Finite Group Databasebeta

← #38741226

Subgroups — #38741226

Order 25216.

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

normal subgroups only

This group has 346 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#121×D197C211yesmaximal, characteristic
2#27161041C211yesmaximal, characteristic, complemented
3#27160926C211yesmaximal, characteristic, complemented
4#27160814C211yesmaximal, characteristic, complemented
5#27160813C211yesmaximal, characteristic, complemented
6#27160812C211yesmaximal, characteristic, complemented
7#327×C197C211yesmaximal, characteristic, complemented
8C2×D4×D197—21no—
9C2×D4×D197C411yescharacteristic, complemented
10C22×C4×D197C2211yescharacteristic
11C2×#15691341—21no—
12#20263301C411yescharacteristic, complemented
13#20263300C2211yescharacteristic
14#20263299—21no—
15#20263298—21no—
16#20263297C411yescharacteristic, complemented
17C4⋊C4×D197C411yescharacteristic, complemented
18C2×#15691335—21no—
19C22⋊C4×D197—21no—
20C2×C4×Dic197—21no—
21C22⋊#13096802C2211yescharacteristic, complemented
22#20263238—21no—
23#20263233C2211yescharacteristic, complemented
24#121×C197C2211yescharacteristic
25C22⋊C8×C197C2211yescharacteristic, complemented
26#95×C197C2211yescharacteristic, complemented
27C23×D197—21no—
28C23×D197D411yescharacteristic
29C2×D788—21no—
30C2×D788C2×C411yescharacteristic
31C2×C22⋊D197—21no—
32C2×C22⋊D197—21no—
33C2×C22⋊D197—21no—
34D4×D197—42no—
35D4×D197—42no—
36D4×D197—42no—
37C22×Dic197—21no—
38C22×Dic197D411yescharacteristic
39C2×C4×D197—21no—
40C2×C4×D197—42no—
41C2×C4×D197C2×C411yescharacteristic
42C2×C4×D197C2×C411yescharacteristic
43C2×C4×D197D411yescharacteristic
44C2×C4×D197D411yescharacteristic
45#15691341—42no—
46C2×#13096802—21no—
47C4⋊Dic197C2×C411yescharacteristic
48#15691337—21no—
49C22⋊Dic197—21no—
50#15691335—24no—
51#15691335—21no—
52#15691335—21no—
53C4×Dic197—42no—
54C2×D4×C197—21no—
55C2×D4×C197C2×C411yescharacteristic
56C22×C4×C197—21no—
57C22×C4×C197C2311yescharacteristic
58C2×C8×C197—21no—
59C4⋊C4×C197C2×C411yescharacteristic
60C22⋊C4×C197—21no—
61C22×D197—21no—
62C22×D197—21no—
63C22×D197—22no—
64C22×D197—21no—
65C22×D197—41no—
66C22×D197—42no—
67C22×D197—42no—
68C22×D197—42no—
69C22×D197C22⋊C411yescharacteristic
70C22×D197C22⋊C411yescharacteristic
71D788—41no—
72D788—42no—
73C22⋊D197—42no—
74C22⋊D197—41no—
75C22⋊D197—41no—
76C22⋊D197—42no—
77C2×Dic197—22no—
78C2×Dic197—22no—
79C2×Dic197—21no—
80C2×Dic197—41no—
81C2×Dic197—41no—
82C2×Dic197C22⋊C411yescharacteristic
83C2×Dic197C22⋊C411yescharacteristic
84C4×D197—22no—
85C4×D197—22no—
86C4×D197—41no—
87C4×D197—41no—
88C4×D197—42no—
89#13096802—41no—
90C23×C197—21no—
91C23×C197C2×D411yescharacteristic
92D4×C197—41no—
93D4×C197—41no—
94D4×C197—41no—
95C2×C4×C197—22no—
96C2×C4×C197—21no—
97C2×C4×C197—41no—
98C2×C4×C197—41no—
99C2×C4×C197—41no—
100C2×C4×C197C22×C411yescharacteristic
101C2×C4×C197C2×D411yescharacteristic
102C8×C197—41no—
103C22×C197—21no—
104C22×C197—21no—
105C22×C197—41no—
106C22×C197—41no—
107C22×C197—41no—
108C22×C197C2×C22⋊C411yescharacteristic
109C22×C197AGL(1,ℤ/8)11yescharacteristic
110C22×C197AGL(1,ℤ/8)11yescharacteristic
111C2×D197—21no—
112C2×D197—21no—
113C2×D197—21no—
114C2×D197—21no—
115C2×D197—21no—
116C2×D197—42no—
117C2×D197—42no—
118C2×D197—41no—
119C2×D197—41no—
120C2×D197—41no—
121C2×D197—84no—
122C2×D197#10412yes—
123C4×C197—21no—
124C4×C197—21no—
125C4×C197—42no—
126C4×C197—41no—
127Dic197—21no—
128Dic197—42no—
129Dic197—41no—
130Dic197#9912yes—
131C2×C197—21no—
132C2×C197—21no—
133C2×C197—81no—
134C2×C197C2×#9911yescharacteristic
135C2×C197#41811yescharacteristic
136C2×C197C2×#10411yescharacteristic
137D197—22no—
138D197—41no—
139D197—81no—
140C197#150711yescharacteristic, complemented, Sylow
141#1507—1971nomaximal, Sylow
142C2×#121—1971no—
143#407—1971no—
144C4×C22⋊C4—1971no—
145#327—1971no—
146#327—1971no—
147#327—1971no—
148#327D19711yescharacteristic, complemented
149C22×D4—3941no—
150C22×D4—1971no—
151C23×C4—1971no—
152C2×M16—3941no—
153#121—3941no—
154#121—3941no—
155#121—1971no—
156#121—1971no—
157#121—1971no—
158#121C2×D19711yescharacteristic
159C2×C4⋊C4—1971no—
160C2×C22⋊C4—3941no—
161C2×C22⋊C4—3941no—
162C2×C42—3941no—
163C22⋊C8—3941no—
164C22⋊C8—1971no—
165C22⋊C8C2×D19711yescharacteristic, complemented
166#95—1971no—
167#95C2×D19711yescharacteristic, complemented
168C24—3941no—
169C24—1971no—
170C2×D4—21no—
171C2×D4—7882no—
172C2×D4—7882no—
173C2×D4—7882no—
174C2×D4—3941no—
175C2×D4—3941no—
176C2×D4—3941no—
177C2×D4—3941no—
178C2×D4—1971no—
179C2×D4C4×D19711yescharacteristic
180C22×C4—21no—
181C22×C4—7882no—
182C22×C4—3941no—
183C22×C4—3941no—
184C22×C4—1971no—
185C22×C4—1971no—
186C22×C4—1971no—
187C22×C4—1971no—
188C22×C4—1971no—
189C22×C4C22×D19711yescharacteristic
190M16—7882no—
191C2×C8—21no—
192C2×C8—3941no—
193C4⋊C4—3941no—
194C4⋊C4—1971no—
195C4⋊C4C4×D19711yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—3944no—
198C22⋊C4—3941no—
199C22⋊C4—3941no—
200C22⋊C4—3941no—
201C42—7882no—
202C23—21no—
203C23—7881no—
204C23—7882no—
205C23—7882no—
206C23—7882no—
207C23—3942no—
208C23—3941no—
209C23—3941no—
210C23—3941no—
211C23—1971no—
212C23—1971no—
213C23D4×D19711yescharacteristic
214D4—41no—
215D4—41no—
216D4—41no—
217D4—7881no—
218D4—7882no—
219D4—7882no—
220D4—7881no—
221D4—7881no—
222D4—7882no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—7881no—
229C2×C4—7881no—
230C2×C4—7882no—
231C2×C4—7881no—
232C2×C4—7881no—
233C2×C4—3942no—
234C2×C4—3942no—
235C2×C4—3942no—
236C2×C4—3941no—
237C2×C4—3942no—
238C2×C4—1971no—
239C2×C4—1971no—
240C2×C4C2×C4×D19711yescharacteristic
241C2×C4D4×D19711yescharacteristic
242C8—41no—
243C8—7881no—
244C22—21no—
245C22—21no—
246C22—41no—
247C22—41no—
248C22—41no—
249C22—7882no—
250C22—7882no—
251C22—7881no—
252C22—7881no—
253C22—7881no—
254C22—15764no—
255C22—3941no—
256C22—3941no—
257C22—3941no—
258C22—3941no—
259C22—3941no—
260C22—1972no—
261C22C22⋊C4×D19711yescharacteristic
262C22#2026335011yescharacteristic
263C22#2026335311yescharacteristic
264C4—21no—
265C4—21no—
266C4—42no—
267C4—41no—
268C4—7882no—
269C4—7881no—
270C4—3941no—
271C4—1972no—
272C2—21no—
273C2—21no—
274C2—81no—
275C2—7881no—
276C2—15761no—
277C2—3942no—
278C2#99×D19711yescharacteristic
279C2#2716108211yescharacteristic
280C2#104×D19711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D197
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#327×C197C2
socleC22×C197C2×C22⋊C4
prime radical#121×D197C2
cosocleC22×C4×C197C23
hypercenter#327D197
hypocenterC197#1507
LCSG ▷ C2×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #27161041C2semidirect
3. #27160926C2semidirect
4. #27160814C2semidirect
5. #27160813C2semidirect
6. #27160812C2semidirect
7. #327×C197C2semidirect
9. C2×D4×D197C4semidirect
12. #20263301C4semidirect
16. #20263297C4semidirect
17. C4⋊C4×D197C4semidirect
21. C22⋊#13096802C22semidirect
23. #20263233C22semidirect
25. C22⋊C8×C197C22semidirect
26. #95×C197C22semidirect
140. C197#1507semidirect
148. #327D197semidirect
165. C22⋊C8C2×D197semidirect
167. #95C2×D197semidirect

Maximal subgroups

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

subgroupindex
#121×D1972
#271610412
#271609262
#271608142
#271608132
#271608122
#327×C1972
#1507197