Finite Group Databasebeta

← #38754088

Subgroups — #38754088

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#27161616C211yesmaximal, characteristic
3#27161385C212yesmaximal, complemented
4#27161366C211yesmaximal, characteristic, complemented
5#27161336C212yesmaximal, complemented
6#27161326C212yesmaximal
7#27161304C212yesmaximal, complemented
8#27161289C211yesmaximal, characteristic, complemented
9#27161285C211yesmaximal, characteristic, complemented
10#27161101C211yesmaximal, characteristic, complemented
11#477×C197C211yesmaximal, characteristic, complemented
12C2×#15691348C2212yes—
13C2×#15691347C2212yes—
14C2×#15691341C2212yes—
15#20263326C2211yescharacteristic
16#20263321—21no—
17#20263317—21no—
18#20263317C2211yescharacteristic
19#20263315C2211yescharacteristic
20#20263311—21no—
21#20263306C2212yes—
22#20263302—21no—
23#20263302C2211yescharacteristic
24#20263301C2211yescharacteristic
25Q8×Dic197—21no—
26#20263291C2212yes—
27C4×D788—21no—
28C4×D788C2211yescharacteristic
29#20263284—21no—
30#20263283—21no—
31C42×D197C2211yescharacteristic
32#20263257C2212yes—
33C4⋊#13096802C2211yescharacteristic
34#20263255C2212yes—
35#20263251C2212yes—
36#20263249C2212yes—
37#20263246C2212yes—
38C4⋊Q8×C197C2211yescharacteristic
39C4×Q8×C197C2211yescharacteristic
40#107×C197C2212yes—
41C4⋊C8×C197C2211yescharacteristic
42#103×C197C2212yes—
43C2×Dic394C2312yes—
44C2×D788—21no—
45C2×D788C2311yescharacteristic
46C2×C4×D197—21no—
47C2×C4×D197D411yescharacteristic
48C2×C4×D197D411yescharacteristic
49C2×C4×D197C2311yescharacteristic
50#15691348—42no—
51#15691347—42no—
52#15691341—42no—
53C2×#13096802C2312yes—
54C4⋊Dic197—21no—
55C4⋊Dic197—22no—
56C4⋊Dic197—21no—
57C4⋊Dic197—21no—
58C4⋊Dic197C2311yescharacteristic
59#15691337—21no—
60#15691337—21no—
61#15691337—22no—
62#15691335—22no—
63#15691335—21no—
64#15691335—21no—
65#15691335—21no—
66#15691335—21no—
67#15691335—21no—
68C4×Dic197—22no—
69C4×Dic197D411yescharacteristic
70C4×Dic197D411yescharacteristic
71C4×Dic197C2311yescharacteristic
72C2×Q8×C197—21no—
73C2×Q8×C197C2311yescharacteristic
74C2×C8×C197C2312yes—
75C4⋊C4×C197—21no—
76C4⋊C4×C197—21no—
77C4⋊C4×C197C2312yes—
78C4⋊C4×C197C2311yescharacteristic
79C42×C197—21no—
80C42×C197C2311yescharacteristic
81C22×D197—21no—
82C22×D197C2×D411yescharacteristic
83Dic394—22no—
84Dic394—22no—
85D788—21no—
86D788—21no—
87D788—42no—
88C2×Dic197—22no—
89C2×Dic197—21no—
90C2×Dic197C2×D411yescharacteristic
91C2×Dic197C2×D411yescharacteristic
92C2×Dic197C2×D411yescharacteristic
93C4×D197—22no—
94C4×D197—21no—
95C4×D197—21no—
96C4×D197—41no—
97C4×D197—42no—
98#13096802—22no—
99Q8×C197—21no—
100Q8×C197—21no—
101Q8×C197—42no—
102C2×C4×C197—21no—
103C2×C4×C197—21no—
104C2×C4×C197—22no—
105C2×C4×C197C2×D411yescharacteristic
106C2×C4×C197C2×D411yescharacteristic
107C2×C4×C197C2411yescharacteristic
108C8×C197—22no—
109C22×C197C22×D411yescharacteristic
110C2×D197—21no—
111C2×D197—21no—
112C2×D197—21no—
113C2×D197—41no—
114C2×D197—41no—
115C2×D197—41no—
116C4×C197—21no—
117C4×C197—41no—
118C4×C197—42no—
119C4×C197—41no—
120C4×C197#13712yes—
121C4×C197H(5,𝔽2)11yescharacteristic
122C4×C1972−1+411yescharacteristic
123Dic197—21no—
124Dic197—21no—
125Dic197—21no—
126Dic197—42no—
127Dic197—41no—
128C2×C197#53711yescharacteristic
129C2×C197C2×#13711yescharacteristic
130C2×C197#57511yescharacteristic
131D197—41no—
132D197—81no—
133C197#325211yescharacteristic, complemented, Sylow
134#3252—1971nomaximal, Sylow
135#533—1971no—
136#529—1971no—
137#479—1972no—
138#477D19711yescharacteristic, complemented
139#476—1971no—
140#470—1972no—
141#468—1972no—
142#464—1972no—
143#462—1971no—
144#461—1971no—
145#423—1971no—
146C2×Q16—1972no—
147C2×SD16—1972no—
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—1971no—
158C22⋊Q8—1972no—
159#121—1971no—
160C4×Q8—3941no—
161C4×Q8—1972no—
162C4×Q8C2×D19711yescharacteristic
163C4×D4—3941no—
164C4×D4—1971no—
165#117—3941no—
166#117—3941no—
167C2×C42—1971no—
168#107C2×D19712yes—
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—21no—
178C2×Q8—1972no—
179C2×Q8C22×D19711yescharacteristic
180C2×D4—3941no—
181C2×D4—1971no—
182C22×C4—3941no—
183C22×C4—1971no—
184C22×C4—1971no—
185C22×C4—1971no—
186Q16—7882no—
187SD16—7882no—
188M16—7882no—
189C2×C8—1972no—
190C2×C8C22×D19712yes—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—3941no—
194C4⋊C4—3941no—
195C4⋊C4—3942no—
196C4⋊C4—3941no—
197C4⋊C4—3942no—
198C4⋊C4—3941no—
199C4⋊C4—3941no—
200C4⋊C4—1971no—
201C4⋊C4C22×D19712yes—
202C4⋊C4C22×D19711yescharacteristic
203C22⋊C4—3942no—
204C22⋊C4—3941no—
205C22⋊C4—3941no—
206C22⋊C4—3941no—
207C22⋊C4—3941no—
208C22⋊C4—3941no—
209C42—21no—
210C42—3942no—
211C42—1971no—
212C42—1971no—
213C42—1971no—
214C42C22×D19711yescharacteristic
215C23—3941no—
216C23—1971no—
217Q8—21no—
218Q8—21no—
219Q8—42no—
220Q8—3942no—
221Q8—3942no—
222D4—7882no—
223D4—3941no—
224D4—3941no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—7881no—
229C2×C4—7882no—
230C2×C4—3942no—
231C2×C4—3941no—
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—7881no—
244C22—7881no—
245C22—7881no—
246C22—3941no—
247C22—3941no—
248C22—3941no—
249C22C2×D4×D19711yescharacteristic
250C4—21no—
251C4—41no—
252C4—42no—
253C4—41no—
254C4—7881no—
255C4—7882no—
256C4—3941no—
257C4—3941no—
258C4—3941no—
259C4#2026335812yes—
260C4#2026337211yescharacteristic
261C4#2026337411yescharacteristic
262C2—7881no—
263C2—15761no—
264C2#2716165911yescharacteristic
265C2C2×#2026335811yescharacteristic
266C2#2716187511yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#477×C197C2
socleC22×C197C22×D4
prime radical#20263301C22
hypercenter#477D197
hypocenterC197#3252
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
1. #27161634C2semidirect
3. #27161385C2semidirect
4. #27161366C2semidirect
5. #27161336C2semidirect
7. #27161304C2semidirect
8. #27161289C2semidirect
9. #27161285C2semidirect
10. #27161101C2semidirect
11. #477×C197C2semidirect
133. C197#3252semidirect
138. #477D197semidirect

Maximal subgroups

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

subgroupindex
#271616342
#271616162
#271613852
#271613662
#271613362
#271613262
#271613042
#271612892
#271612852
#271611012
#477×C1972
#3252197