Finite Group Databasebeta

← C2×C127×#1785 #44584263

Subgroups — C2×C127×#1785 #44584263

Order 32512.

3926 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#41×C127C211yesmaximal, characteristic
2C22×#101×C127C212yesmaximal, complemented
3C22×#100×C127C212yesmaximal, complemented
4C2×#407×C127C212yesmaximal, complemented
5C127×#1785C218yesmaximal, direct product factor
6C22×#41×C127—28no—
7C22×#41×C127—28no—
8C22×#41×C127—24no—
9C22×#41×C127—24no—
10C22×#41×C127C414yes—
11C22×#41×C127C2214yes—
12C23×Q8×C127C2211yescharacteristic
13C23×D4×C127C412yes—
14C23×D4×C127C2211yescharacteristic
15C24×C4×C127C412yes—
16C24×C4×C127C2211yescharacteristic
17C22×M16×C127C2214yes—
18C2×#101×C127—28no—
19C2×#101×C127C2218yescomplemented
20C2×#100×C127—28no—
21C2×#100×C127C2218yescomplemented
22#407×C127—24no—
23#407×C127C2218yescomplemented
24C2×C22⋊C8×C127—24no—
25C25×C127C2×C412yes—
26C25×C127C2×C411yescharacteristic
27C2×#41×C127—264no—
28C2×#41×C127—264no—
29C2×#41×C127—216no—
30C2×#41×C127—216no—
31C2×#41×C127—216no—
32C2×#41×C127—216no—
33C2×#41×C127—216no—
34C2×#41×C127—216no—
35C22×Q8×C127—24no—
36C22×Q8×C127—22no—
37C22×Q8×C127—22no—
38C22×Q8×C127C2×C412yes—
39C22×Q8×C127D418yes—
40C22×Q8×C127C2312yes—
41C22×D4×C127—28no—
42C22×D4×C127—24no—
43C22×D4×C127—24no—
44C22×D4×C127—28no—
45C22×D4×C127—24no—
46C22×D4×C127—22no—
47C22×D4×C127—22no—
48C22×D4×C127C2×C418yes—
49C22×D4×C127C2×C412yes—
50C22×D4×C127D418yes—
51C22×D4×C127C2312yes—
52C23×C4×C127—28no—
53C23×C4×C127—28no—
54C23×C4×C127—24no—
55C23×C4×C127—24no—
56C23×C4×C127—24no—
57C23×C4×C127—24no—
58C23×C4×C127C2×C418yes—
59C23×C4×C127C2×C411yescharacteristic
60C23×C4×C127C2×C412yes—
61C23×C4×C127C2×C412yes—
62C23×C4×C127D414yes—
63C23×C4×C127D414yes—
64C23×C4×C127C2312yes—
65C23×C4×C127C2311yescharacteristic
66C2×M16×C127—24no—
67C2×M16×C127—28no—
68C2×M16×C127—28no—
69C2×M16×C127C2318yes—
70C22×C8×C127—24no—
71#101×C127—216no—
72#100×C127—216no—
73C22⋊C8×C127—216no—
74C24×C127—22no—
75C24×C127—22no—
76C24×C127—24no—
77C24×C127—22no—
78C24×C127—22no—
79C24×C127—44no—
80C24×C127—48no—
81C24×C127C22⋊C412yes—
82C24×C127C22⋊C418yes—
83C24×C127C22⋊C412yes—
84C24×C127C22×C411yescharacteristic
85C24×C127C22×C414yes—
86C24×C127C22×C412yes—
87#41×C127—2128no—
88#41×C127—2128no—
89C2×Q8×C127—24no—
90C2×Q8×C127—24no—
91C2×Q8×C127—216no—
92C2×Q8×C127—216no—
93C2×Q8×C127—24no—
94C2×Q8×C127—24no—
95C2×Q8×C127C22⋊C418yes—
96C2×Q8×C127C2×D418yes—
97C2×D4×C127—232no—
98C2×D4×C127—232no—
99C2×D4×C127—216no—
100C2×D4×C127—216no—
101C2×D4×C127—216no—
102C2×D4×C127—24no—
103C2×D4×C127—24no—
104C2×D4×C127—216no—
105C2×D4×C127—216no—
106C2×D4×C127—24no—
107C2×D4×C127—24no—
108C2×D4×C127C22⋊C418yes—
109C2×D4×C127C2×D418yes—
110C22×C4×C127—232no—
111C22×C4×C127—232no—
112C22×C4×C127—22no—
113C22×C4×C127—22no—
114C22×C4×C127—24no—
115C22×C4×C127—22no—
116C22×C4×C127—216no—
117C22×C4×C127—22no—
118C22×C4×C127—216no—
119C22×C4×C127—216no—
120C22×C4×C127—24no—
121C22×C4×C127—24no—
122C22×C4×C127—216no—
123C22×C4×C127—216no—
124C22×C4×C127—24no—
125C22×C4×C127—24no—
126C22×C4×C127C22⋊C412yes—
127C22×C4×C127C22⋊C418yes—
128C22×C4×C127C22⋊C418yes—
129C22×C4×C127C22⋊C412yes—
130C22×C4×C127C22×C412yes—
131C22×C4×C127C22×C414yes—
132C22×C4×C127C22×C411yescharacteristic
133C22×C4×C127C2×D418yes—
134C22×C4×C127C2×D414yes—
135C22×C4×C127C2×D414yes—
136C22×C4×C127C2×D412yes—
137C22×C4×C127C2×D412yes—
138C22×C4×C127C2411yescharacteristic
139M16×C127—28no—
140M16×C127—28no—
141M16×C127—48no—
142C2×C8×C127—28no—
143C2×C8×C127—48no—
144C23×C127—24no—
145C23×C127—22no—
146C23×C127—24no—
147C23×C127—22no—
148C23×C127—24no—
149C23×C127—28no—
150C23×C127—24no—
151C23×C127—24no—
152C23×C127—28no—
153C23×C127—28no—
154C23×C127—44no—
155C23×C127—44no—
156C23×C127—416no—
157C23×C127—48no—
158C23×C127—48no—
159C23×C127—48no—
160C23×C127—48no—
161C23×C127—416no—
162C23×C127—48no—
163C23×C127C2×C22⋊C414yes—
164C23×C127C2×C22⋊C412yes—
165C23×C127C2×C22⋊C411yescharacteristic
166C23×C127C2×C22⋊C412yes—
167C23×C127C2×C22⋊C418yes—
168C23×C127C2×C22⋊C411yescharacteristic
169C23×C127C23×C411yescharacteristic
170Q8×C127—216no—
171Q8×C127—216no—
172D4×C127—264no—
173D4×C127—216no—
174D4×C127—216no—
175C2×C4×C127—264no—
176C2×C4×C127—22no—
177C2×C4×C127—22no—
178C2×C4×C127—24no—
179C2×C4×C127—24no—
180C2×C4×C127—24no—
181C2×C4×C127—24no—
182C2×C4×C127—28no—
183C2×C4×C127—28no—
184C2×C4×C127—216no—
185C2×C4×C127—216no—
186C2×C4×C127C2×C22⋊C414yes—
187C2×C4×C127C2×C22⋊C418yes—
188C2×C4×C127C2×C22⋊C412yes—
189C2×C4×C127C2×C22⋊C412yes—
190C2×C4×C127C2×C22⋊C412yes—
191C2×C4×C127#12018yes—
192C2×C4×C127#12018yes—
193C2×C4×C127C22×D412yes—
194C2×C4×C127C22×D412yes—
195C2×C4×C127C22×D412yes—
196C8×C127—48no—
197C22×C127—22no—
198C22×C127—22no—
199C22×C127—24no—
200C22×C127—24no—
201C22×C127—28no—
202C22×C127—24no—
203C22×C127—28no—
204C22×C127—48no—
205C22×C127—44no—
206C22×C127—44no—
207C22×C127—48no—
208C22×C127—48no—
209C22×C127—416no—
210C22×C127—48no—
211C22×C127—48no—
212C22×C127—416no—
213C22×C127#37912yes—
214C22×C127#37912yes—
215C22×C127#41314yes—
216C22×C127C22×C22⋊C411yescharacteristic
217C22×C127C22×C22⋊C411yescharacteristic
218C22×C127C22×C22⋊C411yescharacteristic
219C4×C127—24no—
220C4×C127—28no—
221C4×C127#37914yes—
222C4×C127C2×#12014yes—
223C2×#1785C12711yesmaximal, characteristic, direct product factor, Sylow
224C2×C127—21no—
225C2×C127—21no—
226C2×C127—22no—
227C2×C127—44no—
228C2×C127—48no—
229C2×C127#178514yesdirect product factor
230C2×C127C2×#37911yescharacteristic
231C2×C127C2×#41312yes—
232C23×#41C2×C12711yescharacteristic
233C22×#101C2×C12712yescomplemented
234C22×#100C2×C12712yescomplemented
235C2×#407C2×C12712yescomplemented
236#1785C2×C12718yesdirect product factor
237C127C2×#178511yescharacteristic, direct product factor, Sylow
238C22×#41—28no—
239C22×#41—28no—
240C22×#41—24no—
241C22×#41—24no—
242C22×#41C4×C12714yes—
243C22×#41C22×C12714yes—
244C23×Q8C22×C12711yescharacteristic
245C23×D4C4×C12712yes—
246C23×D4C22×C12711yescharacteristic
247C24×C4C4×C12712yes—
248C24×C4C22×C12711yescharacteristic
249C22×M16C22×C12714yes—
250C2×#101—28no—
251C2×#101C22×C12718yescomplemented
252C2×#100—28no—
253C2×#100C22×C12718yescomplemented
254#407—24no—
255#407C22×C12718yescomplemented
256C2×C22⋊C8—24no—
257C25C2×C4×C12712yes—
258C25C2×C4×C12711yescharacteristic
259C2×#41—264no—
260C2×#41—264no—
261C2×#41—216no—
262C2×#41—216no—
263C2×#41—216no—
264C2×#41—216no—
265C2×#41—216no—
266C2×#41—216no—
267C22×Q8—22no—
268C22×Q8—24no—
269C22×Q8—22no—
270C22×Q8C2×C4×C12712yes—
271C22×Q8D4×C12718yes—
272C22×Q8C23×C12712yes—
273C22×D4—24no—
274C22×D4—28no—
275C22×D4—24no—
276C22×D4—28no—
277C22×D4—22no—
278C22×D4—24no—
279C22×D4—22no—
280C22×D4C2×C4×C12718yes—
281C22×D4C2×C4×C12712yes—
282C22×D4D4×C12718yes—
283C22×D4C23×C12712yes—
284C23×C4—28no—
285C23×C4—28no—
286C23×C4—24no—
287C23×C4—24no—
288C23×C4—24no—
289C23×C4—24no—
290C23×C4C2×C4×C12718yes—
291C23×C4C2×C4×C12711yescharacteristic
292C23×C4C2×C4×C12712yes—
293C23×C4C2×C4×C12712yes—
294C23×C4D4×C12714yes—
295C23×C4D4×C12714yes—
296C23×C4C23×C12712yes—
297C23×C4C23×C12711yescharacteristic
298C2×M16—24no—
299C2×M16—28no—
300C2×M16—28no—
301C2×M16C23×C12718yes—
302C22×C8—24no—
303#101—216no—
304#100—216no—
305C22⋊C8—216no—
306C24—22no—
307C24—22no—
308C24—24no—
309C24—22no—
310C24—22no—
311C24—44no—
312C24—48no—
313C24C22⋊C4×C12712yes—
314C24C22⋊C4×C12718yes—
315C24C22⋊C4×C12712yes—
316C24C22×C4×C12711yescharacteristic
317C24C22×C4×C12714yes—
318C24C22×C4×C12712yes—
319#41—2128no—
320#41—2128no—
321C2×Q8—24no—
322C2×Q8—24no—
323C2×Q8—216no—
324C2×Q8—216no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8C22⋊C4×C12718yes—
328C2×Q8C2×D4×C12718yes—
329C2×D4—232no—
330C2×D4—232no—
331C2×D4—216no—
332C2×D4—216no—
333C2×D4—216no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—216no—
337C2×D4—216no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4C22⋊C4×C12718yes—
341C2×D4C2×D4×C12718yes—
342C22×C4—232no—
343C22×C4—232no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—216no—
350C22×C4—216no—
351C22×C4—216no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—216no—
355C22×C4—216no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4C22⋊C4×C12712yes—
359C22×C4C22⋊C4×C12712yes—
360C22×C4C22⋊C4×C12718yes—
361C22×C4C22⋊C4×C12718yes—
362C22×C4C22×C4×C12712yes—
363C22×C4C22×C4×C12714yes—
364C22×C4C22×C4×C12711yescharacteristic
365C22×C4C2×D4×C12718yes—
366C22×C4C2×D4×C12712yes—
367C22×C4C2×D4×C12714yes—
368C22×C4C2×D4×C12714yes—
369C22×C4C2×D4×C12712yes—
370C22×C4C24×C12711yescharacteristic
371M16—28no—
372M16—28no—
373M16—48no—
374C2×C8—28no—
375C2×C8—48no—
376C23—24no—
377C23—22no—
378C23—24no—
379C23—22no—
380C23—24no—
381C23—28no—
382C23—24no—
383C23—24no—
384C23—28no—
385C23—28no—
386C23—44no—
387C23—44no—
388C23—416no—
389C23—48no—
390C23—48no—
391C23—48no—
392C23—48no—
393C23—416no—
394C23—48no—
395C23C2×C22⋊C4×C12714yes—
396C23C2×C22⋊C4×C12712yes—
397C23C2×C22⋊C4×C12711yescharacteristic
398C23C2×C22⋊C4×C12712yes—
399C23C2×C22⋊C4×C12718yes—
400C23C2×C22⋊C4×C12711yescharacteristic
401C23C23×C4×C12711yescharacteristic
402Q8—216no—
403Q8—216no—
404D4—264no—
405D4—216no—
406D4—216no—
407C2×C4—264no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—28no—
415C2×C4—28no—
416C2×C4—216no—
417C2×C4—216no—
418C2×C4C2×C22⋊C4×C12714yes—
419C2×C4C2×C22⋊C4×C12718yes—
420C2×C4C2×C22⋊C4×C12712yes—
421C2×C4C2×C22⋊C4×C12712yes—
422C2×C4C2×C22⋊C4×C12712yes—
423C2×C4#120×C12718yes—
424C2×C4#120×C12718yes—
425C2×C4C22×D4×C12712yes—
426C2×C4C22×D4×C12712yes—
427C2×C4C22×D4×C12712yes—
428C8—48no—
429C22—22no—
430C22—22no—
431C22—24no—
432C22—24no—
433C22—28no—
434C22—24no—
435C22—28no—
436C22—48no—
437C22—44no—
438C22—44no—
439C22—48no—
440C22—48no—
441C22—416no—
442C22—48no—
443C22—48no—
444C22—416no—
445C22#379×C12712yes—
446C22#379×C12712yes—
447C22#413×C12714yes—
448C22C22×C22⋊C4×C12711yescharacteristic
449C22C22×C22⋊C4×C12711yescharacteristic
450C22C22×C22⋊C4×C12711yescharacteristic
451C4—24no—
452C4—28no—
453C4#379×C12714yes—
454C4C2×#120×C12714yes—
455C2—21no—
456C2—21no—
457C2—22no—
458C2—44no—
459C2—48no—
460C2C127×#178514yesdirect product factor
461C2C2×#379×C12711yescharacteristic
462C2C2×#413×C12712yes—
subgroupG/H
center, socleC23×C127C2×C22⋊C4
commutatorC23C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC23×#41×C127C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C24×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×#101×C127C2semidirect
3. C22×#100×C127C2semidirect
4. C2×#407×C127C2semidirect
5. C127×#1785C2direct (N × H)
19. C2×#101×C127C22semidirect
21. C2×#100×C127C22semidirect
23. #407×C127C22semidirect
223. C2×#1785C127direct (N × H)
229. C2×C127#1785direct (N × H)
233. C22×#101C2×C127semidirect
234. C22×#100C2×C127semidirect
235. C2×#407C2×C127semidirect
236. #1785C2×C127direct (N × H)
237. C127C2×#1785direct (N × H)
251. C2×#101C22×C127semidirect
253. C2×#100C22×C127semidirect
255. #407C22×C127semidirect
460. C2C127×#1785direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#41×C1272
C22×#101×C1272
C22×#100×C1272
C2×#407×C1272
C127×#17852
C2×#1785127