Finite Group Databasebeta

← C2×C127×#1701 #44582813

Subgroups — C2×C127×#1701 #44582813

Order 32512.

810 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#486×C127C211yesmaximal, characteristic, complemented
2C2×#461×C127C211yesmaximal, characteristic, complemented
3C2×#459×C127C211yesmaximal, characteristic
4C2×#436×C127C211yesmaximal, characteristic, complemented
5C127×#1701C218yesmaximal, direct product factor
6C2×#334×C127C211yesmaximal, characteristic, complemented
7C2×#332×C127C211yesmaximal, characteristic, complemented
8C2×#331×C127C211yesmaximal, characteristic, complemented
9C22×SD16×C127—21no—
10C22×D8×C127—21no—
11C22×M16×C127—21no—
12C2×C4⋊Q8×C127C2211yescharacteristic
13C2×#127×C127C2211yescharacteristic
14C2×C4×D4×C127C2211yescharacteristic
15#486×C127—22no—
16#486×C127C2214yescomplemented
17#461×C127—22no—
18#461×C127C2214yescomplemented
19#459×C127—22no—
20#459×C127C2214yes—
21#436×C127—22no—
22#436×C127C2214yescomplemented
23C2×C4⋊C8×C127C2211yescharacteristic
24C2×C4⋊C8×C127C2211yescharacteristic
25C2×C4⋊C8×C127C2211yescharacteristic
26C2×#103×C127—21no—
27C2×#102×C127—21no—
28C2×#102×C127—21no—
29C2×#102×C127—21no—
30C2×C22⋊C8×C127—21no—
31C2×C4×C8×C127C2211yescharacteristic
32#334×C127C2214yescomplemented
33#332×C127C2214yescomplemented
34#331×C127C2214yescomplemented
35C22×Q8×C127—21no—
36C22×D4×C127—21no—
37C22×D4×C127—21no—
38C22×D4×C127—21no—
39C22×D4×C127D411yescharacteristic
40C23×C4×C127—21no—
41C2×SD16×C127—24no—
42C2×SD16×C127—44no—
43C2×D8×C127—24no—
44C2×D8×C127—44no—
45C2×M16×C127—24no—
46C2×M16×C127—44no—
47C22×C8×C127—21no—
48C22×C8×C127—21no—
49C22×C8×C127—21no—
50C22×C8×C127D411yescharacteristic
51C22×C8×C127D411yescharacteristic
52C4⋊Q8×C127—21no—
53C4⋊Q8×C127—21no—
54C4⋊Q8×C127—21no—
55C4⋊Q8×C127C2312yes—
56#127×C127—21no—
57#127×C127—21no—
58#127×C127—21no—
59#127×C127C2312yes—
60C4×D4×C127—21no—
61C4×D4×C127—21no—
62C4×D4×C127D412yes—
63C4×D4×C127C2312yes—
64C2×C4⋊C4×C127—21no—
65C2×C4⋊C4×C127—21no—
66C2×C4⋊C4×C127D411yescharacteristic
67C2×C22⋊C4×C127—21no—
68C2×C42×C127C2311yescharacteristic
69C4⋊C8×C127—21no—
70C4⋊C8×C127—21no—
71C4⋊C8×C127—21no—
72C4⋊C8×C127C2312yes—
73C4⋊C8×C127C2312yes—
74C4⋊C8×C127C2312yes—
75#103×C127—24no—
76#102×C127—24no—
77#102×C127—24no—
78#102×C127—24no—
79C22⋊C8×C127—24no—
80C4×C8×C127D412yes—
81C4×C8×C127C2312yes—
82C24×C127—21no—
83C24×C127—41no—
84C2×Q8×C127—22no—
85C2×Q8×C127—41no—
86C2×Q8×C127—42no—
87C2×Q8×C127—42no—
88C2×D4×C127—21no—
89C2×D4×C127—22no—
90C2×D4×C127—22no—
91C2×D4×C127—22no—
92C2×D4×C127—22no—
93C2×D4×C127—22no—
94C2×D4×C127—41no—
95C2×D4×C127—41no—
96C2×D4×C127—41no—
97C2×D4×C127—42no—
98C2×D4×C127—42no—
99C2×D4×C127—42no—
100C2×D4×C127—42no—
101C2×D4×C127—82no—
102C2×D4×C127C2×D412yes—
103C22×C4×C127—21no—
104C22×C4×C127—22no—
105C22×C4×C127—22no—
106C22×C4×C127—42no—
107C22×C4×C127—42no—
108C22×C4×C127—41no—
109C22×C4×C127C2×D411yescharacteristic
110C22×C4×C127C2×D411yescharacteristic
111C22×C4×C127#4111yescharacteristic
112SD16×C127—48no—
113D8×C127—48no—
114M16×C127—48no—
115C2×C8×C127—22no—
116C2×C8×C127—22no—
117C2×C8×C127—22no—
118C2×C8×C127—22no—
119C2×C8×C127—22no—
120C2×C8×C127—42no—
121C2×C8×C127—42no—
122C2×C8×C127—42no—
123C2×C8×C127C2×D412yes—
124C2×C8×C127C2×D412yes—
125C4⋊C4×C127—21no—
126C4⋊C4×C127—22no—
127C4⋊C4×C127—22no—
128C4⋊C4×C127—41no—
129C4⋊C4×C127—41no—
130C4⋊C4×C127C2×D412yes—
131C22⋊C4×C127—22no—
132C22⋊C4×C127—22no—
133C42×C127C2×D411yescharacteristic
134C42×C127C2×D411yescharacteristic
135C42×C127#4111yescharacteristic
136C42×C127C2411yescharacteristic
137C23×C127—22no—
138C23×C127—42no—
139C23×C127—42no—
140C23×C127—42no—
141C23×C127—42no—
142C23×C127—82no—
143C23×C127—82no—
144C23×C127—82no—
145C23×C127#12111yescharacteristic
146Q8×C127—42no—
147Q8×C127—42no—
148Q8×C127—82no—
149D4×C127—22no—
150D4×C127—22no—
151D4×C127—42no—
152D4×C127—42no—
153D4×C127—42no—
154D4×C127—42no—
155D4×C127—42no—
156D4×C127—82no—
157D4×C127—82no—
158D4×C127—82no—
159D4×C127—82no—
160C2×C4×C127—22no—
161C2×C4×C127—22no—
162C2×C4×C127—42no—
163C2×C4×C127—42no—
164C2×C4×C127—42no—
165C2×C4×C127—44no—
166C2×C4×C127—42no—
167C2×C4×C127—82no—
168C2×C4×C127#12111yescharacteristic
169C2×C4×C127#12111yescharacteristic
170C2×C4×C127#12111yescharacteristic
171C2×C4×C127#13512yes—
172C2×C4×C127AGL(1,ℤ/8)12yes—
173C2×C4×C127AGL(1,ℤ/8)12yes—
174C2×C4×C127AGL(1,ℤ/8)12yes—
175C2×C4×C127C22×D411yescharacteristic
176C2×C4×C127C22×D411yescharacteristic
177C2×C4×C127C2×#4111yescharacteristic
178C8×C127—22no—
179C8×C127—22no—
180C8×C127—42no—
181C8×C127—42no—
182C8×C127—42no—
183C22×C127—42no—
184C22×C127—42no—
185C22×C127—42no—
186C22×C127—84no—
187C22×C127—82no—
188C22×C127—82no—
189C22×C127—82no—
190C22×C127—164no—
191C22×C127#46312yes—
192C22×C127#46912yes—
193C22×C127#47112yes—
194C22×C127C2×#12111yescharacteristic
195C4×C127—21no—
196C4×C127—21no—
197C4×C127—42no—
198C4×C127—82no—
199C4×C127#46312yes—
200C4×C127#46912yes—
201C4×C127C2×#13511yescharacteristic
202C4×C127C2×AGL(1,ℤ/8)11yescharacteristic
203C4×C127C2×AGL(1,ℤ/8)11yescharacteristic
204C4×C127C2×AGL(1,ℤ/8)11yescharacteristic
205C2×#1701C12711yesmaximal, characteristic, direct product factor, Sylow
206C2×C127—82no—
207C2×C127—162no—
208C2×C127#170114yesdirect product factor
209C2×C127C2×#46311yescharacteristic
210C2×C127C2×#46911yescharacteristic
211C2×C127C2×#47111yescharacteristic
212C2×#486C2×C12711yescharacteristic, complemented
213C2×#461C2×C12711yescharacteristic, complemented
214C2×#459C2×C12711yescharacteristic
215C2×#436C2×C12711yescharacteristic, complemented
216#1701C2×C12718yesdirect product factor
217C2×#334C2×C12711yescharacteristic, complemented
218C2×#332C2×C12711yescharacteristic, complemented
219C2×#331C2×C12711yescharacteristic, complemented
220C127C2×#170111yescharacteristic, direct product factor, Sylow
221C22×SD16—21no—
222C22×D8—21no—
223C22×M16—21no—
224C2×C4⋊Q8C22×C12711yescharacteristic
225C2×#127C22×C12711yescharacteristic
226C2×C4×D4C22×C12711yescharacteristic
227#486—22no—
228#486C22×C12714yescomplemented
229#461—22no—
230#461C22×C12714yescomplemented
231#459—22no—
232#459C22×C12714yes—
233#436—22no—
234#436C22×C12714yescomplemented
235C2×C4⋊C8C22×C12711yescharacteristic
236C2×C4⋊C8C22×C12711yescharacteristic
237C2×C4⋊C8C22×C12711yescharacteristic
238C2×#103—21no—
239C2×#102—21no—
240C2×#102—21no—
241C2×#102—21no—
242C2×C22⋊C8—21no—
243C2×C4×C8C22×C12711yescharacteristic
244#334C22×C12714yescomplemented
245#332C22×C12714yescomplemented
246#331C22×C12714yescomplemented
247C22×Q8—21no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4D4×C12711yescharacteristic
252C23×C4—21no—
253C2×SD16—24no—
254C2×SD16—44no—
255C2×D8—24no—
256C2×D8—44no—
257C2×M16—24no—
258C2×M16—44no—
259C22×C8—21no—
260C22×C8—21no—
261C22×C8—21no—
262C22×C8D4×C12711yescharacteristic
263C22×C8D4×C12711yescharacteristic
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4⋊Q8—21no—
267C4⋊Q8C23×C12712yes—
268#127—21no—
269#127—21no—
270#127—21no—
271#127C23×C12712yes—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4D4×C12712yes—
275C4×D4C23×C12712yes—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4D4×C12711yescharacteristic
279C2×C22⋊C4—21no—
280C2×C42C23×C12711yescharacteristic
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8C23×C12712yes—
285C4⋊C8C23×C12712yes—
286C4⋊C8C23×C12712yes—
287#103—24no—
288#102—24no—
289#102—24no—
290#102—24no—
291C22⋊C8—24no—
292C4×C8D4×C12712yes—
293C4×C8C23×C12712yes—
294C24—21no—
295C24—41no—
296C2×Q8—22no—
297C2×Q8—41no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×D4—21no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—82no—
314C2×D4C2×D4×C12712yes—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4C2×D4×C12711yescharacteristic
322C22×C4C2×D4×C12711yescharacteristic
323C22×C4#41×C12711yescharacteristic
324SD16—48no—
325D8—48no—
326M16—48no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8C2×D4×C12712yes—
336C2×C8C2×D4×C12712yes—
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4C2×D4×C12712yes—
343C22⋊C4—22no—
344C22⋊C4—22no—
345C42C2×D4×C12711yescharacteristic
346C42C2×D4×C12711yescharacteristic
347C42#41×C12711yescharacteristic
348C42C24×C12711yescharacteristic
349C23—22no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—82no—
355C23—82no—
356C23—82no—
357C23#121×C12711yescharacteristic
358Q8—42no—
359Q8—42no—
360Q8—82no—
361D4—22no—
362D4—22no—
363D4—42no—
364D4—42no—
365D4—42no—
366D4—42no—
367D4—42no—
368D4—82no—
369D4—82no—
370D4—82no—
371D4—82no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—82no—
380C2×C4#121×C12711yescharacteristic
381C2×C4#121×C12711yescharacteristic
382C2×C4#121×C12711yescharacteristic
383C2×C4#135×C12712yes—
384C2×C4AGL(1,ℤ/8)×C12712yes—
385C2×C4AGL(1,ℤ/8)×C12712yes—
386C2×C4AGL(1,ℤ/8)×C12712yes—
387C2×C4C22×D4×C12711yescharacteristic
388C2×C4C22×D4×C12711yescharacteristic
389C2×C4C2×#41×C12711yescharacteristic
390C8—22no—
391C8—22no—
392C8—42no—
393C8—42no—
394C8—42no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—84no—
399C22—82no—
400C22—82no—
401C22—82no—
402C22—164no—
403C22#463×C12712yes—
404C22#469×C12712yes—
405C22#471×C12712yes—
406C22C2×#121×C12711yescharacteristic
407C4—21no—
408C4—21no—
409C4—42no—
410C4—82no—
411C4#463×C12712yes—
412C4#469×C12712yes—
413C4C2×#135×C12711yescharacteristic
414C4C2×AGL(1,ℤ/8)×C12711yescharacteristic
415C4C2×AGL(1,ℤ/8)×C12711yescharacteristic
416C4C2×AGL(1,ℤ/8)×C12711yescharacteristic
417C2—82no—
418C2—162no—
419C2C127×#170114yesdirect product factor
420C2C2×#463×C12711yescharacteristic
421C2C2×#469×C12711yescharacteristic
422C2C2×#471×C12711yescharacteristic
subgroupG/H
center, socleC23×C127#121
commutator, Frattini, cosocleC42C24×C127
prime radicalC2×#459×C127C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#486×C127C2semidirect
2. C2×#461×C127C2semidirect
4. C2×#436×C127C2semidirect
5. C127×#1701C2direct (N × H)
6. C2×#334×C127C2semidirect
7. C2×#332×C127C2semidirect
8. C2×#331×C127C2semidirect
16. #486×C127C22semidirect
18. #461×C127C22semidirect
22. #436×C127C22semidirect
32. #334×C127C22semidirect
33. #332×C127C22semidirect
34. #331×C127C22semidirect
205. C2×#1701C127direct (N × H)
208. C2×C127#1701direct (N × H)
212. C2×#486C2×C127semidirect
213. C2×#461C2×C127semidirect
215. C2×#436C2×C127semidirect
216. #1701C2×C127direct (N × H)
217. C2×#334C2×C127semidirect
218. C2×#332C2×C127semidirect
219. C2×#331C2×C127semidirect
220. C127C2×#1701direct (N × H)
228. #486C22×C127semidirect
230. #461C22×C127semidirect
234. #436C22×C127semidirect
244. #334C22×C127semidirect
245. #332C22×C127semidirect
246. #331C22×C127semidirect
419. C2C127×#1701direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#486×C1272
C2×#461×C1272
C2×#459×C1272
C2×#436×C1272
C127×#17012
C2×#334×C1272
C2×#332×C1272
C2×#331×C1272
C2×#1701127