Finite Group Databasebeta

← #38744351

Subgroups — #38744351

Order 25216.

264 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 65 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#27161586C211yesmaximal, characteristic
2#27161063C211yesmaximal, characteristic, complemented
3#27160868C212yesmaximal, complemented
4#27160865C212yesmaximal, complemented
5#410×C197C211yesmaximal, characteristic, complemented
6C22×C4×D197C411yescharacteristic, complemented
7C2×#15691342—21no—
8#20263307C411yescharacteristic, complemented
9#20263307C2211yescharacteristic
10#20263302—21no—
11#20263297—21no—
12#20263294C411yescharacteristic, complemented
13#20263294C2211yescharacteristic
14C4×C22⋊D197—21no—
15C4×C22⋊D197—21no—
16#20263285C411yescharacteristic, complemented
17#20263245—21no—
18#20263243C2212yes—
19C2×#41×C197C2211yescharacteristic
20#117×C197—21no—
21#99×C197—21no—
22#99×C197C2212yes—
23C23×D197C2×C411yescharacteristic
24C2×C22⋊D197—21no—
25C2×C22⋊D197—21no—
26C22×Dic197C2×C411yescharacteristic
27C2×C4×D197—21no—
28C2×C4×D197—21no—
29C2×C4×D197—41no—
30C2×C4×D197—41no—
31#15691342—22no—
32#15691342—41no—
33C2×#13096802—41no—
34C4⋊Dic197—21no—
35#15691337—21no—
36#15691337—21no—
37C22⋊Dic197—21no—
38C22⋊Dic197—21no—
39C22⋊Dic197C2×C411yescharacteristic
40C22⋊Dic197C2×C411yescharacteristic
41#15691335—21no—
42#15691335—21no—
43C4×Dic197—21no—
44#41×C197—41no—
45#41×C197—41no—
46C2×Q8×C197D411yescharacteristic
47C2×D4×C197C2×C411yescharacteristic
48C2×D4×C197D411yescharacteristic
49C2×D4×C197C2311yescharacteristic
50C22×C4×C197C2×C411yescharacteristic
51C22×C4×C197D411yescharacteristic
52C22×C4×C197D411yescharacteristic
53C4⋊C4×C197—41no—
54C22⋊C4×C197—22no—
55C42×C197—41no—
56C22×D197—21no—
57C22×D197—42no—
58C22×D197—41no—
59C22⋊D197—42no—
60C22⋊D197—42no—
61C2×Dic197—21no—
62C2×Dic197—21no—
63C2×Dic197—21no—
64C2×Dic197—41no—
65C4×D197—42no—
66C4×D197—41no—
67C4×D197—41no—
68#13096802—42no—
69C23×C197C22⋊C411yescharacteristic
70C23×C197C22×C411yescharacteristic
71C23×C197C2×D411yescharacteristic
72Q8×C197—41no—
73D4×C197—41no—
74D4×C197—41no—
75D4×C197—41no—
76C2×C4×C197—21no—
77C2×C4×C197—21no—
78C2×C4×C197—21no—
79C2×C4×C197—21no—
80C2×C4×C197—21no—
81C2×C4×C197—21no—
82C2×C4×C197—42no—
83C2×C4×C197C22⋊C411yescharacteristic
84C2×C4×C197C22⋊C411yescharacteristic
85C2×C4×C197C22⋊C411yescharacteristic
86C2×C4×C197C2×D411yescharacteristic
87C22×C197—21no—
88C22×C197—21no—
89C22×C197—21no—
90C22×C197—41no—
91C22×C197—41no—
92C22×C197—41no—
93C22×C197C2×C22⋊C411yescharacteristic
94C2×D197—22no—
95C2×D197—41no—
96C2×D197—42no—
97C2×D197—41no—
98C2×D197—42no—
99C4×C197—21no—
100C4×C197—21no—
101C4×C197—21no—
102C4×C197—82no—
103C4×C197#9911yescharacteristic
104C4×C197#9911yescharacteristic
105Dic197—41no—
106Dic197—41no—
107Dic197—41no—
108C2×C197—21no—
109C2×C197—41no—
110C2×C197—41no—
111C2×C197—41no—
112C2×C197C2×#9911yescharacteristic
113D197—42no—
114C197#212011yescharacteristic, complemented, Sylow
115#2120—1971nomaximal, Sylow
116#525—1971no—
117#413—1971no—
118#410D19711yescharacteristic, complemented
119#352—1972no—
120AΣL(1,𝔽16)—1972no—
121C2×#41C2×D19711yescharacteristic
122C23×C4—1971no—
123C2×M16—3941no—
124#123—1971no—
125#123—1971no—
126C22⋊Q8—3941no—
127#121—3941no—
128#120—1971no—
129#120—1971no—
130C4×D4—3941no—
131C4×D4—3941no—
132#117—21no—
133#117—1971no—
134#101—3941no—
135#100—1972no—
136#99—21no—
137#99C2×D19712yes—
138C24—1971no—
139#41—41no—
140#41—41no—
141C2×Q8D78811yescharacteristic
142C2×D4—3941no—
143C2×D4—3941no—
144C2×D4C4×D19711yescharacteristic
145C2×D4C22⋊D19711yescharacteristic
146C2×D4C22×D19711yescharacteristic
147C22×C4—7881no—
148C22×C4—7881no—
149C22×C4—3941no—
150C22×C4—3941no—
151C22×C4—1971no—
152C22×C4C4×D19711yescharacteristic
153C22×C4C22⋊D19711yescharacteristic
154C22×C4D78811yescharacteristic
155M16—7881no—
156M16—3942no—
157C2×C8—7881no—
158C4⋊C4—41no—
159C4⋊C4—3941no—
160C4⋊C4—3941no—
161C4⋊C4—3941no—
162C22⋊C4—22no—
163C22⋊C4—3941no—
164C22⋊C4—3941no—
165C22⋊C4—3941no—
166C22⋊C4—3941no—
167C22⋊C4—1971no—
168C22⋊C4—1971no—
169C42—41no—
170C42—3941no—
171C23—7881no—
172C23—7882no—
173C23—3941no—
174C23#1569133511yescharacteristic
175C23C2×C4×D19711yescharacteristic
176C23C2×C22⋊D19711yescharacteristic
177Q8—41no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—7882no—
182D4—7882no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—7882no—
191C2×C4—7881no—
192C2×C4—7881no—
193C2×C4—7881no—
194C2×C4—3941no—
195C2×C4—3941no—
196C2×C4—3941no—
197C2×C4#1569133511yescharacteristic
198C2×C4#1569133511yescharacteristic
199C2×C4#1569133511yescharacteristic
200C2×C4C2×D78811yescharacteristic
201C8—7882no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—7881no—
209C22—7882no—
210C22—7882no—
211C22—7881no—
212C22—3942no—
213C22C2×#1569133511yescharacteristic
214C4—21no—
215C4—21no—
216C4—21no—
217C4—82no—
218C4—7881no—
219C4—7881no—
220C4—7881no—
221C4#2026324011yescharacteristic
222C4#2026324011yescharacteristic
223C2—21no—
224C2—41no—
225C2—41no—
226C2—41no—
227C2—7882no—
228C2C2×#2026324011yescharacteristic
subgroupG/H
centerC4#20263240
commutatorC23×C197C22×C4
FrattiniC2×D4C22×D197
Fitting#410×C197C2
socleC2×C197C2×#99
prime radical#27161586C2
cosocleC2×D4×C197C23
hypercenter#410D197
hypocenterC197#2120
LCSG ▷ C23×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #27161063C2semidirect
3. #27160868C2semidirect
4. #27160865C2semidirect
5. #410×C197C2semidirect
6. C22×C4×D197C4semidirect
8. #20263307C4semidirect
12. #20263294C4semidirect
16. #20263285C4semidirect
114. C197#2120semidirect
118. #410D197semidirect

Maximal subgroups

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

subgroupindex
#271615862
#271610632
#271608682
#271608652
#410×C1972
#2120197