Finite Group Databasebeta

← C127×#10329 #44574002

Subgroups — C127×#10329 #44574002

Order 32512.

1044 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 1044 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3193C211yesmaximal, characteristic, complemented
2C127×#2435C211yesmaximal, characteristic, complemented
3C127×#2433C211yesmaximal, characteristic
4C127×#1783C211yesmaximal, characteristic, complemented
5C127×#1508C211yesmaximal, characteristic, complemented
6C127×#1507C211yesmaximal, characteristic, complemented
7C127×#1297C211yesmaximal, characteristic, complemented
8C23×D4×C127C2211yescharacteristic
9C2×#127×C127—21no—
10C2×#124×C127—21no—
11C2×C22⋊Q8×C127C2211yescharacteristic
12C2×#121×C127C2211yescharacteristic
13C2×#120×C127—22no—
14#484×C127—21no—
15#480×C127—21no—
16#468×C127—22no—
17#450×C127—21no—
18#448×C127—21no—
19C2×#100×C127—22no—
20#407×C127C2211yescharacteristic
21#407×C127C2211yescharacteristic
22#394×C127—22no—
23#392×C127—22no—
24#390×C127—21no—
25#390×C127—21no—
26#379×C127—22no—
27C4×C22⋊C4×C127C2211yescharacteristic
28C4×C22⋊C4×C127C2211yescharacteristic
29#328×C127—22no—
30#327×C127—22no—
31C25×C127D412yes—
32C22×Q8×C127D411yescharacteristic, complemented
33C22×D4×C127—21no—
34C22×D4×C127—24no—
35C22×D4×C127—21no—
36C22×D4×C127—21no—
37C22×D4×C127—21no—
38C22×D4×C127—21no—
39C22×D4×C127—21no—
40C22×D4×C127—21no—
41C22×D4×C127—44no—
42C22×D4×C127D411yescharacteristic
43C23×C4×C127C2311yescharacteristic
44C2×SD16×C127—22no—
45C2×M16×C127—21no—
46C2×M16×C127D412yescomplemented
47#127×C127—44no—
48#124×C127—44no—
49C22⋊Q8×C127—21no—
50C22⋊Q8×C127—21no—
51C22⋊Q8×C127—21no—
52C22⋊Q8×C127—21no—
53#121×C127—21no—
54#121×C127—21no—
55#121×C127—21no—
56#121×C127—21no—
57#120×C127—48no—
58C2×C4⋊C4×C127—21no—
59C2×C4⋊C4×C127D411yescharacteristic
60C2×C4⋊C4×C127D411yescharacteristic, complemented
61C2×C22⋊C4×C127—21no—
62C2×C22⋊C4×C127—24no—
63C2×C22⋊C4×C127—24no—
64C2×C22⋊C4×C127—21no—
65C2×C22⋊C4×C127—21no—
66C2×C22⋊C4×C127—21no—
67C2×C22⋊C4×C127—24no—
68C2×C42×C127—21no—
69C2×C42×C127—21no—
70#107×C127—22no—
71#103×C127—22no—
72#102×C127—22no—
73#100×C127—44no—
74C22⋊C8×C127—21no—
75C22⋊C8×C127—21no—
76C22⋊C8×C127—21no—
77C22⋊C8×C127—21no—
78#95×C127D412yes—
79#95×C127D412yescomplemented
80C24×C127—22no—
81C24×C127—22no—
82C24×C127—22no—
83C24×C127—21no—
84C24×C127—44no—
85C24×C127—44no—
86C24×C127—44no—
87C24×C127C2×D411yescharacteristic
88C2×Q8×C127—21no—
89C2×Q8×C127—21no—
90C2×Q8×C127—42no—
91C2×D4×C127—24no—
92C2×D4×C127—24no—
93C2×D4×C127—24no—
94C2×D4×C127—21no—
95C2×D4×C127—21no—
96C2×D4×C127—42no—
97C2×D4×C127—42no—
98C2×D4×C127—42no—
99C2×D4×C127—42no—
100C2×D4×C127—48no—
101C2×D4×C127—44no—
102C2×D4×C127—44no—
103C2×D4×C127—44no—
104C2×D4×C127—44no—
105C2×D4×C127—48no—
106C2×D4×C127—41no—
107C2×D4×C127—41no—
108C2×D4×C127—41no—
109C2×D4×C127—41no—
110C2×D4×C127—41no—
111C2×D4×C127—41no—
112C2×D4×C127—84no—
113C22×C4×C127—22no—
114C22×C4×C127—22no—
115C22×C4×C127—21no—
116C22×C4×C127—21no—
117C22×C4×C127—21no—
118C22×C4×C127—42no—
119C22×C4×C127C2×D411yescharacteristic
120C22×C4×C127C2×D411yescharacteristic
121C22×C4×C127C2×D411yescharacteristic
122C22×C4×C127C2×D411yescharacteristic
123C22×C4×C127C2×D411yescharacteristic
124C22×C4×C127#4111yescharacteristic
125SD16×C127—82no—
126M16×C127—42no—
127M16×C127—42no—
128C2×C8×C127—22no—
129C2×C8×C127—41no—
130C4⋊C4×C127—21no—
131C4⋊C4×C127—21no—
132C4⋊C4×C127—21no—
133C4⋊C4×C127—21no—
134C4⋊C4×C127—41no—
135C4⋊C4×C127—41no—
136C22⋊C4×C127—24no—
137C22⋊C4×C127—48no—
138C22⋊C4×C127—42no—
139C22⋊C4×C127—48no—
140C22⋊C4×C127—41no—
141C22⋊C4×C127—41no—
142C22⋊C4×C127—41no—
143C22⋊C4×C127—41no—
144C22⋊C4×C127—84no—
145C42×C127—42no—
146C42×C127—42no—
147C23×C127—22no—
148C23×C127—22no—
149C23×C127—24no—
150C23×C127—24no—
151C23×C127—24no—
152C23×C127—41no—
153C23×C127—44no—
154C23×C127—42no—
155C23×C127—42no—
156C23×C127—42no—
157C23×C127—44no—
158C23×C127—42no—
159C23×C127—42no—
160C23×C127—42no—
161C23×C127—44no—
162C23×C127—44no—
163C23×C127—41no—
164C23×C127—42no—
165C23×C127—82no—
166C23×C127—82no—
167C23×C127—816no—
168C23×C127#12011yescharacteristic
169C23×C127#12011yescharacteristic
170C23×C127#12111yescharacteristic
171Q8×C127—42no—
172Q8×C127—81no—
173D4×C127—44no—
174D4×C127—44no—
175D4×C127—44no—
176D4×C127—44no—
177D4×C127—42no—
178D4×C127—81no—
179D4×C127—816no—
180D4×C127—82no—
181D4×C127—82no—
182C2×C4×C127—22no—
183C2×C4×C127—22no—
184C2×C4×C127—22no—
185C2×C4×C127—24no—
186C2×C4×C127—41no—
187C2×C4×C127—41no—
188C2×C4×C127—41no—
189C2×C4×C127—42no—
190C2×C4×C127—42no—
191C2×C4×C127—42no—
192C2×C4×C127—42no—
193C2×C4×C127—42no—
194C2×C4×C127—42no—
195C2×C4×C127—41no—
196C2×C4×C127—41no—
197C2×C4×C127—41no—
198C2×C4×C127—42no—
199C2×C4×C127—42no—
200C2×C4×C127#12011yescharacteristic
201C2×C4×C127#12111yescharacteristic
202C2×C4×C127#12111yescharacteristic
203C2×C4×C127#12711yescharacteristic
204C8×C127—42no—
205C8×C127—81no—
206C22×C127—21no—
207C22×C127—21no—
208C22×C127—21no—
209C22×C127—21no—
210C22×C127—21no—
211C22×C127—21no—
212C22×C127—21no—
213C22×C127—44no—
214C22×C127—48no—
215C22×C127—48no—
216C22×C127—44no—
217C22×C127—84no—
218C22×C127—84no—
219C22×C127—84no—
220C22×C127—84no—
221C22×C127—84no—
222C22×C127—82no—
223C22×C127—82no—
224C22×C127—81no—
225C22×C127—81no—
226C22×C127—81no—
227C22×C127#39211yescharacteristic
228C22×C127#45312yes—
229C22×C127#45512yes—
230C4×C127—21no—
231C4×C127—21no—
232C4×C127—21no—
233C4×C127—44no—
234C4×C127—82no—
235C4×C127—81no—
236C4×C127—81no—
237C4×C127—81no—
238C4×C127B(4,𝔽2)12yes—
239#10329C12711yesmaximal, characteristic, direct product factor, Sylow
240C2×C127—22no—
241C2×C127—22no—
242C2×C127—41no—
243C2×C127—84no—
244C2×C127—161no—
245C2×C127#200211yescharacteristic
246C2×C127#200411yescharacteristic
247C2×C127#202111yescharacteristic
248#3193C2×C12711yescharacteristic, complemented
249#2435C2×C12711yescharacteristic, complemented
250#2433C2×C12711yescharacteristic
251#1783C2×C12711yescharacteristic, complemented
252#1508C2×C12711yescharacteristic, complemented
253#1507C2×C12711yescharacteristic, complemented
254#1297C2×C12711yescharacteristic, complemented
255C127#1032911yescharacteristic, direct product factor, Sylow
256C23×D4C22×C12711yescharacteristic
257C2×#127—21no—
258C2×#124—21no—
259C2×C22⋊Q8C22×C12711yescharacteristic
260C2×#121C22×C12711yescharacteristic
261C2×#120—22no—
262#484—21no—
263#480—21no—
264#468—22no—
265#450—21no—
266#448—21no—
267C2×#100—22no—
268#407C22×C12711yescharacteristic
269#407C22×C12711yescharacteristic
270#394—22no—
271#392—22no—
272#390—21no—
273#390—21no—
274#379—22no—
275C4×C22⋊C4C22×C12711yescharacteristic
276C4×C22⋊C4C22×C12711yescharacteristic
277#328—22no—
278#327—22no—
279C25D4×C12712yes—
280C22×Q8D4×C12711yescharacteristic, complemented
281C22×D4—21no—
282C22×D4—24no—
283C22×D4—21no—
284C22×D4—21no—
285C22×D4—21no—
286C22×D4—21no—
287C22×D4—21no—
288C22×D4—21no—
289C22×D4—44no—
290C22×D4D4×C12711yescharacteristic
291C23×C4C23×C12711yescharacteristic
292C2×SD16—22no—
293C2×M16—21no—
294C2×M16D4×C12712yescomplemented
295#127—44no—
296#124—44no—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C22⋊Q8—21no—
300C22⋊Q8—21no—
301#121—21no—
302#121—21no—
303#121—21no—
304#121—21no—
305#120—48no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4D4×C12711yescharacteristic
308C2×C4⋊C4D4×C12711yescharacteristic, complemented
309C2×C22⋊C4—21no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—24no—
316C2×C42—21no—
317C2×C42—21no—
318#107—22no—
319#103—22no—
320#102—22no—
321#100—44no—
322C22⋊C8—21no—
323C22⋊C8—21no—
324C22⋊C8—21no—
325C22⋊C8—21no—
326#95D4×C12712yes—
327#95D4×C12712yescomplemented
328C24—22no—
329C24—22no—
330C24—22no—
331C24—21no—
332C24—44no—
333C24—44no—
334C24—44no—
335C24C2×D4×C12711yescharacteristic
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—42no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—48no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—48no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—84no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—42no—
367C22×C4C2×D4×C12711yescharacteristic
368C22×C4C2×D4×C12711yescharacteristic
369C22×C4C2×D4×C12711yescharacteristic
370C22×C4C2×D4×C12711yescharacteristic
371C22×C4C2×D4×C12711yescharacteristic
372C22×C4#41×C12711yescharacteristic
373SD16—82no—
374M16—42no—
375M16—42no—
376C2×C8—22no—
377C2×C8—41no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C22⋊C4—24no—
385C22⋊C4—48no—
386C22⋊C4—42no—
387C22⋊C4—48no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—84no—
393C42—42no—
394C42—42no—
395C23—22no—
396C23—22no—
397C23—24no—
398C23—24no—
399C23—24no—
400C23—41no—
401C23—44no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—44no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—44no—
410C23—44no—
411C23—41no—
412C23—42no—
413C23—82no—
414C23—82no—
415C23—816no—
416C23#120×C12711yescharacteristic
417C23#120×C12711yescharacteristic
418C23#121×C12711yescharacteristic
419Q8—42no—
420Q8—81no—
421D4—44no—
422D4—44no—
423D4—44no—
424D4—44no—
425D4—42no—
426D4—81no—
427D4—816no—
428D4—82no—
429D4—82no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4#120×C12711yescharacteristic
449C2×C4#121×C12711yescharacteristic
450C2×C4#121×C12711yescharacteristic
451C2×C4#127×C12711yescharacteristic
452C8—42no—
453C8—81no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—44no—
462C22—48no—
463C22—48no—
464C22—44no—
465C22—84no—
466C22—84no—
467C22—84no—
468C22—84no—
469C22—84no—
470C22—82no—
471C22—82no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22#392×C12711yescharacteristic
476C22#453×C12712yes—
477C22#455×C12712yes—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—44no—
482C4—82no—
483C4—81no—
484C4—81no—
485C4—81no—
486C4B(4,𝔽2)×C12712yes—
487C2—22no—
488C2—22no—
489C2—41no—
490C2—84no—
491C2—161no—
492C2C127×#200211yescharacteristic
493C2C127×#200411yescharacteristic
494C2C127×#202111yescharacteristic
subgroupG/H
center, socleC22×C127#392
commutator, Frattini, cosocleC23×C4C23×C127
prime radicalC127×#2433C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3193C2semidirect
2. C127×#2435C2semidirect
4. C127×#1783C2semidirect
5. C127×#1508C2semidirect
6. C127×#1507C2semidirect
7. C127×#1297C2semidirect
32. C22×Q8×C127D4semidirect
46. C2×M16×C127D4semidirect
60. C2×C4⋊C4×C127D4semidirect
79. #95×C127D4semidirect
239. #10329C127direct (N × H)
248. #3193C2×C127semidirect
249. #2435C2×C127semidirect
251. #1783C2×C127semidirect
252. #1508C2×C127semidirect
253. #1507C2×C127semidirect
254. #1297C2×C127semidirect
255. C127#10329direct (N × H)
280. C22×Q8D4×C127semidirect
294. C2×M16D4×C127semidirect
308. C2×C4⋊C4D4×C127semidirect
327. #95D4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#31932
C127×#24352
C127×#24332
C127×#17832
C127×#15082
C127×#15072
C127×#12972
#10329127