Finite Group Databasebeta

← #38751174

Subgroups — #38751174

Order 25216.

716 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 209 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
1C2×#20263373C211yesmaximal, characteristic
2#27161556C212yesmaximal, complemented
3#27161053C211yesmaximal, characteristic, complemented
4#27161052C212yesmaximal, complemented
5#27161051C218yesmaximal, complemented
6#410×C197C211yesmaximal, characteristic, complemented
7#20263373—24no—
8#20263373—24no—
9C2×#15691358C412yescomplemented
10C2×#15691358C2211yescharacteristic
11C2×#15691357C412yescomplemented
12C2×#15691357C2214yes—
13C22×Dic394C412yescomplemented
14C22×Dic394C2211yescharacteristic
15C2×Q8×D197C412yescomplemented
16C4×Dic394—24no—
17#20263291—24no—
18#20263285C2212yes—
19#20263282C2218yes—
20#20263241C2214yes—
21#20263240C2218yes—
22C2×#41×C197C2211yescharacteristic
23#117×C197C2212yes—
24#99×C197C2214yes—
25#15691358—24no—
26#15691358—24no—
27#15691358—22no—
28#15691358—22no—
29#15691357—28no—
30#15691357—216no—
31C2×Dic394—24no—
32C2×Dic394—24no—
33C2×Dic394C2×C414yes—
34C2×Dic394C2×C411yescharacteristic
35C2×Dic394C2×C412yes—
36C2×Dic394C2×C412yes—
37C2×Dic394D414yes—
38C2×Dic394D414yes—
39Q8×D197—28no—
40C2×D788C2×C411yescharacteristic
41C2×C22⋊D197C2×C414yes—
42C2×C22⋊D197C2312yes—
43C22×Dic197C2×C414yes—
44C22×Dic197C2312yes—
45C2×C4×D197C2×C412yes—
46C2×C4×D197C2×C414yes—
47C4⋊Dic197—22no—
48#15691337—28no—
49C22⋊Dic197C2314yes—
50C4×Dic197—28no—
51C4×Dic197—22no—
52#41×C197—22no—
53#41×C197—22no—
54C2×Q8×C197C2×C411yescharacteristic
55C2×D4×C197C2×C411yescharacteristic
56C2×D4×C197C2312yes—
57C22×C4×C197C2×C412yes—
58C22×C4×C197C2311yescharacteristic
59C4⋊C4×C197—22no—
60C22⋊C4×C197C2314yes—
61C42×C197—22no—
62C22×D197C22×C412yes—
63Dic394—21no—
64Dic394—21no—
65Dic394—28no—
66Dic394C22⋊C418yes—
67Dic394C22⋊C414yes—
68Dic394C22⋊C414yes—
69D788—21no—
70D788—21no—
71C22⋊D197—24no—
72C22⋊D197—28no—
73C2×Dic197—28no—
74C2×Dic197—24no—
75C2×Dic197C22×C412yes—
76C2×Dic197C22×C414yes—
77C2×Dic197C2×D418yes—
78C4×D197—24no—
79C4×D197—28no—
80C23×C197C22×C412yes—
81C23×C197C2411yescharacteristic
82Q8×C197—22no—
83D4×C197—24no—
84D4×C197—22no—
85C2×C4×C197—22no—
86C2×C4×C197—22no—
87C2×C4×C197—24no—
88C2×C4×C197C22×C412yes—
89C2×C4×C197C22×C411yescharacteristic
90C2×C4×C197C22×C411yescharacteristic
91C2×C4×C197C2×D412yes—
92C2×C4×C197C2×D412yes—
93C22×C197—22no—
94C22×C197—41no—
95C22×C197—42no—
96C22×C197C23×C411yescharacteristic
97C22×C197C22×D412yes—
98C2×D197—22no—
99C2×D197—42no—
100C4×C197—22no—
101C4×C197—44no—
102C4×C197C2×C22⋊C411yescharacteristic
103C4×C197C2×C22⋊C411yescharacteristic
104C4×C197C2×C22⋊C412yes—
105Dic197—22no—
106Dic197—44no—
107Dic197C2×C22⋊C418yes—
108C2×C197—22no—
109C2×C197—21no—
110C2×C197—42no—
111C2×C197C22×C22⋊C411yescharacteristic
112D197—42no—
113C197#288411yescharacteristic, complemented, Sylow
114#2884—1971nomaximal, Sylow
115C2×2−1+4—1971no—
116#519—1972no—
117#410—1971no—
118#410—1972no—
119#410—1978no—
120#410D19711yescharacteristic, complemented
1212−1+4—3944no—
1222−1+4—3944no—
123C2×#41—1972no—
124C2×#41—1974no—
125C2×#41—1972no—
126C2×#41—1971no—
127C2×#41C2×D19711yescharacteristic
128C22×Q8—1972no—
129C22×Q8—1972no—
130C22×Q8—1971no—
131C4×Q8—3944no—
132C4×Q8—3944no—
133#117—1972no—
134#117—1978no—
135#117C2×D19712yes—
136#99—1974no—
137#99—1978no—
138#99C2×D19714yes—
139#41—22no—
140#41—22no—
141#41—3944no—
142#41—3944no—
143#41—3948no—
144#41—39416no—
145#41—3942no—
146#41—3942no—
147C2×Q8—3944no—
148C2×Q8—3944no—
149C2×Q8—3948no—
150C2×Q8—1971no—
151C2×Q8—1974no—
152C2×Q8—1972no—
153C2×Q8—1972no—
154C2×Q8—1974no—
155C2×Q8—1974no—
156C2×Q8C4×D19711yescharacteristic
157C2×D4—1974no—
158C2×D4—1972no—
159C2×D4—1971no—
160C2×D4C4×D19711yescharacteristic
161C2×D4C22×D19712yes—
162C22×C4—1974no—
163C22×C4—1974no—
164C22×C4—1972no—
165C22×C4—1972no—
166C22×C4C4×D19712yes—
167C22×C4C22×D19711yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—3948no—
170C4⋊C4—3942no—
171C22⋊C4—1974no—
172C22⋊C4C22×D19714yes—
173C42—22no—
174C42—3948no—
175C42—3942no—
176C23—1972no—
177C23C2×C4×D19712yes—
178C23C23×D19711yescharacteristic
179Q8—22no—
180Q8—3948no—
181Q8—3941no—
182Q8—3941no—
183Q8—1974no—
184Q8—1974no—
185Q8—1978no—
186D4—24no—
187D4—22no—
188D4—3944no—
189D4—3948no—
190D4—3941no—
191D4—3941no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—3948no—
196C2×C4—3944no—
197C2×C4—3948no—
198C2×C4—3944no—
199C2×C4—1972no—
200C2×C4—1974no—
201C2×C4—1978no—
202C2×C4C2×C4×D19712yes—
203C2×C4C2×C4×D19711yescharacteristic
204C2×C4C2×C4×D19711yescharacteristic
205C2×C4D4×D19712yes—
206C2×C4D4×D19712yes—
207C22—22no—
208C22—41no—
209C22—42no—
210C22—7882no—
211C22—3942no—
212C22C22×C4×D19711yescharacteristic
213C22C2×D4×D19712yes—
214C4—22no—
215C4—44no—
216C4—7884no—
217C4—3942no—
218C4—1978no—
219C4C22⋊C4×D19711yescharacteristic
220C4C22⋊C4×D19711yescharacteristic
221C4C22⋊C4×D19712yes—
222C2—22no—
223C2—21no—
224C2—42no—
225C2—7882no—
226C2C2×C22⋊C4×D19711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D197
commutatorC22×C197C23×C4
FrattiniC23C23×D197
Fitting#410×C197C2
socleC2×C197C22×C22⋊C4
prime radicalC2×#20263373C2
cosocleC23×C197C24
hypercenter#410D197
hypocenterC197#2884
LCSG ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #27161556C2semidirect
3. #27161053C2semidirect
4. #27161052C2semidirect
5. #27161051C2semidirect
6. #410×C197C2semidirect
9. C2×#15691358C4semidirect
11. C2×#15691357C4semidirect
13. C22×Dic394C4semidirect
15. C2×Q8×D197C4semidirect
113. C197#2884semidirect
120. #410D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633732
#271615562
#271610532
#271610522
#271610512
#410×C1972
#2884197