Finite Group Databasebeta

← C127×#20992 #44584665

Subgroups — C127×#20992 #44584665

Order 32512.

1738 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#568×C127C211yesmaximal, characteristic, complemented
2C2×#537×C127C211yesmaximal, characteristic
3C2×#413×C127C212yesmaximal, complemented
4C127×#2879C211yesmaximal, characteristic, complemented
5C127×#1901C214yesmaximal, complemented
6C127×#1900C214yesmaximal, complemented
7C2×#343×C127C212yesmaximal, complemented
8C22×#41×C127C2211yescharacteristic
9C22×#41×C127C2211yescharacteristic
10#568×C127—28no—
11C2×#131×C127—24no—
12C22×M16×C127C2212yes—
13C22×M16×C127C2214yes—
14C22×M16×C127C2212yes—
15#537×C127—24no—
16#537×C127—24no—
17C2×C22⋊Q8×C127C412yes—
18C2×C22⋊Q8×C127C2212yes—
19C2×#121×C127C412yes—
20C2×#121×C127C414yes—
21C2×#121×C127C2212yes—
22C22×C4⋊C4×C127C2211yescharacteristic
23#413×C127—28no—
24C2×#101×C127C2214yes—
25C2×#100×C127C2214yes—
26#408×C127—24no—
27#407×C127C2214yes—
28#343×C127C2218yes—
29C2×#41×C127—24no—
30C2×#41×C127—22no—
31C2×#41×C127—22no—
32C2×#41×C127—24no—
33C2×#41×C127—24no—
34C2×#41×C127—24no—
35C2×#41×C127D414yes—
36C2×#41×C127D414yes—
37C22×Q8×C127C2×C411yescharacteristic
38C22×Q8×C127C2311yescharacteristic
39C22×D4×C127C2×C412yes—
40C22×D4×C127C2×C412yes—
41C22×D4×C127C2×C411yescharacteristic
42C22×D4×C127C2×C414yes—
43C22×D4×C127C2311yescharacteristic
44C23×C4×C127C2×C412yes—
45C23×C4×C127C2×C412yes—
46C23×C4×C127C2312yes—
47C23×C4×C127C2311yescharacteristic
48#131×C127—232no—
49C2×M16×C127—24no—
50C2×M16×C127—22no—
51C2×M16×C127—24no—
52C2×M16×C127—216no—
53C2×M16×C127—22no—
54C2×M16×C127—24no—
55C2×M16×C127—24no—
56C2×M16×C127D418yescomplemented
57C2×M16×C127C2314yes—
58C2×M16×C127C2314yes—
59C22×C8×C127—24no—
60C22×C8×C127—22no—
61C22×C8×C127—22no—
62C22⋊Q8×C127—28no—
63C22⋊Q8×C127—24no—
64C22⋊Q8×C127—24no—
65#121×C127—28no—
66#121×C127—216no—
67#121×C127—24no—
68#121×C127—24no—
69C2×C4⋊C4×C127—22no—
70C2×C4⋊C4×C127—22no—
71C2×C4⋊C4×C127C2×C412yes—
72C2×C4⋊C4×C127C2×C414yes—
73C2×C4⋊C4×C127C2312yes—
74C2×C22⋊C4×C127C2×C414yes—
75C2×C22⋊C4×C127C2×C414yes—
76#101×C127—28no—
77#100×C127—28no—
78C22⋊C8×C127—28no—
79C24×C127C22×C412yes—
80C24×C127C22×C412yes—
81C24×C127C22×C411yescharacteristic
82#41×C127—28no—
83#41×C127—28no—
84#41×C127—28no—
85#41×C127—28no—
86#41×C127—44no—
87#41×C127—44no—
88#41×C127—44no—
89#41×C127—44no—
90C2×Q8×C127—24no—
91C2×Q8×C127—22no—
92C2×Q8×C127—21no—
93C2×Q8×C127—21no—
94C2×Q8×C127—22no—
95C2×Q8×C127C22⋊C414yes—
96C2×D4×C127—28no—
97C2×D4×C127—24no—
98C2×D4×C127—28no—
99C2×D4×C127—28no—
100C2×D4×C127—24no—
101C2×D4×C127—22no—
102C2×D4×C127—21no—
103C2×D4×C127—21no—
104C2×D4×C127—22no—
105C2×D4×C127—84no—
106C2×D4×C127C22⋊C418yes—
107C2×D4×C127C22⋊C414yes—
108C22×C4×C127—24no—
109C22×C4×C127—24no—
110C22×C4×C127—24no—
111C22×C4×C127—24no—
112C22×C4×C127—28no—
113C22×C4×C127—82no—
114C22×C4×C127C22×C412yes—
115C22×C4×C127C22×C412yes—
116C22×C4×C127C22×C414yes—
117C22×C4×C127C22×C411yescharacteristic
118C22×C4×C127C2×D414yes—
119C22×C4×C127C2×D414yes—
120C22×C4×C127C2×D412yes—
121C22×C4×C127C2×D412yes—
122C22×C4×C127C2×D414yes—
123C22×C4×C127#4114yes—
124C22×C4×C127C2411yescharacteristic
125M16×C127—216no—
126M16×C127—28no—
127M16×C127—24no—
128M16×C127—24no—
129M16×C127—24no—
130M16×C127—44no—
131M16×C127C2×D418yes—
132C2×C8×C127—216no—
133C2×C8×C127—28no—
134C2×C8×C127—24no—
135C2×C8×C127—24no—
136C2×C8×C127—44no—
137C4⋊C4×C127—24no—
138C4⋊C4×C127—28no—
139C4⋊C4×C127—22no—
140C4⋊C4×C127—22no—
141C22⋊C4×C127—28no—
142C22⋊C4×C127—28no—
143C23×C127—24no—
144C23×C127—22no—
145C23×C127—24no—
146C23×C127—41no—
147C23×C127—41no—
148C23×C127—42no—
149C23×C127—42no—
150C23×C127—42no—
151C23×C127—42no—
152C23×C127C2×C22⋊C414yes—
153C23×C127C2×C22⋊C412yes—
154C23×C127C4×D414yes—
155C23×C127C23×C411yescharacteristic
156Q8×C127—28no—
157Q8×C127—41no—
158Q8×C127—42no—
159Q8×C127—41no—
160D4×C127—216no—
161D4×C127—28no—
162D4×C127—48no—
163D4×C127—41no—
164D4×C127—42no—
165D4×C127—41no—
166D4×C127—88no—
167C2×C4×C127—24no—
168C2×C4×C127—24no—
169C2×C4×C127—28no—
170C2×C4×C127—24no—
171C2×C4×C127—24no—
172C2×C4×C127—22no—
173C2×C4×C127—24no—
174C2×C4×C127—48no—
175C2×C4×C127—44no—
176C2×C4×C127—84no—
177C2×C4×C127C2×C22⋊C414yes—
178C2×C4×C127C2×C22⋊C412yes—
179C2×C4×C127C4×D414yes—
180C2×C4×C127#12014yes—
181C2×C4×C127#12114yes—
182C2×C4×C127#12114yes—
183C2×C4×C127#12314yes—
184C2×C4×C127C22×D412yes—
185C2×C4×C127C22×D411yescharacteristic
186C2×C4×C127C22×D411yescharacteristic
187C2×C4×C127C2×#4112yes—
188C8×C127—28no—
189C8×C127—44no—
190C22×C127—21no—
191C22×C127—22no—
192C22×C127—21no—
193C22×C127—22no—
194C22×C127—24no—
195C22×C127—24no—
196C22×C127—24no—
197C22×C127—44no—
198C22×C127—42no—
199C22×C127—42no—
200C22×C127—42no—
201C22×C127—84no—
202C22×C127—84no—
203C22×C127#38614yes—
204C22×C127C22×C22⋊C411yescharacteristic
205C22×C127C2×C4×D412yes—
206C4×C127—24no—
207C4×C127—44no—
208C4×C127—42no—
209C4×C127#38614yes—
210C4×C127C2×#12011yescharacteristic
211C4×C127C2×#12112yes—
212C4×C127C2×#12311yescharacteristic
213#20992C12711yesmaximal, characteristic, direct product factor, Sylow
214C2×C127—22no—
215C2×C127—24no—
216C2×C127—44no—
217C2×C127—82no—
218C2×C127C2×#38611yescharacteristic
219C2×C127#288811yescharacteristic
220C2×C127#288911yescharacteristic
221C2×#568C2×C12711yescharacteristic, complemented
222C2×#537C2×C12711yescharacteristic
223C2×#413C2×C12712yescomplemented
224#2879C2×C12711yescharacteristic, complemented
225#1901C2×C12714yescomplemented
226#1900C2×C12714yescomplemented
227C2×#343C2×C12712yescomplemented
228C127#2099211yescharacteristic, direct product factor, Sylow
229C22×#41C22×C12711yescharacteristic
230C22×#41C22×C12711yescharacteristic
231#568—28no—
232C2×#131—24no—
233C22×M16C22×C12712yes—
234C22×M16C22×C12714yes—
235C22×M16C22×C12712yes—
236#537—24no—
237#537—24no—
238C2×C22⋊Q8C4×C12712yes—
239C2×C22⋊Q8C22×C12712yes—
240C2×#121C4×C12712yes—
241C2×#121C4×C12714yes—
242C2×#121C22×C12712yes—
243C22×C4⋊C4C22×C12711yescharacteristic
244#413—28no—
245C2×#101C22×C12714yes—
246C2×#100C22×C12714yes—
247#408—24no—
248#407C22×C12714yes—
249#343C22×C12718yes—
250C2×#41—24no—
251C2×#41—22no—
252C2×#41—22no—
253C2×#41—24no—
254C2×#41—24no—
255C2×#41—24no—
256C2×#41D4×C12714yes—
257C2×#41D4×C12714yes—
258C22×Q8C2×C4×C12711yescharacteristic
259C22×Q8C23×C12711yescharacteristic
260C22×D4C2×C4×C12712yes—
261C22×D4C2×C4×C12712yes—
262C22×D4C2×C4×C12711yescharacteristic
263C22×D4C2×C4×C12714yes—
264C22×D4C23×C12711yescharacteristic
265C23×C4C2×C4×C12712yes—
266C23×C4C2×C4×C12712yes—
267C23×C4C23×C12712yes—
268C23×C4C23×C12711yescharacteristic
269#131—232no—
270C2×M16—24no—
271C2×M16—22no—
272C2×M16—24no—
273C2×M16—216no—
274C2×M16—22no—
275C2×M16—24no—
276C2×M16—24no—
277C2×M16D4×C12718yescomplemented
278C2×M16C23×C12714yes—
279C2×M16C23×C12714yes—
280C22×C8—24no—
281C22×C8—22no—
282C22×C8—22no—
283C22⋊Q8—28no—
284C22⋊Q8—24no—
285C22⋊Q8—24no—
286#121—28no—
287#121—216no—
288#121—24no—
289#121—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4C2×C4×C12712yes—
293C2×C4⋊C4C2×C4×C12714yes—
294C2×C4⋊C4C23×C12712yes—
295C2×C22⋊C4C2×C4×C12714yes—
296C2×C22⋊C4C2×C4×C12714yes—
297#101—28no—
298#100—28no—
299C22⋊C8—28no—
300C24C22×C4×C12712yes—
301C24C22×C4×C12712yes—
302C24C22×C4×C12711yescharacteristic
303#41—28no—
304#41—28no—
305#41—28no—
306#41—28no—
307#41—44no—
308#41—44no—
309#41—44no—
310#41—44no—
311C2×Q8—24no—
312C2×Q8—22no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—22no—
316C2×Q8C22⋊C4×C12714yes—
317C2×D4—28no—
318C2×D4—24no—
319C2×D4—28no—
320C2×D4—28no—
321C2×D4—24no—
322C2×D4—22no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—22no—
326C2×D4—84no—
327C2×D4C22⋊C4×C12718yes—
328C2×D4C22⋊C4×C12714yes—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—28no—
334C22×C4—82no—
335C22×C4C22×C4×C12712yes—
336C22×C4C22×C4×C12712yes—
337C22×C4C22×C4×C12714yes—
338C22×C4C22×C4×C12711yescharacteristic
339C22×C4C2×D4×C12714yes—
340C22×C4C2×D4×C12714yes—
341C22×C4C2×D4×C12712yes—
342C22×C4C2×D4×C12712yes—
343C22×C4C2×D4×C12714yes—
344C22×C4#41×C12714yes—
345C22×C4C24×C12711yescharacteristic
346M16—216no—
347M16—28no—
348M16—24no—
349M16—24no—
350M16—24no—
351M16—44no—
352M16C2×D4×C12718yes—
353C2×C8—216no—
354C2×C8—28no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—44no—
358C4⋊C4—24no—
359C4⋊C4—28no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C22⋊C4—28no—
363C22⋊C4—28no—
364C23—24no—
365C23—22no—
366C23—24no—
367C23—41no—
368C23—41no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23C2×C22⋊C4×C12714yes—
374C23C2×C22⋊C4×C12712yes—
375C23C4×D4×C12714yes—
376C23C23×C4×C12711yescharacteristic
377Q8—28no—
378Q8—41no—
379Q8—42no—
380Q8—41no—
381D4—216no—
382D4—28no—
383D4—48no—
384D4—41no—
385D4—42no—
386D4—41no—
387D4—88no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—28no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—48no—
396C2×C4—44no—
397C2×C4—84no—
398C2×C4C2×C22⋊C4×C12714yes—
399C2×C4C2×C22⋊C4×C12712yes—
400C2×C4C4×D4×C12714yes—
401C2×C4#120×C12714yes—
402C2×C4#121×C12714yes—
403C2×C4#121×C12714yes—
404C2×C4#123×C12714yes—
405C2×C4C22×D4×C12712yes—
406C2×C4C22×D4×C12711yescharacteristic
407C2×C4C22×D4×C12711yescharacteristic
408C2×C4C2×#41×C12712yes—
409C8—28no—
410C8—44no—
411C22—21no—
412C22—22no—
413C22—21no—
414C22—22no—
415C22—24no—
416C22—24no—
417C22—24no—
418C22—44no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—84no—
423C22—84no—
424C22#386×C12714yes—
425C22C22×C22⋊C4×C12711yescharacteristic
426C22C2×C4×D4×C12712yes—
427C4—24no—
428C4—44no—
429C4—42no—
430C4#386×C12714yes—
431C4C2×#120×C12711yescharacteristic
432C4C2×#121×C12712yes—
433C4C2×#123×C12711yescharacteristic
434C2—22no—
435C2—24no—
436C2—44no—
437C2—82no—
438C2C2×#386×C12711yescharacteristic
439C2C127×#288811yescharacteristic
440C2C127×#288911yescharacteristic
subgroupG/H
center, socleC22×C127C22×C22⋊C4
commutatorC23C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC2×#537×C127C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×#41×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#568×C127C2semidirect
3. C2×#413×C127C2semidirect
4. C127×#2879C2semidirect
5. C127×#1901C2semidirect
6. C127×#1900C2semidirect
7. C2×#343×C127C2semidirect
56. C2×M16×C127D4semidirect
213. #20992C127direct (N × H)
221. C2×#568C2×C127semidirect
223. C2×#413C2×C127semidirect
224. #2879C2×C127semidirect
225. #1901C2×C127semidirect
226. #1900C2×C127semidirect
227. C2×#343C2×C127semidirect
228. C127#20992direct (N × H)
277. C2×M16D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#568×C1272
C2×#537×C1272
C2×#413×C1272
C127×#28792
C127×#19012
C127×#19002
C2×#343×C1272
#20992127