Finite Group Databasebeta

← C127×#13092 #44576765

Subgroups — C127×#13092 #44576765

Order 32512.

702 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#101×C127C211yesmaximal, characteristic, complemented
2C127×#2436C211yesmaximal, characteristic
3C2×#355×C127C212yesmaximal, complemented
4C127×#1784C211yesmaximal, characteristic, complemented
5C127×#1394C212yesmaximal, complemented
6C23×Q8×C127C2211yescharacteristic
7C22×M16×C127—21no—
8C2×#124×C127C411yescharacteristic
9C2×#124×C127C2211yescharacteristic
10C2×#121×C127C411yescharacteristic
11C2×#101×C127—21no—
12C2×#101×C127—24no—
13C2×#101×C127—21no—
14C2×#101×C127C2212yes—
15C2×#101×C127C2212yes—
16#407×C127—21no—
17#393×C127—21no—
18#393×C127—21no—
19#391×C127C411yescharacteristic
20#391×C127C2211yescharacteristic
21#386×C127—21no—
22#386×C127—21no—
23C4×C22⋊C4×C127C411yescharacteristic
24#355×C127—24no—
25C22×Q8×C127—22no—
26C22×Q8×C127—22no—
27C22×Q8×C127—41no—
28C22×Q8×C127—41no—
29C22×Q8×C127—42no—
30C22×Q8×C127C2×C411yescharacteristic
31C22×Q8×C127D411yescharacteristic
32C22×Q8×C127D411yescharacteristic
33C22×Q8×C127C2311yescharacteristic
34C22×D4×C127—21no—
35C22×D4×C127C2×C411yescharacteristic
36C23×C4×C127C2×C411yescharacteristic
37C23×C4×C127D411yescharacteristic
38C23×C4×C127D411yescharacteristic
39C2×M16×C127—24no—
40C2×M16×C127—22no—
41C2×M16×C127—22no—
42C2×M16×C127—41no—
43C2×M16×C127—42no—
44C22×C8×C127—41no—
45#124×C127—22no—
46#124×C127—22no—
47#124×C127—42no—
48#121×C127—41no—
49#121×C127—41no—
50C2×C4⋊C4×C127C2×C411yescharacteristic
51C2×C22⋊C4×C127—21no—
52C2×C22⋊C4×C127—21no—
53C2×C22⋊C4×C127—21no—
54C2×C22⋊C4×C127—21no—
55C2×C22⋊C4×C127—21no—
56C2×C22⋊C4×C127—21no—
57C2×C42×C127C2×C411yescharacteristic
58C2×C42×C127C2×C411yescharacteristic
59#101×C127—24no—
60#101×C127—22no—
61#101×C127—22no—
62#101×C127—22no—
63#101×C127—42no—
64#101×C127D414yescomplemented
65C22⋊C8×C127—41no—
66C22⋊C8×C127—41no—
67#95×C127—21no—
68#95×C127—21no—
69#95×C127—21no—
70#95×C127—21no—
71#95×C127—21no—
72C24×C127—21no—
73C24×C127C22⋊C411yescharacteristic
74C2×Q8×C127—22no—
75C2×Q8×C127—22no—
76C2×Q8×C127—22no—
77C2×Q8×C127—21no—
78C2×Q8×C127—22no—
79C2×Q8×C127—21no—
80C2×Q8×C127—41no—
81C2×Q8×C127—41no—
82C2×Q8×C127—41no—
83C2×Q8×C127—41no—
84C2×Q8×C127—41no—
85C2×Q8×C127—41no—
86C2×Q8×C127—44no—
87C2×Q8×C127—44no—
88C2×Q8×C127—44no—
89C2×Q8×C127—44no—
90C2×Q8×C127C2×D412yes—
91C2×Q8×C127#4112yes—
92C2×D4×C127—22no—
93C2×D4×C127—41no—
94C2×D4×C127—41no—
95C2×D4×C127—44no—
96C2×D4×C127—41no—
97C2×D4×C127—41no—
98C22×C4×C127—21no—
99C22×C4×C127—21no—
100C22×C4×C127—22no—
101C22×C4×C127—21no—
102C22×C4×C127—21no—
103C22×C4×C127—21no—
104C22×C4×C127—21no—
105C22×C4×C127—21no—
106C22×C4×C127—21no—
107C22×C4×C127—22no—
108C22×C4×C127—22no—
109C22×C4×C127—21no—
110C22×C4×C127—21no—
111C22×C4×C127—21no—
112C22×C4×C127—41no—
113C22×C4×C127—41no—
114C22×C4×C127C22⋊C411yescharacteristic
115C22×C4×C127C22⋊C411yescharacteristic
116C22×C4×C127C22⋊C411yescharacteristic
117C22×C4×C127C22×C411yescharacteristic
118C22×C4×C127C2×D411yescharacteristic
119C22×C4×C127C2×D411yescharacteristic
120M16×C127—24no—
121M16×C127—44no—
122M16×C127—42no—
123M16×C127—42no—
124M16×C127—42no—
125C2×C8×C127—44no—
126C2×C8×C127—42no—
127C2×C8×C127—42no—
128C4⋊C4×C127—41no—
129C22⋊C4×C127—42no—
130C22⋊C4×C127—42no—
131C22⋊C4×C127—42no—
132C22⋊C4×C127—42no—
133C22⋊C4×C127—41no—
134C22⋊C4×C127—42no—
135C22⋊C4×C127—41no—
136C42×C127—22no—
137C42×C127—41no—
138C23×C127—22no—
139C23×C127—21no—
140C23×C127—21no—
141C23×C127—41no—
142C23×C127—42no—
143C23×C127—42no—
144C23×C127—41no—
145C23×C127—41no—
146C23×C127—41no—
147C23×C127#9911yescharacteristic
148C23×C127#9911yescharacteristic
149C23×C127C2×C22⋊C411yescharacteristic
150Q8×C127—42no—
151Q8×C127—42no—
152Q8×C127—44no—
153Q8×C127—44no—
154Q8×C127—42no—
155Q8×C127—42no—
156D4×C127—82no—
157D4×C127—82no—
158D4×C127—82no—
159C2×C4×C127—22no—
160C2×C4×C127—22no—
161C2×C4×C127—22no—
162C2×C4×C127—22no—
163C2×C4×C127—21no—
164C2×C4×C127—21no—
165C2×C4×C127—21no—
166C2×C4×C127—21no—
167C2×C4×C127—21no—
168C2×C4×C127—21no—
169C2×C4×C127—21no—
170C2×C4×C127—21no—
171C2×C4×C127—21no—
172C2×C4×C127—21no—
173C2×C4×C127—21no—
174C2×C4×C127—21no—
175C2×C4×C127—21no—
176C2×C4×C127—21no—
177C2×C4×C127—42no—
178C2×C4×C127—42no—
179C2×C4×C127—42no—
180C2×C4×C127—42no—
181C2×C4×C127—41no—
182C2×C4×C127—41no—
183C2×C4×C127—41no—
184C2×C4×C127—41no—
185C2×C4×C127—42no—
186C2×C4×C127—41no—
187C2×C4×C127—41no—
188C2×C4×C127—41no—
189C2×C4×C127#9912yes—
190C2×C4×C127C4×D412yes—
191C2×C4×C127#12011yescharacteristic
192C2×C4×C127#12112yes—
193C2×C4×C127#12311yescharacteristic
194C8×C127—44no—
195C8×C127—82no—
196C22×C127—21no—
197C22×C127—21no—
198C22×C127—21no—
199C22×C127—21no—
200C22×C127—21no—
201C22×C127—21no—
202C22×C127—21no—
203C22×C127—42no—
204C22×C127—42no—
205C22×C127—82no—
206C22×C127—82no—
207C22×C127—81no—
208C22×C127—81no—
209C22×C127—81no—
210C22×C127#35512yescomplemented
211C22×C127#38611yescharacteristic
212C22×C127C2×#9911yescharacteristic
213C22×C127C2×#9911yescharacteristic
214C4×C127—21no—
215C4×C127—22no—
216C4×C127—22no—
217C4×C127—21no—
218C4×C127—22no—
219C4×C127—41no—
220C4×C127—41no—
221C4×C127—81no—
222C4×C127—81no—
223C4×C127—81no—
224#13092C12711yesmaximal, characteristic, direct product factor, Sylow
225C2×C127—21no—
226C2×C127—21no—
227C2×C127—22no—
228C2×C127—41no—
229C2×C127—161no—
230C2×C127#189811yescharacteristic
231C2×C127C2×#35511yescharacteristic
232C2×C127#213311yescharacteristic
233C22×#101C2×C12711yescharacteristic, complemented
234#2436C2×C12711yescharacteristic
235C2×#355C2×C12712yescomplemented
236#1784C2×C12711yescharacteristic, complemented
237#1394C2×C12712yescomplemented
238C127#1309211yescharacteristic, direct product factor, Sylow
239C23×Q8C22×C12711yescharacteristic
240C22×M16—21no—
241C2×#124C4×C12711yescharacteristic
242C2×#124C22×C12711yescharacteristic
243C2×#121C4×C12711yescharacteristic
244C2×#101—21no—
245C2×#101—24no—
246C2×#101—21no—
247C2×#101C22×C12712yes—
248C2×#101C22×C12712yes—
249#407—21no—
250#393—21no—
251#393—21no—
252#391C4×C12711yescharacteristic
253#391C22×C12711yescharacteristic
254#386—21no—
255#386—21no—
256C4×C22⋊C4C4×C12711yescharacteristic
257#355—24no—
258C22×Q8—22no—
259C22×Q8—22no—
260C22×Q8—41no—
261C22×Q8—41no—
262C22×Q8—42no—
263C22×Q8C2×C4×C12711yescharacteristic
264C22×Q8D4×C12711yescharacteristic
265C22×Q8D4×C12711yescharacteristic
266C22×Q8C23×C12711yescharacteristic
267C22×D4—21no—
268C22×D4C2×C4×C12711yescharacteristic
269C23×C4C2×C4×C12711yescharacteristic
270C23×C4D4×C12711yescharacteristic
271C23×C4D4×C12711yescharacteristic
272C2×M16—24no—
273C2×M16—22no—
274C2×M16—22no—
275C2×M16—41no—
276C2×M16—42no—
277C22×C8—41no—
278#124—22no—
279#124—22no—
280#124—42no—
281#121—41no—
282#121—41no—
283C2×C4⋊C4C2×C4×C12711yescharacteristic
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C42C2×C4×C12711yescharacteristic
291C2×C42C2×C4×C12711yescharacteristic
292#101—24no—
293#101—22no—
294#101—22no—
295#101—22no—
296#101—42no—
297#101D4×C12714yescomplemented
298C22⋊C8—41no—
299C22⋊C8—41no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305C24—21no—
306C24C22⋊C4×C12711yescharacteristic
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—21no—
311C2×Q8—22no—
312C2×Q8—21no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—44no—
322C2×Q8—44no—
323C2×Q8C2×D4×C12712yes—
324C2×Q8#41×C12712yes—
325C2×D4—22no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—44no—
329C2×D4—41no—
330C2×D4—41no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4C22⋊C4×C12711yescharacteristic
348C22×C4C22⋊C4×C12711yescharacteristic
349C22×C4C22⋊C4×C12711yescharacteristic
350C22×C4C22×C4×C12711yescharacteristic
351C22×C4C2×D4×C12711yescharacteristic
352C22×C4C2×D4×C12711yescharacteristic
353M16—24no—
354M16—44no—
355M16—42no—
356M16—42no—
357M16—42no—
358C2×C8—44no—
359C2×C8—42no—
360C2×C8—42no—
361C4⋊C4—41no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—41no—
367C22⋊C4—42no—
368C22⋊C4—41no—
369C42—22no—
370C42—41no—
371C23—22no—
372C23—21no—
373C23—21no—
374C23—41no—
375C23—42no—
376C23—42no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23#99×C12711yescharacteristic
381C23#99×C12711yescharacteristic
382C23C2×C22⋊C4×C12711yescharacteristic
383Q8—42no—
384Q8—42no—
385Q8—44no—
386Q8—44no—
387Q8—42no—
388Q8—42no—
389D4—82no—
390D4—82no—
391D4—82no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4#99×C12712yes—
423C2×C4C4×D4×C12712yes—
424C2×C4#120×C12711yescharacteristic
425C2×C4#121×C12712yes—
426C2×C4#123×C12711yescharacteristic
427C8—44no—
428C8—82no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—42no—
437C22—42no—
438C22—82no—
439C22—82no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22#355×C12712yescomplemented
444C22#386×C12711yescharacteristic
445C22C2×#99×C12711yescharacteristic
446C22C2×#99×C12711yescharacteristic
447C4—21no—
448C4—22no—
449C4—22no—
450C4—21no—
451C4—22no—
452C4—41no—
453C4—41no—
454C4—81no—
455C4—81no—
456C4—81no—
457C2—21no—
458C2—21no—
459C2—22no—
460C2—41no—
461C2—161no—
462C2C127×#189811yescharacteristic
463C2C2×#355×C12711yescharacteristic
464C2C127×#213311yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC22×Q8C23×C127
prime radicalC2×#121×C127C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C24×C127 ◁ C23×Q8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×#101×C127C2semidirect
3. C2×#355×C127C2semidirect
4. C127×#1784C2semidirect
5. C127×#1394C2semidirect
64. #101×C127D4semidirect
210. C22×C127#355semidirect
224. #13092C127direct (N × H)
233. C22×#101C2×C127semidirect
235. C2×#355C2×C127semidirect
236. #1784C2×C127semidirect
237. #1394C2×C127semidirect
238. C127#13092direct (N × H)
297. #101D4×C127semidirect
443. C22#355×C127semidirect

Maximal subgroups

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

subgroupindex
C22×#101×C1272
C127×#24362
C2×#355×C1272
C127×#17842
C127×#13942
#13092127