Finite Group Databasebeta

← #38741202

Subgroups — #38741202

Order 25216.

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

normal subgroups only

This group has 304 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
2C22⋊C8×D197C211yesmaximal, characteristic, complemented
3#27160926C211yesmaximal, characteristic, complemented
4#27160816C211yesmaximal, characteristic, complemented
5#27160814C211yesmaximal, characteristic, complemented
6#27160812C211yesmaximal, characteristic, complemented
7#328×C197C211yesmaximal, characteristic, complemented
8C2×#15691359C411yescharacteristic, complemented
9C22×C4×D197C2211yescharacteristic
10C2×C8×D197—21no—
11#20263311—21no—
12#20263302C411yescharacteristic, complemented
13#20263301C411yescharacteristic, complemented
14#20263300C2211yescharacteristic
15#20263296—21no—
16#20263288—21no—
17#20263286—21no—
18#20263284C411yescharacteristic, complemented
19C2×#15691335—21no—
20C2×C4×Dic197—21no—
21C22⋊#13096802C2211yescharacteristic
22#20263238—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
32C2×C22⋊D197—21no—
33C22×Dic197—21no—
34C22×Dic197D411yescharacteristic
35C2×C4×D197—21no—
36C2×C4×D197—42no—
37C2×C4×D197C2×C411yescharacteristic
38C2×C4×D197C2×C411yescharacteristic
39C2×C4×D197D411yescharacteristic
40C2×C4×D197D411yescharacteristic
41C2×#13096802—21no—
42C8×D197—42no—
43C4⋊Dic197C2×C411yescharacteristic
44#15691337—21no—
45#15691335—24no—
46#15691335—21no—
47#15691335—21no—
48#15691335—21no—
49#15691335—21no—
50C4×Dic197—21no—
51C4×Dic197—42no—
52C2×Q8×C197C2×C411yescharacteristic
53C22×C4×C197—21no—
54C22×C4×C197C2311yescharacteristic
55C2×C8×C197—21no—
56C4⋊C4×C197—21no—
57C4⋊C4×C197C2×C411yescharacteristic
58C22⋊C4×C197—21no—
59C22×D197—21no—
60C22×D197—21no—
61C22×D197—22no—
62C22×D197—21no—
63C22×D197—41no—
64C22×D197C22⋊C411yescharacteristic
65C22×D197C22⋊C411yescharacteristic
66D788—41no—
67D788—42no—
68D788—42no—
69C22⋊D197—41no—
70C22⋊D197—41no—
71C2×Dic197—22no—
72C2×Dic197—22no—
73C2×Dic197—21no—
74C2×Dic197—41no—
75C2×Dic197—41no—
76C2×Dic197C22⋊C411yescharacteristic
77C2×Dic197C22⋊C411yescharacteristic
78C4×D197—22no—
79C4×D197—22no—
80C4×D197—41no—
81C4×D197—41no—
82C4×D197—42no—
83#13096802—41no—
84C23×C197C2×D411yescharacteristic
85Q8×C197—41no—
86C2×C4×C197—22no—
87C2×C4×C197—21no—
88C2×C4×C197—21no—
89C2×C4×C197—41no—
90C2×C4×C197—41no—
91C2×C4×C197—41no—
92C2×C4×C197C22×C411yescharacteristic
93C2×C4×C197C2×D411yescharacteristic
94C8×C197—41no—
95C22×C197—21no—
96C22×C197—21no—
97C22×C197C2×C22⋊C411yescharacteristic
98C22×C197#13511yescharacteristic
99C22×C197#13511yescharacteristic
100C2×D197—21no—
101C2×D197—21no—
102C2×D197—21no—
103C2×D197—21no—
104C2×D197—21no—
105C2×D197—42no—
106C2×D197—42no—
107C2×D197—41no—
108C2×D197—41no—
109C2×D197—41no—
110C2×D197#10412yes—
111C4×C197—21no—
112C4×C197—21no—
113C4×C197—42no—
114C4×C197—41no—
115C4×C197—41no—
116Dic197—21no—
117Dic197—42no—
118Dic197—41no—
119Dic197#9912yes—
120C2×C197—21no—
121C2×C197—21no—
122C2×C197C2×#9911yescharacteristic
123C2×C197#41611yescharacteristic
124C2×C197C2×#10411yescharacteristic
125D197—22no—
126D197—41no—
127D197—81no—
128C197#150111yescharacteristic, complemented, Sylow
129#1501—1971nomaximal, Sylow
130#525—1971no—
131C2×C22⋊C8—1971no—
132C4×C22⋊C4—1971no—
133#328—1971no—
134#328D19711yescharacteristic, complemented
135#327—1971no—
136#327—1971no—
137C2×#41—1971no—
138C23×C4—1971no—
139C22×C8—3941no—
140#123—3941no—
141C22⋊Q8—1971no—
142C22⋊Q8C2×D19711yescharacteristic
143#121—1971no—
144#121—1971no—
145#120—3941no—
146C4×D4—3941no—
147C4×D4—3941no—
148#117—1971no—
149C2×C22⋊C4—3941no—
150C2×C42—3941no—
151C22⋊C8—3941no—
152C22⋊C8—1971no—
153C22⋊C8C2×D19711yescharacteristic
154#95—1971no—
155#95C2×D19711yescharacteristic
156C24—1971no—
157#41—7882no—
158C2×Q8C4×D19711yescharacteristic
159C2×D4—3941no—
160C2×D4—3941no—
161C2×D4—3941no—
162C2×D4—1971no—
163C22×C4—21no—
164C22×C4—7882no—
165C22×C4—3941no—
166C22×C4—3941no—
167C22×C4—1971no—
168C22×C4—1971no—
169C22×C4—1971no—
170C22×C4—1971no—
171C22×C4—1971no—
172C22×C4C22×D19711yescharacteristic
173C2×C8—21no—
174C2×C8—7882no—
175C2×C8—3941no—
176C4⋊C4—21no—
177C4⋊C4—3941no—
178C4⋊C4—1971no—
179C4⋊C4C4×D19711yescharacteristic
180C22⋊C4—21no—
181C22⋊C4—3944no—
182C22⋊C4—3941no—
183C22⋊C4—3941no—
184C22⋊C4—3941no—
185C22⋊C4—3941no—
186C42—7882no—
187C42—3941no—
188C23—7881no—
189C23—3942no—
190C23—3941no—
191C23—3941no—
192C23—3941no—
193C23—1971no—
194C23—1971no—
195C23D4×D19711yescharacteristic
196Q8—41no—
197D4—7881no—
198D4—7882no—
199D4—7882no—
200D4—7881no—
201D4—7881no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—7881no—
209C2×C4—7881no—
210C2×C4—7882no—
211C2×C4—7881no—
212C2×C4—7881no—
213C2×C4—3942no—
214C2×C4—3942no—
215C2×C4—3942no—
216C2×C4—3941no—
217C2×C4—3942no—
218C2×C4—1971no—
219C2×C4—1971no—
220C2×C4C2×C4×D19711yescharacteristic
221C2×C4D4×D19711yescharacteristic
222C8—41no—
223C8—7881no—
224C22—21no—
225C22—21no—
226C22—7882no—
227C22—7882no—
228C22—7881no—
229C22—7881no—
230C22—7881no—
231C22—3941no—
232C22—3941no—
233C22—3941no—
234C22—3941no—
235C22—3941no—
236C22—1972no—
237C22C22⋊C4×D19711yescharacteristic
238C22#2026334711yescharacteristic
239C22#2026334911yescharacteristic
240C4—21no—
241C4—21no—
242C4—42no—
243C4—41no—
244C4—41no—
245C4—7882no—
246C4—7881no—
247C4—3941no—
248C4—1972no—
249C2—21no—
250C2—21no—
251C2—7881no—
252C2—15761no—
253C2—3942no—
254C2#99×D19711yescharacteristic
255C2#2716107411yescharacteristic
256C2#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#1501
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. C22⋊C8×D197C2semidirect
3. #27160926C2semidirect
4. #27160816C2semidirect
5. #27160814C2semidirect
6. #27160812C2semidirect
7. #328×C197C2semidirect
8. C2×#15691359C4semidirect
12. #20263302C4semidirect
13. #20263301C4semidirect
18. #20263284C4semidirect
128. C197#1501semidirect
134. #328D197semidirect

Maximal subgroups

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

subgroupindex
#271615922
C22⋊C8×D1972
#271609262
#271608162
#271608142
#271608122
#328×C1972
#1501197