Finite Group Databasebeta

← C2×C127×#1697 #44582811

Subgroups — C2×C127×#1697 #44582811

Order 32512.

778 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
1C2×#486×C127C211yesmaximal, characteristic, complemented
2C2×#476×C127C211yesmaximal, characteristic, complemented
3C2×#459×C127C211yesmaximal, characteristic
4C2×#436×C127C211yesmaximal, characteristic, complemented
5C127×#1697C218yesmaximal, direct product factor
6C2×#335×C127C211yesmaximal, characteristic, complemented
7C2×#333×C127C211yesmaximal, characteristic, complemented
8C2×#331×C127C211yesmaximal, characteristic, complemented
9C22×D8×C127—21no—
10C22×M16×C127—21no—
11C2×C4⋊Q8×C127C2211yescharacteristic
12C2×#127×C127C2211yescharacteristic
13C2×C4×D4×C127C2211yescharacteristic
14#486×C127—22no—
15#486×C127C2214yescomplemented
16#476×C127—22no—
17#476×C127C2214yescomplemented
18#459×C127—22no—
19#459×C127C2214yes—
20#436×C127—22no—
21#436×C127C2214yescomplemented
22C2×#106×C127—21no—
23C2×C4⋊C8×C127C2211yescharacteristic
24C2×C4⋊C8×C127C2211yescharacteristic
25C2×C4⋊C8×C127C2211yescharacteristic
26C2×#102×C127—21no—
27C2×#102×C127—21no—
28C2×#102×C127—21no—
29C2×#102×C127—21no—
30C2×C22⋊C8×C127—21no—
31C2×C4×C8×C127C2211yescharacteristic
32#335×C127C2214yescomplemented
33#333×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×D8×C127—24no—
42C2×D8×C127—44no—
43C2×M16×C127—24no—
44C2×M16×C127—44no—
45C22×C8×C127—21no—
46C22×C8×C127—21no—
47C22×C8×C127—21no—
48C22×C8×C127D411yescharacteristic
49C22×C8×C127D411yescharacteristic
50C4⋊Q8×C127—21no—
51C4⋊Q8×C127—21no—
52C4⋊Q8×C127—21no—
53C4⋊Q8×C127C2312yes—
54#127×C127—21no—
55#127×C127—21no—
56#127×C127—21no—
57#127×C127C2312yes—
58C4×D4×C127—21no—
59C4×D4×C127—21no—
60C4×D4×C127D412yes—
61C4×D4×C127C2312yes—
62C2×C4⋊C4×C127—21no—
63C2×C4⋊C4×C127—21no—
64C2×C4⋊C4×C127D411yescharacteristic
65C2×C22⋊C4×C127—21no—
66C2×C42×C127C2311yescharacteristic
67#106×C127—24no—
68C4⋊C8×C127—21no—
69C4⋊C8×C127—21no—
70C4⋊C8×C127—21no—
71C4⋊C8×C127C2312yes—
72C4⋊C8×C127C2312yes—
73C4⋊C8×C127C2312yes—
74#102×C127—24no—
75#102×C127—24no—
76#102×C127—24no—
77#102×C127—24no—
78C22⋊C8×C127—24no—
79C4×C8×C127D412yes—
80C4×C8×C127C2312yes—
81C24×C127—21no—
82C24×C127—41no—
83C2×Q8×C127—22no—
84C2×Q8×C127—41no—
85C2×Q8×C127—82no—
86C2×D4×C127—21no—
87C2×D4×C127—22no—
88C2×D4×C127—22no—
89C2×D4×C127—22no—
90C2×D4×C127—22no—
91C2×D4×C127—22no—
92C2×D4×C127—41no—
93C2×D4×C127—41no—
94C2×D4×C127—41no—
95C2×D4×C127—42no—
96C2×D4×C127—42no—
97C2×D4×C127—42no—
98C2×D4×C127—42no—
99C2×D4×C127—82no—
100C2×D4×C127C2×D412yes—
101C22×C4×C127—21no—
102C22×C4×C127—22no—
103C22×C4×C127—22no—
104C22×C4×C127—42no—
105C22×C4×C127—42no—
106C22×C4×C127—41no—
107C22×C4×C127C2×D411yescharacteristic
108C22×C4×C127C2×D411yescharacteristic
109C22×C4×C127#4111yescharacteristic
110D8×C127—48no—
111M16×C127—48no—
112C2×C8×C127—22no—
113C2×C8×C127—22no—
114C2×C8×C127—22no—
115C2×C8×C127—22no—
116C2×C8×C127—22no—
117C2×C8×C127—42no—
118C2×C8×C127—42no—
119C2×C8×C127—42no—
120C2×C8×C127C2×D412yes—
121C2×C8×C127C2×D412yes—
122C4⋊C4×C127—21no—
123C4⋊C4×C127—22no—
124C4⋊C4×C127—22no—
125C4⋊C4×C127—41no—
126C4⋊C4×C127—41no—
127C4⋊C4×C127C2×D412yes—
128C22⋊C4×C127—22no—
129C22⋊C4×C127—22no—
130C42×C127C2×D411yescharacteristic
131C42×C127C2×D411yescharacteristic
132C42×C127#4111yescharacteristic
133C42×C127C2411yescharacteristic
134C23×C127—22no—
135C23×C127—42no—
136C23×C127—42no—
137C23×C127—42no—
138C23×C127—42no—
139C23×C127—82no—
140C23×C127—82no—
141C23×C127—82no—
142C23×C127#12111yescharacteristic
143Q8×C127—82no—
144Q8×C127—82no—
145D4×C127—22no—
146D4×C127—22no—
147D4×C127—42no—
148D4×C127—42no—
149D4×C127—42no—
150D4×C127—42no—
151D4×C127—42no—
152D4×C127—82no—
153D4×C127—82no—
154D4×C127—82no—
155D4×C127—82no—
156C2×C4×C127—22no—
157C2×C4×C127—22no—
158C2×C4×C127—42no—
159C2×C4×C127—42no—
160C2×C4×C127—42no—
161C2×C4×C127—44no—
162C2×C4×C127—42no—
163C2×C4×C127—82no—
164C2×C4×C127#12111yescharacteristic
165C2×C4×C127#12111yescharacteristic
166C2×C4×C127#12111yescharacteristic
167C2×C4×C127#13512yes—
168C2×C4×C127AGL(1,ℤ/8)12yes—
169C2×C4×C127AGL(1,ℤ/8)12yes—
170C2×C4×C127#13712yes—
171C2×C4×C127C22×D411yescharacteristic
172C2×C4×C127C22×D411yescharacteristic
173C2×C4×C127C2×#4111yescharacteristic
174C8×C127—22no—
175C8×C127—22no—
176C8×C127—42no—
177C8×C127—42no—
178C8×C127—42no—
179C22×C127—42no—
180C22×C127—42no—
181C22×C127—42no—
182C22×C127—84no—
183C22×C127—82no—
184C22×C127—82no—
185C22×C127—82no—
186C22×C127—164no—
187C22×C127#46312yes—
188C22×C127#46812yes—
189C22×C127#47212yes—
190C22×C127C2×#12111yescharacteristic
191C4×C127—21no—
192C4×C127—21no—
193C4×C127—42no—
194C4×C127—82no—
195C4×C127#46312yes—
196C4×C127#46812yes—
197C4×C127C2×#13511yescharacteristic
198C4×C127C2×AGL(1,ℤ/8)11yescharacteristic
199C4×C127C2×AGL(1,ℤ/8)11yescharacteristic
200C4×C127C2×#13711yescharacteristic
201C2×#1697C12711yesmaximal, characteristic, direct product factor, Sylow
202C2×C127—82no—
203C2×C127—162no—
204C2×C127#169714yesdirect product factor
205C2×C127C2×#46311yescharacteristic
206C2×C127C2×#46811yescharacteristic
207C2×C127C2×#47211yescharacteristic
208C2×#486C2×C12711yescharacteristic, complemented
209C2×#476C2×C12711yescharacteristic, complemented
210C2×#459C2×C12711yescharacteristic
211C2×#436C2×C12711yescharacteristic, complemented
212#1697C2×C12718yesdirect product factor
213C2×#335C2×C12711yescharacteristic, complemented
214C2×#333C2×C12711yescharacteristic, complemented
215C2×#331C2×C12711yescharacteristic, complemented
216C127C2×#169711yescharacteristic, direct product factor, Sylow
217C22×D8—21no—
218C22×M16—21no—
219C2×C4⋊Q8C22×C12711yescharacteristic
220C2×#127C22×C12711yescharacteristic
221C2×C4×D4C22×C12711yescharacteristic
222#486—22no—
223#486C22×C12714yescomplemented
224#476—22no—
225#476C22×C12714yescomplemented
226#459—22no—
227#459C22×C12714yes—
228#436—22no—
229#436C22×C12714yescomplemented
230C2×#106—21no—
231C2×C4⋊C8C22×C12711yescharacteristic
232C2×C4⋊C8C22×C12711yescharacteristic
233C2×C4⋊C8C22×C12711yescharacteristic
234C2×#102—21no—
235C2×#102—21no—
236C2×#102—21no—
237C2×#102—21no—
238C2×C22⋊C8—21no—
239C2×C4×C8C22×C12711yescharacteristic
240#335C22×C12714yescomplemented
241#333C22×C12714yescomplemented
242#331C22×C12714yescomplemented
243C22×Q8—21no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4D4×C12711yescharacteristic
248C23×C4—21no—
249C2×D8—24no—
250C2×D8—44no—
251C2×M16—24no—
252C2×M16—44no—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8D4×C12711yescharacteristic
257C22×C8D4×C12711yescharacteristic
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C4⋊Q8C23×C12712yes—
262#127—21no—
263#127—21no—
264#127—21no—
265#127C23×C12712yes—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4D4×C12712yes—
269C4×D4C23×C12712yes—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4D4×C12711yescharacteristic
273C2×C22⋊C4—21no—
274C2×C42C23×C12711yescharacteristic
275#106—24no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8C23×C12712yes—
280C4⋊C8C23×C12712yes—
281C4⋊C8C23×C12712yes—
282#102—24no—
283#102—24no—
284#102—24no—
285#102—24no—
286C22⋊C8—24no—
287C4×C8D4×C12712yes—
288C4×C8C23×C12712yes—
289C24—21no—
290C24—41no—
291C2×Q8—22no—
292C2×Q8—41no—
293C2×Q8—82no—
294C2×D4—21no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—82no—
308C2×D4C2×D4×C12712yes—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4C2×D4×C12711yescharacteristic
316C22×C4C2×D4×C12711yescharacteristic
317C22×C4#41×C12711yescharacteristic
318D8—48no—
319M16—48no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8C2×D4×C12712yes—
329C2×C8C2×D4×C12712yes—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4C2×D4×C12712yes—
336C22⋊C4—22no—
337C22⋊C4—22no—
338C42C2×D4×C12711yescharacteristic
339C42C2×D4×C12711yescharacteristic
340C42#41×C12711yescharacteristic
341C42C24×C12711yescharacteristic
342C23—22no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—82no—
348C23—82no—
349C23—82no—
350C23#121×C12711yescharacteristic
351Q8—82no—
352Q8—82no—
353D4—22no—
354D4—22no—
355D4—42no—
356D4—42no—
357D4—42no—
358D4—42no—
359D4—42no—
360D4—82no—
361D4—82no—
362D4—82no—
363D4—82no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—82no—
372C2×C4#121×C12711yescharacteristic
373C2×C4#121×C12711yescharacteristic
374C2×C4#121×C12711yescharacteristic
375C2×C4#135×C12712yes—
376C2×C4AGL(1,ℤ/8)×C12712yes—
377C2×C4AGL(1,ℤ/8)×C12712yes—
378C2×C4#137×C12712yes—
379C2×C4C22×D4×C12711yescharacteristic
380C2×C4C22×D4×C12711yescharacteristic
381C2×C4C2×#41×C12711yescharacteristic
382C8—22no—
383C8—22no—
384C8—42no—
385C8—42no—
386C8—42no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—84no—
391C22—82no—
392C22—82no—
393C22—82no—
394C22—164no—
395C22#463×C12712yes—
396C22#468×C12712yes—
397C22#472×C12712yes—
398C22C2×#121×C12711yescharacteristic
399C4—21no—
400C4—21no—
401C4—42no—
402C4—82no—
403C4#463×C12712yes—
404C4#468×C12712yes—
405C4C2×#135×C12711yescharacteristic
406C4C2×AGL(1,ℤ/8)×C12711yescharacteristic
407C4C2×AGL(1,ℤ/8)×C12711yescharacteristic
408C4C2×#137×C12711yescharacteristic
409C2—82no—
410C2—162no—
411C2C127×#169714yesdirect product factor
412C2C2×#463×C12711yescharacteristic
413C2C2×#468×C12711yescharacteristic
414C2C2×#472×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×#476×C127C2semidirect
4. C2×#436×C127C2semidirect
5. C127×#1697C2direct (N × H)
6. C2×#335×C127C2semidirect
7. C2×#333×C127C2semidirect
8. C2×#331×C127C2semidirect
15. #486×C127C22semidirect
17. #476×C127C22semidirect
21. #436×C127C22semidirect
32. #335×C127C22semidirect
33. #333×C127C22semidirect
34. #331×C127C22semidirect
201. C2×#1697C127direct (N × H)
204. C2×C127#1697direct (N × H)
208. C2×#486C2×C127semidirect
209. C2×#476C2×C127semidirect
211. C2×#436C2×C127semidirect
212. #1697C2×C127direct (N × H)
213. C2×#335C2×C127semidirect
214. C2×#333C2×C127semidirect
215. C2×#331C2×C127semidirect
216. C127C2×#1697direct (N × H)
223. #486C22×C127semidirect
225. #476C22×C127semidirect
229. #436C22×C127semidirect
240. #335C22×C127semidirect
241. #333C22×C127semidirect
242. #331C22×C127semidirect
411. C2C127×#1697direct (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×#476×C1272
C2×#459×C1272
C2×#436×C1272
C127×#16972
C2×#335×C1272
C2×#333×C1272
C2×#331×C1272
C2×#1697127