Finite Group Databasebeta

← C127×#21768 #44585441

Subgroups — C127×#21768 #44585441

Order 32512.

1466 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8×C127C211yesmaximal, characteristic
2C2×#407×C127C211yesmaximal, characteristic
3C127×#2280C211yesmaximal, characteristic
4C127×#1894C214yesmaximal, complemented
5C127×#1873C214yesmaximal, complemented
6C127×#1824C212yesmaximal, complemented
7C127×#1787C212yesmaximal, complemented
8C23×Q8×C127C2211yescharacteristic
9C24×C4×C127C2211yescharacteristic
10C22×M16×C127C2212yes—
11C2×C22⋊Q8×C127—24no—
12C2×C22⋊Q8×C127—21no—
13C2×C22⋊Q8×C127—21no—
14C2×C22⋊Q8×C127C418yescomplemented
15C2×C22⋊Q8×C127C2212yes—
16C2×C22⋊Q8×C127C2212yes—
17C2×#117×C127C2212yes—
18C22×C4⋊C4×C127—21no—
19C22×C4⋊C4×C127C2211yescharacteristic
20C22×C22⋊C4×C127—21no—
21C2×#103×C127C2218yescomplemented
22#407×C127—24no—
23C2×C22⋊C8×C127C2212yes—
24C2×C22⋊C8×C127C2212yes—
25#381×C127—21no—
26#380×C127C2212yes—
27#380×C127C2212yes—
28#379×C127—21no—
29C4×C22⋊C4×C127—21no—
30C4×C22⋊C4×C127—21no—
31#340×C127C2218yescomplemented
32C25×C127D411yescharacteristic
33C22×Q8×C127—22no—
34C22×Q8×C127—21no—
35C22×Q8×C127—44no—
36C22×Q8×C127C2×C414yes—
37C22×Q8×C127C2312yes—
38C23×C4×C127—21no—
39C23×C4×C127—21no—
40C23×C4×C127—44no—
41C23×C4×C127C2×C414yes—
42C23×C4×C127D411yescharacteristic
43C23×C4×C127D411yescharacteristic
44C23×C4×C127D411yescharacteristic
45C23×C4×C127D414yes—
46C23×C4×C127C2311yescharacteristic
47C23×C4×C127C2311yescharacteristic
48C23×C4×C127C2311yescharacteristic
49C2×M16×C127—24no—
50C2×M16×C127—44no—
51C22×C8×C127C2314yes—
52C22⋊Q8×C127—28no—
53C22⋊Q8×C127—28no—
54C22⋊Q8×C127—24no—
55C22⋊Q8×C127—24no—
56C22⋊Q8×C127C2×C418yes—
57C22⋊Q8×C127C2×C418yes—
58#117×C127—44no—
59C2×C4⋊C4×C127—28no—
60C2×C4⋊C4×C127—22no—
61C2×C4⋊C4×C127—22no—
62C2×C4⋊C4×C127—21no—
63C2×C4⋊C4×C127—22no—
64C2×C4⋊C4×C127C2×C414yes—
65C2×C4⋊C4×C127C2314yes—
66C2×C4⋊C4×C127C2312yes—
67C2×C22⋊C4×C127—28no—
68C2×C22⋊C4×C127—22no—
69C2×C22⋊C4×C127—22no—
70C2×C22⋊C4×C127—22no—
71C2×C22⋊C4×C127—22no—
72C2×C22⋊C4×C127—22no—
73C2×C42×C127—22no—
74#103×C127—48no—
75C22⋊C8×C127—24no—
76C22⋊C8×C127—24no—
77#95×C127—22no—
78#95×C127—22no—
79C24×C127—24no—
80C24×C127—42no—
81C24×C127—42no—
82C24×C127C22⋊C414yes—
83C24×C127C2×D411yescharacteristic
84C24×C127C2×D411yescharacteristic
85C24×C127C2×D411yescharacteristic
86C2×Q8×C127—24no—
87C2×Q8×C127—22no—
88C2×Q8×C127—28no—
89C2×Q8×C127—48no—
90C2×Q8×C127—44no—
91C2×Q8×C127—48no—
92C2×Q8×C127C22×C414yes—
93C22×C4×C127—24no—
94C22×C4×C127—24no—
95C22×C4×C127—24no—
96C22×C4×C127—24no—
97C22×C4×C127—24no—
98C22×C4×C127—22no—
99C22×C4×C127—22no—
100C22×C4×C127—24no—
101C22×C4×C127—48no—
102C22×C4×C127—48no—
103C22×C4×C127—42no—
104C22×C4×C127—42no—
105C22×C4×C127—42no—
106C22×C4×C127—44no—
107C22×C4×C127—44no—
108C22×C4×C127C22⋊C414yes—
109C22×C4×C127C22⋊C414yes—
110C22×C4×C127C22⋊C414yes—
111C22×C4×C127C22×C412yes—
112C22×C4×C127C22×C412yes—
113C22×C4×C127C22×C412yes—
114C22×C4×C127C2×D411yescharacteristic
115C22×C4×C127C2×D411yescharacteristic
116C22×C4×C127C2×D411yescharacteristic
117C22×C4×C127C2×D412yes—
118C22×C4×C127C2×D412yes—
119C22×C4×C127C2×D412yes—
120C22×C4×C127C2411yescharacteristic
121M16×C127—48no—
122C2×C8×C127—24no—
123C2×C8×C127—28no—
124C4⋊C4×C127—28no—
125C4⋊C4×C127—28no—
126C4⋊C4×C127—24no—
127C4⋊C4×C127—22no—
128C4⋊C4×C127—28no—
129C4⋊C4×C127C22×C414yes—
130C22⋊C4×C127—28no—
131C22⋊C4×C127—28no—
132C22⋊C4×C127—44no—
133C22⋊C4×C127—44no—
134C22⋊C4×C127—44no—
135C42×C127—44no—
136C23×C127—22no—
137C23×C127—22no—
138C23×C127—22no—
139C23×C127—22no—
140C23×C127—22no—
141C23×C127—22no—
142C23×C127—22no—
143C23×C127—22no—
144C23×C127—28no—
145C23×C127—44no—
146C23×C127—44no—
147C23×C127—416no—
148C23×C127C2×C22⋊C412yes—
149C23×C127C2×C22⋊C412yes—
150C23×C127C2×C22⋊C412yes—
151C23×C127#13714yes—
152C23×C127C22×D411yescharacteristic
153Q8×C127—28no—
154Q8×C127—48no—
155Q8×C127—44no—
156C2×C4×C127—22no—
157C2×C4×C127—22no—
158C2×C4×C127—22no—
159C2×C4×C127—24no—
160C2×C4×C127—24no—
161C2×C4×C127—24no—
162C2×C4×C127—24no—
163C2×C4×C127—24no—
164C2×C4×C127—24no—
165C2×C4×C127—24no—
166C2×C4×C127—24no—
167C2×C4×C127—44no—
168C2×C4×C127—44no—
169C2×C4×C127—44no—
170C2×C4×C127—48no—
171C2×C4×C127—48no—
172C2×C4×C127—44no—
173C2×C4×C127—44no—
174C2×C4×C127—44no—
175C2×C4×C127—44no—
176C2×C4×C127—44no—
177C2×C4×C127C2×C22⋊C412yes—
178C2×C4×C127C2×C22⋊C412yes—
179C2×C4×C127C2×C22⋊C412yes—
180C2×C4×C127C23×C411yescharacteristic
181C2×C4×C127C22×D411yescharacteristic
182C2×C4×C127H(5,𝔽2)14yes—
183C8×C127—44no—
184C22×C127—22no—
185C22×C127—22no—
186C22×C127—28no—
187C22×C127—22no—
188C22×C127—22no—
189C22×C127—22no—
190C22×C127—22no—
191C22×C127—22no—
192C22×C127—22no—
193C22×C127—416no—
194C22×C127—44no—
195C22×C127—44no—
196C22×C127#41914yes—
197C22×C127C22×C22⋊C411yescharacteristic
198C22×C127#53412yes—
199C22×C127C2×#13712yes—
200C22×C127#57512yes—
201C4×C127—22no—
202C4×C127—22no—
203C4×C127—22no—
204C4×C127—44no—
205C4×C127—44no—
206C4×C127—84no—
207C4×C127#51812yes—
208C4×C127#53412yes—
209#21768C12711yesmaximal, characteristic, direct product factor, Sylow
210C2×C127—24no—
211C2×C127—42no—
212C2×C127—42no—
213C2×C127#230811yescharacteristic
214C2×C127C2×#41911yescharacteristic
215C2×C127#289611yescharacteristic
216C2×C127#319614yes—
217C22×C22⋊Q8C2×C12711yescharacteristic
218C2×#407C2×C12711yescharacteristic
219#2280C2×C12711yescharacteristic
220#1894C2×C12714yescomplemented
221#1873C2×C12714yescomplemented
222#1824C2×C12712yescomplemented
223#1787C2×C12712yescomplemented
224C127#2176811yescharacteristic, direct product factor, Sylow
225C23×Q8C22×C12711yescharacteristic
226C24×C4C22×C12711yescharacteristic
227C22×M16C22×C12712yes—
228C2×C22⋊Q8—24no—
229C2×C22⋊Q8—21no—
230C2×C22⋊Q8—21no—
231C2×C22⋊Q8C4×C12718yescomplemented
232C2×C22⋊Q8C22×C12712yes—
233C2×C22⋊Q8C22×C12712yes—
234C2×#117C22×C12712yes—
235C22×C4⋊C4—21no—
236C22×C4⋊C4C22×C12711yescharacteristic
237C22×C22⋊C4—21no—
238C2×#103C22×C12718yescomplemented
239#407—24no—
240C2×C22⋊C8C22×C12712yes—
241C2×C22⋊C8C22×C12712yes—
242#381—21no—
243#380C22×C12712yes—
244#380C22×C12712yes—
245#379—21no—
246C4×C22⋊C4—21no—
247C4×C22⋊C4—21no—
248#340C22×C12718yescomplemented
249C25D4×C12711yescharacteristic
250C22×Q8—22no—
251C22×Q8—21no—
252C22×Q8—44no—
253C22×Q8C2×C4×C12714yes—
254C22×Q8C23×C12712yes—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—44no—
258C23×C4C2×C4×C12714yes—
259C23×C4D4×C12711yescharacteristic
260C23×C4D4×C12711yescharacteristic
261C23×C4D4×C12711yescharacteristic
262C23×C4D4×C12714yes—
263C23×C4C23×C12711yescharacteristic
264C23×C4C23×C12711yescharacteristic
265C23×C4C23×C12711yescharacteristic
266C2×M16—24no—
267C2×M16—44no—
268C22×C8C23×C12714yes—
269C22⋊Q8—28no—
270C22⋊Q8—28no—
271C22⋊Q8—24no—
272C22⋊Q8—24no—
273C22⋊Q8C2×C4×C12718yes—
274C22⋊Q8C2×C4×C12718yes—
275#117—44no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4C2×C4×C12714yes—
282C2×C4⋊C4C23×C12714yes—
283C2×C4⋊C4C23×C12712yes—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C42—22no—
291#103—48no—
292C22⋊C8—24no—
293C22⋊C8—24no—
294#95—22no—
295#95—22no—
296C24—24no—
297C24—42no—
298C24—42no—
299C24C22⋊C4×C12714yes—
300C24C2×D4×C12711yescharacteristic
301C24C2×D4×C12711yescharacteristic
302C24C2×D4×C12711yescharacteristic
303C2×Q8—24no—
304C2×Q8—22no—
305C2×Q8—28no—
306C2×Q8—48no—
307C2×Q8—44no—
308C2×Q8—48no—
309C2×Q8C22×C4×C12714yes—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—48no—
319C22×C4—48no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4C22⋊C4×C12714yes—
326C22×C4C22⋊C4×C12714yes—
327C22×C4C22⋊C4×C12714yes—
328C22×C4C22×C4×C12712yes—
329C22×C4C22×C4×C12712yes—
330C22×C4C22×C4×C12712yes—
331C22×C4C2×D4×C12711yescharacteristic
332C22×C4C2×D4×C12711yescharacteristic
333C22×C4C2×D4×C12711yescharacteristic
334C22×C4C2×D4×C12712yes—
335C22×C4C2×D4×C12712yes—
336C22×C4C2×D4×C12712yes—
337C22×C4C24×C12711yescharacteristic
338M16—48no—
339C2×C8—24no—
340C2×C8—28no—
341C4⋊C4—28no—
342C4⋊C4—28no—
343C4⋊C4—24no—
344C4⋊C4—22no—
345C4⋊C4—28no—
346C4⋊C4C22×C4×C12714yes—
347C22⋊C4—28no—
348C22⋊C4—28no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C22⋊C4—44no—
352C42—44no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—28no—
362C23—44no—
363C23—44no—
364C23—416no—
365C23C2×C22⋊C4×C12712yes—
366C23C2×C22⋊C4×C12712yes—
367C23C2×C22⋊C4×C12712yes—
368C23#137×C12714yes—
369C23C22×D4×C12711yescharacteristic
370Q8—28no—
371Q8—48no—
372Q8—44no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—48no—
388C2×C4—48no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4C2×C22⋊C4×C12712yes—
395C2×C4C2×C22⋊C4×C12712yes—
396C2×C4C2×C22⋊C4×C12712yes—
397C2×C4C23×C4×C12711yescharacteristic
398C2×C4C22×D4×C12711yescharacteristic
399C2×C4H(5,𝔽2)×C12714yes—
400C8—44no—
401C22—22no—
402C22—28no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—416no—
411C22—44no—
412C22—44no—
413C22#419×C12714yes—
414C22C22×C22⋊C4×C12711yescharacteristic
415C22#534×C12712yes—
416C22C2×#137×C12712yes—
417C22#575×C12712yes—
418C4—22no—
419C4—22no—
420C4—22no—
421C4—44no—
422C4—44no—
423C4—84no—
424C4#518×C12712yes—
425C4#534×C12712yes—
426C2—24no—
427C2—42no—
428C2—42no—
429C2C127×#230811yescharacteristic
430C2C2×#419×C12711yescharacteristic
431C2C127×#289611yescharacteristic
432C2C127×#319614yes—
subgroupG/H
center, socleC23×C127C22×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC25×C127D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C24×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1894C2semidirect
5. C127×#1873C2semidirect
6. C127×#1824C2semidirect
7. C127×#1787C2semidirect
14. C2×C22⋊Q8×C127C4semidirect
21. C2×#103×C127C22semidirect
31. #340×C127C22semidirect
209. #21768C127direct (N × H)
220. #1894C2×C127semidirect
221. #1873C2×C127semidirect
222. #1824C2×C127semidirect
223. #1787C2×C127semidirect
224. C127#21768direct (N × H)
231. C2×C22⋊Q8C4×C127semidirect
238. C2×#103C22×C127semidirect
248. #340C22×C127semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q8×C1272
C2×#407×C1272
C127×#22802
C127×#18942
C127×#18732
C127×#18242
C127×#17872
#21768127