Finite Group Databasebeta

← #38754086

Subgroups — #38754086

Order 25216.

348 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 121 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#27161636C211yesmaximal, characteristic, complemented
2#27161613C211yesmaximal, characteristic
3#27161387C212yesmaximal, complemented
4#27161375C211yesmaximal, characteristic, complemented
5#27161370C211yesmaximal, characteristic, complemented
6#27161337C212yesmaximal, complemented
7#27161325C212yesmaximal
8#27161302C212yesmaximal, complemented
9#27161290C211yesmaximal, characteristic, complemented
10#27161285C211yesmaximal, characteristic, complemented
11#423×C197C211yesmaximal, characteristic, complemented
12C2×Dic788C2212yes—
13C2×#15691347C2212yes—
14#20263326C2211yescharacteristic
15#20263326C2211yescharacteristic
16#20263320—21no—
17#20263318—21no—
18#20263318C2211yescharacteristic
19#20263314C2211yescharacteristic
20#20263309—21no—
21#20263305—21no—
22#20263305C2211yescharacteristic
23#20263305C2212yes—
24#20263300C2211yescharacteristic
25C4×Dic394—21no—
26C4×Dic394C2211yescharacteristic
27C4×Dic394C2212yes—
28C4×D788C2211yescharacteristic
29C4×C22⋊D197—21no—
30#20263285—21no—
31#20263283—21no—
32#20263260C2212yes—
33#20263258C2212yes—
34#20263251C2212yes—
35#20263251C2212yes—
36#20263251C2212yes—
37#20263248C2212yes—
38C2×M16×C197C2212yes—
39C2×C42×C197C2211yescharacteristic
40C4⋊C8×C197C2211yescharacteristic
41C4⋊C8×C197C2211yescharacteristic
42C4⋊C8×C197C2212yes—
43C2×Dic394—21no—
44C2×Dic394C2311yescharacteristic
45C2×Dic394C2312yes—
46C2×D788C2311yescharacteristic
47C2×C22⋊D197—21no—
48C2×C4×D197—21no—
49Dic788—42no—
50#15691347—42no—
51C4⋊Dic197—21no—
52C4⋊Dic197—21no—
53C4⋊Dic197—21no—
54C4⋊Dic197C2312yes—
55C4⋊Dic197C2311yescharacteristic
56C4⋊Dic197C2311yescharacteristic
57#15691337—21no—
58#15691337—21no—
59#15691337—21no—
60#15691337—22no—
61#15691337—22no—
62#15691337—21no—
63C22⋊Dic197—22no—
64C22⋊Dic197—21no—
65#15691335—21no—
66#15691335—21no—
67#15691335—21no—
68#15691335—21no—
69C4×Dic197—21no—
70C4×Dic197—22no—
71C22×C4×C197D411yescharacteristic
72C22×C4×C197D411yescharacteristic
73C22×C4×C197C2311yescharacteristic
74M16×C197—42no—
75C2×C8×C197C2312yes—
76C2×C8×C197C2312yes—
77C42×C197D411yescharacteristic
78C42×C197D411yescharacteristic
79C42×C197C2311yescharacteristic
80C42×C197C2311yescharacteristic
81C22×D197—21no—
82Dic394—22no—
83Dic394—21no—
84Dic394—21no—
85Dic394—22no—
86Dic394—42no—
87D788—21no—
88D788—21no—
89C22⋊D197—42no—
90C2×Dic197—21no—
91C2×Dic197—21no—
92C2×Dic197—22no—
93C2×Dic197—22no—
94C2×Dic197—21no—
95C4×D197—42no—
96C23×C197C2×D411yescharacteristic
97C2×C4×C197—22no—
98C2×C4×C197—21no—
99C2×C4×C197—21no—
100C2×C4×C197—41no—
101C2×C4×C197C2×D411yescharacteristic
102C2×C4×C197C2×D411yescharacteristic
103C2×C4×C197C2×D411yescharacteristic
104C2×C4×C197C2×D411yescharacteristic
105C2×C4×C197C2×D411yescharacteristic
106C2×C4×C197C2411yescharacteristic
107C8×C197—22no—
108C8×C197—22no—
109C22×C197—21no—
110C22×C197—21no—
111C22×C197—21no—
112C22×C197C22×D411yescharacteristic
113C2×D197—41no—
114C2×D197—41no—
115C2×D197—41no—
116C4×C197—21no—
117C4×C197—21no—
118C4×C197—21no—
119C4×C197—21no—
120C4×C197#13712yes—
121C4×C197H(5,𝔽2)11yescharacteristic
122C4×C1972−1+411yescharacteristic
123Dic197—41no—
124Dic197—42no—
125Dic197—41no—
126Dic197—41no—
127Dic197—42no—
128C2×C197—41no—
129C2×C197#53711yescharacteristic
130C2×C197C2×#13711yescharacteristic
131C2×C197#57511yescharacteristic
132D197—81no—
133C197#325211yescharacteristic, complemented, Sylow
134#3252—1971nomaximal, Sylow
135#533—1971no—
136#529—1971no—
137#479—1972no—
138#477—1971no—
139#476—1971no—
140#470—1972no—
141#468—1972no—
142#464—1972no—
143#462—1971no—
144#461—1971no—
145#423D19711yescharacteristic, complemented
146C2×Q16—1972no—
147C2×SD16—1972no—
148C2×M16C2×D19712yes—
149C4⋊Q8—1971no—
150C4⋊Q8—1971no—
151#126—3941no—
152#125—3941no—
153#125—1971no—
154#124—1971no—
155#123—3941no—
156C22⋊Q8—3941no—
157C22⋊Q8—1971no—
158C22⋊Q8—1972no—
159#121—1971no—
160C4×Q8—3941no—
161C4×Q8—1971no—
162C4×Q8—1972no—
163C4×D4—3941no—
164C4×D4—1971no—
165#117—3941no—
166#117—3941no—
167C2×C42C2×D19711yescharacteristic
168#107—1972no—
169#106—1972no—
170C4⋊C8C2×D19711yescharacteristic
171C4⋊C8C2×D19711yescharacteristic
172C4⋊C8C2×D19712yes—
173#103—1972no—
174#103—1972no—
175#103—1972no—
176#102—1972no—
177C2×Q8—3941no—
178C2×Q8—1971no—
179C2×Q8—1972no—
180C2×D4—3941no—
181C2×D4—1971no—
182C22×C4—3941no—
183C22×C4D78811yescharacteristic
184C22×C4D78811yescharacteristic
185C22×C4C22×D19711yescharacteristic
186Q16—7882no—
187SD16—7882no—
188M16—42no—
189C2×C8C22×D19712yes—
190C2×C8C22×D19712yes—
191C4⋊C4—3941no—
192C4⋊C4—3941no—
193C4⋊C4—3941no—
194C4⋊C4—3942no—
195C4⋊C4—3942no—
196C4⋊C4—3941no—
197C4⋊C4—3941no—
198C4⋊C4—3941no—
199C4⋊C4—3941no—
200C4⋊C4—1972no—
201C4⋊C4—1971no—
202C4⋊C4—1971no—
203C22⋊C4—3942no—
204C22⋊C4—3941no—
205C22⋊C4—3941no—
206C22⋊C4—3941no—
207C22⋊C4—3941no—
208C22⋊C4—3941no—
209C42—3941no—
210C42—3942no—
211C42D78811yescharacteristic
212C42D78811yescharacteristic
213C42C22×D19711yescharacteristic
214C42C22×D19711yescharacteristic
215C23—3941no—
216C23C2×D78811yescharacteristic
217Q8—7882no—
218Q8—3942no—
219Q8—3941no—
220Q8—3941no—
221Q8—3942no—
222D4—7882no—
223D4—3941no—
224D4—3941no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—41no—
229C2×C4—7882no—
230C2×C4—3941no—
231C2×C4—3942no—
232C2×C4—3941no—
233C2×C4—3942no—
234C2×C4—3941no—
235C2×C4C2×D78811yescharacteristic
236C2×C4C2×D78811yescharacteristic
237C2×C4C2×D78811yescharacteristic
238C2×C4C2×D78811yescharacteristic
239C2×C4C2×D78811yescharacteristic
240C2×C4C23×D19711yescharacteristic
241C8—22no—
242C8—22no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—7881no—
247C22—7881no—
248C22—7881no—
249C22C22×D78811yescharacteristic
250C4—21no—
251C4—21no—
252C4—21no—
253C4—21no—
254C4—7881no—
255C4—7881no—
256C4—7882no—
257C4—7881no—
258C4—7882no—
259C4#2026335912yes—
260C4#2026337111yescharacteristic
261C4#2026337411yescharacteristic
262C2—41no—
263C2—15761no—
264C2#2716166011yescharacteristic
265C2C2×#2026335911yescharacteristic
266C2#2716187911yescharacteristic
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#423×C197C2
socleC22×C197C22×D4
prime radical#20263300C22
hypercenter#423D197
hypocenterC197#3252
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
1. #27161636C2semidirect
3. #27161387C2semidirect
4. #27161375C2semidirect
5. #27161370C2semidirect
6. #27161337C2semidirect
8. #27161302C2semidirect
9. #27161290C2semidirect
10. #27161285C2semidirect
11. #423×C197C2semidirect
133. C197#3252semidirect
145. #423D197semidirect

Maximal subgroups

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

subgroupindex
#271616362
#271616132
#271613872
#271613752
#271613702
#271613372
#271613252
#271613022
#271612902
#271612852
#423×C1972
#3252197