Finite Group Databasebeta

← C127×#8349 #44572022

Subgroups — C127×#8349 #44572022

Order 32512.

720 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3493C211yesmaximal, characteristic
2C127×#1750C211yesmaximal, characteristic, complemented
3C127×#1532C212yesmaximal, complemented
4C127×#1494C212yesmaximal, complemented
5C127×#1464C211yesmaximal, characteristic, complemented
6#556×C127—22no—
7#555×C127—22no—
8#553×C127—22no—
9#552×C127—22no—
10#544×C127C411yescharacteristic
11#538×C127—21no—
12#538×C127C412yes—
13#537×C127C411yescharacteristic
14#529×C127—21no—
15#529×C127C2212yes—
16#525×C127—21no—
17#520×C127—21no—
18C2×#117×C127C2211yescharacteristic
19#432×C127C2212yes—
20C2×C22⋊C8×C127—21no—
21C2×#97×C127C2211yescharacteristic
22C2×C4×C8×C127C2211yescharacteristic
23#336×C127—21no—
24#330×C127—22no—
25#329×C127—22no—
26Q8⋊C8×C127—22no—
27D4⋊C8×C127—22no—
28#324×C127—21no—
29#323×C127—21no—
30C2×#41×C127—21no—
31C23×C4×C127D411yescharacteristic
32C22×C8×C127—21no—
33C22×C8×C127—21no—
34C4⋊Q8×C127—22no—
35#126×C127—22no—
36#126×C127—22no—
37#126×C127—22no—
38#126×C127—22no—
39#125×C127—21no—
40#125×C127—22no—
41#125×C127D412yes—
42#124×C127—21no—
43#124×C127—22no—
44#124×C127—22no—
45#124×C127D412yes—
46#123×C127—22no—
47#123×C127—22no—
48#123×C127—22no—
49C22⋊Q8×C127—21no—
50C22⋊Q8×C127—21no—
51C22⋊Q8×C127—21no—
52C22⋊Q8×C127—22no—
53C22⋊Q8×C127C2×C412yes—
54#121×C127—21no—
55#121×C127—21no—
56#121×C127—22no—
57#121×C127—22no—
58#121×C127—22no—
59#121×C127—21no—
60#121×C127C2×C412yes—
61#120×C127—22no—
62C4×Q8×C127—22no—
63C4×Q8×C127—21no—
64C4×Q8×C127D412yes—
65C4×D4×C127—22no—
66C4×D4×C127—22no—
67C4×D4×C127—22no—
68C4×D4×C127—21no—
69C4×D4×C127—22no—
70C4×D4×C127—22no—
71C4×D4×C127D412yes—
72#117×C127—22no—
73#117×C127—22no—
74#117×C127—21no—
75#117×C127—22no—
76#117×C127—22no—
77#117×C127—22no—
78C2×C4⋊C4×C127—22no—
79C2×C4⋊C4×C127C2×C411yescharacteristic
80C2×C4⋊C4×C127D411yescharacteristic
81C2×C22⋊C4×C127C2×C411yescharacteristic
82C2×C22⋊C4×C127D411yescharacteristic
83C2×C42×C127D411yescharacteristic
84C2×C42×C127C2311yescharacteristic
85C4⋊C8×C127—22no—
86C22⋊C8×C127—24no—
87C22⋊C8×C127—22no—
88#97×C127—22no—
89#97×C127—22no—
90C4×C8×C127—22no—
91C24×C127C22⋊C411yescharacteristic
92#41×C127—42no—
93#41×C127—42no—
94C2×Q8×C127—21no—
95C2×Q8×C127—21no—
96C2×Q8×C127C22⋊C412yes—
97C2×D4×C127—21no—
98C2×D4×C127—22no—
99C2×D4×C127—22no—
100C2×D4×C127—22no—
101C2×D4×C127—21no—
102C2×D4×C127C22⋊C412yes—
103C22×C4×C127—22no—
104C22×C4×C127—22no—
105C22×C4×C127—22no—
106C22×C4×C127—21no—
107C22×C4×C127—21no—
108C22×C4×C127—22no—
109C22×C4×C127—41no—
110C22×C4×C127C22⋊C411yescharacteristic
111C22×C4×C127C22⋊C411yescharacteristic
112C22×C4×C127C22⋊C411yescharacteristic
113C22×C4×C127C22×C411yescharacteristic
114C22×C4×C127C2×D411yescharacteristic
115C22×C4×C127C2×D411yescharacteristic
116C2×C8×C127—22no—
117C2×C8×C127—22no—
118C2×C8×C127—41no—
119C2×C8×C127—41no—
120C2×C8×C127—41no—
121C2×C8×C127—41no—
122C2×C8×C127—42no—
123C4⋊C4×C127—21no—
124C4⋊C4×C127—22no—
125C4⋊C4×C127—21no—
126C4⋊C4×C127—22no—
127C4⋊C4×C127—21no—
128C4⋊C4×C127—21no—
129C4⋊C4×C127—22no—
130C4⋊C4×C127—22no—
131C4⋊C4×C127—22no—
132C4⋊C4×C127—22no—
133C4⋊C4×C127—22no—
134C4⋊C4×C127—22no—
135C4⋊C4×C127C22⋊C412yes—
136C4⋊C4×C127C22⋊C412yes—
137C22⋊C4×C127—22no—
138C22⋊C4×C127—22no—
139C22⋊C4×C127—22no—
140C22⋊C4×C127—22no—
141C22⋊C4×C127—22no—
142C22⋊C4×C127—22no—
143C22⋊C4×C127—22no—
144C22⋊C4×C127—22no—
145C22⋊C4×C127—22no—
146C22⋊C4×C127—22no—
147C22⋊C4×C127—22no—
148C42×C127—22no—
149C42×C127—22no—
150C42×C127—21no—
151C42×C127—21no—
152C42×C127—22no—
153C42×C127C2×D412yes—
154C42×C127C2×D412yes—
155C23×C127—21no—
156C23×C127—22no—
157C23×C127—21no—
158C23×C127—21no—
159C23×C127—41no—
160C23×C127—41no—
161C23×C127C2×C22⋊C411yescharacteristic
162Q8×C127—42no—
163Q8×C127—42no—
164Q8×C127—81no—
165D4×C127—42no—
166D4×C127—42no—
167D4×C127—42no—
168D4×C127—42no—
169D4×C127—82no—
170D4×C127—81no—
171D4×C127—82no—
172C2×C4×C127—21no—
173C2×C4×C127—21no—
174C2×C4×C127—21no—
175C2×C4×C127—21no—
176C2×C4×C127—21no—
177C2×C4×C127—22no—
178C2×C4×C127—21no—
179C2×C4×C127—22no—
180C2×C4×C127—22no—
181C2×C4×C127—41no—
182C2×C4×C127—41no—
183C2×C4×C127—42no—
184C2×C4×C127—42no—
185C2×C4×C127—44no—
186C2×C4×C127—41no—
187C2×C4×C127—41no—
188C2×C4×C127—82no—
189C2×C4×C127—82no—
190C2×C4×C127#10412yes—
191C2×C4×C127C2×C22⋊C412yes—
192C2×C4×C127#12012yes—
193C2×C4×C127#12011yescharacteristic
194C2×C4×C127#12011yescharacteristic
195C8×C127—42no—
196C8×C127—42no—
197C8×C127—82no—
198C22×C127—21no—
199C22×C127—21no—
200C22×C127—42no—
201C22×C127—42no—
202C22×C127—41no—
203C22×C127—41no—
204C22×C127—42no—
205C22×C127—42no—
206C22×C127—41no—
207C22×C127—81no—
208C22×C127#37911yescharacteristic
209C22×C127C2×#10411yescharacteristic
210C22×C127#42111yescharacteristic
211C4×C127—22no—
212C4×C127—21no—
213C4×C127—21no—
214C4×C127—22no—
215C4×C127—22no—
216C4×C127—41no—
217C4×C127—42no—
218C4×C127—42no—
219C4×C127—41no—
220C4×C127—42no—
221#8349C12711yesmaximal, characteristic, direct product factor, Sylow
222C2×C127—21no—
223C2×C127—21no—
224C2×C127—82no—
225C2×C127—81no—
226C2×C127#178911yescharacteristic
227C2×C127#179111yescharacteristic
228C2×C127#179411yescharacteristic
229#3493C2×C12711yescharacteristic
230#1750C2×C12711yescharacteristic, complemented
231#1532C2×C12712yescomplemented
232#1494C2×C12712yescomplemented
233#1464C2×C12711yescharacteristic, complemented
234C127#834911yescharacteristic, direct product factor, Sylow
235#556—22no—
236#555—22no—
237#553—22no—
238#552—22no—
239#544C4×C12711yescharacteristic
240#538—21no—
241#538C4×C12712yes—
242#537C4×C12711yescharacteristic
243#529—21no—
244#529C22×C12712yes—
245#525—21no—
246#520—21no—
247C2×#117C22×C12711yescharacteristic
248#432C22×C12712yes—
249C2×C22⋊C8—21no—
250C2×#97C22×C12711yescharacteristic
251C2×C4×C8C22×C12711yescharacteristic
252#336—21no—
253#330—22no—
254#329—22no—
255Q8⋊C8—22no—
256D4⋊C8—22no—
257#324—21no—
258#323—21no—
259C2×#41—21no—
260C23×C4D4×C12711yescharacteristic
261C22×C8—21no—
262C22×C8—21no—
263C4⋊Q8—22no—
264#126—22no—
265#126—22no—
266#126—22no—
267#126—22no—
268#125—21no—
269#125—22no—
270#125D4×C12712yes—
271#124—21no—
272#124—22no—
273#124—22no—
274#124D4×C12712yes—
275#123—22no—
276#123—22no—
277#123—22no—
278C22⋊Q8—21no—
279C22⋊Q8—21no—
280C22⋊Q8—21no—
281C22⋊Q8—22no—
282C22⋊Q8C2×C4×C12712yes—
283#121—21no—
284#121—21no—
285#121—22no—
286#121—22no—
287#121—22no—
288#121—21no—
289#121C2×C4×C12712yes—
290#120—22no—
291C4×Q8—22no—
292C4×Q8—21no—
293C4×Q8D4×C12712yes—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—21no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4D4×C12712yes—
301#117—22no—
302#117—22no—
303#117—21no—
304#117—22no—
305#117—22no—
306#117—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4C2×C4×C12711yescharacteristic
309C2×C4⋊C4D4×C12711yescharacteristic
310C2×C22⋊C4C2×C4×C12711yescharacteristic
311C2×C22⋊C4D4×C12711yescharacteristic
312C2×C42D4×C12711yescharacteristic
313C2×C42C23×C12711yescharacteristic
314C4⋊C8—22no—
315C22⋊C8—24no—
316C22⋊C8—22no—
317#97—22no—
318#97—22no—
319C4×C8—22no—
320C24C22⋊C4×C12711yescharacteristic
321#41—42no—
322#41—42no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8C22⋊C4×C12712yes—
326C2×D4—21no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—21no—
331C2×D4C22⋊C4×C12712yes—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—41no—
339C22×C4C22⋊C4×C12711yescharacteristic
340C22×C4C22⋊C4×C12711yescharacteristic
341C22×C4C22⋊C4×C12711yescharacteristic
342C22×C4C22×C4×C12711yescharacteristic
343C22×C4C2×D4×C12711yescharacteristic
344C22×C4C2×D4×C12711yescharacteristic
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—42no—
352C4⋊C4—21no—
353C4⋊C4—22no—
354C4⋊C4—21no—
355C4⋊C4—22no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4C22⋊C4×C12712yes—
365C4⋊C4C22⋊C4×C12712yes—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C42—22no—
378C42—22no—
379C42—21no—
380C42—21no—
381C42—22no—
382C42C2×D4×C12712yes—
383C42C2×D4×C12712yes—
384C23—21no—
385C23—22no—
386C23—21no—
387C23—21no—
388C23—41no—
389C23—41no—
390C23C2×C22⋊C4×C12711yescharacteristic
391Q8—42no—
392Q8—42no—
393Q8—81no—
394D4—42no—
395D4—42no—
396D4—42no—
397D4—42no—
398D4—81no—
399D4—82no—
400D4—82no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4#104×C12712yes—
420C2×C4C2×C22⋊C4×C12712yes—
421C2×C4#120×C12711yescharacteristic
422C2×C4#120×C12711yescharacteristic
423C2×C4#120×C12712yes—
424C8—42no—
425C8—42no—
426C8—82no—
427C22—21no—
428C22—21no—
429C22—42no—
430C22—41no—
431C22—42no—
432C22—42no—
433C22—41no—
434C22—42no—
435C22—41no—
436C22—81no—
437C22#379×C12711yescharacteristic
438C22C2×#104×C12711yescharacteristic
439C22#421×C12711yescharacteristic
440C4—21no—
441C4—21no—
442C4—22no—
443C4—22no—
444C4—22no—
445C4—42no—
446C4—41no—
447C4—42no—
448C4—41no—
449C4—42no—
450C2—21no—
451C2—21no—
452C2—81no—
453C2—82no—
454C2C127×#178911yescharacteristic
455C2C127×#179111yescharacteristic
456C2C127×#179411yescharacteristic
subgroupG/H
center, socleC22×C127#379
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC127×#3493C2
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. C127×#1750C2semidirect
3. C127×#1532C2semidirect
4. C127×#1494C2semidirect
5. C127×#1464C2semidirect
221. #8349C127direct (N × H)
230. #1750C2×C127semidirect
231. #1532C2×C127semidirect
232. #1494C2×C127semidirect
233. #1464C2×C127semidirect
234. C127#8349direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#34932
C127×#17502
C127×#15322
C127×#14942
C127×#14642
#8349127