Finite Group Databasebeta

← C127×#8343 #44572016

Subgroups — C127×#8343 #44572016

Order 32512.

802 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#533×C127C211yesmaximal, characteristic
2C127×#1797C211yesmaximal, characteristic
3C127×#1797C211yesmaximal, characteristic
4C2×#333×C127C212yesmaximal, complemented
5C2×#332×C127C212yesmaximal, complemented
6#533×C127—24no—
7#533×C127—24no—
8C2×C4⋊Q8×C127C2211yescharacteristic
9C2×C4⋊Q8×C127C2211yescharacteristic
10C2×#125×C127—21no—
11C2×C22⋊Q8×C127—21no—
12C2×C22⋊Q8×C127C412yes—
13C2×C4×Q8×C127—21no—
14C2×C4×Q8×C127C412yes—
15C2×#117×C127—21no—
16C22×C42×C127C2211yescharacteristic
17C2×C4⋊C8×C127C2212yes—
18C2×C4⋊C8×C127C2212yes—
19C2×C22⋊C8×C127—21no—
20C2×C22⋊C8×C127—21no—
21#336×C127—21no—
22#336×C127—21no—
23#333×C127—24no—
24#332×C127—24no—
25C22×Q8×C127—21no—
26C22×Q8×C127C2×C412yes—
27C23×C4×C127C2×C411yescharacteristic
28C23×C4×C127D411yescharacteristic
29C23×C4×C127D411yescharacteristic
30C22×C8×C127—22no—
31C22×C8×C127—22no—
32C4⋊Q8×C127—22no—
33C4⋊Q8×C127—22no—
34C4⋊Q8×C127D414yes—
35C4⋊Q8×C127D414yes—
36#125×C127—28no—
37C22⋊Q8×C127—28no—
38C22⋊Q8×C127—24no—
39C22⋊Q8×C127—24no—
40C4×Q8×C127—28no—
41C4×Q8×C127—24no—
42C4×Q8×C127—24no—
43#117×C127—28no—
44C2×C4⋊C4×C127—22no—
45C2×C4⋊C4×C127—22no—
46C2×C4⋊C4×C127—21no—
47C2×C4⋊C4×C127—21no—
48C2×C4⋊C4×C127—21no—
49C2×C4⋊C4×C127C2×C412yes—
50C2×C22⋊C4×C127—22no—
51C2×C42×C127—22no—
52C2×C42×C127—22no—
53C2×C42×C127—22no—
54C2×C42×C127C2×C411yescharacteristic
55C2×C42×C127D411yescharacteristic
56C2×C42×C127D411yescharacteristic
57C2×C42×C127C2311yescharacteristic
58C4⋊C8×C127—22no—
59C4⋊C8×C127—22no—
60C4⋊C8×C127—22no—
61C4⋊C8×C127—22no—
62C22⋊C8×C127—42no—
63C22⋊C8×C127—42no—
64C24×C127C22⋊C411yescharacteristic
65C2×Q8×C127—22no—
66C2×Q8×C127—24no—
67C2×Q8×C127—42no—
68C2×Q8×C127—42no—
69C2×Q8×C127—42no—
70C2×Q8×C127—41no—
71C2×Q8×C127—41no—
72C2×Q8×C127C22⋊C414yes—
73C22×C4×C127—22no—
74C22×C4×C127—22no—
75C22×C4×C127—22no—
76C22×C4×C127—22no—
77C22×C4×C127—21no—
78C22×C4×C127—21no—
79C22×C4×C127—22no—
80C22×C4×C127—22no—
81C22×C4×C127—21no—
82C22×C4×C127—21no—
83C22×C4×C127—41no—
84C22×C4×C127—41no—
85C22×C4×C127C22⋊C411yescharacteristic
86C22×C4×C127C22⋊C411yescharacteristic
87C22×C4×C127C22⋊C411yescharacteristic
88C22×C4×C127SD1612yes—
89C22×C4×C127Q1612yes—
90C22×C4×C127C22×C411yescharacteristic
91C22×C4×C127C2×D411yescharacteristic
92C22×C4×C127C2×D411yescharacteristic
93C2×C8×C127—42no—
94C2×C8×C127—42no—
95C2×C8×C127—42no—
96C2×C8×C127—42no—
97C2×C8×C127—42no—
98C2×C8×C127—42no—
99C4⋊C4×C127—28no—
100C4⋊C4×C127—28no—
101C4⋊C4×C127—22no—
102C4⋊C4×C127—24no—
103C4⋊C4×C127—24no—
104C4⋊C4×C127—24no—
105C4⋊C4×C127C22⋊C414yes—
106C22⋊C4×C127—28no—
107C42×C127—28no—
108C42×C127—21no—
109C42×C127—21no—
110C42×C127—22no—
111C42×C127—22no—
112C42×C127C2×D412yes—
113C42×C127C2×D412yes—
114C23×C127—21no—
115C23×C127—21no—
116C23×C127—21no—
117C23×C127—21no—
118C23×C127—21no—
119C23×C127—21no—
120C23×C127#10312yes—
121C23×C127C2×C22⋊C411yescharacteristic
122Q8×C127—44no—
123Q8×C127—44no—
124Q8×C127—44no—
125Q8×C127—44no—
126C2×C4×C127—21no—
127C2×C4×C127—21no—
128C2×C4×C127—21no—
129C2×C4×C127—21no—
130C2×C4×C127—24no—
131C2×C4×C127—24no—
132C2×C4×C127—21no—
133C2×C4×C127—21no—
134C2×C4×C127—24no—
135C2×C4×C127—24no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127—42no—
141C2×C4×C127—42no—
142C2×C4×C127—42no—
143C2×C4×C127—42no—
144C2×C4×C127—42no—
145C2×C4×C127—42no—
146C2×C4×C127—42no—
147C2×C4×C127—42no—
148C2×C4×C127#10312yes—
149C2×C4×C127C2×C22⋊C411yescharacteristic
150C2×C4×C127C2×C22⋊C411yescharacteristic
151C2×C4×C127#12012yes—
152C2×C4×C127#12012yes—
153C2×C4×C127C2×SD1611yescharacteristic
154C2×C4×C127C2×Q1611yescharacteristic
155C2×C4×C127#13512yes—
156C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C127#13712yes—
158C2×C4×C127#13711yescharacteristic
159C8×C127—82no—
160C8×C127—82no—
161C22×C127—22no—
162C22×C127—22no—
163C22×C127—22no—
164C22×C127—41no—
165C22×C127—41no—
166C22×C127—41no—
167C22×C127—41no—
168C22×C127#37911yescharacteristic
169C22×C127#41311yescharacteristic
170C22×C127#41311yescharacteristic
171C22×C127C2×#10311yescharacteristic
172C22×C127#41611yescharacteristic
173C22×C127#41711yescharacteristic
174C22×C127#41911yescharacteristic
175C4×C127—21no—
176C4×C127—21no—
177C4×C127—22no—
178C4×C127—41no—
179C4×C127—44no—
180C4×C127—41no—
181C4×C127—44no—
182C4×C127#44812yes—
183C4×C127#44912yes—
184C4×C127#45112yes—
185C4×C127#45212yes—
186#8343C12711yesmaximal, characteristic, direct product factor, Sylow
187C2×C127—42no—
188C2×C127#154211yescharacteristic
189C2×C127#155011yescharacteristic
190C2×C127#155411yescharacteristic
191C2×C127#155911yescharacteristic
192C2×C127#178511yescharacteristic
193C2×C127#178711yescharacteristic
194C2×C127#178811yescharacteristic
195C2×#533C2×C12711yescharacteristic
196#1797C2×C12711yescharacteristic
197#1797C2×C12711yescharacteristic
198C2×#333C2×C12712yescomplemented
199C2×#332C2×C12712yescomplemented
200C127#834311yescharacteristic, direct product factor, Sylow
201#533—24no—
202#533—24no—
203C2×C4⋊Q8C22×C12711yescharacteristic
204C2×C4⋊Q8C22×C12711yescharacteristic
205C2×#125—21no—
206C2×C22⋊Q8—21no—
207C2×C22⋊Q8C4×C12712yes—
208C2×C4×Q8—21no—
209C2×C4×Q8C4×C12712yes—
210C2×#117—21no—
211C22×C42C22×C12711yescharacteristic
212C2×C4⋊C8C22×C12712yes—
213C2×C4⋊C8C22×C12712yes—
214C2×C22⋊C8—21no—
215C2×C22⋊C8—21no—
216#336—21no—
217#336—21no—
218#333—24no—
219#332—24no—
220C22×Q8—21no—
221C22×Q8C2×C4×C12712yes—
222C23×C4C2×C4×C12711yescharacteristic
223C23×C4D4×C12711yescharacteristic
224C23×C4D4×C12711yescharacteristic
225C22×C8—22no—
226C22×C8—22no—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C4⋊Q8D4×C12714yes—
230C4⋊Q8D4×C12714yes—
231#125—28no—
232C22⋊Q8—28no—
233C22⋊Q8—24no—
234C22⋊Q8—24no—
235C4×Q8—28no—
236C4×Q8—24no—
237C4×Q8—24no—
238#117—28no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4C2×C4×C12712yes—
245C2×C22⋊C4—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42C2×C4×C12711yescharacteristic
250C2×C42D4×C12711yescharacteristic
251C2×C42D4×C12711yescharacteristic
252C2×C42C23×C12711yescharacteristic
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C22⋊C8—42no—
258C22⋊C8—42no—
259C24C22⋊C4×C12711yescharacteristic
260C2×Q8—22no—
261C2×Q8—24no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8C22⋊C4×C12714yes—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4C22⋊C4×C12711yescharacteristic
281C22×C4C22⋊C4×C12711yescharacteristic
282C22×C4C22⋊C4×C12711yescharacteristic
283C22×C4SD16×C12712yes—
284C22×C4Q16×C12712yes—
285C22×C4C22×C4×C12711yescharacteristic
286C22×C4C2×D4×C12711yescharacteristic
287C22×C4C2×D4×C12711yescharacteristic
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C4⋊C4—28no—
295C4⋊C4—28no—
296C4⋊C4—22no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4C22⋊C4×C12714yes—
301C22⋊C4—28no—
302C42—28no—
303C42—21no—
304C42—21no—
305C42—22no—
306C42—22no—
307C42C2×D4×C12712yes—
308C42C2×D4×C12712yes—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23#103×C12712yes—
316C23C2×C22⋊C4×C12711yescharacteristic
317Q8—44no—
318Q8—44no—
319Q8—44no—
320Q8—44no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4#103×C12712yes—
344C2×C4C2×C22⋊C4×C12711yescharacteristic
345C2×C4C2×C22⋊C4×C12711yescharacteristic
346C2×C4#120×C12712yes—
347C2×C4#120×C12712yes—
348C2×C4C2×SD16×C12711yescharacteristic
349C2×C4C2×Q16×C12711yescharacteristic
350C2×C4#135×C12712yes—
351C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
352C2×C4#137×C12712yes—
353C2×C4#137×C12711yescharacteristic
354C8—82no—
355C8—82no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22#379×C12711yescharacteristic
364C22#413×C12711yescharacteristic
365C22#413×C12711yescharacteristic
366C22C2×#103×C12711yescharacteristic
367C22#416×C12711yescharacteristic
368C22#417×C12711yescharacteristic
369C22#419×C12711yescharacteristic
370C4—21no—
371C4—21no—
372C4—22no—
373C4—41no—
374C4—44no—
375C4—41no—
376C4—44no—
377C4#448×C12712yes—
378C4#449×C12712yes—
379C4#451×C12712yes—
380C4#452×C12712yes—
381C2—42no—
382C2C127×#154211yescharacteristic
383C2C127×#155011yescharacteristic
384C2C127×#155411yescharacteristic
385C2C127×#155911yescharacteristic
386C2C127×#178511yescharacteristic
387C2C127×#178711yescharacteristic
388C2C127×#178811yescharacteristic
subgroupG/H
center, socleC23×C127C2×C22⋊C4
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC24×C127C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C127 ◁ C22×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C2×#333×C127C2semidirect
5. C2×#332×C127C2semidirect
186. #8343C127direct (N × H)
198. C2×#333C2×C127semidirect
199. C2×#332C2×C127semidirect
200. C127#8343direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#533×C1272
C127×#17972
C2×#333×C1272
C2×#332×C1272
#8343127