Finite Group Databasebeta

← #38744350

Subgroups — #38744350

Order 25216.

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

normal subgroups only

This group has 264 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#27161585C211yesmaximal, characteristic
2#27161060C211yesmaximal, characteristic, complemented
3#27161051C211yesmaximal, characteristic, complemented
4#27160870C211yesmaximal, characteristic, complemented
5#27160868C211yesmaximal, characteristic, complemented
6#27160866C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C197C211yesmaximal, characteristic, complemented
8C2×#15691357C2211yescharacteristic
9C23×Dic197C411yescharacteristic, complemented
10M16×D197—21no—
11#20263310C411yescharacteristic, complemented
12#20263307C2211yescharacteristic
13#20263303—21no—
14#20263298—21no—
15#20263295C411yescharacteristic, complemented
16D4×Dic197—21no—
17#20263286—21no—
18#20263282—21no—
19#20263282C411yescharacteristic, complemented
20#20263244—21no—
21#20263243C2211yescharacteristic
22#20263241C2211yescharacteristic
23#20263240—21no—
24#120×C197C2211yescharacteristic
25#100×C197C2211yescharacteristic
26#99×C197C2211yescharacteristic
27#15691357—41no—
28#15691357—41no—
29C2×Dic394D411yescharacteristic
30C2×C22⋊D197—21no—
31C2×C22⋊D197C2×C411yescharacteristic
32C2×C22⋊D197D411yescharacteristic
33C22×Dic197—21no—
34C22×Dic197—21no—
35C22×Dic197—41no—
36C22×Dic197—41no—
37C22×Dic197C2×C411yescharacteristic
38C22×Dic197C2×C411yescharacteristic
39C22×Dic197D411yescharacteristic
40C2×C4×D197D411yescharacteristic
41#15691342—21no—
42#15691341—41no—
43C8×D197—41no—
44C4⋊Dic197—21no—
45#15691337—21no—
46#15691337—21no—
47#15691337—41no—
48C22⋊Dic197—21no—
49C22⋊Dic197—21no—
50C22⋊Dic197—21no—
51C22⋊Dic197C2×C411yescharacteristic
52#15691335—21no—
53C4×Dic197—21no—
54C4×Dic197—41no—
55C24×C197C2×C411yescharacteristic
56C2×D4×C197—21no—
57C2×D4×C197C2311yescharacteristic
58M16×C197—21no—
59C22⋊C4×C197—21no—
60C22⋊C4×C197—21no—
61C22⋊C4×C197C2×C411yescharacteristic
62C22×D197C22⋊C411yescharacteristic
63Dic394—41no—
64C22⋊D197—41no—
65C22⋊D197—41no—
66C22⋊D197—42no—
67C2×Dic197—21no—
68C2×Dic197—21no—
69C2×Dic197—21no—
70C2×Dic197—21no—
71C2×Dic197—21no—
72C2×Dic197—21no—
73C2×Dic197—42no—
74C2×Dic197—41no—
75C2×Dic197—41no—
76C2×Dic197—41no—
77C2×Dic197—41no—
78C2×Dic197C22⋊C411yescharacteristic
79C2×Dic197C22⋊C411yescharacteristic
80C2×Dic197C22⋊C411yescharacteristic
81C4×D197—21no—
82C4×D197—21no—
83#13096802—41no—
84C23×C197—21no—
85C23×C197—42no—
86C23×C197—41no—
87C23×C197C22×C411yescharacteristic
88C23×C197C2×D411yescharacteristic
89D4×C197—41no—
90D4×C197—42no—
91C2×C4×C197—21no—
92C2×C4×C197—41no—
93C2×C4×C197C2×D411yescharacteristic
94C8×C197—41no—
95C22×C197—22no—
96C22×C197—21no—
97C22×C197—21no—
98C22×C197—41no—
99C22×C197—41no—
100C22×C197—41no—
101C22×C197—42no—
102C22×C197—42no—
103C22×C197—41no—
104C22×C197C2×C22⋊C411yescharacteristic
105C2×D197—21no—
106C2×D197—41no—
107C4×C197—21no—
108C4×C197—41no—
109C4×C197—81no—
110Dic197—21no—
111Dic197—21no—
112Dic197—41no—
113Dic197—41no—
114Dic197—81no—
115Dic197#9911yescharacteristic
116Dic197#9911yescharacteristic
117C2×C197—21no—
118C2×C197—41no—
119C2×C197—42no—
120C2×C197—41no—
121C2×C197C2×#9911yescharacteristic
122D197—41no—
123C197#212011yescharacteristic, complemented, Sylow
124#2120—1971nomaximal, Sylow
125#525—1971no—
126#413—1971no—
127#410—1971no—
128#352—1971no—
129#352—1971no—
130AΣL(1,𝔽16)—1971no—
131AΣL(1,𝔽16)D19711yescharacteristic, complemented
132C2×#41—1971no—
133C23×C4—1971no—
134C2×M16—3941no—
135#123—1971no—
136#123—1971no—
137C22⋊Q8—3941no—
138#121—3941no—
139#120—1971no—
140#120C2×D19711yescharacteristic
141C4×D4—3941no—
142C4×D4—3941no—
143#117—3941no—
144#117—1971no—
145#101—3941no—
146#100—1971no—
147#100C2×D19711yescharacteristic
148#99—3941no—
149#99—1971no—
150#99C2×D19711yescharacteristic
151C24C4×D19711yescharacteristic
152#41—7881no—
153#41—7881no—
154C2×Q8—1971no—
155C2×D4—21no—
156C2×D4—3941no—
157C2×D4—1971no—
158C2×D4—1971no—
159C2×D4C22×D19711yescharacteristic
160C22×C4—7881no—
161C22×C4—7881no—
162C22×C4—3941no—
163C22×C4—3941no—
164C22×C4—1971no—
165C22×C4—1971no—
166C22×C4—1971no—
167C22×C4—1971no—
168M16—21no—
169M16—7881no—
170M16—3941no—
171C2×C8—7881no—
172C4⋊C4—7881no—
173C4⋊C4—3941no—
174C4⋊C4—3941no—
175C4⋊C4—3941no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—3941no—
179C22⋊C4—3941no—
180C22⋊C4—3941no—
181C22⋊C4—3941no—
182C22⋊C4—1971no—
183C22⋊C4C4×D19711yescharacteristic
184C42—7881no—
185C42—3941no—
186C23—21no—
187C23—41no—
188C23—42no—
189C23—1971no—
190C23C2×C4×D19711yescharacteristic
191C23D4×D19711yescharacteristic
192Q8—7881no—
193D4—41no—
194D4—42no—
195D4—7882no—
196D4—7881no—
197D4—7881no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—7882no—
201C2×C4—7881no—
202C2×C4—7881no—
203C2×C4—7881no—
204C2×C4—7881no—
205C2×C4—3941no—
206C2×C4—3941no—
207C2×C4—3941no—
208C2×C4—3941no—
209C2×C4—3941no—
210C2×C4—3941no—
211C2×C4—3941no—
212C2×C4—3941no—
213C2×C4—1971no—
214C2×C4—1971no—
215C2×C4—1971no—
216C2×C4D4×D19711yescharacteristic
217C8—41no—
218C8—7881no—
219C22—22no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—42no—
226C22—42no—
227C22—41no—
228C22—7881no—
229C22—3941no—
230C22C22⋊C4×D19711yescharacteristic
231C4—21no—
232C4—41no—
233C4—81no—
234C4—7881no—
235C4—7881no—
236C4—15761no—
237C4—3941no—
238C4—3941no—
239C4—1971no—
240C4—1971no—
241C2—21no—
242C2—41no—
243C2—42no—
244C2—41no—
245C2—7881no—
246C2#99×D19711yescharacteristic
subgroupG/H
centerC2#99×D197
commutatorC23×C197C22×C4
FrattiniC2×D4C22×D197
FittingAΣL(1,𝔽16)×C197C2
socleC2×C197C2×#99
prime radical#27161585C2
cosocleC2×D4×C197C23
hypercenterAΣL(1,𝔽16)D197
hypocenterC197#2120
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. #27161060C2semidirect
3. #27161051C2semidirect
4. #27160870C2semidirect
5. #27160868C2semidirect
6. #27160866C2semidirect
7. AΣL(1,𝔽16)×C197C2semidirect
9. C23×Dic197C4semidirect
11. #20263310C4semidirect
15. #20263295C4semidirect
19. #20263282C4semidirect
123. C197#2120semidirect
131. AΣL(1,𝔽16)D197semidirect

Maximal subgroups

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

subgroupindex
#271615852
#271610602
#271610512
#271608702
#271608682
#271608662
AΣL(1,𝔽16)×C1972
#2120197