Finite Group Databasebeta

← C2×#27161094 #38751252

Subgroups — C2×#27161094 #38751252

Order 25216.

772 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D197C211yesmaximal, characteristic
2C2×M16×D197C211yesmaximal, characteristic, complemented
3C2×#20263283C211yesmaximal, characteristic, complemented
4#27161094C218yesmaximal, direct product factor
5C2×#20263254C211yesmaximal, characteristic, complemented
6C2×#20263253C211yesmaximal, characteristic, complemented
7C2×#20263252C211yesmaximal, characteristic, complemented
8C2×#104×C197C211yesmaximal, characteristic, complemented
9C2×#15691359C411yescharacteristic, complemented
10C2×#15691358—21no—
11C2×#15691358C2211yescharacteristic
12C2×#15691357—21no—
13C2×#15691357C411yescharacteristic, complemented
14#41×D197—22no—
15#41×D197—22no—
16#41×D197C414yescomplemented
17#41×D197C2214yes—
18C2×Q8×D197C411yescharacteristic, complemented
19C2×D4×D197—21no—
20C2×D4×D197C411yescharacteristic, complemented
21C22×C4×D197—21no—
22C22×C4×D197C2211yescharacteristic
23C2×#15691342C2211yescharacteristic, complemented
24C2×#15691341—21no—
25M16×D197—22no—
26M16×D197C2214yescomplemented
27C2×C8×D197—21no—
28#20263283—22no—
29#20263283C2214yescomplemented
30C2×#15691337—21no—
31C2×#15691335—21no—
32C2×C4×Dic197C2211yescharacteristic, complemented
33#20263254C2214yescomplemented
34#20263253C2214yescomplemented
35#20263252C2214yescomplemented
36C2×#41×C197C2211yescharacteristic
37C2×M16×C197C2211yescharacteristic, complemented
38C2×C42×C197C2211yescharacteristic, complemented
39#104×C197C2214yescomplemented
40C23×D197—21no—
41C23×D197D411yescharacteristic
42#15691359—22no—
43#15691359C2×C414yescomplemented
44#15691358—24no—
45#15691358—24no—
46#15691358—21no—
47#15691358—21no—
48#15691358C2×C412yes—
49#15691358C2312yes—
50#15691357—24no—
51#15691357—24no—
52#15691357—22no—
53#15691357C2×C414yescomplemented
54C2×Dic394—21no—
55C2×Dic394C2×C411yescharacteristic
56Q8×D197—22no—
57Q8×D197C2×C414yescomplemented
58C2×D788—21no—
59C2×D788C2×C411yescharacteristic
60C2×C22⋊D197—21no—
61C2×C22⋊D197—21no—
62C2×C22⋊D197—21no—
63D4×D197—24no—
64D4×D197—24no—
65D4×D197—22no—
66D4×D197C2×C414yescomplemented
67C22×Dic197—21no—
68C22×Dic197—21no—
69C22×Dic197D411yescharacteristic
70C2×C4×D197—21no—
71C2×C4×D197—21no—
72C2×C4×D197—24no—
73C2×C4×D197—24no—
74C2×C4×D197—21no—
75C2×C4×D197C2×C411yescharacteristic
76C2×C4×D197C2×C412yes—
77C2×C4×D197C2×C411yescharacteristic
78C2×C4×D197D412yes—
79C2×C4×D197D411yescharacteristic
80C2×C4×D197D412yes—
81C2×C4×D197D411yescharacteristic
82C2×C4×D197C2312yes—
83#15691342—21no—
84#15691342C2312yescomplemented
85#15691341—24no—
86C2×#13096802—21no—
87C8×D197—24no—
88#15691337—24no—
89#15691335—24no—
90C4×Dic197—21no—
91C4×Dic197C2312yescomplemented
92#41×C197—21no—
93#41×C197—21no—
94#41×C197C2×C412yes—
95#41×C197C2312yes—
96C2×Q8×C197C2×C411yescharacteristic
97C2×D4×C197—21no—
98C2×D4×C197C2×C411yescharacteristic
99C22×C4×C197—21no—
100C22×C4×C197—21no—
101C22×C4×C197C2311yescharacteristic
102M16×C197—21no—
103M16×C197C2312yescomplemented
104C2×C8×C197—21no—
105C42×C197—21no—
106C42×C197C2312yescomplemented
107C22×D197—21no—
108C22×D197—24no—
109C22×D197—42no—
110C22×D197—44no—
111C22×D197C22⋊C411yescharacteristic
112C22×D197C22⋊C411yescharacteristic
113C22×D197C22⋊C412yes—
114C22×D197C2×D412yes—
115Dic394—21no—
116Dic394—24no—
117Dic394C22×C412yes—
118D788—24no—
119D788—21no—
120D788C22×C412yes—
121C22⋊D197—22no—
122C22⋊D197—22no—
123C22⋊D197—24no—
124C22⋊D197—24no—
125C2×Dic197—21no—
126C2×Dic197—24no—
127C2×Dic197—22no—
128C2×Dic197—42no—
129C2×Dic197C22⋊C411yescharacteristic
130C2×Dic197C22⋊C411yescharacteristic
131C2×Dic197C22⋊C412yes—
132C2×Dic197C2×D412yes—
133C4×D197—24no—
134C4×D197—22no—
135C4×D197—22no—
136C4×D197—24no—
137C4×D197C22⋊C412yes—
138C4×D197C22⋊C412yes—
139C4×D197C22⋊C412yes—
140C4×D197C22⋊C412yes—
141C4×D197C22×C412yes—
142C4×D197C22×C412yes—
143C4×D197C2×D412yes—
144C4×D197C2×D412yes—
145#13096802—22no—
146C23×C197—21no—
147C23×C197C2×D411yescharacteristic
148Q8×C197—21no—
149Q8×C197C22×C412yes—
150D4×C197—22no—
151D4×C197—22no—
152D4×C197—21no—
153D4×C197C22×C412yes—
154C2×C4×C197—21no—
155C2×C4×C197—22no—
156C2×C4×C197—22no—
157C2×C4×C197—22no—
158C2×C4×C197—42no—
159C2×C4×C197C22×C411yescharacteristic
160C2×C4×C197C22×C411yescharacteristic
161C2×C4×C197C2×D411yescharacteristic
162C2×C4×C197C2×D411yescharacteristic
163C2×C4×C197C2×D411yescharacteristic
164C2×C4×C197C2411yescharacteristic
165C8×C197—22no—
166C22×C197—22no—
167C22×C197—22no—
168C22×C197—42no—
169C22×C197C2×C22⋊C411yescharacteristic
170C22×C197C2×C22⋊C411yescharacteristic
171C22×C197C22×D411yescharacteristic
172C2×D197—22no—
173C2×D197—22no—
174C2×D197—22no—
175C2×D197—42no—
176C2×D197—42no—
177C2×D197—48no—
178C2×D197—42no—
179C2×D197C2×C22⋊C412yes—
180C2×D197C2×C22⋊C412yes—
181C4×C197—22no—
182C4×C197—42no—
183C4×C197C2×C22⋊C411yescharacteristic
184C4×C197C2×C22⋊C411yescharacteristic
185C4×C197C23×C411yescharacteristic
186C4×C197C22×D411yescharacteristic
187Dic197—22no—
188Dic197—42no—
189Dic197C2×C22⋊C412yes—
190Dic197C2×C22⋊C412yes—
191C2×C197—21no—
192C2×C197—21no—
193C2×C197—42no—
194C2×C197#42112yescomplemented
195C2×C197C22×C22⋊C411yescharacteristic
196D197—22no—
197D197—22no—
198D197—42no—
199C197C2×#42111yescharacteristic, complemented, Sylow
200C2×#421—1971nomaximal, Sylow
201C22×#41—1971no—
202C22×M16—1971no—
203C2×#117—1971no—
204#421—1978no—
205C2×#104—1971no—
206C2×#104—1971no—
207C2×#104—1971no—
208C2×#104D19711yescharacteristic, complemented
209C2×#41—3941no—
210C2×#41—3941no—
211C2×#41—3942no—
212C2×#41—3942no—
213C2×#41—1974no—
214C2×#41—1974no—
215C2×#41—1971no—
216C2×#41—1971no—
217C2×#41—1971no—
218C2×#41C2×D19711yescharacteristic
219C22×Q8—1971no—
220C22×D4—3941no—
221C22×D4—1971no—
222C23×C4—3941no—
223C23×C4—1971no—
224C2×M16—3941no—
225C2×M16—3942no—
226C2×M16—1971no—
227C2×M16—1974no—
228C2×M16C2×D19711yescharacteristic, complemented
229C22×C8—3941no—
230#117—3942no—
231#117—1974no—
232C2×C4⋊C4—3941no—
233C2×C22⋊C4—3941no—
234C2×C42—1971no—
235C2×C42C2×D19711yescharacteristic, complemented
236#104—1974no—
237#104—1974no—
238#104—1974no—
239#104C2×D19714yescomplemented
240C24—3941no—
241C24—1971no—
242#41—21no—
243#41—21no—
244#41—3944no—
245#41—3944no—
246#41—3944no—
247#41—3944no—
248#41—3942no—
249#41—3942no—
250#41—3941no—
251#41—3941no—
252#41—1974no—
253#41—1974no—
254#41—1972no—
255#41—1972no—
256#41C4×D19712yes—
257#41C22×D19712yes—
258C2×Q8—3941no—
259C2×Q8—3942no—
260C2×Q8—1974no—
261C2×Q8—1971no—
262C2×Q8C4×D19711yescharacteristic
263C2×D4—21no—
264C2×D4—3941no—
265C2×D4—3944no—
266C2×D4—3944no—
267C2×D4—3941no—
268C2×D4—3941no—
269C2×D4—3942no—
270C2×D4—3941no—
271C2×D4—1974no—
272C2×D4—1971no—
273C2×D4C4×D19711yescharacteristic
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—3941no—
277C22×C4—3941no—
278C22×C4—3944no—
279C22×C4—3944no—
280C22×C4—3941no—
281C22×C4—3941no—
282C22×C4—3941no—
283C22×C4—1971no—
284C22×C4—1971no—
285C22×C4—1971no—
286C22×C4—1972no—
287C22×C4—1972no—
288C22×C4—1972no—
289C22×C4—1971no—
290C22×C4—1971no—
291C22×C4—1972no—
292C22×C4C22×D19711yescharacteristic
293M16—21no—
294M16—3944no—
295M16—3941no—
296M16—1972no—
297M16C22×D19712yescomplemented
298C2×C8—21no—
299C2×C8—3944no—
300C2×C8—3941no—
301C4⋊C4—3944no—
302C22⋊C4—3944no—
303C42—21no—
304C42—3941no—
305C42—1972no—
306C42C22×D19712yescomplemented
307C23—21no—
308C23—7882no—
309C23—7884no—
310C23—3944no—
311C23—3941no—
312C23—1972no—
313C23—1972no—
314C23—1971no—
315C23—1971no—
316C23D4×D19711yescharacteristic
317Q8—21no—
318Q8—3944no—
319Q8—3941no—
320Q8—1972no—
321Q8C2×C4×D19712yes—
322D4—22no—
323D4—22no—
324D4—21no—
325D4—3944no—
326D4—3944no—
327D4—3942no—
328D4—3942no—
329D4—3944no—
330D4—3941no—
331D4—1972no—
332D4C2×C4×D19712yes—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—42no—
338C2×C4—7882no—
339C2×C4—3944no—
340C2×C4—3944no—
341C2×C4—3942no—
342C2×C4—3941no—
343C2×C4—3944no—
344C2×C4—3942no—
345C2×C4—3942no—
346C2×C4—1971no—
347C2×C4—1971no—
348C2×C4—1972no—
349C2×C4—1972no—
350C2×C4—1972no—
351C2×C4—1972no—
352C2×C4—1972no—
353C2×C4—1972no—
354C2×C4—1972no—
355C2×C4—1972no—
356C2×C4—1972no—
357C2×C4—1972no—
358C2×C4C2×C4×D19711yescharacteristic
359C2×C4C2×C4×D19711yescharacteristic
360C2×C4D4×D19711yescharacteristic
361C2×C4D4×D19711yescharacteristic
362C2×C4D4×D19711yescharacteristic
363C2×C4C23×D19711yescharacteristic
364C8—22no—
365C8—3942no—
366C22—22no—
367C22—22no—
368C22—42no—
369C22—7882no—
370C22—7882no—
371C22—7888no—
372C22—7882no—
373C22—3942no—
374C22—3942no—
375C22—3942no—
376C22—1972no—
377C22—1972no—
378C22C22⋊C4×D19711yescharacteristic
379C22C22⋊C4×D19711yescharacteristic
380C22C2×D4×D19711yescharacteristic
381C4—22no—
382C4—42no—
383C4—7882no—
384C4—3942no—
385C4—1972no—
386C4—1972no—
387C4C22⋊C4×D19711yescharacteristic
388C4C22⋊C4×D19711yescharacteristic
389C4C22×C4×D19711yescharacteristic
390C4C2×D4×D19711yescharacteristic
391C2—21no—
392C2—21no—
393C2—42no—
394C2—7882no—
395C2—3942no—
396C2—3942no—
397C2#2716109412yesdirect product factor
398C2C2×C22⋊C4×D19711yescharacteristic
subgroupG/H
centerC2×C4D4×D197
commutatorC4×C197C23×C4
FrattiniC2×C4C23×D197
FittingC2×#104×C197C2
socleC22×C197C2×C22⋊C4
prime radicalC2×#41×D197C2
cosocleC2×C4×C197C24
hypercenterC2×#104D197
hypocenterC197C2×#421
LCSG ▷ 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×M16×D197C2semidirect
3. C2×#20263283C2semidirect
4. #27161094C2direct (N × H)
5. C2×#20263254C2semidirect
6. C2×#20263253C2semidirect
7. C2×#20263252C2semidirect
8. C2×#104×C197C2semidirect
9. C2×#15691359C4semidirect
13. C2×#15691357C4semidirect
16. #41×D197C4semidirect
18. C2×Q8×D197C4semidirect
20. C2×D4×D197C4semidirect
23. C2×#15691342C22semidirect
26. M16×D197C22semidirect
29. #20263283C22semidirect
32. C2×C4×Dic197C22semidirect
33. #20263254C22semidirect
34. #20263253C22semidirect
35. #20263252C22semidirect
37. C2×M16×C197C22semidirect
38. C2×C42×C197C22semidirect
39. #104×C197C22semidirect
43. #15691359C2×C4semidirect
53. #15691357C2×C4semidirect
57. Q8×D197C2×C4semidirect
66. D4×D197C2×C4semidirect
84. #15691342C23semidirect
91. C4×Dic197C23semidirect
103. M16×C197C23semidirect
106. C42×C197C23semidirect
194. C2×C197#421semidirect
199. C197C2×#421semidirect
208. C2×#104D197semidirect
228. C2×M16C2×D197semidirect
235. C2×C42C2×D197semidirect
239. #104C2×D197semidirect
297. M16C22×D197semidirect
306. C42C22×D197semidirect
397. C2#27161094direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#41×D1972
C2×M16×D1972
C2×#202632832
#271610942
C2×#202632542
C2×#202632532
C2×#202632522
C2×#104×C1972
C2×#421197