Finite Group Databasebeta

← #38745404

Subgroups — #38745404

Order 25216.

668 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263299C211yesmaximal, characteristic
2C2×C4×C22⋊D197C211yesmaximal, characteristic, complemented
3C2×#20263284C211yesmaximal, characteristic, complemented
4#27160980C211yesmaximal, characteristic, complemented
5#27160979C211yesmaximal, characteristic, complemented
6#27160979C211yesmaximal, characteristic, complemented
7#27160974C211yesmaximal, characteristic, complemented
8#27160972C211yesmaximal, characteristic, complemented
9#27160971C211yesmaximal, characteristic, complemented
10#27160958C211yesmaximal, characteristic, complemented
11#27160957C211yesmaximal, characteristic, complemented
12#27160950C211yesmaximal, characteristic, complemented
13#27160938C211yesmaximal, characteristic, complemented
14#27160930C211yesmaximal, characteristic, complemented
15#385×C197C211yesmaximal, characteristic, complemented
16C22×D788C2211yescharacteristic
17C22×C22⋊D197C2211yescharacteristic
18C22×C22⋊D197C2211yescharacteristic
19C22×C4×D197C2211yescharacteristic
20C22×C4×D197C2211yescharacteristic
21#20263299C418yescomplemented
22C4×C22⋊D197—24no—
23#20263284—24no—
24C2×C4⋊Dic197C2211yescharacteristic
25C2×#15691337C2211yescharacteristic
26C2×#15691337C2211yescharacteristic
27C2×#15691337C2211yescharacteristic
28C2×#15691337C2211yescharacteristic
29C2×C22⋊Dic197C2211yescharacteristic
30C2×C22⋊Dic197C2211yescharacteristic
31C2×#15691335C2211yescharacteristic
32C2×#15691335C2211yescharacteristic
33C2×#15691335C2211yescharacteristic
34C2×#15691335C2211yescharacteristic
35C2×#15691335C2211yescharacteristic
36C2×#15691335C2211yescharacteristic
37C2×#15691335C2211yescharacteristic
38C2×#15691335C2211yescharacteristic
39C2×C4×Dic197C2211yescharacteristic
40C2×C4×Dic197C2211yescharacteristic
41C2×C4×Dic197C2211yescharacteristic
42C2×C4×Dic197C2211yescharacteristic
43#20263233C2211yescharacteristic
44#20263233C2211yescharacteristic
45#20263233C2211yescharacteristic
46#20263233C2211yescharacteristic
47C23×C4×C197C2211yescharacteristic
48C2×C4⋊C4×C197C2211yescharacteristic
49C2×C4⋊C4×C197C2211yescharacteristic
50C2×C22⋊C4×C197C2211yescharacteristic
51C2×C22⋊C4×C197C2211yescharacteristic
52#95×C197C2211yescharacteristic
53#95×C197C2211yescharacteristic
54C23×D197C2311yescharacteristic
55C23×D197C2311yescharacteristic
56C2×D788—41no—
57C2×D788—41no—
58C2×D788C2×C414yescomplemented
59C2×C22⋊D197—22no—
60C2×C22⋊D197—22no—
61C2×C22⋊D197—41no—
62C2×C22⋊D197—41no—
63C2×C22⋊D197C2×C414yes—
64C2×C22⋊D197C2×C414yescomplemented
65C22×Dic197C2311yescharacteristic
66C22×Dic197C2311yescharacteristic
67C22×Dic197C2311yescharacteristic
68C22×Dic197C2311yescharacteristic
69C22×Dic197C2311yescharacteristic
70C22×Dic197C2311yescharacteristic
71C2×C4×D197—22no—
72C2×C4×D197—22no—
73C2×C4×D197—22no—
74C2×C4×D197—41no—
75C2×C4×D197—41no—
76C2×C4×D197C2×C414yescomplemented
77C4⋊Dic197—22no—
78#15691337—22no—
79#15691337—22no—
80#15691337—41no—
81#15691337C2×C414yescomplemented
82C22⋊Dic197—22no—
83C22⋊Dic197—41no—
84#15691335—22no—
85#15691335—22no—
86#15691335—22no—
87#15691335—41no—
88#15691335—41no—
89#15691335—41no—
90#15691335—41no—
91#15691335C2×C414yescomplemented
92C4×Dic197—22no—
93C4×Dic197—41no—
94C4×Dic197—41no—
95C4×Dic197D414yescomplemented
96C24×C197C2311yescharacteristic
97C22×C4×C197—22no—
98C22×C4×C197—41no—
99C22×C4×C197—41no—
100C22×C4×C197C2311yescharacteristic
101C22×C4×C197C2311yescharacteristic
102C22×C4×C197C2311yescharacteristic
103C22×C4×C197C2311yescharacteristic
104C22×C4×C197C2311yescharacteristic
105C22×C4×C197C2311yescharacteristic
106C4⋊C4×C197—22no—
107C4⋊C4×C197—41no—
108C22⋊C4×C197—41no—
109C22⋊C4×C197C2×C414yescomplemented
110C22×D197—21no—
111C22×D197—21no—
112C22×D197—21no—
113C22×D197—21no—
114C22×D197—21no—
115C22×D197—21no—
116C22×D197—21no—
117C22×D197—21no—
118C22×D197—21no—
119C22×D197—21no—
120C22×D197—21no—
121C22×D197—21no—
122C22×D197C22×C412yes—
123C22×D197C22×C412yes—
124D788—44no—
125C22⋊D197—28no—
126C22⋊D197—44no—
127C2×Dic197—21no—
128C2×Dic197—21no—
129C2×Dic197—21no—
130C2×Dic197—21no—
131C2×Dic197—21no—
132C2×Dic197—21no—
133C2×Dic197—21no—
134C2×Dic197—21no—
135C2×Dic197—21no—
136C2×Dic197—21no—
137C2×Dic197—21no—
138C2×Dic197—21no—
139C2×Dic197—21no—
140C2×Dic197C22×C412yes—
141C2×Dic197C22×C412yes—
142C2×Dic197C2×D412yes—
143C2×Dic197C2×D412yes—
144C2×Dic197#4112yes—
145C4×D197—28no—
146C4×D197—44no—
147C23×C197—21no—
148C23×C197—21no—
149C23×C197—21no—
150C23×C197—21no—
151C23×C197—21no—
152C23×C197—21no—
153C23×C197C22×C412yes—
154C23×C197C2411yescharacteristic
155C2×C4×C197—21no—
156C2×C4×C197—21no—
157C2×C4×C197—21no—
158C2×C4×C197—21no—
159C2×C4×C197—21no—
160C2×C4×C197—21no—
161C2×C4×C197—21no—
162C2×C4×C197—21no—
163C2×C4×C197—21no—
164C2×C4×C197—21no—
165C2×C4×C197—21no—
166C2×C4×C197—21no—
167C2×C4×C197—21no—
168C2×C4×C197—21no—
169C2×C4×C197—21no—
170C2×C4×C197—44no—
171C2×C4×C197C22×C412yes—
172C2×C4×C197C22×C412yes—
173C2×C4×C197C2×D412yes—
174C22×C197—22no—
175C22×C197—22no—
176C22×C197—22no—
177C22×C197—41no—
178C22×C197—41no—
179C22×C197—41no—
180C22×C197—41no—
181C22×C197C23×C411yescharacteristic
182C22×C197C22×D411yescharacteristic
183C22×C197C2×#4111yescharacteristic
184C22×C197H(5,𝔽2)11yescharacteristic
185C22×C197H(5,𝔽2)11yescharacteristic
186C22×C197H(5,𝔽2)11yescharacteristic
187C22×C1972−1+411yescharacteristic
188C2×D197—22no—
189C2×D197—22no—
190C2×D197—22no—
191C2×D197—22no—
192C2×D197—22no—
193C2×D197—22no—
194C2×D197—41no—
195C2×D197—41no—
196C2×D197—41no—
197C2×D197—41no—
198C2×D197—41no—
199C2×D197—41no—
200C2×D197—41no—
201C2×D197—41no—
202C4×C197—22no—
203C4×C197—22no—
204C4×C197—22no—
205C4×C197—41no—
206C4×C197—41no—
207C4×C197—41no—
208Dic197—22no—
209Dic197—22no—
210Dic197—41no—
211Dic197—41no—
212Dic197—41no—
213Dic197C4×D414yes—
214C2×C197—42no—
215C2×C197C2×C4×D411yescharacteristic
216C2×C197#51811yescharacteristic
217C2×C197#52011yescharacteristic
218C2×C197#53511yescharacteristic
219C2×C197#53711yescharacteristic
220C2×C197#54011yescharacteristic
221C2×C197#54211yescharacteristic
222D197—42no—
223D197—42no—
224C197#232411yescharacteristic, complemented, Sylow
225#2324—1971nomaximal, Sylow
226C2×#121—1971no—
227C2×C4×D4—1971no—
228C2×#117—1971no—
229#390—1971no—
230#390—1971no—
231#390—1971no—
232#389—1971no—
233#388—1971no—
234#388—1971no—
235#386—1971no—
236#386—1971no—
237#385—1971no—
238#385D19711yescharacteristic, complemented
239#380—1971no—
240C4×C4⋊C4—1971no—
241C22×D4—1971no—
242C22×D4—1971no—
243C22×D4—1971no—
244C23×C4—1971no—
245C23×C4—1971no—
246C23×C4C2×D19711yescharacteristic
247#121—1978no—
248C4×D4—3944no—
249#117—3944no—
250C2×C4⋊C4—1971no—
251C2×C4⋊C4—1971no—
252C2×C4⋊C4—1971no—
253C2×C4⋊C4—1971no—
254C2×C4⋊C4—1971no—
255C2×C4⋊C4C2×D19711yescharacteristic
256C2×C4⋊C4C2×D19711yescharacteristic
257C2×C22⋊C4—1971no—
258C2×C22⋊C4—1971no—
259C2×C22⋊C4—1971no—
260C2×C22⋊C4—1971no—
261C2×C22⋊C4—1971no—
262C2×C22⋊C4—1971no—
263C2×C22⋊C4—1971no—
264C2×C22⋊C4—1971no—
265C2×C22⋊C4—1971no—
266C2×C22⋊C4—1971no—
267C2×C22⋊C4C2×D19711yescharacteristic
268C2×C22⋊C4C2×D19711yescharacteristic
269C2×C42—1971no—
270C2×C42—1971no—
271C2×C42—1971no—
272C2×C42—1971no—
273#95—1971no—
274#95—1971no—
275#95—1971no—
276#95—1971no—
277#95C2×D19711yescharacteristic
278#95C2×D19711yescharacteristic
279C24—1971no—
280C24—1971no—
281C24C22×D19711yescharacteristic
282C2×D4—7881no—
283C2×D4—7881no—
284C2×D4—7881no—
285C2×D4—7881no—
286C2×D4—3942no—
287C2×D4—3942no—
288C2×D4—1974no—
289C2×D4—1974no—
290C2×D4—1974no—
291C22×C4—22no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—7881no—
295C22×C4—7881no—
296C22×C4—3942no—
297C22×C4—3942no—
298C22×C4—3942no—
299C22×C4—1971no—
300C22×C4—1971no—
301C22×C4—1971no—
302C22×C4—1971no—
303C22×C4—1971no—
304C22×C4—1974no—
305C22×C4—1971no—
306C22×C4C22×D19711yescharacteristic
307C22×C4C22×D19711yescharacteristic
308C22×C4C22×D19711yescharacteristic
309C22×C4C22×D19711yescharacteristic
310C22×C4C22×D19711yescharacteristic
311C22×C4C22×D19711yescharacteristic
312C4⋊C4—22no—
313C4⋊C4—41no—
314C4⋊C4—7881no—
315C4⋊C4—3942no—
316C4⋊C4—3942no—
317C4⋊C4—3942no—
318C4⋊C4—1974no—
319C22⋊C4—41no—
320C22⋊C4—7881no—
321C22⋊C4—7881no—
322C22⋊C4—7881no—
323C22⋊C4—7881no—
324C22⋊C4—7881no—
325C22⋊C4—3942no—
326C22⋊C4—3942no—
327C22⋊C4—3942no—
328C22⋊C4—3942no—
329C22⋊C4—1974no—
330C22⋊C4C4×D19714yescomplemented
331C42—7881no—
332C42—7881no—
333C42—3942no—
334C42—1974no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—3941no—
342C23—3941no—
343C23—3941no—
344C23—3941no—
345C23—3941no—
346C23—3941no—
347C23—3941no—
348C23—3941no—
349C23—3941no—
350C23—3941no—
351C23—3941no—
352C23—3941no—
353C23—1972no—
354C23—1972no—
355C23C2×C4×D19712yes—
356C23C23×D19711yescharacteristic
357D4—7884no—
358D4—7884no—
359D4—3948no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—44no—
376C2×C4—7884no—
377C2×C4—3948no—
378C2×C4—3941no—
379C2×C4—3941no—
380C2×C4—3941no—
381C2×C4—3941no—
382C2×C4—3941no—
383C2×C4—3941no—
384C2×C4—3941no—
385C2×C4—3941no—
386C2×C4—3941no—
387C2×C4—3941no—
388C2×C4—3941no—
389C2×C4—3941no—
390C2×C4—3941no—
391C2×C4—1972no—
392C2×C4—1972no—
393C2×C4—1972no—
394C2×C4—1972no—
395C2×C4—1972no—
396C2×C4C2×C4×D19712yes—
397C2×C4C2×C4×D19712yes—
398C2×C4D4×D19712yes—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—7881no—
407C22—7881no—
408C22—7881no—
409C22—7881no—
410C22—7881no—
411C22—7881no—
412C22—7881no—
413C22—7881no—
414C22—3942no—
415C22—3942no—
416C22—3942no—
417C22—3942no—
418C22—3942no—
419C22—3942no—
420C22C22×C4×D19711yescharacteristic
421C22C2×D4×D19711yescharacteristic
422C22#41×D19711yescharacteristic
423C22#2026337111yescharacteristic
424C22#2026337111yescharacteristic
425C22#2026337211yescharacteristic
426C22#2026337411yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—7881no—
434C4—7881no—
435C4—7881no—
436C4—3942no—
437C4—3942no—
438C4—1974no—
439C2—42no—
440C2—7882no—
441C2—7882no—
442C2C4×D4×D19711yescharacteristic
443C2#2716155111yescharacteristic
444C2#2716156011yescharacteristic
445C2#2716164411yescharacteristic
446C2#2716165911yescharacteristic
447C2#2716168111yescharacteristic
448C2#2716169911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#385×C197C2
socle, cosocleC23×C197C24
prime radicalC2×#20263299C2
hypercenter#385D197
hypocenterC197#2324
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D197C2semidirect
3. C2×#20263284C2semidirect
4. #27160980C2semidirect
5. #27160979C2semidirect
6. #27160979C2semidirect
7. #27160974C2semidirect
8. #27160972C2semidirect
9. #27160971C2semidirect
10. #27160958C2semidirect
11. #27160957C2semidirect
12. #27160950C2semidirect
13. #27160938C2semidirect
14. #27160930C2semidirect
15. #385×C197C2semidirect
21. #20263299C4semidirect
58. C2×D788C2×C4semidirect
64. C2×C22⋊D197C2×C4semidirect
76. C2×C4×D197C2×C4semidirect
81. #15691337C2×C4semidirect
91. #15691335C2×C4semidirect
95. C4×Dic197D4semidirect
109. C22⋊C4×C197C2×C4semidirect
224. C197#2324semidirect
238. #385D197semidirect
330. C22⋊C4C4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202632992
C2×C4×C22⋊D1972
C2×#202632842
#271609802
#271609792
#271609742
#271609722
#271609712
#271609582
#271609572
#271609502
#271609382
#271609302
#385×C1972
#2324197