Finite Group Databasebeta

← #38742810

Subgroups — #38742810

Order 25216.

306 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 79 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#117×D197C211yesmaximal, characteristic
2#27161042C211yesmaximal, characteristic
3C4×#15691335C211yesmaximal, characteristic
4#27160840C211yesmaximal, characteristic, complemented
5#27160839C211yesmaximal, characteristic, complemented
6#27160838C211yesmaximal, characteristic, complemented
7#339×C197C211yesmaximal, characteristic, complemented
8C22×C4×D197C2211yescharacteristic
9C2×#15691342C2211yescharacteristic
10#20263285C2211yescharacteristic
11#20263284C411yescharacteristic, complemented
12#20263284C411yescharacteristic, complemented
13#20263283—21no—
14#20263282—21no—
15C4⋊C4×D197C411yescharacteristic, complemented
16C4⋊C4×D197C411yescharacteristic, complemented
17C2×#15691335—21no—
18C22⋊C4×D197—21no—
19C2×C4×Dic197C2211yescharacteristic
20C42×D197—21no—
21#20263238—21no—
22#20263238—21no—
23#20263233—21no—
24C2×M16×C197C2211yescharacteristic
25#117×C197C2211yescharacteristic
26C2×C42×C197C2211yescharacteristic
27C23×D197D411yescharacteristic
28C22×Dic197—21no—
29C22×Dic197D411yescharacteristic
30C2×C4×D197—21no—
31C2×C4×D197—21no—
32C2×C4×D197—21no—
33C2×C4×D197—21no—
34C2×C4×D197—21no—
35C2×C4×D197—21no—
36C2×C4×D197C2×C411yescharacteristic
37C2×C4×D197C2×C411yescharacteristic
38C2×C4×D197D411yescharacteristic
39C2×C4×D197D411yescharacteristic
40#15691342—21no—
41#15691342—21no—
42C2×#13096802—21no—
43C4⋊Dic197C2×C411yescharacteristic
44C4⋊Dic197C2×C411yescharacteristic
45#15691337—21no—
46#15691337—21no—
47C22⋊Dic197—21no—
48#15691335—24no—
49#15691335—21no—
50#15691335—21no—
51C4×Dic197—21no—
52C4×Dic197—21no—
53C4×Dic197—21no—
54C4×Dic197—21no—
55C4×Dic197—21no—
56C22×C4×C197—21no—
57C22×C4×C197C2311yescharacteristic
58M16×C197—21no—
59M16×C197—21no—
60C2×C8×C197—21no—
61C4⋊C4×C197C2×C411yescharacteristic
62C4⋊C4×C197C2×C411yescharacteristic
63C22⋊C4×C197—21no—
64C42×C197—21no—
65C42×C197—21no—
66C42×C197—21no—
67C22×D197—21no—
68C22×D197—21no—
69C22×D197—22no—
70C22×D197—41no—
71C22×D197C22⋊C411yescharacteristic
72C22×D197C22⋊C411yescharacteristic
73C2×Dic197—21no—
74C2×Dic197—21no—
75C2×Dic197—22no—
76C2×Dic197—21no—
77C2×Dic197—21no—
78C2×Dic197—41no—
79C2×Dic197—41no—
80C2×Dic197C22⋊C411yescharacteristic
81C2×Dic197C22⋊C411yescharacteristic
82C4×D197—21no—
83C4×D197—21no—
84C4×D197—21no—
85C4×D197—21no—
86C4×D197—21no—
87C4×D197—21no—
88C4×D197—21no—
89C4×D197—21no—
90C4×D197—42no—
91C4×D197—42no—
92#13096802—41no—
93C23×C197C2×D411yescharacteristic
94C2×C4×C197—22no—
95C2×C4×C197—21no—
96C2×C4×C197—21no—
97C2×C4×C197—41no—
98C2×C4×C197—41no—
99C2×C4×C197C22×C411yescharacteristic
100C2×C4×C197C2×D411yescharacteristic
101C2×C4×C197#4111yescharacteristic
102C2×C4×C197#4111yescharacteristic
103C2×C4×C197#4111yescharacteristic
104C2×C4×C197#4111yescharacteristic
105C8×C197—41no—
106C22×C197—21no—
107C22×C197—21no—
108C22×C197C2×C22⋊C411yescharacteristic
109C22×C197#12311yescharacteristic
110C22×C197#12311yescharacteristic
111C2×D197—21no—
112C2×D197—21no—
113C2×D197—21no—
114C2×D197—21no—
115C2×D197—21no—
116C2×D197—42no—
117C2×D197—42no—
118C2×D197#10412yes—
119C4×C197—42no—
120C4×C197—41no—
121C4×C197—41no—
122C4×C197#11711yescharacteristic
123C4×C197#11711yescharacteristic
124C4×C197#12311yescharacteristic
125C4×C197#12311yescharacteristic
126Dic197—21no—
127Dic197—21no—
128Dic197—42no—
129Dic197—41no—
130Dic197—41no—
131C2×C197—21no—
132C2×C197—21no—
133C2×C197#38111yescharacteristic
134C2×C197C2×#10411yescharacteristic
135C2×C197#42111yescharacteristic
136D197—22no—
137D197—41no—
138C197#182611yescharacteristic, complemented, Sylow
139#1826—1971nomaximal, Sylow
140C2×#117—1971no—
141#407—1971no—
142C4×C22⋊C4—1971no—
143#339—1971no—
144#339—1971no—
145#339—1971no—
146#339D19711yescharacteristic, complemented
147C23×C4—1971no—
148C2×M16—1971no—
149C2×M16C2×D19711yescharacteristic
150#117—3941no—
151#117—3941no—
152#117—1971no—
153#117—1971no—
154#117—1971no—
155#117C2×D19711yescharacteristic
156C2×C4⋊C4—1971no—
157C2×C4⋊C4—1971no—
158C2×C22⋊C4—3941no—
159C2×C22⋊C4—3941no—
160C2×C42—3941no—
161C2×C42—1971no—
162C2×C42C2×D19711yescharacteristic
163C22⋊C8—3941no—
164C22⋊C8—3941no—
165#95—3941no—
166C24—1971no—
167C22×C4—21no—
168C22×C4—3941no—
169C22×C4—3941no—
170C22×C4—3941no—
171C22×C4—3941no—
172C22×C4—3941no—
173C22×C4—3941no—
174C22×C4—3941no—
175C22×C4—1971no—
176C22×C4—1971no—
177C22×C4—1971no—
178C22×C4—1971no—
179C22×C4—1971no—
180C22×C4C22×D19711yescharacteristic
181M16—21no—
182M16—21no—
183M16—3941no—
184M16—3941no—
185C2×C8—21no—
186C2×C8—3941no—
187C4⋊C4—3941no—
188C4⋊C4—3941no—
189C4⋊C4—1971no—
190C4⋊C4—1971no—
191C4⋊C4C4×D19711yescharacteristic
192C4⋊C4C4×D19711yescharacteristic
193C22⋊C4—21no—
194C22⋊C4—3944no—
195C22⋊C4—3941no—
196C22⋊C4—3941no—
197C22⋊C4—3941no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—3941no—
202C42—3941no—
203C42—3941no—
204C42—3941no—
205C42—3941no—
206C23—7881no—
207C23—3942no—
208C23—3941no—
209C23—3941no—
210C23—1971no—
211C23—1971no—
212C23D4×D19711yescharacteristic
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—7882no—
219C2×C4—7882no—
220C2×C4—7881no—
221C2×C4—7881no—
222C2×C4—3942no—
223C2×C4—3941no—
224C2×C4—3941no—
225C2×C4—3941no—
226C2×C4—3941no—
227C2×C4—3941no—
228C2×C4—3941no—
229C2×C4—3941no—
230C2×C4—3941no—
231C2×C4—3941no—
232C2×C4—3941no—
233C2×C4—3941no—
234C2×C4—3941no—
235C2×C4—1971no—
236C2×C4—1971no—
237C2×C4C2×C4×D19711yescharacteristic
238C2×C4D4×D19711yescharacteristic
239C2×C4#1569135711yescharacteristic
240C2×C4#1569135811yescharacteristic
241C2×C4#1569135811yescharacteristic
242C2×C4#1569135911yescharacteristic
243C8—41no—
244C8—7881no—
245C22—21no—
246C22—21no—
247C22—7882no—
248C22—7882no—
249C22—3941no—
250C22—3941no—
251C22—3941no—
252C22—3941no—
253C22—3941no—
254C22—1972no—
255C22C22⋊C4×D19711yescharacteristic
256C22#2026330811yescharacteristic
257C22#2026331111yescharacteristic
258C4—42no—
259C4—41no—
260C4—41no—
261C4—7882no—
262C4—7881no—
263C4—7881no—
264C4—3941no—
265C4—3941no—
266C4#2026328311yescharacteristic
267C4#2026328411yescharacteristic
268C4#2026330811yescharacteristic
269C4#2026331111yescharacteristic
270C2—21no—
271C2—21no—
272C2—7881no—
273C2—3942no—
274C2#2716094111yescharacteristic
275C2#104×D19711yescharacteristic
276C2#2716109411yescharacteristic
subgroupG/H
centerC2×C4D4×D197
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#339×C197C2
socleC22×C197C2×C22⋊C4
prime radicalC23×D197D4
cosocleC22×C4×C197C23
hypercenter#339D197
hypocenterC197#1826
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
4. #27160840C2semidirect
5. #27160839C2semidirect
6. #27160838C2semidirect
7. #339×C197C2semidirect
11. #20263284C4semidirect
12. #20263284C4semidirect
15. C4⋊C4×D197C4semidirect
16. C4⋊C4×D197C4semidirect
138. C197#1826semidirect
146. #339D197semidirect

Maximal subgroups

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

subgroupindex
#117×D1972
#271610422
C4×#156913352
#271608402
#271608392
#271608382
#339×C1972
#1826197