Finite Group Databasebeta

← C127×#8369 #44572042

Subgroups — C127×#8369 #44572042

Order 32512.

998 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
1C2×#548×C127C211yesmaximal, characteristic
2C4×#100×C127C211yesmaximal, characteristic, complemented
3C127×#1527C212yesmaximal, complemented
4C127×#1491C212yesmaximal, complemented
5C127×#1466C211yesmaximal, characteristic, complemented
6C22×#41×C127C411yescharacteristic
7C23×Q8×C127C411yescharacteristic
8#548×C127—24no—
9#548×C127—24no—
10C2×#124×C127—21no—
11C2×#124×C127C2211yescharacteristic
12C2×C22⋊Q8×C127—21no—
13C2×C22⋊Q8×C127C411yescharacteristic
14C2×#121×C127—21no—
15C2×#121×C127C411yescharacteristic
16C2×C4×Q8×C127C2211yescharacteristic
17C2×C4×D4×C127—21no—
18C2×C4×D4×C127C2211yescharacteristic
19#432×C127C2212yes—
20C2×#100×C127—21no—
21C4×M16×C127C2212yes—
22#343×C127—21no—
23#329×C127—24no—
24Q8⋊C8×C127—24no—
25#323×C127—21no—
26#323×C127—21no—
27C2×#41×C127—22no—
28C2×#41×C127—22no—
29C2×#41×C127—44no—
30C22×Q8×C127—22no—
31C22×Q8×C127—22no—
32C22×Q8×C127—21no—
33C22×Q8×C127—44no—
34C22×Q8×C127C2×C411yescharacteristic
35C22×Q8×C127C2×C411yescharacteristic
36C22×D4×C127—21no—
37C22×D4×C127—21no—
38C22×D4×C127C2×C411yescharacteristic
39C22×D4×C127D411yescharacteristic
40C23×C4×C127—21no—
41C23×C4×C127—21no—
42C23×C4×C127C2×C411yescharacteristic
43C23×C4×C127C2×C411yescharacteristic
44C2×M16×C127—22no—
45#124×C127—28no—
46#124×C127—22no—
47#124×C127D414yes—
48C22⋊Q8×C127—28no—
49C22⋊Q8×C127—22no—
50C22⋊Q8×C127—22no—
51#121×C127—28no—
52#121×C127—22no—
53#121×C127—22no—
54C4×Q8×C127—22no—
55C4×Q8×C127D414yes—
56C4×D4×C127—28no—
57C4×D4×C127—22no—
58C4×D4×C127—22no—
59C2×C4⋊C4×C127—21no—
60C2×C4⋊C4×C127—21no—
61C2×C4⋊C4×C127C2×C411yescharacteristic
62C2×C4⋊C4×C127D411yescharacteristic
63C2×C22⋊C4×C127—21no—
64C2×C22⋊C4×C127—21no—
65C2×C22⋊C4×C127—21no—
66C2×C22⋊C4×C127—21no—
67C2×C22⋊C4×C127D411yescharacteristic
68C2×C22⋊C4×C127D411yescharacteristic
69C2×C42×C127—21no—
70C2×C42×C127C2311yescharacteristic
71C4⋊C8×C127—22no—
72#100×C127—24no—
73C22⋊C8×C127—22no—
74#97×C127—22no—
75#97×C127—22no—
76C4×C8×C127—22no—
77C24×C127—21no—
78C24×C127C22⋊C411yescharacteristic
79C24×C127C22⋊C411yescharacteristic
80#41×C127—48no—
81#41×C127—44no—
82#41×C127—44no—
83C2×Q8×C127—24no—
84C2×Q8×C127—24no—
85C2×Q8×C127—21no—
86C2×Q8×C127—21no—
87C2×Q8×C127—42no—
88C2×Q8×C127—44no—
89C2×Q8×C127—48no—
90C2×Q8×C127—44no—
91C2×Q8×C127—44no—
92C2×Q8×C127C22⋊C412yes—
93C2×Q8×C127C22⋊C412yes—
94C2×D4×C127—24no—
95C2×D4×C127—24no—
96C2×D4×C127—22no—
97C2×D4×C127—21no—
98C2×D4×C127—22no—
99C2×D4×C127—41no—
100C2×D4×C127—44no—
101C2×D4×C127—42no—
102C2×D4×C127—42no—
103C2×D4×C127—42no—
104C2×D4×C127C22⋊C412yes—
105C22×C4×C127—21no—
106C22×C4×C127—24no—
107C22×C4×C127—24no—
108C22×C4×C127—21no—
109C22×C4×C127—21no—
110C22×C4×C127—21no—
111C22×C4×C127—21no—
112C22×C4×C127—21no—
113C22×C4×C127—21no—
114C22×C4×C127—44no—
115C22×C4×C127—44no—
116C22×C4×C127—42no—
117C22×C4×C127—42no—
118C22×C4×C127C22⋊C411yescharacteristic
119C22×C4×C127C22⋊C411yescharacteristic
120C22×C4×C127C22×C411yescharacteristic
121C22×C4×C127C2×D411yescharacteristic
122C22×C4×C127C2×D411yescharacteristic
123M16×C127—24no—
124M16×C127—42no—
125C2×C8×C127—42no—
126C2×C8×C127—42no—
127C4⋊C4×C127—24no—
128C4⋊C4×C127—21no—
129C4⋊C4×C127—24no—
130C4⋊C4×C127—22no—
131C4⋊C4×C127C22⋊C412yes—
132C22⋊C4×C127—24no—
133C22⋊C4×C127—24no—
134C22⋊C4×C127—24no—
135C22⋊C4×C127—24no—
136C22⋊C4×C127—22no—
137C22⋊C4×C127—22no—
138C42×C127—24no—
139C42×C127C2×D412yes—
140C42×C127C2×D412yes—
141C23×C127—21no—
142C23×C127—21no—
143C23×C127—21no—
144C23×C127—22no—
145C23×C127—21no—
146C23×C127—21no—
147C23×C127—41no—
148C23×C127—41no—
149C23×C127—42no—
150C23×C127—42no—
151C23×C127—41no—
152C23×C127—41no—
153C23×C127—41no—
154C23×C127—41no—
155C23×C127#10412yes—
156C23×C127C2×C22⋊C411yescharacteristic
157Q8×C127—24no—
158Q8×C127—22no—
159Q8×C127—42no—
160Q8×C127—48no—
161Q8×C127—42no—
162Q8×C127#10014yescomplemented
163D4×C127—44no—
164D4×C127—42no—
165D4×C127—48no—
166D4×C127—48no—
167D4×C127—42no—
168C2×C4×C127—24no—
169C2×C4×C127—22no—
170C2×C4×C127—21no—
171C2×C4×C127—21no—
172C2×C4×C127—22no—
173C2×C4×C127—22no—
174C2×C4×C127—22no—
175C2×C4×C127—21no—
176C2×C4×C127—21no—
177C2×C4×C127—41no—
178C2×C4×C127—48no—
179C2×C4×C127—42no—
180C2×C4×C127—48no—
181C2×C4×C127—42no—
182C2×C4×C127—42no—
183C2×C4×C127—41no—
184C2×C4×C127—41no—
185C2×C4×C127—42no—
186C2×C4×C127—42no—
187C2×C4×C127—41no—
188C2×C4×C127—41no—
189C2×C4×C127C2×C22⋊C411yescharacteristic
190C2×C4×C127C2×C22⋊C411yescharacteristic
191C2×C4×C127#12012yes—
192C2×C4×C127#12012yes—
193C8×C127—44no—
194C8×C127—82no—
195C22×C127—21no—
196C22×C127—21no—
197C22×C127—22no—
198C22×C127—42no—
199C22×C127—42no—
200C22×C127—41no—
201C22×C127—41no—
202C22×C127—41no—
203C22×C127—42no—
204C22×C127—41no—
205C22×C127—42no—
206C22×C127—41no—
207C22×C127—42no—
208C22×C127—81no—
209C22×C127—81no—
210C22×C127—82no—
211C22×C127—82no—
212C22×C127#37911yescharacteristic
213C22×C127C2×#10411yescharacteristic
214C22×C127#42111yescharacteristic
215C4×C127—22no—
216C4×C127—24no—
217C4×C127—21no—
218C4×C127—22no—
219C4×C127—41no—
220C4×C127—42no—
221C4×C127—41no—
222C4×C127—42no—
223C4×C127C2×#10012yes—
224#8369C12711yesmaximal, characteristic, direct product factor, Sylow
225C2×C127—21no—
226C2×C127—21no—
227C2×C127—42no—
228C2×C127—81no—
229C2×C127—82no—
230C2×C127#178311yescharacteristic
231C2×C127#178911yescharacteristic
232C2×C127#179611yescharacteristic
233C2×#548C2×C12711yescharacteristic
234C4×#100C2×C12711yescharacteristic, complemented
235#1527C2×C12712yescomplemented
236#1491C2×C12712yescomplemented
237#1466C2×C12711yescharacteristic, complemented
238C127#836911yescharacteristic, direct product factor, Sylow
239C22×#41C4×C12711yescharacteristic
240C23×Q8C4×C12711yescharacteristic
241#548—24no—
242#548—24no—
243C2×#124—21no—
244C2×#124C22×C12711yescharacteristic
245C2×C22⋊Q8—21no—
246C2×C22⋊Q8C4×C12711yescharacteristic
247C2×#121—21no—
248C2×#121C4×C12711yescharacteristic
249C2×C4×Q8C22×C12711yescharacteristic
250C2×C4×D4—21no—
251C2×C4×D4C22×C12711yescharacteristic
252#432C22×C12712yes—
253C2×#100—21no—
254C4×M16C22×C12712yes—
255#343—21no—
256#329—24no—
257Q8⋊C8—24no—
258#323—21no—
259#323—21no—
260C2×#41—22no—
261C2×#41—22no—
262C2×#41—44no—
263C22×Q8—22no—
264C22×Q8—22no—
265C22×Q8—21no—
266C22×Q8—44no—
267C22×Q8C2×C4×C12711yescharacteristic
268C22×Q8C2×C4×C12711yescharacteristic
269C22×D4—21no—
270C22×D4—21no—
271C22×D4C2×C4×C12711yescharacteristic
272C22×D4D4×C12711yescharacteristic
273C23×C4—21no—
274C23×C4—21no—
275C23×C4C2×C4×C12711yescharacteristic
276C23×C4C2×C4×C12711yescharacteristic
277C2×M16—22no—
278#124—28no—
279#124—22no—
280#124D4×C12714yes—
281C22⋊Q8—28no—
282C22⋊Q8—22no—
283C22⋊Q8—22no—
284#121—28no—
285#121—22no—
286#121—22no—
287C4×Q8—22no—
288C4×Q8D4×C12714yes—
289C4×D4—28no—
290C4×D4—22no—
291C4×D4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4C2×C4×C12711yescharacteristic
295C2×C4⋊C4D4×C12711yescharacteristic
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4D4×C12711yescharacteristic
301C2×C22⋊C4D4×C12711yescharacteristic
302C2×C42—21no—
303C2×C42C23×C12711yescharacteristic
304C4⋊C8—22no—
305#100—24no—
306C22⋊C8—22no—
307#97—22no—
308#97—22no—
309C4×C8—22no—
310C24—21no—
311C24C22⋊C4×C12711yescharacteristic
312C24C22⋊C4×C12711yescharacteristic
313#41—48no—
314#41—44no—
315#41—44no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—42no—
321C2×Q8—44no—
322C2×Q8—48no—
323C2×Q8—44no—
324C2×Q8—44no—
325C2×Q8C22⋊C4×C12712yes—
326C2×Q8C22⋊C4×C12712yes—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—22no—
330C2×D4—21no—
331C2×D4—22no—
332C2×D4—41no—
333C2×D4—44no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4C22⋊C4×C12712yes—
338C22×C4—21no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4C22⋊C4×C12711yescharacteristic
352C22×C4C22⋊C4×C12711yescharacteristic
353C22×C4C22×C4×C12711yescharacteristic
354C22×C4C2×D4×C12711yescharacteristic
355C22×C4C2×D4×C12711yescharacteristic
356M16—24no—
357M16—42no—
358C2×C8—42no—
359C2×C8—42no—
360C4⋊C4—24no—
361C4⋊C4—21no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4C22⋊C4×C12712yes—
365C22⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C42—24no—
372C42C2×D4×C12712yes—
373C42C2×D4×C12712yes—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—22no—
378C23—21no—
379C23—21no—
380C23—41no—
381C23—41no—
382C23—42no—
383C23—42no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23#104×C12712yes—
389C23C2×C22⋊C4×C12711yescharacteristic
390Q8—24no—
391Q8—22no—
392Q8—42no—
393Q8—48no—
394Q8—42no—
395Q8#100×C12714yescomplemented
396D4—44no—
397D4—42no—
398D4—48no—
399D4—48no—
400D4—42no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—41no—
411C2×C4—48no—
412C2×C4—42no—
413C2×C4—48no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4C2×C22⋊C4×C12711yescharacteristic
423C2×C4C2×C22⋊C4×C12711yescharacteristic
424C2×C4#120×C12712yes—
425C2×C4#120×C12712yes—
426C8—44no—
427C8—82no—
428C22—21no—
429C22—21no—
430C22—22no—
431C22—42no—
432C22—42no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—42no—
437C22—41no—
438C22—42no—
439C22—42no—
440C22—41no—
441C22—81no—
442C22—81no—
443C22—82no—
444C22—82no—
445C22#379×C12711yescharacteristic
446C22C2×#104×C12711yescharacteristic
447C22#421×C12711yescharacteristic
448C4—24no—
449C4—21no—
450C4—22no—
451C4—22no—
452C4—41no—
453C4—42no—
454C4—41no—
455C4—42no—
456C4C2×#100×C12712yes—
457C2—21no—
458C2—21no—
459C2—42no—
460C2—82no—
461C2—81no—
462C2C127×#178311yescharacteristic
463C2C127×#178911yescharacteristic
464C2C127×#179611yescharacteristic
subgroupG/H
center, socleC22×C127#379
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC2×#548×C127C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×#100×C127C2semidirect
3. C127×#1527C2semidirect
4. C127×#1491C2semidirect
5. C127×#1466C2semidirect
162. Q8×C127#100semidirect
224. #8369C127direct (N × H)
234. C4×#100C2×C127semidirect
235. #1527C2×C127semidirect
236. #1491C2×C127semidirect
237. #1466C2×C127semidirect
238. C127#8369direct (N × H)
395. Q8#100×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#548×C1272
C4×#100×C1272
C127×#15272
C127×#14912
C127×#14662
#8369127