Finite Group Databasebeta

← #38743840

Subgroups — #38743840

Order 25216.

430 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 75 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#27161638C211yesmaximal, characteristic
2C2×#20263309C211yesmaximal, characteristic, complemented
3C2×#20263241C211yesmaximal, characteristic, complemented
4C2×#20263240C211yesmaximal, characteristic, complemented
5#27160942C211yesmaximal, characteristic, complemented
6#27160845C211yesmaximal, characteristic, complemented
7C2×#123×C197C211yesmaximal, characteristic, complemented
8C22×C22⋊D197C2211yescharacteristic
9#20263309—22no—
10#20263309—21no—
11#20263309—22no—
12#20263307—21no—
13#20263298—22no—
14#20263295—21no—
15#20263294—21no—
16C2×#15691337—21no—
17C2×C22⋊Dic197C2211yescharacteristic
18C2×C22⋊Dic197C2211yescharacteristic
19C2×C22⋊Dic197C2211yescharacteristic
20C2×#15691335—21no—
21#20263241—22no—
22#20263240—22no—
23#20263233—22no—
24C22×D4×C197C2211yescharacteristic
25C23×C4×C197C2211yescharacteristic
26#123×C197—22no—
27#123×C197—22no—
28C2×C4⋊C4×C197—21no—
29C2×C22⋊C4×C197—21no—
30C2×C22⋊C4×C197C2211yescharacteristic
31C23×D197D411yescharacteristic
32C2×C22⋊D197—22no—
33C2×C22⋊D197—21no—
34C2×C22⋊D197—21no—
35C2×C22⋊D197—21no—
36C2×C22⋊D197—21no—
37C2×C22⋊D197—21no—
38C2×C22⋊D197—21no—
39C22×Dic197—21no—
40C22×Dic197—21no—
41C22×Dic197D411yescharacteristic
42C22×Dic197D412yes—
43#15691337—24no—
44#15691337—22no—
45C22⋊Dic197—22no—
46C22⋊Dic197—21no—
47C22⋊Dic197—21no—
48C22⋊Dic197—21no—
49C22⋊Dic197—21no—
50C22⋊Dic197—21no—
51C22⋊Dic197D412yescomplemented
52#15691335—24no—
53#15691335—22no—
54C24×C197D411yescharacteristic
55C24×C197C2311yescharacteristic
56C2×D4×C197—21no—
57C2×D4×C197—21no—
58C2×D4×C197—21no—
59C2×D4×C197—21no—
60C2×D4×C197—21no—
61C2×D4×C197—21no—
62C22×C4×C197—21no—
63C22×C4×C197—21no—
64C22×C4×C197—21no—
65C22×C4×C197—21no—
66C22×C4×C197—21no—
67C22×C4×C197—21no—
68C22×C4×C197—21no—
69C22×C4×C197D411yescharacteristic
70C22×C4×C197D412yescomplemented
71C4⋊C4×C197—24no—
72C22⋊C4×C197—24no—
73C22⋊C4×C197—21no—
74C22⋊C4×C197D412yescomplemented
75C22×D197—21no—
76C22×D197—21no—
77C22×D197—21no—
78C22×D197—42no—
79C22⋊D197—41no—
80C22⋊D197—41no—
81C22⋊D197—41no—
82C22⋊D197—41no—
83C22⋊D197—82no—
84C2×Dic197—22no—
85C2×Dic197—22no—
86C2×Dic197—21no—
87C2×Dic197—21no—
88C2×Dic197—21no—
89C2×Dic197—42no—
90C2×Dic197—42no—
91C2×Dic197—42no—
92C2×Dic197—41no—
93C23×C197—21no—
94C23×C197—21no—
95C23×C197—21no—
96C23×C197—42no—
97C23×C197—42no—
98C23×C197C2×D411yescharacteristic
99C23×C197C2×D411yescharacteristic
100C23×C197C2×D411yescharacteristic
101C23×C197C2×D411yescharacteristic
102C23×C197C2×D411yescharacteristic
103C23×C197C2×D411yescharacteristic
104C23×C197#4111yescharacteristic
105D4×C197—41no—
106D4×C197—41no—
107D4×C197—41no—
108D4×C197—41no—
109C2×C4×C197—22no—
110C2×C4×C197—22no—
111C2×C4×C197—21no—
112C2×C4×C197—22no—
113C2×C4×C197—21no—
114C2×C4×C197—22no—
115C2×C4×C197—21no—
116C2×C4×C197—42no—
117C2×C4×C197—41no—
118C2×C4×C197—41no—
119C2×C4×C197—41no—
120C2×C4×C197—41no—
121C22×C197—22no—
122C22×C197—22no—
123C22×C197—22no—
124C22×C197—41no—
125C22×C197—41no—
126C22×C197—41no—
127C22×C197—41no—
128C22×C197—42no—
129C22×C197—41no—
130C22×C197—42no—
131C22×C197—81no—
132C22×C197#12011yescharacteristic
133C22×C197#12011yescharacteristic
134C22×C197#12011yescharacteristic
135C22×C197#12111yescharacteristic
136C22×C197#12111yescharacteristic
137C22×C197#12111yescharacteristic
138C22×C197#12711yescharacteristic
139C2×D197—42no—
140C2×D197—41no—
141C2×D197—42no—
142C2×D197—81no—
143C4×C197—42no—
144C4×C197—42no—
145C4×C197—41no—
146Dic197—42no—
147Dic197—41no—
148Dic197—42no—
149Dic197—81no—
150C2×C197—21no—
151C2×C197—21no—
152C2×C197—21no—
153C2×C197—21no—
154C2×C197—21no—
155C2×C197—21no—
156C2×C197—81no—
157C2×C197#39211yescharacteristic
158C2×C197#45812yes—
159D197—81no—
160C197#202211yescharacteristic, complemented, Sylow
161#2022—1971nomaximal, Sylow
162#534—1971no—
163C2×#123—1971no—
164C2×#123D19711yescharacteristic, complemented
165C2×#99—1971no—
166C2×#99—1971no—
167#381—1971no—
168#342—1971no—
169C22×D4—1971no—
170C22×D4C2×D19711yescharacteristic
171C23×C4C2×D19711yescharacteristic
172#123—22no—
173#123—22no—
174#123—3941no—
175#123—3941no—
176#123—3942no—
177#123—3942no—
178#121—3942no—
179#120—3941no—
180#120—3941no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—3941no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—3941no—
185C2×C22⋊C4—1971no—
186C2×C22⋊C4—1971no—
187C2×C22⋊C4—1971no—
188C2×C22⋊C4C2×D19711yescharacteristic
189#99—3942no—
190#99—3942no—
191#95—3942no—
192C24—1971no—
193C24C22⋊D19711yescharacteristic
194C24C22×D19711yescharacteristic
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—3941no—
202C2×D4—3941no—
203C2×D4—3941no—
204C2×D4—3941no—
205C2×D4—3941no—
206C2×D4—3941no—
207C2×D4—3942no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—3941no—
216C22×C4—3941no—
217C22×C4—1972no—
218C22×C4—1971no—
219C22×C4C22⋊D19711yescharacteristic
220C22×C4C22⋊D19712yescomplemented
221C4⋊C4—24no—
222C4⋊C4—3944no—
223C4⋊C4—3942no—
224C22⋊C4—24no—
225C22⋊C4—21no—
226C22⋊C4—3944no—
227C22⋊C4—3942no—
228C22⋊C4—3941no—
229C22⋊C4—3941no—
230C22⋊C4—3942no—
231C22⋊C4—3941no—
232C22⋊C4—3941no—
233C22⋊C4—3941no—
234C22⋊C4—1972no—
235C22⋊C4C22⋊D19712yescomplemented
236C23—21no—
237C23—21no—
238C23—21no—
239C23—42no—
240C23—42no—
241C23—7882no—
242C23—3941no—
243C23—3941no—
244C23—3941no—
245C23D4×D19711yescharacteristic
246C23D4×D19711yescharacteristic
247C23D4×D19711yescharacteristic
248C23C2×C22⋊D19711yescharacteristic
249C23C2×C22⋊D19711yescharacteristic
250C23C2×C22⋊D19711yescharacteristic
251C23#1569135711yescharacteristic
252D4—41no—
253D4—41no—
254D4—41no—
255D4—41no—
256D4—7881no—
257D4—7881no—
258D4—7881no—
259D4—7881no—
260D4—15762no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—7882no—
274C2×C4—7882no—
275C2×C4—7882no—
276C2×C4—7881no—
277C2×C4—3942no—
278C2×C4—3941no—
279C2×C4—3942no—
280C2×C4—3941no—
281C2×C4—3941no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—42no—
290C22—41no—
291C22—42no—
292C22—81no—
293C22—7882no—
294C22—7881no—
295C22—7882no—
296C22—15761no—
297C22#2026329411yescharacteristic
298C22#2026329411yescharacteristic
299C22#2026329511yescharacteristic
300C22#2026329711yescharacteristic
301C22#2026329811yescharacteristic
302C22#2026329811yescharacteristic
303C22#2026332211yescharacteristic
304C4—42no—
305C4—42no—
306C4—41no—
307C4—7882no—
308C4—7881no—
309C4—7882no—
310C4—15761no—
311C2—21no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2—21no—
317C2—81no—
318C2—15761no—
319C2#2716098711yescharacteristic
320C2#2716126812yes—
subgroupG/H
centerC22#20263294
commutator, cosocleC24×C197C23
FrattiniC24C22×D197
FittingC2×#123×C197C2
socleC22×C197#120
prime radical#27161638C2
hypercenterC2×#123D197
hypocenterC197#2022
LCSG ▷ C24×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C24 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#20263309C2semidirect
3. C2×#20263241C2semidirect
4. C2×#20263240C2semidirect
5. #27160942C2semidirect
6. #27160845C2semidirect
7. C2×#123×C197C2semidirect
51. C22⋊Dic197D4semidirect
70. C22×C4×C197D4semidirect
74. C22⋊C4×C197D4semidirect
160. C197#2022semidirect
164. C2×#123D197semidirect
220. C22×C4C22⋊D197semidirect
235. C22⋊C4C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
#271616382
C2×#202633092
C2×#202632412
C2×#202632402
#271609422
#271608452
C2×#123×C1972
#2022197