Finite Group Databasebeta

← C2×C127×#1852 #44584513

Subgroups — C2×C127×#1852 #44584513

Order 32512.

1966 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊C4×C127C211yesmaximal, characteristic
2C22×#107×C127C212yesmaximal, complemented
3C22×C22⋊C8×C127C211yesmaximal, characteristic
4C2×#380×C127C211yesmaximal, characteristic
5C127×#1852C218yesmaximal, direct product factor
6C2×#340×C127C212yesmaximal, complemented
7C24×C4×C127—21no—
8C24×C4×C127C2211yescharacteristic
9C23×C8×C127C2212yes—
10C22×C4⋊C4×C127—22no—
11C22×C4⋊C4×C127—21no—
12C22×C4⋊C4×C127—28no—
13C22×C4⋊C4×C127C2214yes—
14C22×C4⋊C4×C127C2211yescharacteristic
15C22×C4⋊C4×C127C2211yescharacteristic
16C22×C4⋊C4×C127C2212yes—
17C22×C22⋊C4×C127—21no—
18C2×#107×C127—28no—
19C2×#107×C127C2218yescomplemented
20C2×C22⋊C8×C127C418yescomplemented
21C2×C22⋊C8×C127C2214yes—
22#380×C127—22no—
23#380×C127C2214yes—
24C2×#95×C127—21no—
25#340×C127C2218yescomplemented
26C25×C127D411yescharacteristic
27C23×C4×C127—24no—
28C23×C4×C127—24no—
29C23×C4×C127—28no—
30C23×C4×C127—24no—
31C23×C4×C127—22no—
32C23×C4×C127—21no—
33C23×C4×C127—21no—
34C23×C4×C127—44no—
35C23×C4×C127—44no—
36C23×C4×C127C2×C414yes—
37C23×C4×C127D412yes—
38C23×C4×C127D411yescharacteristic
39C23×C4×C127Q814yes—
40C23×C4×C127C2312yes—
41C23×C4×C127C2311yescharacteristic
42C22×C8×C127—24no—
43C22×C8×C127—24no—
44C22×C8×C127C2×C418yescomplemented
45C22×C8×C127C2314yes—
46C2×C4⋊C4×C127—28no—
47C2×C4⋊C4×C127—216no—
48C2×C4⋊C4×C127—24no—
49C2×C4⋊C4×C127—22no—
50C2×C4⋊C4×C127—24no—
51C2×C4⋊C4×C127—216no—
52C2×C4⋊C4×C127—24no—
53C2×C4⋊C4×C127—24no—
54C2×C4⋊C4×C127—21no—
55C2×C4⋊C4×C127—21no—
56C2×C4⋊C4×C127D418yes—
57C2×C4⋊C4×C127C2312yes—
58C2×C4⋊C4×C127C2312yes—
59C2×C4⋊C4×C127C2314yes—
60C2×C22⋊C4×C127—24no—
61C2×C22⋊C4×C127—44no—
62#107×C127—216no—
63C22⋊C8×C127C2×C4116yescomplemented
64#95×C127—24no—
65C24×C127—22no—
66C24×C127—22no—
67C24×C127—22no—
68C24×C127—22no—
69C24×C127C4⋊C414yes—
70C24×C127D814yes—
71C24×C127Q1614yes—
72C24×C127C2×D412yes—
73C24×C127C2×D411yescharacteristic
74C22×C4×C127—22no—
75C22×C4×C127—22no—
76C22×C4×C127—216no—
77C22×C4×C127—24no—
78C22×C4×C127—216no—
79C22×C4×C127—24no—
80C22×C4×C127—24no—
81C22×C4×C127—24no—
82C22×C4×C127—216no—
83C22×C4×C127—28no—
84C22×C4×C127—24no—
85C22×C4×C127—22no—
86C22×C4×C127—22no—
87C22×C4×C127—48no—
88C22×C4×C127—432no—
89C22×C4×C127—48no—
90C22×C4×C127—44no—
91C22×C4×C127—44no—
92C22×C4×C127—42no—
93C22×C4×C127—42no—
94C22×C4×C127—42no—
95C22×C4×C127—42no—
96C22×C4×C127—42no—
97C22×C4×C127—42no—
98C22×C4×C127C4⋊C414yes—
99C22×C4×C127C4⋊C418yes—
100C22×C4×C127C22×C414yes—
101C22×C4×C127C22×C412yes—
102C22×C4×C127C2×D412yes—
103C22×C4×C127C2×D411yescharacteristic
104C22×C4×C127C2×D414yes—
105C22×C4×C127C2×Q814yes—
106C22×C4×C127C2×Q812yes—
107C22×C4×C127#4114yes—
108C22×C4×C127C2411yescharacteristic
109C2×C8×C127—216no—
110C2×C8×C127—24no—
111C2×C8×C127—24no—
112C2×C8×C127C22×C418yescomplemented
113C4⋊C4×C127—216no—
114C4⋊C4×C127—28no—
115C4⋊C4×C127—24no—
116C4⋊C4×C127—24no—
117C4⋊C4×C127—24no—
118C4⋊C4×C127—42no—
119C4⋊C4×C127—42no—
120C4⋊C4×C127C2×D418yes—
121C22⋊C4×C127—48no—
122C23×C127—22no—
123C23×C127—22no—
124C23×C127—22no—
125C23×C127—22no—
126C23×C127—28no—
127C23×C127—28no—
128C23×C127—28no—
129C23×C127—28no—
130C23×C127—28no—
131C23×C127—28no—
132C23×C127#107116yescomplemented
133C23×C127C2×C4⋊C414yes—
134C23×C127C2×C4⋊C412yes—
135C23×C127#12012yes—
136C23×C127#12312yes—
137C23×C127C2×D814yes—
138C23×C127C2×D812yes—
139C23×C127C2×Q1614yes—
140C23×C127C2×Q1612yes—
141C23×C127AGL(1,ℤ/8)12yes—
142C23×C127#13712yes—
143C23×C127C22×D411yescharacteristic
144C2×C4×C127—22no—
145C2×C4×C127—22no—
146C2×C4×C127—28no—
147C2×C4×C127—28no—
148C2×C4×C127—28no—
149C2×C4×C127—216no—
150C2×C4×C127—28no—
151C2×C4×C127—432no—
152C2×C4×C127—44no—
153C2×C4×C127—44no—
154C2×C4×C127—416no—
155C2×C4×C127—42no—
156C2×C4×C127—42no—
157C2×C4×C127—42no—
158C2×C4×C127—42no—
159C2×C4×C127—42no—
160C2×C4×C127—42no—
161C2×C4×C127—84no—
162C2×C4×C127—84no—
163C2×C4×C127C2×C4⋊C414yes—
164C2×C4×C127C2×C4⋊C412yes—
165C2×C4×C127C4×D418yes—
166C2×C4×C127#12012yes—
167C2×C4×C127C22⋊Q818yes—
168C2×C4×C127#12312yes—
169C2×C4×C127C23×C411yescharacteristic
170C2×C4×C127C22×D412yes—
171C2×C4×C127C22×Q811yescharacteristic
172C2×C4×C127C2×#4112yes—
173C8×C127—28no—
174C22×C127—24no—
175C22×C127—24no—
176C22×C127—232no—
177C22×C127—24no—
178C22×C127—24no—
179C22×C127—24no—
180C22×C127—24no—
181C22×C127#38514yes—
182C22×C127C2×#10714yes—
183C22×C127C2×#10718yescomplemented
184C22×C127#42814yes—
185C22×C127#44714yes—
186C22×C127#45114yes—
187C22×C127#48014yes—
188C22×C127#48414yes—
189C22×C127C22×C4⋊C411yescharacteristic
190C22×C127C2×#12011yescharacteristic
191C22×C127C2×#12311yescharacteristic
192C22×C127C22×D811yescharacteristic
193C22×C127C22×Q1611yescharacteristic
194C22×C127C2×AGL(1,ℤ/8)11yescharacteristic
195C22×C127C2×#13711yescharacteristic
196C4×C127—24no—
197C4×C127—48no—
198C4×C127—82no—
199C4×C127—82no—
200C4×C127#38514yes—
201C4×C127C2×C4×D412yes—
202C4×C127C2×C22⋊Q812yes—
203C2×#1852C12711yesmaximal, characteristic, direct product factor, Sylow
204C2×C127—28no—
205C2×C127#185218yesdirect product factor
206C2×C127C2×#38511yescharacteristic
207C2×C127C22×#10711yescharacteristic
208C2×C127C2×#42811yescharacteristic
209C2×C127C2×#44711yescharacteristic
210C2×C127C2×#45111yescharacteristic
211C2×C127C2×#48011yescharacteristic
212C2×C127C2×#48411yescharacteristic
213C23×C4⋊C4C2×C12711yescharacteristic
214C22×#107C2×C12712yescomplemented
215C22×C22⋊C8C2×C12711yescharacteristic
216C2×#380C2×C12711yescharacteristic
217#1852C2×C12718yesdirect product factor
218C2×#340C2×C12712yescomplemented
219C127C2×#185211yescharacteristic, direct product factor, Sylow
220C24×C4—21no—
221C24×C4C22×C12711yescharacteristic
222C23×C8C22×C12712yes—
223C22×C4⋊C4—21no—
224C22×C4⋊C4—22no—
225C22×C4⋊C4—28no—
226C22×C4⋊C4C22×C12714yes—
227C22×C4⋊C4C22×C12711yescharacteristic
228C22×C4⋊C4C22×C12711yescharacteristic
229C22×C4⋊C4C22×C12712yes—
230C22×C22⋊C4—21no—
231C2×#107—28no—
232C2×#107C22×C12718yescomplemented
233C2×C22⋊C8C4×C12718yescomplemented
234C2×C22⋊C8C22×C12714yes—
235#380—22no—
236#380C22×C12714yes—
237C2×#95—21no—
238#340C22×C12718yescomplemented
239C25D4×C12711yescharacteristic
240C23×C4—24no—
241C23×C4—24no—
242C23×C4—28no—
243C23×C4—24no—
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—44no—
248C23×C4—44no—
249C23×C4C2×C4×C12714yes—
250C23×C4D4×C12711yescharacteristic
251C23×C4D4×C12712yes—
252C23×C4Q8×C12714yes—
253C23×C4C23×C12712yes—
254C23×C4C23×C12711yescharacteristic
255C22×C8—24no—
256C22×C8—24no—
257C22×C8C2×C4×C12718yescomplemented
258C22×C8C23×C12714yes—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—216no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—216no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4D4×C12718yes—
270C2×C4⋊C4C23×C12712yes—
271C2×C4⋊C4C23×C12712yes—
272C2×C4⋊C4C23×C12714yes—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—44no—
275#107—216no—
276C22⋊C8C2×C4×C127116yescomplemented
277#95—24no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24C4⋊C4×C12714yes—
283C24D8×C12714yes—
284C24Q16×C12714yes—
285C24C2×D4×C12712yes—
286C24C2×D4×C12711yescharacteristic
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—216no—
290C22×C4—24no—
291C22×C4—216no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—216no—
296C22×C4—28no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—48no—
301C22×C4—432no—
302C22×C4—48no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4C4⋊C4×C12714yes—
312C22×C4C4⋊C4×C12718yes—
313C22×C4C22×C4×C12714yes—
314C22×C4C22×C4×C12712yes—
315C22×C4C2×D4×C12712yes—
316C22×C4C2×D4×C12711yescharacteristic
317C22×C4C2×D4×C12714yes—
318C22×C4C2×Q8×C12714yes—
319C22×C4C2×Q8×C12712yes—
320C22×C4#41×C12714yes—
321C22×C4C24×C12711yescharacteristic
322C2×C8—216no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8C22×C4×C12718yescomplemented
326C4⋊C4—216no—
327C4⋊C4—28no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4C2×D4×C12718yes—
334C22⋊C4—48no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—28no—
340C23—28no—
341C23—28no—
342C23—28no—
343C23—28no—
344C23—28no—
345C23#107×C127116yescomplemented
346C23C2×C4⋊C4×C12714yes—
347C23C2×C4⋊C4×C12712yes—
348C23#120×C12712yes—
349C23#123×C12712yes—
350C23C2×D8×C12714yes—
351C23C2×D8×C12712yes—
352C23C2×Q16×C12714yes—
353C23C2×Q16×C12712yes—
354C23AGL(1,ℤ/8)×C12712yes—
355C23#137×C12712yes—
356C23C22×D4×C12711yescharacteristic
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—28no—
360C2×C4—28no—
361C2×C4—28no—
362C2×C4—216no—
363C2×C4—28no—
364C2×C4—432no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—416no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—84no—
375C2×C4—84no—
376C2×C4C2×C4⋊C4×C12714yes—
377C2×C4C2×C4⋊C4×C12712yes—
378C2×C4C4×D4×C12718yes—
379C2×C4#120×C12712yes—
380C2×C4C22⋊Q8×C12718yes—
381C2×C4#123×C12712yes—
382C2×C4C23×C4×C12711yescharacteristic
383C2×C4C22×D4×C12712yes—
384C2×C4C22×Q8×C12711yescharacteristic
385C2×C4C2×#41×C12712yes—
386C8—28no—
387C22—24no—
388C22—24no—
389C22—232no—
390C22—24no—
391C22—24no—
392C22—24no—
393C22—24no—
394C22#385×C12714yes—
395C22C2×#107×C12714yes—
396C22C2×#107×C12718yescomplemented
397C22#428×C12714yes—
398C22#447×C12714yes—
399C22#451×C12714yes—
400C22#480×C12714yes—
401C22#484×C12714yes—
402C22C22×C4⋊C4×C12711yescharacteristic
403C22C2×#120×C12711yescharacteristic
404C22C2×#123×C12711yescharacteristic
405C22C22×D8×C12711yescharacteristic
406C22C22×Q16×C12711yescharacteristic
407C22C2×AGL(1,ℤ/8)×C12711yescharacteristic
408C22C2×#137×C12711yescharacteristic
409C4—24no—
410C4—48no—
411C4—82no—
412C4—82no—
413C4#385×C12714yes—
414C4C2×C4×D4×C12712yes—
415C4C2×C22⋊Q8×C12712yes—
416C2—28no—
417C2C127×#185218yesdirect product factor
418C2C2×#385×C12711yescharacteristic
419C2C22×#107×C12711yescharacteristic
420C2C2×#428×C12711yescharacteristic
421C2C2×#447×C12711yescharacteristic
422C2C2×#451×C12711yescharacteristic
423C2C2×#480×C12711yescharacteristic
424C2C2×#484×C12711yescharacteristic
subgroupG/H
center, socleC24×C127C2×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC25×C127D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24×C127 ◁ C24×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×#107×C127C2semidirect
5. C127×#1852C2direct (N × H)
6. C2×#340×C127C2semidirect
19. C2×#107×C127C22semidirect
20. C2×C22⋊C8×C127C4semidirect
25. #340×C127C22semidirect
44. C22×C8×C127C2×C4semidirect
63. C22⋊C8×C127C2×C4semidirect
112. C2×C8×C127C22×C4semidirect
132. C23×C127#107semidirect
183. C22×C127C2×#107semidirect
203. C2×#1852C127direct (N × H)
205. C2×C127#1852direct (N × H)
214. C22×#107C2×C127semidirect
217. #1852C2×C127direct (N × H)
218. C2×#340C2×C127semidirect
219. C127C2×#1852direct (N × H)
232. C2×#107C22×C127semidirect
233. C2×C22⋊C8C4×C127semidirect
238. #340C22×C127semidirect
257. C22×C8C2×C4×C127semidirect
276. C22⋊C8C2×C4×C127semidirect
325. C2×C8C22×C4×C127semidirect
345. C23#107×C127semidirect
396. C22C2×#107×C127semidirect
417. C2C127×#1852direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4⋊C4×C1272
C22×#107×C1272
C22×C22⋊C8×C1272
C2×#380×C1272
C127×#18522
C2×#340×C1272
C2×#1852127