Finite Group Databasebeta

← C2×#27161677 #38756064

Subgroups — C2×#27161677 #38756064

Order 25216.

1144 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161677C2116yesmaximal, direct product factor
2C2×#20263321C211yesmaximal, characteristic, complemented
3C2×#20263320C211yesmaximal, characteristic, complemented
4C2×#20263316C211yesmaximal, characteristic, complemented
5C2×#20263311C211yesmaximal, characteristic, complemented
6C2×#20263311C211yesmaximal, characteristic, complemented
7C2×#20263311C211yesmaximal, characteristic, complemented
8C2×#20263311C211yesmaximal, characteristic, complemented
9C2×#20263306C211yesmaximal, characteristic, complemented
10C2×#20263304C211yesmaximal, characteristic, complemented
11C2×#20263304C211yesmaximal, characteristic, complemented
12C2×#20263301C211yesmaximal, characteristic
13C2×C4×D788C211yesmaximal, characteristic, complemented
14C2×#20263288C211yesmaximal, characteristic, complemented
15C2×C4⋊C4×D197C211yesmaximal, characteristic, complemented
16C2×#125×C197C211yesmaximal, characteristic, complemented
17C22×Dic394C2211yescharacteristic
18C22×D788C2211yescharacteristic
19C22×D788C2211yescharacteristic
20C22×D788C2211yescharacteristic
21C22×C4×D197C2211yescharacteristic
22C22×C4×D197C2211yescharacteristic
23C22×C4×D197C2211yescharacteristic
24C22×C4×D197C2211yescharacteristic
25C22×C4×D197C2211yescharacteristic
26#20263321C2218yescomplemented
27#20263320C2218yescomplemented
28#20263316C2218yescomplemented
29#20263311C2218yescomplemented
30#20263311C2218yescomplemented
31#20263311C2218yescomplemented
32#20263311C2218yescomplemented
33#20263306C2218yescomplemented
34#20263304C2218yescomplemented
35#20263304C2218yescomplemented
36#20263301C2218yes—
37C4×D788C2218yescomplemented
38#20263288C2218yescomplemented
39C2×C4⋊Dic197C2211yescharacteristic
40C2×C4⋊Dic197C2211yescharacteristic
41C2×#15691337C2211yescharacteristic
42C2×#15691337C2211yescharacteristic
43C2×#15691337C2211yescharacteristic
44C2×#15691337C2211yescharacteristic
45C2×#15691337C2211yescharacteristic
46C2×#15691337C2211yescharacteristic
47C4⋊C4×D197C2218yescomplemented
48C2×#15691335C2211yescharacteristic
49C2×#15691335C2211yescharacteristic
50C2×#15691335C2211yescharacteristic
51C2×#15691335C2211yescharacteristic
52C2×#15691335C2211yescharacteristic
53C2×#15691335C2211yescharacteristic
54C2×#15691335C2211yescharacteristic
55C2×#15691335C2211yescharacteristic
56C2×#15691335C2211yescharacteristic
57C2×#15691335C2211yescharacteristic
58C2×C4×Dic197C2211yescharacteristic
59#125×C197C2218yescomplemented
60C2×C4⋊C4×C197C2211yescharacteristic
61C2×C4⋊C4×C197C2211yescharacteristic
62C2×C4⋊C4×C197C2211yescharacteristic
63C2×C4⋊C4×C197C2211yescharacteristic
64C2×C4⋊C4×C197C2211yescharacteristic
65C2×C4⋊C4×C197C2211yescharacteristic
66C2×C42×C197C2211yescharacteristic
67C23×D197C2311yescharacteristic
68C23×D197C2311yescharacteristic
69C23×D197C2311yescharacteristic
70C2×Dic394—42no—
71C2×Dic394C2314yes—
72C2×D788—24no—
73C2×D788—24no—
74C2×D788—42no—
75C2×D788C2314yes—
76C2×D788C2314yes—
77C2×D788C2314yes—
78C22×Dic197C2311yescharacteristic
79C22×Dic197C2311yescharacteristic
80C22×Dic197C2311yescharacteristic
81C22×Dic197C2311yescharacteristic
82C22×Dic197C2311yescharacteristic
83C2×C4×D197—24no—
84C2×C4×D197—42no—
85C2×C4×D197—42no—
86C2×C4×D197—42no—
87C2×C4×D197—42no—
88C2×C4×D197C2314yes—
89C2×C4×D197C2314yes—
90C2×C4×D197C2314yes—
91C2×C4×D197C2314yes—
92C2×C4×D197C2314yes—
93C4⋊Dic197C2314yes—
94C4⋊Dic197C2314yes—
95#15691337C2314yes—
96#15691337C2314yes—
97#15691337C2314yes—
98#15691337C2314yes—
99#15691337C2314yes—
100#15691337C2314yes—
101#15691335C2314yes—
102#15691335C2314yes—
103#15691335C2314yes—
104#15691335C2314yes—
105#15691335C2314yes—
106#15691335C2314yes—
107#15691335C2314yes—
108#15691335C2314yes—
109#15691335C2314yes—
110#15691335C2314yes—
111C4×Dic197C2314yes—
112C22×C4×C197C2311yescharacteristic
113C22×C4×C197C2311yescharacteristic
114C22×C4×C197C2311yescharacteristic
115C22×C4×C197C2311yescharacteristic
116C22×C4×C197C2311yescharacteristic
117C22×C4×C197C2311yescharacteristic
118C22×C4×C197C2311yescharacteristic
119C4⋊C4×C197C2314yes—
120C4⋊C4×C197C2314yes—
121C4⋊C4×C197C2314yes—
122C4⋊C4×C197C2314yes—
123C4⋊C4×C197C2314yes—
124C4⋊C4×C197C2314yes—
125C42×C197C2314yes—
126C22×D197—22no—
127C22×D197—22no—
128C22×D197—22no—
129C22×D197—22no—
130C22×D197—22no—
131C22×D197—22no—
132C22×D197—22no—
133C22×D197—22no—
134C22×D197#4114yes—
135C22×D197C2412yes—
136C22×D197C2412yes—
137C22×D197C2412yes—
138Dic394—44no—
139D788—28no—
140D788—28no—
141D788—44no—
142C2×Dic197—22no—
143C2×Dic197—22no—
144C2×Dic197—22no—
145C2×Dic197—22no—
146C2×Dic197—22no—
147C2×Dic197C2412yes—
148C2×Dic197C2412yes—
149C2×Dic197C2412yes—
150C2×Dic197C2412yes—
151C2×Dic197C2412yes—
152C4×D197—28no—
153C4×D197—44no—
154C4×D197—44no—
155C4×D197—44no—
156C4×D197—44no—
157C23×C197C2411yescharacteristic
158C2×C4×C197—22no—
159C2×C4×C197—22no—
160C2×C4×C197—22no—
161C2×C4×C197—22no—
162C2×C4×C197—22no—
163C2×C4×C197—22no—
164C2×C4×C197—22no—
165C2×C4×C197C2412yes—
166C2×C4×C197C2412yes—
167C2×C4×C197C2412yes—
168C2×C4×C197C2412yes—
169C2×C4×C197C2412yes—
170C2×C4×C197C2412yes—
171C2×C4×C197C2412yes—
172C22×C197C2×#4112yes—
173C22×C197H(5,𝔽2)12yes—
174C22×C1972−1+412yes—
175C22×C197C2511yescharacteristic
176C2×D197—28no—
177C2×D197—22no—
178C2×D197—22no—
179C2×D197—22no—
180C2×D197—22no—
181C2×D197—22no—
182C2×D197—22no—
183C2×D197—22no—
184C2×D197—22no—
185C2×D197—44no—
186C2×D197—44no—
187C2×D197C2×#4114yes—
188C4×C197—22no—
189C4×C197—22no—
190C4×C197—22no—
191C4×C197—22no—
192C4×C197—22no—
193C4×C197—22no—
194C4×C197—22no—
195Dic197—22no—
196Dic197—22no—
197Dic197—22no—
198Dic197—22no—
199Dic197—22no—
200C2×C197#53914yescomplemented
201C2×C197C22×#4111yescharacteristic
202C2×C197C2×H(5,𝔽2)11yescharacteristic
203C2×C197C2×2−1+411yescharacteristic
204D197—24no—
205D197—42no—
206D197—42no—
207C197C2×#53911yescharacteristic, complemented, Sylow
208C2×#539—1971nomaximal, Sylow
209#539—19716no—
210C2×#126—1971no—
211C2×#126—1971no—
212C2×#125—1971no—
213C2×#125D19711yescharacteristic, complemented
214C2×#123—1971no—
215C2×#123—1971no—
216C2×#123—1971no—
217C2×#123—1971no—
218C2×C22⋊Q8—1971no—
219C2×C22⋊Q8—1971no—
220C2×C22⋊Q8—1971no—
221C2×#121—1971no—
222C2×C4×D4—1971no—
223C2×C4×D4—1971no—
224C22×C4⋊C4—1971no—
225C22×Q8—1971no—
226C22×D4—1971no—
227C22×D4—1971no—
228C22×D4—1971no—
229C23×C4—1971no—
230C23×C4—1971no—
231C23×C4—1971no—
232C23×C4—1971no—
233C23×C4—1971no—
234#126—1978no—
235#126—1978no—
236#125—1978no—
237#125C2×D19718yescomplemented
238#123—1978no—
239#123—1978no—
240#123—1978no—
241#123—1978no—
242C22⋊Q8—1978no—
243C22⋊Q8—1978no—
244C22⋊Q8—1978no—
245#121—1978no—
246C4×D4—1978no—
247C4×D4—1978no—
248C2×C4⋊C4—1971no—
249C2×C4⋊C4—1971no—
250C2×C4⋊C4—1971no—
251C2×C4⋊C4—1971no—
252C2×C4⋊C4—1971no—
253C2×C4⋊C4—1971no—
254C2×C4⋊C4—1971no—
255C2×C4⋊C4—1971no—
256C2×C4⋊C4—1978no—
257C2×C4⋊C4C2×D19711yescharacteristic
258C2×C4⋊C4C2×D19711yescharacteristic
259C2×C4⋊C4C2×D19711yescharacteristic
260C2×C4⋊C4C2×D19711yescharacteristic
261C2×C4⋊C4C2×D19711yescharacteristic
262C2×C4⋊C4C2×D19711yescharacteristic
263C2×C22⋊C4—1971no—
264C2×C22⋊C4—1971no—
265C2×C22⋊C4—1971no—
266C2×C22⋊C4—1971no—
267C2×C22⋊C4—1971no—
268C2×C22⋊C4—1971no—
269C2×C22⋊C4—1971no—
270C2×C22⋊C4—1971no—
271C2×C22⋊C4—1971no—
272C2×C22⋊C4—1971no—
273C2×C42—1971no—
274C2×C42C2×D19711yescharacteristic
275C24—1971no—
276C24—1971no—
277C24—1971no—
278C2×Q8—7882no—
279C2×Q8—1974no—
280C2×D4—7882no—
281C2×D4—3944no—
282C2×D4—3944no—
283C2×D4—1974no—
284C2×D4—1974no—
285C2×D4—1974no—
286C22×C4—7882no—
287C22×C4—7882no—
288C22×C4—7882no—
289C22×C4—7882no—
290C22×C4—3944no—
291C22×C4—1974no—
292C22×C4—1974no—
293C22×C4—1974no—
294C22×C4—1974no—
295C22×C4—1971no—
296C22×C4—1971no—
297C22×C4—1971no—
298C22×C4—1971no—
299C22×C4—1971no—
300C22×C4—1974no—
301C22×C4C22×D19711yescharacteristic
302C22×C4C22×D19711yescharacteristic
303C22×C4C22×D19711yescharacteristic
304C22×C4C22×D19711yescharacteristic
305C22×C4C22×D19711yescharacteristic
306C22×C4C22×D19711yescharacteristic
307C22×C4C22×D19711yescharacteristic
308C4⋊C4—1974no—
309C4⋊C4—1974no—
310C4⋊C4—1974no—
311C4⋊C4—1974no—
312C4⋊C4—1974no—
313C4⋊C4—1974no—
314C4⋊C4—1974no—
315C4⋊C4—1974no—
316C4⋊C4C22×D19714yes—
317C4⋊C4C22×D19714yes—
318C4⋊C4C22×D19714yes—
319C4⋊C4C22×D19714yes—
320C4⋊C4C22×D19714yes—
321C4⋊C4C22×D19714yes—
322C22⋊C4—1974no—
323C22⋊C4—1974no—
324C22⋊C4—1974no—
325C22⋊C4—1974no—
326C22⋊C4—1974no—
327C22⋊C4—1974no—
328C22⋊C4—1974no—
329C22⋊C4—1974no—
330C22⋊C4—1974no—
331C22⋊C4—1974no—
332C42—1974no—
333C42C22×D19714yes—
334C23—3942no—
335C23—3942no—
336C23—3942no—
337C23—3942no—
338C23—3942no—
339C23—3942no—
340C23—3942no—
341C23—3942no—
342C23—1972no—
343C23—1972no—
344C23—1974no—
345C23—1972no—
346C23C23×D19711yescharacteristic
347Q8—7884no—
348D4—7884no—
349D4—3948no—
350D4—3948no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—7884no—
359C2×C4—7884no—
360C2×C4—7884no—
361C2×C4—7884no—
362C2×C4—3948no—
363C2×C4—3942no—
364C2×C4—3942no—
365C2×C4—3942no—
366C2×C4—3942no—
367C2×C4—3942no—
368C2×C4—1972no—
369C2×C4—1972no—
370C2×C4—1972no—
371C2×C4—1972no—
372C2×C4—1972no—
373C2×C4C23×D19712yes—
374C2×C4C23×D19712yes—
375C2×C4C23×D19712yes—
376C2×C4C23×D19712yes—
377C2×C4C23×D19712yes—
378C2×C4C23×D19712yes—
379C2×C4C23×D19712yes—
380C22—7884no—
381C22—7884no—
382C22—3942no—
383C22—3942no—
384C22—3942no—
385C22—3942no—
386C22—3942no—
387C22—3942no—
388C22—3942no—
389C22—3942no—
390C22—3948no—
391C22—1974no—
392C22#41×D19712yes—
393C22#2026337212yes—
394C22#2026337412yes—
395C22C24×D19711yescharacteristic
396C4—22no—
397C4—22no—
398C4—22no—
399C4—22no—
400C4—22no—
401C4—22no—
402C4—22no—
403C4—3942no—
404C4—3942no—
405C4—3942no—
406C4—3942no—
407C4—3942no—
408C2—7882no—
409C2—7882no—
410C2—3944no—
411C2#2716167714yesdirect product factor
412C2C2×#41×D19711yescharacteristic
413C2C2×#2026337211yescharacteristic
414C2C2×#2026337411yescharacteristic
subgroupG/H
centerC23C23×D197
commutator, cosocleC22×C197C25
FrattiniC22C24×D197
FittingC2×#125×C197C2
socleC23×C197C24
prime radicalC2×#20263301C2
hypercenterC2×#125D197
hypocenterC197C2×#539
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
1. #27161677C2direct (N × H)
2. C2×#20263321C2semidirect
3. C2×#20263320C2semidirect
4. C2×#20263316C2semidirect
5. C2×#20263311C2semidirect
6. C2×#20263311C2semidirect
7. C2×#20263311C2semidirect
8. C2×#20263311C2semidirect
9. C2×#20263306C2semidirect
10. C2×#20263304C2semidirect
11. C2×#20263304C2semidirect
13. C2×C4×D788C2semidirect
14. C2×#20263288C2semidirect
15. C2×C4⋊C4×D197C2semidirect
16. C2×#125×C197C2semidirect
26. #20263321C22semidirect
27. #20263320C22semidirect
28. #20263316C22semidirect
29. #20263311C22semidirect
30. #20263311C22semidirect
31. #20263311C22semidirect
32. #20263311C22semidirect
33. #20263306C22semidirect
34. #20263304C22semidirect
35. #20263304C22semidirect
37. C4×D788C22semidirect
38. #20263288C22semidirect
47. C4⋊C4×D197C22semidirect
59. #125×C197C22semidirect
200. C2×C197#539semidirect
207. C197C2×#539semidirect
213. C2×#125D197semidirect
237. #125C2×D197semidirect
411. C2#27161677direct (N × H)

Maximal subgroups

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

subgroupindex
#271616772
C2×#202633212
C2×#202633202
C2×#202633162
C2×#202633112
C2×#202633062
C2×#202633042
C2×#202633012
C2×C4×D7882
C2×#202632882
C2×C4⋊C4×D1972
C2×#125×C1972
C2×#539197