Finite Group Databasebeta

← #38745951

Subgroups — #38745951

Order 25216.

696 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C2×D4×Dic197C211yesmaximal, characteristic, complemented
2C2×#20263286C212yesmaximal, complemented
3C2×#20263284C211yesmaximal, characteristic, complemented
4#27161009C211yesmaximal, characteristic, complemented
5#27161006C211yesmaximal, characteristic, complemented
6#27161002C212yesmaximal, complemented
7#27160994C212yesmaximal, complemented
8#27160987C211yesmaximal, characteristic
9#27160930C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic197C212yesmaximal, complemented
11#394×C197C211yesmaximal, characteristic, complemented
12C22×C22⋊D197C2212yes—
13C23×Dic197C2212yes—
14C22×C4×D197C2211yescharacteristic
15D4×Dic197—24no—
16#20263286—28no—
17#20263284—24no—
18C2×C4⋊Dic197C2211yescharacteristic
19C2×C4⋊Dic197C2211yescharacteristic
20C2×#15691337C2212yes—
21C2×#15691337C2212yes—
22C2×C22⋊Dic197C2212yes—
23C2×C22⋊Dic197C2212yes—
24C2×#15691335C2211yescharacteristic
25C2×#15691335C2211yescharacteristic
26C2×#15691335C2212yes—
27C2×C4×Dic197C2211yescharacteristic
28C2×C4×Dic197C2212yes—
29C2×C4×Dic197C2211yescharacteristic
30#20263233C2212yes—
31#20263233C2212yes—
32#20263233C2211yescharacteristic
33C22×D4×C197C2211yescharacteristic
34C2×C4⋊C4×C197C2211yescharacteristic
35C2×C22⋊C4×C197C2212yes—
36C2×C22⋊C4×C197C2212yes—
37#95×C197C2211yescharacteristic
38C23×D197C2311yescharacteristic
39C2×C22⋊D197—24no—
40C2×C22⋊D197—24no—
41C2×C22⋊D197—24no—
42C22×Dic197—24no—
43C22×Dic197—24no—
44C22×Dic197—24no—
45C22×Dic197C2311yescharacteristic
46C22×Dic197C2312yes—
47C22×Dic197C2311yescharacteristic
48C22×Dic197C2312yes—
49C22×Dic197C2311yescharacteristic
50C2×C4×D197—22no—
51C2×C4×D197—22no—
52C2×C4×D197D414yescomplemented
53C4⋊Dic197—22no—
54C4⋊Dic197—22no—
55#15691337—24no—
56#15691337—24no—
57C22⋊Dic197—24no—
58C22⋊Dic197—24no—
59#15691335—24no—
60#15691335—22no—
61#15691335—22no—
62C4×Dic197—22no—
63C4×Dic197—24no—
64C4×Dic197D414yescomplemented
65C24×C197C2312yes—
66C2×D4×C197—24no—
67C2×D4×C197—22no—
68C22×C4×C197C2312yes—
69C22×C4×C197C2311yescharacteristic
70C22×C4×C197C2311yescharacteristic
71C22×C4×C197C2311yescharacteristic
72C4⋊C4×C197—22no—
73C22⋊C4×C197—24no—
74C22⋊C4×C197—24no—
75C22×D197—21no—
76C22×D197—21no—
77C22×D197—22no—
78C22×D197—22no—
79C22×D197C2×D412yes—
80C22⋊D197—216no—
81C2×Dic197—22no—
82C2×Dic197—22no—
83C2×Dic197—22no—
84C2×Dic197—21no—
85C2×Dic197—22no—
86C2×Dic197—22no—
87C2×Dic197—22no—
88C2×Dic197—22no—
89C2×Dic197—48no—
90C2×Dic197C2×D412yes—
91C2×Dic197C2×D412yes—
92C2×Dic197C2×D412yes—
93C2×Dic197#4114yes—
94C2×Dic197#4112yes—
95C4×D197—28no—
96C23×C197—22no—
97C23×C197—22no—
98C23×C197—22no—
99C23×C197—22no—
100C23×C197—22no—
101C23×C197—22no—
102C23×C197#4114yes—
103C23×C197C2411yescharacteristic
104D4×C197—28no—
105C2×C4×C197—21no—
106C2×C4×C197—22no—
107C2×C4×C197—21no—
108C2×C4×C197—22no—
109C2×C4×C197—21no—
110C2×C4×C197—22no—
111C2×C4×C197—22no—
112C2×C4×C197—22no—
113C2×C4×C197C2×D412yes—
114C2×C4×C197C2×D412yes—
115C22×C197—24no—
116C22×C197—24no—
117C22×C197—24no—
118C22×C197—42no—
119C22×C197—42no—
120C22×C197—42no—
121C22×C197—42no—
122C22×C197C22×D411yescharacteristic
123C22×C197C22×D411yescharacteristic
124C22×C197C2×#4111yescharacteristic
125C22×C197C2×#4112yes—
126C22×C197C2×#4112yes—
127C2×D197—22no—
128C2×D197—24no—
129C2×D197—41no—
130C2×D197—42no—
131C2×D197—41no—
132C4×C197—22no—
133C4×C197—22no—
134C4×C197—41no—
135C4×C197—42no—
136Dic197—24no—
137Dic197—22no—
138Dic197—42no—
139Dic197—41no—
140Dic197#12114yes—
141C2×C197—44no—
142C2×C197C2×#12111yescharacteristic
143C2×C197#52511yescharacteristic
144C2×C197#52512yes—
145C2×C197#52912yes—
146C2×C197#53211yescharacteristic
147D197—42no—
148C197#239211yescharacteristic, complemented, Sylow
149#2392—1971nomaximal, Sylow
150C2×C4×D4—1971no—
151C2×C4×D4—1972no—
152C2×#117—1971no—
153#398—1971no—
154#397—1971no—
155#396—1972no—
156#394—1972no—
157#394D19711yescharacteristic, complemented
158#392—1971no—
159C4×C4⋊C4—1971no—
160C4×C22⋊C4—1972no—
161C22×D4—1972no—
162C22×D4C2×D19711yescharacteristic
163C23×C4—1972no—
164C23×C4—1971no—
165C4×D4—3948no—
166C4×D4—3944no—
167#117—3944no—
168C2×C4⋊C4—1971no—
169C2×C4⋊C4—1972no—
170C2×C4⋊C4—1971no—
171C2×C4⋊C4—1972no—
172C2×C4⋊C4C2×D19711yescharacteristic
173C2×C22⋊C4—1972no—
174C2×C22⋊C4—1972no—
175C2×C22⋊C4—1971no—
176C2×C22⋊C4—1971no—
177C2×C22⋊C4—1972no—
178C2×C22⋊C4C2×D19712yes—
179C2×C22⋊C4C2×D19712yes—
180C2×C42—1971no—
181C2×C42—1972no—
182C2×C42—1971no—
183#95—1972no—
184#95—1972no—
185#95—1971no—
186#95C2×D19711yescharacteristic
187C24—1971no—
188C24C22×D19712yes—
189C2×D4—24no—
190C2×D4—22no—
191C2×D4—3944no—
192C2×D4—3944no—
193C2×D4—3944no—
194C22×C4—3942no—
195C22×C4—3942no—
196C22×C4—3944no—
197C22×C4—3944no—
198C22×C4—3944no—
199C22×C4—1971no—
200C22×C4—1972no—
201C22×C4—1972no—
202C22×C4—1971no—
203C22×C4—1974no—
204C22×C4—1971no—
205C22×C4C22×D19711yescharacteristic
206C22×C4C22×D19712yes—
207C22×C4C22×D19711yescharacteristic
208C22×C4C22×D19711yescharacteristic
209C4⋊C4—22no—
210C4⋊C4—3944no—
211C4⋊C4—3942no—
212C4⋊C4—3944no—
213C4⋊C4—3942no—
214C22⋊C4—24no—
215C22⋊C4—24no—
216C22⋊C4—3944no—
217C22⋊C4—3942no—
218C22⋊C4—3944no—
219C22⋊C4—3944no—
220C22⋊C4—3942no—
221C42—3942no—
222C42—3944no—
223C42—1974no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—3941no—
231C23—3941no—
232C23—3942no—
233C23—3942no—
234C23—1972no—
235C23#1569135714yes—
236C23C23×D19711yescharacteristic
237D4—28no—
238D4—39416no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—7888no—
248C2×C4—3948no—
249C2×C4—3941no—
250C2×C4—3942no—
251C2×C4—3942no—
252C2×C4—3942no—
253C2×C4—3942no—
254C2×C4—3942no—
255C2×C4—3942no—
256C2×C4—3942no—
257C2×C4—1972no—
258C2×C4—1974no—
259C2×C4—1972no—
260C2×C4—1972no—
261C2×C4—1972no—
262C2×C4D4×D19712yes—
263C2×C4D4×D19712yes—
264C22—24no—
265C22—24no—
266C22—24no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—7882no—
272C22—7881no—
273C22—7881no—
274C22—3942no—
275C22—3944no—
276C22C2×D4×D19711yescharacteristic
277C22C2×D4×D19711yescharacteristic
278C22#41×D19711yescharacteristic
279C22#41×D19712yes—
280C22C2×#1569135712yes—
281C4—22no—
282C4—22no—
283C4—41no—
284C4—42no—
285C4—7881no—
286C4—7882no—
287C4—3942no—
288C4—3944no—
289C4—1974no—
290C2—44no—
291C2—7882no—
292C2#121×D19711yescharacteristic
293C2#2716158511yescharacteristic
294C2#2716158912yes—
295C2#2716160612yes—
296C2#2716162811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutator, socle, cosocleC23×C197C24
Fitting#394×C197C2
prime radical#27160987C2
hypercenter#394D197
hypocenterC197#2392
LCSG ▷ C23×C197 ▷ C197
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic197C2semidirect
2. C2×#20263286C2semidirect
3. C2×#20263284C2semidirect
4. #27161009C2semidirect
5. #27161006C2semidirect
6. #27161002C2semidirect
7. #27160994C2semidirect
9. #27160930C2semidirect
10. C22⋊C4×Dic197C2semidirect
11. #394×C197C2semidirect
52. C2×C4×D197D4semidirect
64. C4×Dic197D4semidirect
148. C197#2392semidirect
157. #394D197semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1972
C2×#202632862
C2×#202632842
#271610092
#271610062
#271610022
#271609942
#271609872
#271609302
C22⋊C4×Dic1972
#394×C1972
#2392197