Finite Group Databasebeta

← C127×#19626 #44583299

Subgroups — C127×#19626 #44583299

Order 32512.

720 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
1C127×#3226C211yesmaximal, characteristic
2C127×#3209C211yesmaximal, characteristic, complemented
3C127×#3167C211yesmaximal, characteristic
4C2×#498×C127C211yesmaximal, characteristic, complemented
5C127×#3141C211yesmaximal, characteristic
6C127×#3139C211yesmaximal, characteristic, complemented
7C127×#1720C212yesmaximal, complemented
8C127×#1718C212yesmaximal, complemented
9C127×#1708C212yesmaximal, complemented
10C127×#1707C212yesmaximal, complemented
11C8×M16×C127C211yesmaximal, characteristic, complemented
12#533×C127C2211yescharacteristic
13#532×C127C2211yescharacteristic
14C2×C4⋊Q8×C127C2211yescharacteristic
15C2×C4⋊Q8×C127C2211yescharacteristic
16#500×C127C2212yes—
17#498×C127—22no—
18#498×C127C2212yes—
19#498×C127C2212yes—
20#498×C127C2211yescharacteristic
21#494×C127C2212yes—
22#492×C127C2211yescharacteristic
23#491×C127—22no—
24#489×C127—22no—
25#487×C127C2212yes—
26#487×C127C2212yes—
27#487×C127C2212yes—
28#487×C127C2211yescharacteristic
29#487×C127C2212yes—
30#486×C127C2212yes—
31#486×C127C2211yescharacteristic
32#486×C127C2212yes—
33#480×C127—22no—
34#474×C127—22no—
35#465×C127—22no—
36#462×C127—22no—
37#451×C127—22no—
38C4×SD16×C127—22no—
39#431×C127C2211yescharacteristic
40C2×#106×C127—22no—
41#419×C127—22no—
42#417×C127—22no—
43C4×M16×C127C2211yescharacteristic
44C2×C4×C8×C127C2211yescharacteristic
45#322×C127C2212yes—
46C82×C127C2212yes—
47C2×#41×C127—21no—
48C22×Q8×C127—21no—
49C22×Q8×C127—21no—
50C2×Q16×C127—24no—
51C2×SD16×C127—22no—
52C2×M16×C127D412yes—
53C22×C8×C127D412yes—
54C4⋊Q8×C127—21no—
55C4⋊Q8×C127—21no—
56C4⋊Q8×C127—21no—
57C4⋊Q8×C127—21no—
58C4⋊Q8×C127—21no—
59C4⋊Q8×C127—21no—
60C4⋊Q8×C127C2312yes—
61C4⋊Q8×C127C2312yes—
62C4⋊Q8×C127C2311yescharacteristic
63C4⋊Q8×C127C2311yescharacteristic
64C4⋊Q8×C127C2311yescharacteristic
65#127×C127C2311yescharacteristic
66#125×C127—21no—
67#124×C127—21no—
68C22⋊Q8×C127—21no—
69C22⋊Q8×C127—21no—
70#121×C127—21no—
71#121×C127—21no—
72C4×Q8×C127—21no—
73C4×Q8×C127—21no—
74C4×D4×C127—21no—
75C4×D4×C127—21no—
76#117×C127—21no—
77C2×C4⋊C4×C127—21no—
78C2×C4⋊C4×C127—21no—
79C2×C4⋊C4×C127—21no—
80C2×C4⋊C4×C127—21no—
81C2×C42×C127C2311yescharacteristic
82#107×C127—24no—
83#106×C127—24no—
84#106×C127—24no—
85#106×C127—22no—
86C4⋊C8×C127D412yes—
87#103×C127—24no—
88#103×C127—24no—
89#103×C127—24no—
90#103×C127—22no—
91#103×C127—24no—
92#102×C127—24no—
93#102×C127—22no—
94#102×C127—24no—
95C22⋊C8×C127D412yes—
96#97×C127D412yes—
97C4×C8×C127D412yes—
98C4×C8×C127C2311yescharacteristic
99C4×C8×C127C2311yescharacteristic
100C4×C8×C127C2312yes—
101C4×C8×C127C2312yes—
102#41×C127—44no—
103C2×Q8×C127—21no—
104C2×Q8×C127—22no—
105C2×Q8×C127—21no—
106C2×Q8×C127—22no—
107C2×Q8×C127—21no—
108C2×Q8×C127—41no—
109C2×Q8×C127—41no—
110C2×Q8×C127—44no—
111C2×Q8×C127—82no—
112C2×D4×C127—21no—
113C2×D4×C127—21no—
114C2×D4×C127—21no—
115C2×D4×C127—41no—
116C22×C4×C127—41no—
117C22×C4×C127—41no—
118C22×C4×C127—41no—
119C22×C4×C127C2×D411yescharacteristic
120C22×C4×C127C2×D411yescharacteristic
121C22×C4×C127C2×D411yescharacteristic
122Q16×C127—84no—
123SD16×C127—44no—
124M16×C127—24no—
125C2×C8×C127—24no—
126C2×C8×C127C2×D412yes—
127C2×C8×C127C2×D414yes—
128C2×C8×C127C2×D414yes—
129C2×C8×C127C2×D412yes—
130C4⋊C4×C127—22no—
131C4⋊C4×C127—21no—
132C4⋊C4×C127—22no—
133C4⋊C4×C127—21no—
134C4⋊C4×C127—21no—
135C4⋊C4×C127—21no—
136C4⋊C4×C127—21no—
137C4⋊C4×C127—22no—
138C4⋊C4×C127—22no—
139C4⋊C4×C127—21no—
140C4⋊C4×C127—41no—
141C4⋊C4×C127—41no—
142C4⋊C4×C127—41no—
143C4⋊C4×C127—41no—
144C4⋊C4×C127—41no—
145C4⋊C4×C127—41no—
146C22⋊C4×C127—41no—
147C22⋊C4×C127—41no—
148C22⋊C4×C127—41no—
149C42×C127—41no—
150C42×C127C2×D411yescharacteristic
151C42×C127C2×D411yescharacteristic
152C42×C127C2×D411yescharacteristic
153C42×C127C2411yescharacteristic
154C23×C127—41no—
155C23×C127#12711yescharacteristic
156Q8×C127—44no—
157Q8×C127—42no—
158Q8×C127—42no—
159Q8×C127—42no—
160Q8×C127—44no—
161Q8×C127—82no—
162Q8×C127—82no—
163D4×C127—42no—
164D4×C127—42no—
165D4×C127—42no—
166D4×C127—82no—
167C2×C4×C127—22no—
168C2×C4×C127—22no—
169C2×C4×C127—41no—
170C2×C4×C127—41no—
171C2×C4×C127—41no—
172C2×C4×C127—42no—
173C2×C4×C127—42no—
174C2×C4×C127—82no—
175C2×C4×C127—82no—
176C2×C4×C127—82no—
177C2×C4×C127#12711yescharacteristic
178C2×C4×C127#12711yescharacteristic
179C2×C4×C127#12711yescharacteristic
180C2×C4×C127#13514yes—
181C2×C4×C127C22×D411yescharacteristic
182C2×C4×C127C22×D411yescharacteristic
183C2×C4×C127C22×D411yescharacteristic
184C8×C127—24no—
185C8×C127—22no—
186C8×C127—24no—
187C8×C127#13714yes—
188C22×C127—21no—
189C22×C127—21no—
190C22×C127—81no—
191C22×C127—81no—
192C22×C127—81no—
193C22×C127#49512yes—
194C22×C127C2×#12711yescharacteristic
195C4×C127—21no—
196C4×C127—21no—
197C4×C127—81no—
198C4×C127—82no—
199C4×C127—81no—
200C4×C127—82no—
201C4×C127—81no—
202C4×C127#49512yes—
203C4×C127C2×#13512yes—
204C4×C127C2×#13712yes—
205C4×C127#57712yes—
206#19626C12711yesmaximal, characteristic, direct product factor, Sylow
207C2×C127—22no—
208C2×C127—161no—
209C2×C127C2×#49511yescharacteristic
210C2×C127#315211yescharacteristic
211C2×C127#315411yescharacteristic
212#3226C2×C12711yescharacteristic
213#3209C2×C12711yescharacteristic, complemented
214#3167C2×C12711yescharacteristic
215C2×#498C2×C12711yescharacteristic, complemented
216#3141C2×C12711yescharacteristic
217#3139C2×C12711yescharacteristic, complemented
218#1720C2×C12712yescomplemented
219#1718C2×C12712yescomplemented
220#1708C2×C12712yescomplemented
221#1707C2×C12712yescomplemented
222C8×M16C2×C12711yescharacteristic, complemented
223C127#1962611yescharacteristic, direct product factor, Sylow
224#533C22×C12711yescharacteristic
225#532C22×C12711yescharacteristic
226C2×C4⋊Q8C22×C12711yescharacteristic
227C2×C4⋊Q8C22×C12711yescharacteristic
228#500C22×C12712yes—
229#498—22no—
230#498C22×C12712yes—
231#498C22×C12712yes—
232#498C22×C12711yescharacteristic
233#494C22×C12712yes—
234#492C22×C12711yescharacteristic
235#491—22no—
236#489—22no—
237#487C22×C12712yes—
238#487C22×C12712yes—
239#487C22×C12712yes—
240#487C22×C12711yescharacteristic
241#487C22×C12712yes—
242#486C22×C12712yes—
243#486C22×C12711yescharacteristic
244#486C22×C12712yes—
245#480—22no—
246#474—22no—
247#465—22no—
248#462—22no—
249#451—22no—
250C4×SD16—22no—
251#431C22×C12711yescharacteristic
252C2×#106—22no—
253#419—22no—
254#417—22no—
255C4×M16C22×C12711yescharacteristic
256C2×C4×C8C22×C12711yescharacteristic
257#322C22×C12712yes—
258C82C22×C12712yes—
259C2×#41—21no—
260C22×Q8—21no—
261C22×Q8—21no—
262C2×Q16—24no—
263C2×SD16—22no—
264C2×M16D4×C12712yes—
265C22×C8D4×C12712yes—
266C4⋊Q8—21no—
267C4⋊Q8—21no—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8—21no—
271C4⋊Q8—21no—
272C4⋊Q8C23×C12712yes—
273C4⋊Q8C23×C12712yes—
274C4⋊Q8C23×C12711yescharacteristic
275C4⋊Q8C23×C12711yescharacteristic
276C4⋊Q8C23×C12711yescharacteristic
277#127C23×C12711yescharacteristic
278#125—21no—
279#124—21no—
280C22⋊Q8—21no—
281C22⋊Q8—21no—
282#121—21no—
283#121—21no—
284C4×Q8—21no—
285C4×Q8—21no—
286C4×D4—21no—
287C4×D4—21no—
288#117—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C42C23×C12711yescharacteristic
294#107—24no—
295#106—24no—
296#106—24no—
297#106—22no—
298C4⋊C8D4×C12712yes—
299#103—24no—
300#103—24no—
301#103—24no—
302#103—22no—
303#103—24no—
304#102—24no—
305#102—22no—
306#102—24no—
307C22⋊C8D4×C12712yes—
308#97D4×C12712yes—
309C4×C8D4×C12712yes—
310C4×C8C23×C12711yescharacteristic
311C4×C8C23×C12712yes—
312C4×C8C23×C12712yes—
313C4×C8C23×C12711yescharacteristic
314#41—44no—
315C2×Q8—21no—
316C2×Q8—22no—
317C2×Q8—21no—
318C2×Q8—22no—
319C2×Q8—21no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—44no—
323C2×Q8—82no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4C2×D4×C12711yescharacteristic
332C22×C4C2×D4×C12711yescharacteristic
333C22×C4C2×D4×C12711yescharacteristic
334Q16—84no—
335SD16—44no—
336M16—24no—
337C2×C8—24no—
338C2×C8C2×D4×C12712yes—
339C2×C8C2×D4×C12714yes—
340C2×C8C2×D4×C12714yes—
341C2×C8C2×D4×C12712yes—
342C4⋊C4—22no—
343C4⋊C4—21no—
344C4⋊C4—22no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—21no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C42—41no—
362C42C2×D4×C12711yescharacteristic
363C42C2×D4×C12711yescharacteristic
364C42C2×D4×C12711yescharacteristic
365C42C24×C12711yescharacteristic
366C23—41no—
367C23#127×C12711yescharacteristic
368Q8—44no—
369Q8—42no—
370Q8—42no—
371Q8—42no—
372Q8—44no—
373Q8—82no—
374Q8—82no—
375D4—42no—
376D4—42no—
377D4—42no—
378D4—82no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4#127×C12711yescharacteristic
390C2×C4#127×C12711yescharacteristic
391C2×C4#127×C12711yescharacteristic
392C2×C4#135×C12714yes—
393C2×C4C22×D4×C12711yescharacteristic
394C2×C4C22×D4×C12711yescharacteristic
395C2×C4C22×D4×C12711yescharacteristic
396C8—24no—
397C8—22no—
398C8—24no—
399C8#137×C12714yes—
400C22—21no—
401C22—21no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22#495×C12712yes—
406C22C2×#127×C12711yescharacteristic
407C4—21no—
408C4—21no—
409C4—81no—
410C4—82no—
411C4—81no—
412C4—82no—
413C4—81no—
414C4#495×C12712yes—
415C4C2×#135×C12712yes—
416C4C2×#137×C12712yes—
417C4#577×C12712yes—
418C2—22no—
419C2—161no—
420C2C2×#495×C12711yescharacteristic
421C2C127×#315211yescharacteristic
422C2C127×#315411yescharacteristic
subgroupG/H
center, socleC22×C127C2×#127
commutator, Frattini, cosocleC42C24×C127
prime radical#532×C127C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#3209C2semidirect
4. C2×#498×C127C2semidirect
6. C127×#3139C2semidirect
7. C127×#1720C2semidirect
8. C127×#1718C2semidirect
9. C127×#1708C2semidirect
10. C127×#1707C2semidirect
11. C8×M16×C127C2semidirect
206. #19626C127direct (N × H)
213. #3209C2×C127semidirect
215. C2×#498C2×C127semidirect
217. #3139C2×C127semidirect
218. #1720C2×C127semidirect
219. #1718C2×C127semidirect
220. #1708C2×C127semidirect
221. #1707C2×C127semidirect
222. C8×M16C2×C127semidirect
223. C127#19626direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32262
C127×#32092
C127×#31672
C2×#498×C1272
C127×#31412
C127×#31392
C127×#17202
C127×#17182
C127×#17082
C127×#17072
C8×M16×C1272
#19626127