Finite Group Databasebeta

← #38742654

Subgroups — #38742654

Order 25216.

618 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 103 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×#41×D197C211yesmaximal, characteristic
2C2×#20263254C211yesmaximal, characteristic, complemented
3C2×#20263253C211yesmaximal, characteristic, complemented
4C2×#20263252C211yesmaximal, characteristic, complemented
5#27161042C211yesmaximal, characteristic, complemented
6C4×#15691335C211yesmaximal, characteristic, complemented
7C2×#104×C197C211yesmaximal, characteristic, complemented
8C2×#15691359C411yescharacteristic, complemented
9C2×#15691358—21no—
10C2×#15691358C2211yescharacteristic
11C2×#15691357—21no—
12C2×#15691357C411yescharacteristic, complemented
13#41×D197—22no—
14#41×D197—22no—
15#41×D197—22no—
16#41×D197—22no—
17C2×Q8×D197C411yescharacteristic, complemented
18C2×D4×D197—21no—
19C2×D4×D197C411yescharacteristic, complemented
20C22×C4×D197—21no—
21C22×C4×D197C2211yescharacteristic
22C2×#15691342C2211yescharacteristic, complemented
23C2×#15691335—21no—
24C2×C4×Dic197C2211yescharacteristic, complemented
25#20263254—22no—
26#20263253—22no—
27#20263252—22no—
28#20263238—21no—
29#20263238—21no—
30#20263233—21no—
31C2×#41×C197C2211yescharacteristic
32C2×M16×C197C2211yescharacteristic, complemented
33C2×C42×C197C2211yescharacteristic, complemented
34#104×C197—22no—
35C23×D197—21no—
36C23×D197D411yescharacteristic
37#15691359—22no—
38#15691359—22no—
39#15691358—24no—
40#15691358—24no—
41#15691358—21no—
42#15691358—21no—
43#15691358D412yes—
44#15691358D412yes—
45#15691357—24no—
46#15691357—24no—
47#15691357—22no—
48#15691357—22no—
49C2×Dic394—21no—
50C2×Dic394C2×C411yescharacteristic
51Q8×D197—22no—
52Q8×D197—22no—
53C2×D788—21no—
54C2×D788C2×C411yescharacteristic
55C2×C22⋊D197—21no—
56C2×C22⋊D197—21no—
57C2×C22⋊D197—21no—
58D4×D197—24no—
59D4×D197—24no—
60D4×D197—22no—
61D4×D197—22no—
62C22×Dic197—21no—
63C22×Dic197—21no—
64C22×Dic197D411yescharacteristic
65C2×C4×D197—24no—
66C2×C4×D197—24no—
67C2×C4×D197—21no—
68C2×C4×D197—21no—
69C2×C4×D197—21no—
70C2×C4×D197—21no—
71C2×C4×D197—21no—
72C2×C4×D197—21no—
73C2×C4×D197—21no—
74C2×C4×D197C2×C411yescharacteristic
75C2×C4×D197C2×C411yescharacteristic
76C2×C4×D197D411yescharacteristic
77C2×C4×D197D411yescharacteristic
78#15691342—21no—
79#15691342—21no—
80C2×#13096802—21no—
81#15691335—24no—
82C4×Dic197—21no—
83C4×Dic197—21no—
84#41×C197—21no—
85#41×C197—21no—
86#41×C197D412yes—
87#41×C197D412yes—
88C2×Q8×C197C2×C411yescharacteristic
89C2×D4×C197—21no—
90C2×D4×C197C2×C411yescharacteristic
91C22×C4×C197—21no—
92C22×C4×C197—21no—
93C22×C4×C197C2311yescharacteristic
94M16×C197—21no—
95M16×C197—21no—
96C2×C8×C197—21no—
97C42×C197—21no—
98C42×C197—21no—
99C22×D197—24no—
100C22×D197—21no—
101C22×D197—21no—
102C22×D197—21no—
103C22×D197—22no—
104C22×D197—41no—
105C22×D197—42no—
106C22×D197—42no—
107C22×D197C22⋊C411yescharacteristic
108C22×D197C22⋊C411yescharacteristic
109Dic394—24no—
110Dic394—21no—
111Dic394C22⋊C412yescomplemented
112D788—24no—
113D788—21no—
114D788C22⋊C412yescomplemented
115C22⋊D197—24no—
116C22⋊D197—24no—
117C22⋊D197—22no—
118C22⋊D197—22no—
119C2×Dic197—24no—
120C2×Dic197—21no—
121C2×Dic197—21no—
122C2×Dic197—21no—
123C2×Dic197—22no—
124C2×Dic197—41no—
125C2×Dic197—41no—
126C2×Dic197C22⋊C411yescharacteristic
127C2×Dic197C22⋊C411yescharacteristic
128C4×D197—24no—
129C4×D197—24no—
130C4×D197—21no—
131C4×D197—21no—
132C4×D197—21no—
133C4×D197—21no—
134C4×D197—21no—
135C4×D197—21no—
136C4×D197—22no—
137C4×D197—22no—
138C4×D197—21no—
139C4×D197—21no—
140#13096802—41no—
141C23×C197—21no—
142C23×C197C2×D411yescharacteristic
143Q8×C197—21no—
144Q8×C197C22⋊C412yescomplemented
145D4×C197—22no—
146D4×C197—22no—
147D4×C197—21no—
148D4×C197C22⋊C412yescomplemented
149C2×C4×C197—21no—
150C2×C4×C197—22no—
151C2×C4×C197—22no—
152C2×C4×C197—22no—
153C2×C4×C197—41no—
154C2×C4×C197—41no—
155C2×C4×C197C22×C411yescharacteristic
156C2×C4×C197C2×D411yescharacteristic
157C2×C4×C197C2×D411yescharacteristic
158C2×C4×C197C2×D411yescharacteristic
159C2×C4×C197C2×D411yescharacteristic
160C2×C4×C197C2×D411yescharacteristic
161C8×C197—41no—
162C22×C197—21no—
163C22×C197—21no—
164C22×C197—22no—
165C22×C197—41no—
166C22×C197—41no—
167C22×C197C2×C22⋊C411yescharacteristic
168C22×C197#12011yescharacteristic
169C22×C197#12011yescharacteristic
170C2×D197—21no—
171C2×D197—21no—
172C2×D197—21no—
173C2×D197—22no—
174C2×D197—21no—
175C2×D197—21no—
176C2×D197—42no—
177C2×D197—42no—
178C2×D197—48no—
179C2×D197—41no—
180C2×D197—41no—
181C2×D197#10412yescomplemented
182C4×C197—22no—
183C4×C197—42no—
184C4×C197C2×C22⋊C411yescharacteristic
185C4×C197C2×C22⋊C411yescharacteristic
186C4×C197#12011yescharacteristic
187C4×C197#12011yescharacteristic
188Dic197—21no—
189Dic197—22no—
190Dic197—21no—
191Dic197—42no—
192C2×C197—21no—
193C2×C197—21no—
194C2×C197—42no—
195C2×C197#37911yescharacteristic
196C2×C197C2×#10411yescharacteristic
197C2×C197#42111yescharacteristic
198D197—22no—
199D197—41no—
200D197—42no—
201C197#178911yescharacteristic, complemented, Sylow
202#1789—1971nomaximal, Sylow
203C22×#41—1971no—
204C2×#104—1971no—
205C2×#104—1971no—
206C2×#104—1971no—
207C2×#104D19711yescharacteristic, complemented
208#407—1971no—
209C4×C22⋊C4—1971no—
210C2×#41—3941no—
211C2×#41—3942no—
212C2×#41—3942no—
213C2×#41—3942no—
214C2×#41—3942no—
215C2×#41—3941no—
216C2×#41—1971no—
217C2×#41—1971no—
218C2×#41—1971no—
219C2×#41C2×D19711yescharacteristic
220C22×Q8—1971no—
221C22×D4—3941no—
222C22×D4—1971no—
223C23×C4—3941no—
224C23×C4—1971no—
225C2×M16—1971no—
226C2×M16C2×D19711yescharacteristic, complemented
227C2×C22⋊C4—3941no—
228C2×C42—1971no—
229C2×C42C2×D19711yescharacteristic, complemented
230#104—22no—
231#104—3942no—
232#104—3942no—
233#104—3942no—
234C22⋊C8—3941no—
235C22⋊C8—3941no—
236#95—3941no—
237C24—3941no—
238C24—1971no—
239#41—21no—
240#41—21no—
241#41—3944no—
242#41—3944no—
243#41—3944no—
244#41—3944no—
245#41—3942no—
246#41—3942no—
247#41—3942no—
248#41—3942no—
249#41—3941no—
250#41—3941no—
251#41—1972no—
252#41—1972no—
253#41C22⋊D19712yes—
254#41D78812yes—
255C2×Q8—3941no—
256C2×Q8—3942no—
257C2×Q8—3942no—
258C2×Q8—1971no—
259C2×Q8C4×D19711yescharacteristic
260C2×D4—21no—
261C2×D4—3941no—
262C2×D4—3944no—
263C2×D4—3944no—
264C2×D4—3942no—
265C2×D4—3941no—
266C2×D4—3941no—
267C2×D4—3942no—
268C2×D4—3941no—
269C2×D4—1971no—
270C2×D4C4×D19711yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—3944no—
274C22×C4—3944no—
275C22×C4—3941no—
276C22×C4—3941no—
277C22×C4—3941no—
278C22×C4—3941no—
279C22×C4—3941no—
280C22×C4—3941no—
281C22×C4—3941no—
282C22×C4—3941no—
283C22×C4—3941no—
284C22×C4—1971no—
285C22×C4—1971no—
286C22×C4—1971no—
287C22×C4—1971no—
288C22×C4—1971no—
289C22×C4C22×D19711yescharacteristic
290M16—21no—
291M16—21no—
292M16—3941no—
293M16—3941no—
294C2×C8—21no—
295C2×C8—3941no—
296C22⋊C4—3944no—
297C42—21no—
298C42—21no—
299C42—3941no—
300C42—3941no—
301C23—21no—
302C23—7881no—
303C23—7882no—
304C23—7882no—
305C23—3944no—
306C23—3942no—
307C23—3941no—
308C23—3941no—
309C23—3941no—
310C23—1971no—
311C23—1971no—
312C23D4×D19711yescharacteristic
313Q8—21no—
314Q8—3944no—
315Q8—3941no—
316Q8—1972no—
317Q8#1569133512yescomplemented
318D4—22no—
319D4—22no—
320D4—21no—
321D4—3944no—
322D4—3944no—
323D4—3944no—
324D4—3942no—
325D4—3942no—
326D4—3941no—
327D4—1972no—
328D4#1569133512yescomplemented
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—7881no—
336C2×C4—7881no—
337C2×C4—3944no—
338C2×C4—3944no—
339C2×C4—3942no—
340C2×C4—3944no—
341C2×C4—3941no—
342C2×C4—3941no—
343C2×C4—3941no—
344C2×C4—3941no—
345C2×C4—3941no—
346C2×C4—3941no—
347C2×C4—3941no—
348C2×C4—3941no—
349C2×C4—3942no—
350C2×C4—3941no—
351C2×C4—3941no—
352C2×C4—3942no—
353C2×C4—3941no—
354C2×C4—1971no—
355C2×C4—1971no—
356C2×C4C2×C4×D19711yescharacteristic
357C2×C4D4×D19711yescharacteristic
358C2×C4D4×D19711yescharacteristic
359C2×C4D4×D19711yescharacteristic
360C2×C4C2×C22⋊D19711yescharacteristic
361C2×C4C2×D78811yescharacteristic
362C8—41no—
363C8—7881no—
364C22—21no—
365C22—21no—
366C22—22no—
367C22—41no—
368C22—41no—
369C22—7882no—
370C22—7882no—
371C22—7888no—
372C22—7881no—
373C22—7881no—
374C22—3941no—
375C22—3941no—
376C22—3941no—
377C22—3942no—
378C22—3941no—
379C22—3941no—
380C22—1972no—
381C22C22⋊C4×D19711yescharacteristic
382C22#2026329411yescharacteristic
383C22#2026329611yescharacteristic
384C4—22no—
385C4—42no—
386C4—7882no—
387C4—3941no—
388C4—3942no—
389C4—3941no—
390C4C22⋊C4×D19711yescharacteristic
391C4C2×#1569133511yescharacteristic
392C4#2026329411yescharacteristic
393C4#2026329611yescharacteristic
394C2—21no—
395C2—21no—
396C2—42no—
397C2—7882no—
398C2—7881no—
399C2—3942no—
400C2#2716093411yescharacteristic
401C2#104×D19711yescharacteristic
402C2#2716109411yescharacteristic
subgroupG/H
centerC2×C4D4×D197
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
FittingC2×#104×C197C2
socleC22×C197C2×C22⋊C4
prime radicalC2×#41×D197C2
cosocleC22×C4×C197C23
hypercenterC2×#104D197
hypocenterC197#1789
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#20263254C2semidirect
3. C2×#20263253C2semidirect
4. C2×#20263252C2semidirect
5. #27161042C2semidirect
6. C4×#15691335C2semidirect
7. C2×#104×C197C2semidirect
8. C2×#15691359C4semidirect
12. C2×#15691357C4semidirect
17. C2×Q8×D197C4semidirect
19. C2×D4×D197C4semidirect
22. C2×#15691342C22semidirect
24. C2×C4×Dic197C22semidirect
32. C2×M16×C197C22semidirect
33. C2×C42×C197C22semidirect
111. Dic394C22⋊C4semidirect
114. D788C22⋊C4semidirect
144. Q8×C197C22⋊C4semidirect
148. D4×C197C22⋊C4semidirect
181. C2×D197#104semidirect
201. C197#1789semidirect
207. C2×#104D197semidirect
226. C2×M16C2×D197semidirect
229. C2×C42C2×D197semidirect
317. Q8#15691335semidirect
328. D4#15691335semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1972
C2×#202632542
C2×#202632532
C2×#202632522
#271610422
C4×#156913352
C2×#104×C1972
#1789197