no—
220D4—15761no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—7881no—
224C2×C4—7881no—
225C2×C4—15761no—
226C2×C4—15762no—
227C2×C4—3941no—
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—41no—
240C22—41no—
241C22—41no—
242C22—82no—
243C22—7881no—
244C22—7881no—
245C22—7881no—
246C22—7882no—
247C22—15761no—
248C22—15761no—
249C22—15761no—
250C22—3941no—
251C22—3941no—
252C22—1971no—
253C22—1971no—
254C22C22⋊C4×D19711yescharacteristic
255C4—21no—
256C4—41no—
257C4—81no—
258C4—7881no—
259C4—7881no—
260C4—15761no—
261C4—3941no—
262C2—21no—
263C2—41no—
264C2—41no—
265C2—81no—
266C2—7881no—
267C2—7881no—
268C2—3941no—
269C2—3941no—
270C2#99×D19711yescharacteristic
subgroupG/H
centerC2#99×D197
commutatorC23×C197C22×C4
FrattiniC2×D4C22×D197
FittingAΣL(1,𝔽16)×C197C2
socleC2×C197C2×#99
prime radical#27161639C2
cosocleC2×D4×C197C23
hypercenterAΣL(1,𝔽16)D197
hypocenterC197#2121
LCSG ▷ C23×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
2. #100×D197C2semidirect
3. #99×D197C2semidirect
4. #27160870C2semidirect
5. #27160868C2semidirect
6. #27160866C2semidirect
7. AΣL(1,𝔽16)×C197C2semidirect
8. C22×C22⋊D197C4semidirect
11. #20263310C4semidirect
16. #20263295C4semidirect
19. C22⋊C4×D197C4semidirect
39. C22×Dic197C2×C4semidirect
51. C22⋊Dic197C2×C4semidirect
55. C24×C197C2×C4semidirect
61. C22⋊C4×C197C2×C4semidirect
135. C197#2121semidirect
143. AΣL(1,𝔽16)D197semidirect
165. C24C4×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
#271616392
#100×D1972
#99×D1972
#271608702
#271608682
#271608662
AΣL(1,𝔽16)×C1972
#2121197