Finite Group Databasebeta

← C127×#20708 #44584381

Subgroups — C127×#20708 #44584381

Order 32512.

1260 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3428C211yesmaximal, characteristic
2C2×#413×C127C211yesmaximal, characteristic
3C2×#410×C127C211yesmaximal, characteristic
4C127×#1827C218yesmaximal, complemented
5C127×#1733C212yesmaximal, complemented
6C2×H(3,ℤ/4)×C127C212yesmaximal, complemented
7C22×#41×C127C2211yescharacteristic
8C22×M16×C127C2212yes—
9#520×C127—22no—
10#520×C127—22no—
11#519×C127—21no—
12#518×C127—21no—
13#518×C127—22no—
14C4×#41×C127C414yescomplemented
15C4×#41×C127C2214yes—
16C2×#117×C127C414yescomplemented
17C2×#117×C127C2212yes—
18C2×#117×C127C2211yescharacteristic
19C2×#117×C127C2211yescharacteristic
20#413×C127—22no—
21#413×C127C2214yes—
22C2×#101×C127—21no—
23C2×#100×C127—21no—
24#410×C127—22no—
25#410×C127C2214yes—
26C2×#99×C127—22no—
27#339×C127C2218yescomplemented
28H(3,ℤ/4)×C127C2218yescomplemented
29C2×#41×C127—22no—
30C2×#41×C127—22no—
31C2×#41×C127—22no—
32C2×#41×C127—22no—
33C2×#41×C127C2×C412yes—
34C2×#41×C127D412yes—
35C2×#41×C127D412yes—
36C2×#41×C127C2312yes—
37C22×Q8×C127D411yescharacteristic
38C22×D4×C127D412yes—
39C22×D4×C127D411yescharacteristic
40C23×C4×C127C2×C412yes—
41C23×C4×C127C2311yescharacteristic
42C2×M16×C127—22no—
43C2×M16×C127—22no—
44C2×M16×C127—22no—
45C2×M16×C127—22no—
46C2×M16×C127—22no—
47C2×M16×C127C2314yes—
48C22×C8×C127—22no—
49C4×Q8×C127—24no—
50C4×Q8×C127—24no—
51C4×D4×C127—24no—
52C4×D4×C127—28no—
53C4×D4×C127—28no—
54C4×D4×C127—24no—
55#117×C127—24no—
56#117×C127—24no—
57#117×C127—21no—
58#117×C127—21no—
59#117×C127—21no—
60#117×C127—21no—
61#117×C127—22no—
62#117×C127—22no—
63#117×C127—22no—
64#117×C127C2×C412yes—
65#117×C127C2×C412yes—
66#117×C127C2×C412yes—
67#117×C127C2×C418yescomplemented
68#117×C127C2314yes—
69#117×C127C2312yes—
70C2×C4⋊C4×C127—24no—
71C2×C4⋊C4×C127—21no—
72C2×C4⋊C4×C127—21no—
73C2×C4⋊C4×C127—22no—
74C2×C22⋊C4×C127—24no—
75C2×C22⋊C4×C127—21no—
76C2×C22⋊C4×C127—21no—
77C2×C22⋊C4×C127—22no—
78C2×C42×C127—22no—
79C2×C42×C127C2×C412yes—
80C2×C42×C127C2×C418yescomplemented
81C2×C42×C127C2312yes—
82#101×C127—24no—
83#100×C127—24no—
84#99×C127—28no—
85C24×C127C22⋊C412yes—
86C24×C127C2×D411yescharacteristic
87#41×C127—24no—
88#41×C127—24no—
89#41×C127—24no—
90#41×C127—24no—
91#41×C127—24no—
92#41×C127—24no—
93#41×C127—24no—
94#41×C127—24no—
95C2×Q8×C127—22no—
96C2×Q8×C127—22no—
97C2×Q8×C127C22⋊C412yes—
98C2×Q8×C127C2×D412yes—
99C2×D4×C127—24no—
100C2×D4×C127—24no—
101C2×D4×C127—22no—
102C2×D4×C127—22no—
103C2×D4×C127C22⋊C414yes—
104C2×D4×C127C22⋊C412yes—
105C2×D4×C127C2×D414yes—
106C2×D4×C127C2×D412yes—
107C22×C4×C127—22no—
108C22×C4×C127—22no—
109C22×C4×C127—22no—
110C22×C4×C127—22no—
111C22×C4×C127—22no—
112C22×C4×C127—28no—
113C22×C4×C127—22no—
114C22×C4×C127—42no—
115C22×C4×C127C22⋊C414yes—
116C22×C4×C127C22⋊C412yes—
117C22×C4×C127C22×C414yes—
118C22×C4×C127C22×C411yescharacteristic
119C22×C4×C127C22×C411yescharacteristic
120C22×C4×C127C2×D411yescharacteristic
121C22×C4×C127C2×D411yescharacteristic
122C22×C4×C127C2×D411yescharacteristic
123C22×C4×C127#4112yes—
124C22×C4×C127#4112yes—
125C22×C4×C127#4112yes—
126C22×C4×C127#4112yes—
127C22×C4×C127C2411yescharacteristic
128M16×C127—24no—
129M16×C127—24no—
130M16×C127—42no—
131M16×C127—42no—
132C2×C8×C127—24no—
133C2×C8×C127—42no—
134C2×C8×C127—42no—
135C4⋊C4×C127—22no—
136C4⋊C4×C127—22no—
137C4⋊C4×C127—22no—
138C4⋊C4×C127—28no—
139C4⋊C4×C127—22no—
140C4⋊C4×C127—24no—
141C4⋊C4×C127—42no—
142C22⋊C4×C127—22no—
143C22⋊C4×C127—22no—
144C22⋊C4×C127—22no—
145C22⋊C4×C127—28no—
146C22⋊C4×C127—22no—
147C22⋊C4×C127—24no—
148C22⋊C4×C127—42no—
149C42×C127—22no—
150C42×C127—28no—
151C42×C127—22no—
152C42×C127—24no—
153C42×C127—42no—
154C42×C127C22×C414yes—
155C42×C127C22×C414yes—
156C23×C127—22no—
157C23×C127—22no—
158C23×C127—81no—
159C23×C127—82no—
160C23×C127C2×C22⋊C414yes—
161C23×C127C2×C22⋊C411yescharacteristic
162C23×C127C2×C22⋊C411yescharacteristic
163C23×C127#12312yes—
164C23×C127#12312yes—
165C23×C127C22×D411yescharacteristic
166Q8×C127—24no—
167Q8×C127—24no—
168D4×C127—28no—
169D4×C127—28no—
170D4×C127—24no—
171D4×C127—24no—
172C2×C4×C127—22no—
173C2×C4×C127—22no—
174C2×C4×C127—24no—
175C2×C4×C127—24no—
176C2×C4×C127—24no—
177C2×C4×C127—24no—
178C2×C4×C127—24no—
179C2×C4×C127—24no—
180C2×C4×C127—44no—
181C2×C4×C127—82no—
182C2×C4×C127—82no—
183C2×C4×C127—88no—
184C2×C4×C127—84no—
185C2×C4×C127C2×C22⋊C414yes—
186C2×C4×C127C2×C22⋊C411yescharacteristic
187C2×C4×C127C2×C22⋊C411yescharacteristic
188C2×C4×C127#11711yescharacteristic
189C2×C4×C127#11711yescharacteristic
190C2×C4×C127#11712yes—
191C2×C4×C127#11711yescharacteristic
192C2×C4×C127#11711yescharacteristic
193C2×C4×C127#11712yes—
194C2×C4×C127#12312yes—
195C2×C4×C127#12312yes—
196C2×C4×C127#12312yes—
197C2×C4×C127#12312yes—
198C2×C4×C127#12312yes—
199C2×C4×C127#12312yes—
200C2×C4×C127C23×C411yescharacteristic
201C2×C4×C127C22×D411yescharacteristic
202C2×C4×C127C2×#4112yes—
203C2×C4×C127C2×#4112yes—
204C8×C127—44no—
205C22×C127—24no—
206C22×C127—41no—
207C22×C127—42no—
208C22×C127—41no—
209C22×C127—41no—
210C22×C127—42no—
211C22×C127—44no—
212C22×C127—82no—
213C22×C127—82no—
214C22×C127#38111yescharacteristic
215C22×C127#38111yescharacteristic
216C22×C127#38112yes—
217C22×C127C22×C22⋊C411yescharacteristic
218C22×C127C2×#12312yes—
219C4×C127—24no—
220C4×C127—44no—
221C4×C127—44no—
222C4×C127—84no—
223C4×C127#38111yescharacteristic
224C4×C127#38111yescharacteristic
225C4×C127#38112yes—
226C4×C127C2×#11711yescharacteristic
227C4×C127C2×#11711yescharacteristic
228C4×C127C2×#12312yes—
229#20708C12711yesmaximal, characteristic, direct product factor, Sylow
230C2×C127—21no—
231C2×C127—22no—
232C2×C127—21no—
233C2×C127—21no—
234C2×C127—22no—
235C2×C127—44no—
236C2×C127C2×#38111yescharacteristic
237#3428C2×C12711yescharacteristic
238C2×#413C2×C12711yescharacteristic
239C2×#410C2×C12711yescharacteristic
240#1827C2×C12718yescomplemented
241#1733C2×C12712yescomplemented
242C2×H(3,ℤ/4)C2×C12712yescomplemented
243C127#2070811yescharacteristic, direct product factor, Sylow
244C22×#41C22×C12711yescharacteristic
245C22×M16C22×C12712yes—
246#520—22no—
247#520—22no—
248#519—21no—
249#518—21no—
250#518—22no—
251C4×#41C4×C12714yescomplemented
252C4×#41C22×C12714yes—
253C2×#117C4×C12714yescomplemented
254C2×#117C22×C12712yes—
255C2×#117C22×C12711yescharacteristic
256C2×#117C22×C12711yescharacteristic
257#413—22no—
258#413C22×C12714yes—
259C2×#101—21no—
260C2×#100—21no—
261#410—22no—
262#410C22×C12714yes—
263C2×#99—22no—
264#339C22×C12718yescomplemented
265H(3,ℤ/4)C22×C12718yescomplemented
266C2×#41—22no—
267C2×#41—22no—
268C2×#41—22no—
269C2×#41—22no—
270C2×#41C2×C4×C12712yes—
271C2×#41D4×C12712yes—
272C2×#41D4×C12712yes—
273C2×#41C23×C12712yes—
274C22×Q8D4×C12711yescharacteristic
275C22×D4D4×C12712yes—
276C22×D4D4×C12711yescharacteristic
277C23×C4C2×C4×C12712yes—
278C23×C4C23×C12711yescharacteristic
279C2×M16—22no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16C23×C12714yes—
285C22×C8—22no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×D4—24no—
289C4×D4—28no—
290C4×D4—28no—
291C4×D4—24no—
292#117—24no—
293#117—24no—
294#117—21no—
295#117—21no—
296#117—21no—
297#117—21no—
298#117—22no—
299#117—22no—
300#117—22no—
301#117C2×C4×C12712yes—
302#117C2×C4×C12712yes—
303#117C2×C4×C12712yes—
304#117C2×C4×C12718yescomplemented
305#117C23×C12714yes—
306#117C23×C12712yes—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C42—22no—
316C2×C42C2×C4×C12712yes—
317C2×C42C2×C4×C12718yescomplemented
318C2×C42C23×C12712yes—
319#101—24no—
320#100—24no—
321#99—28no—
322C24C22⋊C4×C12712yes—
323C24C2×D4×C12711yescharacteristic
324#41—24no—
325#41—24no—
326#41—24no—
327#41—24no—
328#41—24no—
329#41—24no—
330#41—24no—
331#41—24no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8C22⋊C4×C12712yes—
335C2×Q8C2×D4×C12712yes—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4C22⋊C4×C12714yes—
341C2×D4C22⋊C4×C12712yes—
342C2×D4C2×D4×C12714yes—
343C2×D4C2×D4×C12712yes—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—28no—
350C22×C4—22no—
351C22×C4—42no—
352C22×C4C22⋊C4×C12714yes—
353C22×C4C22⋊C4×C12712yes—
354C22×C4C22×C4×C12714yes—
355C22×C4C22×C4×C12711yescharacteristic
356C22×C4C22×C4×C12711yescharacteristic
357C22×C4C2×D4×C12711yescharacteristic
358C22×C4C2×D4×C12711yescharacteristic
359C22×C4C2×D4×C12711yescharacteristic
360C22×C4#41×C12712yes—
361C22×C4#41×C12712yes—
362C22×C4#41×C12712yes—
363C22×C4#41×C12712yes—
364C22×C4C24×C12711yescharacteristic
365M16—24no—
366M16—24no—
367M16—42no—
368M16—42no—
369C2×C8—24no—
370C2×C8—42no—
371C2×C8—42no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—28no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—42no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—28no—
383C22⋊C4—22no—
384C22⋊C4—24no—
385C22⋊C4—42no—
386C42—22no—
387C42—28no—
388C42—22no—
389C42—24no—
390C42—42no—
391C42C22×C4×C12714yes—
392C42C22×C4×C12714yes—
393C23—22no—
394C23—22no—
395C23—81no—
396C23—82no—
397C23C2×C22⋊C4×C12714yes—
398C23C2×C22⋊C4×C12711yescharacteristic
399C23C2×C22⋊C4×C12711yescharacteristic
400C23#123×C12712yes—
401C23#123×C12712yes—
402C23C22×D4×C12711yescharacteristic
403Q8—24no—
404Q8—24no—
405D4—28no—
406D4—28no—
407D4—24no—
408D4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—44no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—88no—
421C2×C4—84no—
422C2×C4C2×C22⋊C4×C12714yes—
423C2×C4C2×C22⋊C4×C12711yescharacteristic
424C2×C4C2×C22⋊C4×C12711yescharacteristic
425C2×C4#117×C12711yescharacteristic
426C2×C4#117×C12711yescharacteristic
427C2×C4#117×C12711yescharacteristic
428C2×C4#117×C12711yescharacteristic
429C2×C4#117×C12712yes—
430C2×C4#117×C12712yes—
431C2×C4#123×C12712yes—
432C2×C4#123×C12712yes—
433C2×C4#123×C12712yes—
434C2×C4#123×C12712yes—
435C2×C4#123×C12712yes—
436C2×C4#123×C12712yes—
437C2×C4C23×C4×C12711yescharacteristic
438C2×C4C22×D4×C12711yescharacteristic
439C2×C4C2×#41×C12712yes—
440C2×C4C2×#41×C12712yes—
441C8—44no—
442C22—24no—
443C22—41no—
444C22—42no—
445C22—41no—
446C22—41no—
447C22—42no—
448C22—44no—
449C22—82no—
450C22—82no—
451C22#381×C12711yescharacteristic
452C22#381×C12711yescharacteristic
453C22#381×C12712yes—
454C22C22×C22⋊C4×C12711yescharacteristic
455C22C2×#123×C12712yes—
456C4—24no—
457C4—44no—
458C4—44no—
459C4—84no—
460C4#381×C12711yescharacteristic
461C4#381×C12711yescharacteristic
462C4#381×C12712yes—
463C4C2×#117×C12711yescharacteristic
464C4C2×#117×C12711yescharacteristic
465C4C2×#123×C12712yes—
466C2—21no—
467C2—22no—
468C2—21no—
469C2—21no—
470C2—22no—
471C2—44no—
472C2C2×#381×C12711yescharacteristic
subgroupG/H
centerC4×C127C2×#117
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
socleC2×C127C2×#381
prime radicalC22×#41×C127C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C127 ◁ C2×#117×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1827C2semidirect
5. C127×#1733C2semidirect
6. C2×H(3,ℤ/4)×C127C2semidirect
14. C4×#41×C127C4semidirect
16. C2×#117×C127C4semidirect
27. #339×C127C22semidirect
28. H(3,ℤ/4)×C127C22semidirect
67. #117×C127C2×C4semidirect
80. C2×C42×C127C2×C4semidirect
229. #20708C127direct (N × H)
240. #1827C2×C127semidirect
241. #1733C2×C127semidirect
242. C2×H(3,ℤ/4)C2×C127semidirect
243. C127#20708direct (N × H)
251. C4×#41C4×C127semidirect
253. C2×#117C4×C127semidirect
264. #339C22×C127semidirect
265. H(3,ℤ/4)C22×C127semidirect
304. #117C2×C4×C127semidirect
317. C2×C42C2×C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34282
C2×#413×C1272
C2×#410×C1272
C127×#18272
C127×#17332
C2×H(3,ℤ/4)×C1272
#20708127