Finite Group Databasebeta

← #38745703

Subgroups — #38745703

Order 25216.

652 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263308C211yesmaximal, characteristic
2C2×#20263286C211yesmaximal, characteristic
3C2×#20263284C211yesmaximal, characteristic, complemented
4#27160976C211yesmaximal, characteristic, complemented
5#27160967C211yesmaximal, characteristic, complemented
6#27160962C211yesmaximal, characteristic, complemented
7#27160960C211yesmaximal, characteristic, complemented
8#27160955C211yesmaximal, characteristic
9#27160954C211yesmaximal, characteristic, complemented
10#27160953C211yesmaximal, characteristic, complemented
11C4×C4⋊Dic197C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic197C211yesmaximal, characteristic, complemented
13C4×#15691335C211yesmaximal, characteristic, complemented
14#95×D197C211yesmaximal, characteristic, complemented
15#388×C197C211yesmaximal, characteristic, complemented
16C22×C22⋊D197C2211yescharacteristic
17C23×Dic197C2211yescharacteristic
18C22×C4×D197C2211yescharacteristic
19C22×C4×D197C2211yescharacteristic
20C22×C4×D197C2211yescharacteristic
21#20263308C418yescomplemented
22#20263286—24no—
23#20263284—24no—
24C2×C4⋊Dic197C2211yescharacteristic
25C2×C4⋊Dic197C2211yescharacteristic
26C2×C4⋊Dic197C2211yescharacteristic
27C2×#15691337C2211yescharacteristic
28C2×#15691337C2211yescharacteristic
29C2×#15691337C2211yescharacteristic
30C2×C22⋊Dic197C2211yescharacteristic
31C2×C22⋊Dic197C2211yescharacteristic
32C2×#15691335C2211yescharacteristic
33C2×#15691335C2211yescharacteristic
34C2×#15691335C2211yescharacteristic
35C2×C4×Dic197C2211yescharacteristic
36C2×C4×Dic197C2211yescharacteristic
37C2×C4×Dic197C2211yescharacteristic
38C2×C4×Dic197C2211yescharacteristic
39#20263233C2211yescharacteristic
40#20263233C2211yescharacteristic
41#20263233C2211yescharacteristic
42#20263233C2211yescharacteristic
43#20263233C2211yescharacteristic
44#20263233C2211yescharacteristic
45#20263233C2211yescharacteristic
46#20263233C2211yescharacteristic
47C2×C4⋊C4×C197C2211yescharacteristic
48C2×C22⋊C4×C197C2211yescharacteristic
49C2×C22⋊C4×C197C2211yescharacteristic
50C2×C22⋊C4×C197C2211yescharacteristic
51C2×C42×C197C2211yescharacteristic
52#95×C197C2211yescharacteristic
53#95×C197C2211yescharacteristic
54C23×D197C2311yescharacteristic
55C2×C22⋊D197—22no—
56C2×C22⋊D197—22no—
57C2×C22⋊D197C2×C414yes—
58C22×Dic197—22no—
59C22×Dic197—22no—
60C22×Dic197—41no—
61C22×Dic197C2311yescharacteristic
62C22×Dic197C2311yescharacteristic
63C22×Dic197C2311yescharacteristic
64C22×Dic197C2311yescharacteristic
65C22×Dic197C2311yescharacteristic
66C22×Dic197C2311yescharacteristic
67C22×Dic197C2311yescharacteristic
68C2×C4×D197—22no—
69C2×C4×D197—22no—
70C2×C4×D197—22no—
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—
78C4⋊Dic197—22no—
79C4⋊Dic197C2×C414yescomplemented
80#15691337—22no—
81#15691337—41no—
82#15691337C2×C414yescomplemented
83C22⋊Dic197—41no—
84C22⋊Dic197C2×C414yescomplemented
85#15691335—22no—
86#15691335C2×C414yescomplemented
87#15691335D414yes—
88C4×Dic197—22no—
89C4×Dic197—22no—
90C4×Dic197—22no—
91C4×Dic197—41no—
92C24×C197C2311yescharacteristic
93C22×C4×C197C2311yescharacteristic
94C22×C4×C197C2311yescharacteristic
95C22×C4×C197C2311yescharacteristic
96C22×C4×C197C2311yescharacteristic
97C22×C4×C197C2311yescharacteristic
98C22×C4×C197C2311yescharacteristic
99C4⋊C4×C197—22no—
100C22⋊C4×C197—22no—
101C22⋊C4×C197—22no—
102C22⋊C4×C197C2×C414yescomplemented
103C42×C197—22no—
104C22×D197—21no—
105C22×D197—21no—
106C22×D197—21no—
107C22×D197—21no—
108C22×D197C22×C412yes—
109C22×D197C2×D412yes—
110C22×D197#4112yes—
111C22⋊D197—28no—
112C2×Dic197—21no—
113C2×Dic197—21no—
114C2×Dic197—21no—
115C2×Dic197—21no—
116C2×Dic197—21no—
117C2×Dic197—21no—
118C2×Dic197—21no—
119C2×Dic197—21no—
120C2×Dic197—21no—
121C2×Dic197—21no—
122C2×Dic197—21no—
123C2×Dic197—21no—
124C2×Dic197—21no—
125C2×Dic197—21no—
126C2×Dic197—21no—
127C2×Dic197—21no—
128C2×Dic197—44no—
129C2×Dic197C22×C412yes—
130C2×Dic197C22×C412yes—
131C2×Dic197C22×C412yes—
132C2×Dic197C2×D412yes—
133C2×Dic197#4112yes—
134C4×D197—28no—
135C4×D197—44no—
136C4×D197—44no—
137C23×C197—21no—
138C23×C197—21no—
139C23×C197—21no—
140C23×C197—21no—
141C23×C197—21no—
142C23×C197—21no—
143C23×C197C22×C412yes—
144C23×C197C2411yescharacteristic
145C2×C4×C197—21no—
146C2×C4×C197—21no—
147C2×C4×C197—21no—
148C2×C4×C197—21no—
149C2×C4×C197—21no—
150C2×C4×C197—21no—
151C2×C4×C197—21no—
152C2×C4×C197—21no—
153C2×C4×C197—21no—
154C2×C4×C197—21no—
155C2×C4×C197—21no—
156C2×C4×C197—21no—
157C2×C4×C197—21no—
158C2×C4×C197—21no—
159C2×C4×C197C22×C412yes—
160C2×C4×C197C22×C412yes—
161C2×C4×C197C2×D412yes—
162C2×C4×C197#4112yes—
163C22×C197—22no—
164C22×C197—22no—
165C22×C197—22no—
166C22×C197—41no—
167C22×C197—41no—
168C22×C197—41no—
169C22×C197—41no—
170C22×C197C23×C411yescharacteristic
171C22×C197C22×D411yescharacteristic
172C22×C197C2×#4111yescharacteristic
173C22×C197C2×#4111yescharacteristic
174C22×C197C2×#4111yescharacteristic
175C22×C197H(5,𝔽2)11yescharacteristic
176C22×C1972−1+411yescharacteristic
177C2×D197—22no—
178C2×D197—22no—
179C2×D197—22no—
180C2×D197—22no—
181C2×D197—22no—
182C2×D197—22no—
183C2×D197C4×D414yes—
184C4×C197—22no—
185C4×C197—22no—
186C4×C197—22no—
187C4×C197—22no—
188C4×C197—41no—
189C4×C197—41no—
190Dic197—22no—
191Dic197—22no—
192Dic197—22no—
193Dic197—22no—
194Dic197—22no—
195Dic197—41no—
196Dic197—41no—
197C2×C197—42no—
198C2×C197C2×C4×D411yescharacteristic
199C2×C197C4×#4111yescharacteristic
200C2×C197#52011yescharacteristic
201C2×C197#54611yescharacteristic
202C2×C197#54711yescharacteristic
203C2×C197#55111yescharacteristic
204C2×C197#55211yescharacteristic
205D197—24no—
206C197#235911yescharacteristic, complemented, Sylow
207#2359—1971nomaximal, Sylow
208C2×#123—1971no—
209C2×C4×D4—1971no—
210C2×#117—1971no—
211#389—1971no—
212#388—1971no—
213#388D19711yescharacteristic, complemented
214#387—1971no—
215#387—1971no—
216#386—1971no—
217#385—1971no—
218#385—1971no—
219C4×C4⋊C4—1971no—
220C4×C22⋊C4—1971no—
221C4×C22⋊C4—1971no—
222C2×#95—1971no—
223C22×D4—1971no—
224C23×C4—1971no—
225C23×C4—1971no—
226C23×C4—1971no—
227C23×C4—1971no—
228#123—1978no—
229C4×D4—3944no—
230#117—3944no—
231C2×C4⋊C4—1971no—
232C2×C4⋊C4—1971no—
233C2×C4⋊C4—1971no—
234C2×C4⋊C4—1971no—
235C2×C4⋊C4—1971no—
236C2×C4⋊C4—1971no—
237C2×C4⋊C4C2×D19711yescharacteristic
238C2×C22⋊C4—1971no—
239C2×C22⋊C4—1971no—
240C2×C22⋊C4—1971no—
241C2×C22⋊C4—1971no—
242C2×C22⋊C4—1971no—
243C2×C22⋊C4C2×D19711yescharacteristic
244C2×C22⋊C4C2×D19711yescharacteristic
245C2×C22⋊C4C2×D19711yescharacteristic
246C2×C42—1971no—
247C2×C42—1971no—
248C2×C42—1971no—
249C2×C42—1971no—
250C2×C42C2×D19711yescharacteristic
251#95—1971no—
252#95—1971no—
253#95—1971no—
254#95—1971no—
255#95—1971no—
256#95—1971no—
257#95—1971no—
258#95—1971no—
259#95C2×D19711yescharacteristic
260#95C2×D19711yescharacteristic
261C24—1971no—
262C24C22×D19711yescharacteristic
263C2×D4—3942no—
264C2×D4—3942no—
265C2×D4—1974no—
266C22×C4—7881no—
267C22×C4—7881no—
268C22×C4—7881no—
269C22×C4—3942no—
270C22×C4—3942no—
271C22×C4—3942no—
272C22×C4—3942no—
273C22×C4—3942no—
274C22×C4—3942no—
275C22×C4—3942no—
276C22×C4—3942no—
277C22×C4—1971no—
278C22×C4—1971no—
279C22×C4—1971no—
280C22×C4—1971no—
281C22×C4—1971no—
282C22×C4—1971no—
283C22×C4—1971no—
284C22×C4—1974no—
285C22×C4C22×D19711yescharacteristic
286C22×C4C22×D19711yescharacteristic
287C22×C4C22×D19711yescharacteristic
288C22×C4C22×D19711yescharacteristic
289C22×C4C22×D19711yescharacteristic
290C22×C4C22×D19711yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—7881no—
293C4⋊C4—3942no—
294C4⋊C4—3942no—
295C4⋊C4—3942no—
296C4⋊C4—1974no—
297C4⋊C4—1974no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—7881no—
301C22⋊C4—3942no—
302C22⋊C4—1974no—
303C22⋊C4—1974no—
304C22⋊C4—1974no—
305C22⋊C4C4×D19714yescomplemented
306C42—22no—
307C42—7881no—
308C42—3942no—
309C42—3942no—
310C42—3942no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—3941no—
318C23—3941no—
319C23—3941no—
320C23—3941no—
321C23—1972no—
322C23—1972no—
323C23—1972no—
324C23C2×C4×D19712yes—
325C23C23×D19711yescharacteristic
326D4—3948no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—7884no—
342C2×C4—7884no—
343C2×C4—7884no—
344C2×C4—3948no—
345C2×C4—3941no—
346C2×C4—3941no—
347C2×C4—3941no—
348C2×C4—3941no—
349C2×C4—3941no—
350C2×C4—3941no—
351C2×C4—3941no—
352C2×C4—3941no—
353C2×C4—3941no—
354C2×C4—3941no—
355C2×C4—3941no—
356C2×C4—3941no—
357C2×C4—3941no—
358C2×C4—3941no—
359C2×C4—3941no—
360C2×C4—3941no—
361C2×C4—1972no—
362C2×C4—1972no—
363C2×C4—1972no—
364C2×C4—1972no—
365C2×C4—1972no—
366C2×C4C2×C4×D19712yes—
367C2×C4C2×C4×D19712yes—
368C2×C4D4×D19712yes—
369C2×C4#1569135712yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—3942no—
378C22—3942no—
379C22—3942no—
380C22—3942no—
381C22—3942no—
382C22—3942no—
383C22—1974no—
384C22C22×C4×D19711yescharacteristic
385C22C2×D4×D19711yescharacteristic
386C22#41×D19711yescharacteristic
387C22#41×D19711yescharacteristic
388C22C2×#1569135711yescharacteristic
389C22#2026337211yescharacteristic
390C22#2026337311yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—7881no—
398C4—7881no—
399C4—3942no—
400C4—3942no—
401C4—3942no—
402C4—3942no—
403C4—3942no—
404C2—42no—
405C2—3944no—
406C2C4×D4×D19711yescharacteristic
407C2C4×#1569135711yescharacteristic
408C2#2716155911yescharacteristic
409C2#2716172211yescharacteristic
410C2#2716172911yescharacteristic
411C2#2716175211yescharacteristic
412C2#2716176511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#388×C197C2
socle, cosocleC23×C197C24
prime radicalC22×C22⋊D197C22
hypercenter#388D197
hypocenterC197#2359
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
3. C2×#20263284C2semidirect
4. #27160976C2semidirect
5. #27160967C2semidirect
6. #27160962C2semidirect
7. #27160960C2semidirect
9. #27160954C2semidirect
10. #27160953C2semidirect
11. C4×C4⋊Dic197C2semidirect
12. C22⋊C4×Dic197C2semidirect
13. C4×#15691335C2semidirect
14. #95×D197C2semidirect
15. #388×C197C2semidirect
21. #20263308C4semidirect
76. C2×C4×D197C2×C4semidirect
79. C4⋊Dic197C2×C4semidirect
82. #15691337C2×C4semidirect
84. C22⋊Dic197C2×C4semidirect
86. #15691335C2×C4semidirect
102. C22⋊C4×C197C2×C4semidirect
206. C197#2359semidirect
213. #388D197semidirect
305. C22⋊C4C4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633082
C2×#202632862
C2×#202632842
#271609762
#271609672
#271609622
#271609602
#271609552
#271609542
#271609532
C4×C4⋊Dic1972
C22⋊C4×Dic1972
C4×#156913352
#95×D1972
#388×C1972
#2359197