Finite Group Databasebeta

← C127×#17078 #44580751

Subgroups — C127×#17078 #44580751

Order 32512.

1338 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#395×C127C212yesmaximal
2C2×#391×C127C211yesmaximal, characteristic
3C2×#389×C127C211yesmaximal, characteristic
4C127×#1445C212yesmaximal
5C127×#1444C212yesmaximal
6C127×#1441C212yesmaximal
7C127×#1441C214yesmaximal
8C127×C4⋊#95C211yesmaximal, characteristic
9C23×Q8×C127C2211yescharacteristic
10C22×C4⋊C4×C127C2212yes—
11C22×C4⋊C4×C127C2211yescharacteristic
12C22×C4⋊C4×C127C2214yes—
13C22×C4⋊C4×C127C2211yescharacteristic
14C22×C4⋊C4×C127C2211yescharacteristic
15C22×C4⋊C4×C127C2212yes—
16C22×C42×C127C2211yescharacteristic
17#395×C127—28no—
18#391×C127—24no—
19#389×C127—24no—
20C2×#95×C127C2212yes—
21C2×#95×C127C2214yes—
22C2×#95×C127C2212yes—
23C2×#95×C127C2212yes—
24C2×#95×C127C2212yes—
25C2×#95×C127C2212yes—
26C2×#95×C127C2214yes—
27C2×#95×C127C2214yes—
28C22×Q8×C127—22no—
29C22×Q8×C127—22no—
30C22×Q8×C127—42no—
31C22×Q8×C127—42no—
32C22×Q8×C127D414yes—
33C23×C4×C127C2312yes—
34C23×C4×C127C2312yes—
35C23×C4×C127C2312yes—
36C23×C4×C127C2312yes—
37C23×C4×C127C2314yes—
38C23×C4×C127C2311yescharacteristic
39C23×C4×C127C2311yescharacteristic
40C23×C4×C127C2311yescharacteristic
41C2×C4⋊C4×C127—22no—
42C2×C4⋊C4×C127—22no—
43C2×C4⋊C4×C127—22no—
44C2×C4⋊C4×C127—24no—
45C2×C4⋊C4×C127—22no—
46C2×C4⋊C4×C127—28no—
47C2×C4⋊C4×C127—24no—
48C2×C4⋊C4×C127—24no—
49C2×C4⋊C4×C127—42no—
50C2×C4⋊C4×C127—42no—
51C2×C4⋊C4×C127—42no—
52C2×C4⋊C4×C127—42no—
53C2×C4⋊C4×C127—44no—
54C2×C4⋊C4×C127—44no—
55C2×C4⋊C4×C127Q814yescomplemented
56C2×C4⋊C4×C127Q818yescomplemented
57C2×C42×C127—42no—
58C2×C42×C127D414yes—
59#95×C127—24no—
60#95×C127—24no—
61#95×C127—24no—
62#95×C127—28no—
63#95×C127—24no—
64#95×C127—24no—
65#95×C127—44no—
66#95×C127—44no—
67C24×C127C2411yescharacteristic
68C2×Q8×C127—28no—
69C2×Q8×C127—48no—
70C2×Q8×C127—44no—
71C2×Q8×C127—44no—
72C2×Q8×C127—48no—
73C22×C4×C127—22no—
74C22×C4×C127—22no—
75C22×C4×C127—22no—
76C22×C4×C127—21no—
77C22×C4×C127—22no—
78C22×C4×C127—22no—
79C22×C4×C127—21no—
80C22×C4×C127—22no—
81C22×C4×C127—24no—
82C22×C4×C127—22no—
83C22×C4×C127—22no—
84C22×C4×C127—24no—
85C22×C4×C127—22no—
86C22×C4×C127—22no—
87C22×C4×C127—22no—
88C22×C4×C127—22no—
89C22×C4×C127—24no—
90C22×C4×C127—24no—
91C22×C4×C127—24no—
92C22×C4×C127—24no—
93C22×C4×C127—24no—
94C22×C4×C127—22no—
95C22×C4×C127—24no—
96C22×C4×C127—22no—
97C22×C4×C127—22no—
98C22×C4×C127—22no—
99C22×C4×C127—22no—
100C22×C4×C127—22no—
101C22×C4×C127—24no—
102C22×C4×C127—24no—
103C22×C4×C127—24no—
104C22×C4×C127—24no—
105C22×C4×C127—22no—
106C22×C4×C127C2×D414yes—
107C22×C4×C127C2×D412yes—
108C22×C4×C127C2×D412yes—
109C22×C4×C127C2×D412yes—
110C22×C4×C127C2×D412yes—
111C22×C4×C127C2×Q814yes—
112C22×C4×C127C2×Q818yes—
113C22×C4×C127C2×Q812yes—
114C22×C4×C127C2×Q814yes—
115C22×C4×C127#4112yes—
116C22×C4×C127#4112yes—
117C4⋊C4×C127—28no—
118C4⋊C4×C127—48no—
119C4⋊C4×C127—44no—
120C4⋊C4×C127—44no—
121C4⋊C4×C127—416no—
122C4⋊C4×C127—48no—
123C42×C127—44no—
124C23×C127C22×D411yescharacteristic
125C23×C127C22×D411yescharacteristic
126C23×C127C22×Q811yescharacteristic
127C23×C127C22×Q812yes—
128C23×C127C2×#4111yescharacteristic
129C23×C127C2×#4111yescharacteristic
130C23×C127H(5,𝔽2)12yes—
131C23×C127H(5,𝔽2)12yes—
132C23×C127H(5,𝔽2)11yescharacteristic
133C23×C1272−1+412yes—
134C23×C1272−1+411yescharacteristic
135Q8×C127—416no—
136C2×C4×C127—22no—
137C2×C4×C127—22no—
138C2×C4×C127—22no—
139C2×C4×C127—24no—
140C2×C4×C127—22no—
141C2×C4×C127—22no—
142C2×C4×C127—22no—
143C2×C4×C127—24no—
144C2×C4×C127—24no—
145C2×C4×C127—24no—
146C2×C4×C127—24no—
147C2×C4×C127—24no—
148C2×C4×C127—24no—
149C2×C4×C127—24no—
150C2×C4×C127—28no—
151C2×C4×C127—28no—
152C2×C4×C127—28no—
153C2×C4×C127—24no—
154C2×C4×C127—28no—
155C2×C4×C127—24no—
156C2×C4×C127—42no—
157C2×C4×C127—44no—
158C2×C4×C127—42no—
159C2×C4×C127—42no—
160C2×C4×C127—42no—
161C2×C4×C127—44no—
162C2×C4×C127—44no—
163C2×C4×C127—44no—
164C2×C4×C127—42no—
165C2×C4×C127—42no—
166C2×C4×C127—44no—
167C2×C4×C127—42no—
168C2×C4×C127—42no—
169C2×C4×C127—44no—
170C2×C4×C127—42no—
171C2×C4×C127—42no—
172C2×C4×C127—42no—
173C2×C4×C127—42no—
174C2×C4×C127—44no—
175C2×C4×C127—44no—
176C2×C4×C127—44no—
177C2×C4×C127#12114yes—
178C2×C4×C127C22⋊Q814yes—
179C2×C4×C127C4⋊Q814yes—
180C22×C127C2×#12111yescharacteristic
181C22×C127C2×C22⋊Q811yescharacteristic
182C22×C127C2×C4⋊Q811yescharacteristic
183C22×C127#53411yescharacteristic
184C22×C127#53412yes—
185C22×C127#53511yescharacteristic
186C22×C127#53611yescharacteristic
187C22×C127#53711yescharacteristic
188C22×C127#53812yes—
189C22×C127#53912yes—
190C22×C127#54312yes—
191C22×C127#54412yes—
192C22×C127#54412yes—
193C22×C127D4×Q812yes—
194C22×C127D4×Q811yescharacteristic
195C22×C127#55011yescharacteristic
196C22×C127#55412yes—
197C22×C127#55411yescharacteristic
198C22×C127#55611yescharacteristic
199C22×C127#55712yes—
200C22×C127Q8212yes—
201C22×C127#56212yes—
202C22×C127#56312yes—
203C4×C127—24no—
204C4×C127—24no—
205C4×C127—24no—
206C4×C127—44no—
207C4×C127—48no—
208C4×C127—44no—
209C4×C127—82no—
210C4×C127—82no—
211#17078C12711yesmaximal, characteristic, direct product factor, Sylow
212C2×C127#253811yescharacteristic
213C2×C127#254611yescharacteristic
214C2×C127#256711yescharacteristic
215C2×C127#261711yescharacteristic
216C2×C127#262711yescharacteristic
217C2×C127#265911yescharacteristic
218C2×C127#273212yes—
219C2×C127#273412yes—
220C2×C127#280512yes—
221C2×C127#280612yes—
222C2×C127#282011yescharacteristic
223C2×#395C2×C12712yes—
224C2×#391C2×C12711yescharacteristic
225C2×#389C2×C12711yescharacteristic
226#1445C2×C12712yes—
227#1444C2×C12712yes—
228#1441C2×C12712yes—
229#1441C2×C12714yes—
230C4⋊#95C2×C12711yescharacteristic
231C127#1707811yescharacteristic, direct product factor, Sylow
232C23×Q8C22×C12711yescharacteristic
233C22×C4⋊C4C22×C12712yes—
234C22×C4⋊C4C22×C12711yescharacteristic
235C22×C4⋊C4C22×C12714yes—
236C22×C4⋊C4C22×C12711yescharacteristic
237C22×C4⋊C4C22×C12711yescharacteristic
238C22×C4⋊C4C22×C12712yes—
239C22×C42C22×C12711yescharacteristic
240#395—28no—
241#391—24no—
242#389—24no—
243C2×#95C22×C12712yes—
244C2×#95C22×C12714yes—
245C2×#95C22×C12712yes—
246C2×#95C22×C12712yes—
247C2×#95C22×C12712yes—
248C2×#95C22×C12712yes—
249C2×#95C22×C12714yes—
250C2×#95C22×C12714yes—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—42no—
254C22×Q8—42no—
255C22×Q8D4×C12714yes—
256C23×C4C23×C12712yes—
257C23×C4C23×C12712yes—
258C23×C4C23×C12712yes—
259C23×C4C23×C12712yes—
260C23×C4C23×C12714yes—
261C23×C4C23×C12711yescharacteristic
262C23×C4C23×C12711yescharacteristic
263C23×C4C23×C12711yescharacteristic
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4Q8×C12714yescomplemented
279C2×C4⋊C4Q8×C12718yescomplemented
280C2×C42—42no—
281C2×C42D4×C12714yes—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—28no—
286#95—24no—
287#95—24no—
288#95—44no—
289#95—44no—
290C24C24×C12711yescharacteristic
291C2×Q8—28no—
292C2×Q8—48no—
293C2×Q8—44no—
294C2×Q8—44no—
295C2×Q8—48no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4C2×D4×C12714yes—
330C22×C4C2×D4×C12712yes—
331C22×C4C2×D4×C12712yes—
332C22×C4C2×D4×C12712yes—
333C22×C4C2×D4×C12712yes—
334C22×C4C2×Q8×C12714yes—
335C22×C4C2×Q8×C12718yes—
336C22×C4C2×Q8×C12712yes—
337C22×C4C2×Q8×C12714yes—
338C22×C4#41×C12712yes—
339C22×C4#41×C12712yes—
340C4⋊C4—28no—
341C4⋊C4—48no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—416no—
345C4⋊C4—48no—
346C42—44no—
347C23C22×D4×C12711yescharacteristic
348C23C22×D4×C12711yescharacteristic
349C23C22×Q8×C12711yescharacteristic
350C23C22×Q8×C12712yes—
351C23C2×#41×C12711yescharacteristic
352C23C2×#41×C12711yescharacteristic
353C23H(5,𝔽2)×C12712yes—
354C23H(5,𝔽2)×C12712yes—
355C23H(5,𝔽2)×C12711yescharacteristic
356C232−1+4×C12712yes—
357C232−1+4×C12711yescharacteristic
358Q8—416no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—28no—
374C2×C4—28no—
375C2×C4—28no—
376C2×C4—24no—
377C2×C4—28no—
378C2×C4—24no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4#121×C12714yes—
401C2×C4C22⋊Q8×C12714yes—
402C2×C4C4⋊Q8×C12714yes—
403C22C2×#121×C12711yescharacteristic
404C22C2×C22⋊Q8×C12711yescharacteristic
405C22C2×C4⋊Q8×C12711yescharacteristic
406C22#534×C12711yescharacteristic
407C22#534×C12712yes—
408C22#535×C12711yescharacteristic
409C22#536×C12711yescharacteristic
410C22#537×C12711yescharacteristic
411C22#538×C12712yes—
412C22#539×C12712yes—
413C22#543×C12712yes—
414C22#544×C12712yes—
415C22#544×C12712yes—
416C22D4×Q8×C12712yes—
417C22D4×Q8×C12711yescharacteristic
418C22#550×C12711yescharacteristic
419C22#554×C12712yes—
420C22#554×C12711yescharacteristic
421C22#556×C12711yescharacteristic
422C22#557×C12712yes—
423C22Q82×C12712yes—
424C22#562×C12712yes—
425C22#563×C12712yes—
426C4—24no—
427C4—24no—
428C4—24no—
429C4—44no—
430C4—48no—
431C4—44no—
432C4—82no—
433C4—82no—
434C2C127×#253811yescharacteristic
435C2C127×#254611yescharacteristic
436C2C127×#256711yescharacteristic
437C2C127×#261711yescharacteristic
438C2C127×#262711yescharacteristic
439C2C127×#265911yescharacteristic
440C2C127×#273212yes—
441C2C127×#273412yes—
442C2C127×#280512yes—
443C2C127×#280612yes—
444C2C127×#282011yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C127C24
commutator, Frattini, cosocleC24C24×C127
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
55. C2×C4⋊C4×C127Q8semidirect
56. C2×C4⋊C4×C127Q8semidirect
211. #17078C127direct (N × H)
231. C127#17078direct (N × H)
278. C2×C4⋊C4Q8×C127semidirect
279. C2×C4⋊C4Q8×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#395×C1272
C2×#391×C1272
C2×#389×C1272
C127×#14452
C127×#14442
C127×#14412
C127×C4⋊#952
#17078127