Finite Group Databasebeta

← #38754081

Subgroups — #38754081

Order 25216.

348 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 113 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#27161634C211yesmaximal, characteristic, complemented
2#27161611C211yesmaximal, characteristic
3#27161388C212yesmaximal, complemented
4#27161375C211yesmaximal, characteristic, complemented
5#27161369C211yesmaximal, characteristic, complemented
6#27161336C212yesmaximal, complemented
7#27161322C212yesmaximal
8#27161299C212yesmaximal, complemented
9#27161286C211yesmaximal, characteristic, complemented
10#27161101C211yesmaximal, characteristic, complemented
11#461×C197C211yesmaximal, characteristic, complemented
12C2×#15691348C2212yes—
13C2×#15691341C2212yes—
14#20263326C2211yescharacteristic
15#20263319—21no—
16#20263317—21no—
17#20263317C2211yescharacteristic
18#20263313C2211yescharacteristic
19#20263308—21no—
20#20263306—21no—
21#20263306C2211yescharacteristic
22#20263302C2212yes—
23#20263297C2211yescharacteristic
24Q8×Dic197C2212yes—
25C4×Dic394C2211yescharacteristic
26#20263291—21no—
27C4×C22⋊D197—21no—
28#20263285—21no—
29#20263284—21no—
30C42×D197C2211yescharacteristic
31#20263260C2212yes—
32#20263257C2212yes—
33C4⋊#13096802C2211yescharacteristic
34#20263255C2212yes—
35#20263251C2212yes—
36#20263250C2212yes—
37#20263247C2212yes—
38C2×SD16×C197C2212yes—
39C4⋊Q8×C197C2211yescharacteristic
40C4×D4×C197C2211yescharacteristic
41C4⋊C8×C197C2211yescharacteristic
42#103×C197C2212yes—
43C2×Dic394—21no—
44C2×Dic394C2311yescharacteristic
45C2×C22⋊D197—21no—
46C2×C4×D197D411yescharacteristic
47C2×C4×D197D411yescharacteristic
48C2×C4×D197C2311yescharacteristic
49#15691348—42no—
50#15691341—42no—
51C2×#13096802C2312yes—
52C4⋊Dic197—21no—
53C4⋊Dic197—21no—
54C4⋊Dic197—22no—
55C4⋊Dic197—21no—
56C4⋊Dic197C2312yes—
57C4⋊Dic197C2311yescharacteristic
58#15691337—21no—
59#15691337—22no—
60#15691337—21no—
61C22⋊Dic197—21no—
62C22⋊Dic197—21no—
63C22⋊Dic197—21no—
64#15691335—22no—
65#15691335—21no—
66C4×Dic197—22no—
67C4×Dic197—21no—
68C4×Dic197D411yescharacteristic
69C4×Dic197D411yescharacteristic
70C4×Dic197C2311yescharacteristic
71C2×Q8×C197C2312yes—
72C2×D4×C197C2311yescharacteristic
73C22×C4×C197—21no—
74SD16×C197—42no—
75C2×C8×C197C2312yes—
76C4⋊C4×C197—21no—
77C4⋊C4×C197—21no—
78C4⋊C4×C197C2311yescharacteristic
79C22⋊C4×C197—21no—
80C42×C197C2311yescharacteristic
81C22×D197C2×D411yescharacteristic
82Dic394—21no—
83Dic394—21no—
84Dic394—42no—
85C22⋊D197—42no—
86C2×Dic197—21no—
87C2×Dic197—22no—
88C2×Dic197—21no—
89C2×Dic197C2×D411yescharacteristic
90C2×Dic197C2×D411yescharacteristic
91C2×Dic197C2×D411yescharacteristic
92C4×D197—22no—
93C4×D197—21no—
94C4×D197—21no—
95C4×D197—41no—
96#13096802—22no—
97C23×C197—21no—
98Q8×C197—22no—
99Q8×C197—22no—
100D4×C197—21no—
101D4×C197—21no—
102C2×C4×C197—21no—
103C2×C4×C197—22no—
104C2×C4×C197—42no—
105C2×C4×C197C2×D411yescharacteristic
106C2×C4×C197C2×D411yescharacteristic
107C2×C4×C197C2411yescharacteristic
108C8×C197—22no—
109C22×C197—41no—
110C22×C197—41no—
111C22×C197—41no—
112C22×C197C22×D411yescharacteristic
113C2×D197—21no—
114C2×D197—21no—
115C2×D197—21no—
116C4×C197—21no—
117C4×C197—42no—
118C4×C197—41no—
119C4×C197#13712yes—
120C4×C197H(5,𝔽2)11yescharacteristic
121C4×C1972−1+411yescharacteristic
122Dic197—21no—
123Dic197—21no—
124Dic197—21no—
125Dic197—41no—
126Dic197—41no—
127Dic197—42no—
128C2×C197—81no—
129C2×C197#53711yescharacteristic
130C2×C197C2×#13711yescharacteristic
131C2×C197#57511yescharacteristic
132D197—41no—
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#461D19711yescharacteristic, complemented
145#423—1971no—
146C2×Q16—1972no—
147C2×SD16C2×D19712yes—
148C2×M16—1972no—
149C4⋊Q8—1971no—
150C4⋊Q8C2×D19711yescharacteristic
151#126—3941no—
152#125—3941no—
153#125—1971no—
154#124—1971no—
155#123—3941no—
156C22⋊Q8—3941no—
157C22⋊Q8—1972no—
158C22⋊Q8—1971no—
159#121—1971no—
160C4×Q8—3941no—
161C4×Q8—1972no—
162C4×Q8—1971no—
163C4×D4—3941no—
164C4×D4C2×D19711yescharacteristic
165#117—3941no—
166#117—3941no—
167C2×C42—1971no—
168#107—1972no—
169#106—1972no—
170C4⋊C8—1972no—
171C4⋊C8—1971no—
172C4⋊C8C2×D19711yescharacteristic
173#103—1972no—
174#103—1972no—
175#103C2×D19712yes—
176#102—1972no—
177C2×Q8—3941no—
178C2×Q8—1971no—
179C2×Q8C22×D19712yes—
180C2×D4—3941no—
181C2×D4C22×D19711yescharacteristic
182C22×C4—21no—
183C22×C4—1971no—
184C22×C4—1971no—
185C22×C4—1971no—
186Q16—7882no—
187SD16—42no—
188M16—7882no—
189C2×C8—1972no—
190C2×C8C22×D19712yes—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—3941no—
194C4⋊C4—3942no—
195C4⋊C4—3941no—
196C4⋊C4—3941no—
197C4⋊C4—3942no—
198C4⋊C4—3941no—
199C4⋊C4—3941no—
200C4⋊C4—1972no—
201C4⋊C4—1971no—
202C4⋊C4C22×D19711yescharacteristic
203C22⋊C4—21no—
204C22⋊C4—3942no—
205C22⋊C4—3941no—
206C22⋊C4—3941no—
207C22⋊C4—3941no—
208C22⋊C4—3941no—
209C42—3942no—
210C42—3941no—
211C42—1971no—
212C42—1971no—
213C42—1971no—
214C42C22×D19711yescharacteristic
215C23—21no—
216C23—1971no—
217Q8—22no—
218Q8—22no—
219Q8—7882no—
220Q8—3941no—
221Q8—3941no—
222D4—21no—
223D4—21no—
224D4—7882no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—42no—
228C2×C4—7881no—
229C2×C4—3941no—
230C2×C4—3941no—
231C2×C4—3942no—
232C2×C4—3942no—
233C2×C4—3941no—
234C2×C4—3941no—
235C2×C4—1971no—
236C2×C4—1971no—
237C2×C4—1971no—
238C2×C4D4×D19711yescharacteristic
239C2×C4D4×D19711yescharacteristic
240C2×C4C23×D19711yescharacteristic
241C8—22no—
242C8—3942no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—3941no—
247C22—3941no—
248C22—3941no—
249C22C2×D4×D19711yescharacteristic
250C4—21no—
251C4—41no—
252C4—42no—
253C4—7881no—
254C4—7882no—
255C4—7881no—
256C4—3941no—
257C4—3941no—
258C4—3941no—
259C4#2026335512yes—
260C4#2026337111yescharacteristic
261C4#2026337311yescharacteristic
262C2—81no—
263C2—7881no—
264C2#2716165811yescharacteristic
265C2C2×#2026335511yescharacteristic
266C2#2716187811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#461×C197C2
socleC22×C197C22×D4
prime radical#20263297C22
hypercenter#461D197
hypocenterC197#3252
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #27161634C2semidirect
3. #27161388C2semidirect
4. #27161375C2semidirect
5. #27161369C2semidirect
6. #27161336C2semidirect
8. #27161299C2semidirect
9. #27161286C2semidirect
10. #27161101C2semidirect
11. #461×C197C2semidirect
133. C197#3252semidirect
144. #461D197semidirect

Maximal subgroups

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

subgroupindex
#271616342
#271616112
#271613882
#271613752
#271613692
#271613362
#271613222
#271612992
#271612862
#271611012
#461×C1972
#3252197