Finite Group Databasebeta

← #38741215

Subgroups — #38741215

Order 25216.

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

normal subgroups only

This group has 334 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#27161592C211yesmaximal, characteristic
2#27161041C211yesmaximal, characteristic, complemented
3#95×D197C211yesmaximal, characteristic, complemented
4#27160816C211yesmaximal, characteristic, complemented
5#27160814C211yesmaximal, characteristic, complemented
6#27160812C211yesmaximal, characteristic, complemented
7#328×C197C211yesmaximal, characteristic, complemented
8C2×#15691359C411yescharacteristic, complemented
9C22×C4×D197—21no—
10C22×C4×D197C2211yescharacteristic
11C2×#15691341—21no—
12#20263311—21no—
13#20263302C411yescharacteristic, complemented
14#20263301C411yescharacteristic, complemented
15#20263300C2211yescharacteristic
16#20263296—21no—
17#20263288—21no—
18#20263286—21no—
19#20263284C411yescharacteristic, complemented
20C22⋊#13096802C2211yescharacteristic
21#20263238—21no—
22#20263233—21no—
23#20263233C2211yescharacteristic
24C22⋊Q8×C197C2211yescharacteristic
25C22⋊C8×C197C2211yescharacteristic
26#95×C197C2211yescharacteristic
27C23×D197D411yescharacteristic
28#15691359—42no—
29C2×D788—21no—
30C2×D788—21no—
31C2×D788C2×C411yescharacteristic, complemented
32C2×C22⋊D197—21no—
33C22×Dic197—21no—
34C22×Dic197D411yescharacteristic
35C2×C4×D197—24no—
36C2×C4×D197—21no—
37C2×C4×D197—42no—
38C2×C4×D197—42no—
39C2×C4×D197—42no—
40C2×C4×D197C2×C411yescharacteristic
41C2×C4×D197C2×C411yescharacteristic
42C2×C4×D197D411yescharacteristic
43C2×C4×D197D411yescharacteristic
44#15691341—42no—
45C2×#13096802—21no—
46C4⋊Dic197C2×C411yescharacteristic, complemented
47#15691337—21no—
48#15691335—21no—
49#15691335—21no—
50#15691335—21no—
51#15691335—21no—
52C4×Dic197—21no—
53C2×Q8×C197C2×C411yescharacteristic, complemented
54C22×C4×C197—21no—
55C22×C4×C197C2311yescharacteristic
56C2×C8×C197—21no—
57C4⋊C4×C197—21no—
58C4⋊C4×C197C2×C411yescharacteristic, complemented
59C22⋊C4×C197—21no—
60C22×D197—21no—
61C22×D197—21no—
62C22×D197—22no—
63C22×D197—22no—
64C22×D197—21no—
65C22×D197C22⋊C411yescharacteristic
66C22×D197C22⋊C411yescharacteristic
67D788—41no—
68D788—42no—
69D788—42no—
70C22⋊D197—41no—
71C22⋊D197—41no—
72C2×Dic197—22no—
73C2×Dic197—21no—
74C2×Dic197—41no—
75C2×Dic197—41no—
76C2×Dic197—41no—
77C2×Dic197C22⋊C411yescharacteristic
78C2×Dic197C22⋊C411yescharacteristic
79C4×D197—22no—
80C4×D197—22no—
81C4×D197—41no—
82C4×D197—41no—
83C4×D197—42no—
84C4×D197—48no—
85#13096802—41no—
86C23×C197C2×D411yescharacteristic
87Q8×C197—41no—
88C2×C4×C197—22no—
89C2×C4×C197—21no—
90C2×C4×C197—21no—
91C2×C4×C197—41no—
92C2×C4×C197—41no—
93C2×C4×C197—41no—
94C2×C4×C197C22×C411yescharacteristic
95C2×C4×C197C2×D411yescharacteristic
96C8×C197—41no—
97C22×C197—21no—
98C22×C197—21no—
99C22×C197C2×C22⋊C411yescharacteristic
100C22×C197AGL(1,ℤ/8)11yescharacteristic
101C22×C197#13711yescharacteristic
102C2×D197—24no—
103C2×D197—24no—
104C2×D197—21no—
105C2×D197—21no—
106C2×D197—21no—
107C2×D197—21no—
108C2×D197—41no—
109C2×D197—41no—
110C2×D197—41no—
111C2×D197#9912yes—
112C2×D197#10412yes—
113C4×C197—21no—
114C4×C197—21no—
115C4×C197—42no—
116C4×C197—41no—
117C4×C197—41no—
118Dic197—21no—
119Dic197—21no—
120Dic197—42no—
121Dic197—41no—
122C2×C197—21no—
123C2×C197—21no—
124C2×C197C2×#9911yescharacteristic
125C2×C197#41711yescharacteristic
126C2×C197C2×#10411yescharacteristic
127D197—22no—
128D197—22no—
129D197—81no—
130C197#150411yescharacteristic, complemented, Sylow
131#1504—1971nomaximal, Sylow
132#525—1971no—
133#407—1971no—
134C2×#95—1971no—
135#328—1971no—
136#328D19711yescharacteristic, complemented
137#327—1971no—
138#327—1971no—
139C2×#41—1971no—
140C23×C4—3941no—
141C23×C4—1971no—
142C2×M16—3941no—
143#123—3941no—
144C22⋊Q8—1971no—
145C22⋊Q8C2×D19711yescharacteristic
146#121—1971no—
147#121—1971no—
148#120—3941no—
149C4×D4—3941no—
150C4×D4—3941no—
151#117—1971no—
152C22⋊C8—3941no—
153C22⋊C8—1971no—
154C22⋊C8C2×D19711yescharacteristic
155#95—3941no—
156#95—1971no—
157#95C2×D19711yescharacteristic
158C24—1971no—
159#41—7882no—
160C2×Q8C4×D19711yescharacteristic, complemented
161C2×D4—3941no—
162C2×D4—3941no—
163C2×D4—3941no—
164C2×D4—1971no—
165C22×C4—21no—
166C22×C4—7882no—
167C22×C4—7882no—
168C22×C4—7882no—
169C22×C4—3944no—
170C22×C4—3941no—
171C22×C4—3941no—
172C22×C4—1971no—
173C22×C4—1971no—
174C22×C4—1971no—
175C22×C4—1971no—
176C22×C4—1971no—
177C22×C4C22×D19711yescharacteristic
178M16—7882no—
179C2×C8—21no—
180C2×C8—3941no—
181C4⋊C4—21no—
182C4⋊C4—3941no—
183C4⋊C4—1971no—
184C4⋊C4C4×D19711yescharacteristic, complemented
185C22⋊C4—21no—
186C22⋊C4—3941no—
187C22⋊C4—3941no—
188C22⋊C4—3941no—
189C22⋊C4—3941no—
190C42—3941no—
191C23—3942no—
192C23—3942no—
193C23—3941no—
194C23—3941no—
195C23—3941no—
196C23—1971no—
197C23—1971no—
198C23D4×D19711yescharacteristic
199Q8—41no—
200D4—7881no—
201D4—7882no—
202D4—7882no—
203D4—7881no—
204D4—7881no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—7881no—
212C2×C4—7881no—
213C2×C4—7888no—
214C2×C4—7882no—
215C2×C4—7881no—
216C2×C4—7881no—
217C2×C4—7881no—
218C2×C4—3942no—
219C2×C4—3942no—
220C2×C4—3942no—
221C2×C4—3941no—
222C2×C4—1971no—
223C2×C4—1971no—
224C2×C4C2×C4×D19711yescharacteristic
225C2×C4D4×D19711yescharacteristic
226C8—41no—
227C8—7881no—
228C22—21no—
229C22—21no—
230C22—7881no—
231C22—7881no—
232C22—7881no—
233C22—3944no—
234C22—3944no—
235C22—3941no—
236C22—3941no—
237C22—3941no—
238C22—3941no—
239C22—1972no—
240C22—1972no—
241C22C22⋊C4×D19711yescharacteristic
242C22#2026335311yescharacteristic
243C22#2026335811yescharacteristic
244C4—21no—
245C4—21no—
246C4—42no—
247C4—41no—
248C4—41no—
249C4—7882no—
250C4—7881no—
251C4—3941no—
252C4—3941no—
253C2—21no—
254C2—21no—
255C2—15761no—
256C2—3942no—
257C2—3942no—
258C2#99×D19711yescharacteristic
259C2#2716107911yescharacteristic
260C2#104×D19711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D197
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#328×C197C2
socleC22×C197C2×C22⋊C4
prime radical#27161592C2
cosocleC22×C4×C197C23
hypercenter#328D197
hypocenterC197#1504
LCSG ▷ C2×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #27161041C2semidirect
3. #95×D197C2semidirect
4. #27160816C2semidirect
5. #27160814C2semidirect
6. #27160812C2semidirect
7. #328×C197C2semidirect
8. C2×#15691359C4semidirect
13. #20263302C4semidirect
14. #20263301C4semidirect
19. #20263284C4semidirect
31. C2×D788C2×C4semidirect
46. C4⋊Dic197C2×C4semidirect
53. C2×Q8×C197C2×C4semidirect
58. C4⋊C4×C197C2×C4semidirect
130. C197#1504semidirect
136. #328D197semidirect
160. C2×Q8C4×D197semidirect
184. C4⋊C4C4×D197semidirect

Maximal subgroups

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

subgroupindex
#271615922
#271610412
#95×D1972
#271608162
#271608142
#271608122
#328×C1972
#1504197