Finite Group Databasebeta

← C127×#20857 #44584530

Subgroups — C127×#20857 #44584530

Order 32512.

1330 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#519×C127C211yesmaximal, characteristic
2C127×#2911C212yesmaximal
3C22×#101×C127C211yesmaximal, characteristic
4C127×#2310C211yesmaximal, characteristic
5C127×#1857C218yesmaximal, complemented
6C127×#1733C212yesmaximal
7C23×Q8×C127C2211yescharacteristic
8C22×M16×C127C2212yes—
9#519×C127—22no—
10#519×C127—24no—
11#519×C127C2214yes—
12C2×C4×Q8×C127—24no—
13C2×#117×C127—22no—
14C2×#117×C127C2212yes—
15C2×#117×C127C2211yescharacteristic
16C2×#117×C127C2211yescharacteristic
17#428×C127—24no—
18#428×C127C2218yes—
19#427×C127—24no—
20C2×#101×C127C418yescomplemented
21C2×#101×C127C2214yes—
22#391×C127—22no—
23#391×C127C2214yes—
24#381×C127—22no—
25#339×C127C2218yes—
26C22×Q8×C127—22no—
27C22×Q8×C127—22no—
28C22×Q8×C127—24no—
29C22×Q8×C127C2×C414yes—
30C22×Q8×C127D412yes—
31C22×Q8×C127Q814yes—
32C22×Q8×C127C2312yes—
33C23×C4×C127D412yes—
34C23×C4×C127C2311yescharacteristic
35C2×M16×C127—22no—
36C2×M16×C127—24no—
37C2×M16×C127C2×C418yes—
38C2×M16×C127C2314yes—
39C22×C8×C127—22no—
40C4×Q8×C127—216no—
41C4×Q8×C127—216no—
42#117×C127—28no—
43#117×C127—28no—
44#117×C127—22no—
45#117×C127—22no—
46#117×C127—24no—
47#117×C127—22no—
48#117×C127—22no—
49#117×C127D414yes—
50#117×C127C2314yes—
51C2×C4⋊C4×C127—24no—
52C2×C4⋊C4×C127—21no—
53C2×C4⋊C4×C127D414yes—
54C2×C4⋊C4×C127C2312yes—
55C2×C22⋊C4×C127—22no—
56C2×C22⋊C4×C127—21no—
57C2×C22⋊C4×C127—21no—
58C2×C42×C127—24no—
59C2×C42×C127—22no—
60C2×C42×C127—21no—
61C2×C42×C127C2312yes—
62#107×C127—28no—
63#106×C127—28no—
64#101×C127C2×C4116yescomplemented
65#95×C127—28no—
66C24×C127C2×D411yescharacteristic
67C2×Q8×C127—28no—
68C2×Q8×C127—28no—
69C2×Q8×C127—24no—
70C2×Q8×C127—28no—
71C2×Q8×C127—28no—
72C2×Q8×C127—24no—
73C2×Q8×C127—24no—
74C2×Q8×C127—24no—
75C2×Q8×C127C4⋊C418yes—
76C2×Q8×C127C22×C414yes—
77C2×Q8×C127C2×Q814yes—
78C22×C4×C127—22no—
79C22×C4×C127—22no—
80C22×C4×C127—24no—
81C22×C4×C127—24no—
82C22×C4×C127—22no—
83C22×C4×C127—22no—
84C22×C4×C127C4⋊C418yes—
85C22×C4×C127C22×C412yes—
86C22×C4×C127C2×D414yes—
87C22×C4×C127C2×D411yescharacteristic
88C22×C4×C127C2×D412yes—
89C22×C4×C127C2×D412yes—
90C22×C4×C127C2×Q812yes—
91C22×C4×C127#4114yes—
92C22×C4×C127C2411yescharacteristic
93M16×C127—28no—
94M16×C127—24no—
95M16×C127C22×C418yes—
96C2×C8×C127—28no—
97C2×C8×C127—24no—
98C4⋊C4×C127—216no—
99C4⋊C4×C127—24no—
100C4⋊C4×C127—24no—
101C4⋊C4×C127—42no—
102C4⋊C4×C127C2×D418yes—
103C22⋊C4×C127—24no—
104C22⋊C4×C127—24no—
105C22⋊C4×C127—41no—
106C22⋊C4×C127—41no—
107C22⋊C4×C127—42no—
108C42×C127—216no—
109C42×C127—24no—
110C42×C127—24no—
111C42×C127—24no—
112C42×C127—42no—
113C23×C127—42no—
114C23×C127C2×C4⋊C412yes—
115C23×C127#12011yescharacteristic
116C23×C127#12011yescharacteristic
117C23×C127#12312yes—
118C23×C127C22×D411yescharacteristic
119Q8×C127—28no—
120Q8×C127—28no—
121Q8×C127—216no—
122C2×C4×C127—24no—
123C2×C4×C127—24no—
124C2×C4×C127—28no—
125C2×C4×C127—28no—
126C2×C4×C127—28no—
127C2×C4×C127—48no—
128C2×C4×C127—44no—
129C2×C4×C127—44no—
130C2×C4×C127—42no—
131C2×C4×C127—42no—
132C2×C4×C127C2×C4⋊C418yes—
133C2×C4×C127C2×C4⋊C412yes—
134C2×C4×C127C4×D414yes—
135C2×C4×C127C4×D414yes—
136C2×C4×C127#12012yes—
137C2×C4×C127C22⋊Q814yes—
138C2×C4×C127C22⋊Q814yes—
139C2×C4×C127#12312yes—
140C2×C4×C127C23×C411yescharacteristic
141C2×C4×C127C22×D412yes—
142C2×C4×C127C22×Q811yescharacteristic
143C2×C4×C127C2×#4112yes—
144C8×C127—28no—
145C22×C127—22no—
146C22×C127—22no—
147C22×C127—22no—
148C22×C127—42no—
149C22×C127—42no—
150C22×C127#38512yes—
151C22×C127#38512yes—
152C22×C127C22×C4⋊C411yescharacteristic
153C22×C127C2×#12011yescharacteristic
154C22×C127C2×#12311yescharacteristic
155C4×C127—28no—
156C4×C127—48no—
157C4×C127—82no—
158C4×C127—82no—
159C4×C127#38514yes—
160C4×C127C2×C4×D412yes—
161C4×C127C2×C22⋊Q812yes—
162#20857C12711yesmaximal, characteristic, direct product factor, Sylow
163C2×C127—21no—
164C2×C127—21no—
165C2×C127—22no—
166C2×C127—22no—
167C2×C127C2×#38511yescharacteristic
168C2×C127#302012yes—
169C2×#519C2×C12711yescharacteristic
170#2911C2×C12712yes—
171C22×#101C2×C12711yescharacteristic
172#2310C2×C12711yescharacteristic
173#1857C2×C12718yescomplemented
174#1733C2×C12712yes—
175C127#2085711yescharacteristic, direct product factor, Sylow
176C23×Q8C22×C12711yescharacteristic
177C22×M16C22×C12712yes—
178#519—22no—
179#519—24no—
180#519C22×C12714yes—
181C2×C4×Q8—24no—
182C2×#117—22no—
183C2×#117C22×C12712yes—
184C2×#117C22×C12711yescharacteristic
185C2×#117C22×C12711yescharacteristic
186#428—24no—
187#428C22×C12718yes—
188#427—24no—
189C2×#101C4×C12718yescomplemented
190C2×#101C22×C12714yes—
191#391—22no—
192#391C22×C12714yes—
193#381—22no—
194#339C22×C12718yes—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—24no—
198C22×Q8C2×C4×C12714yes—
199C22×Q8D4×C12712yes—
200C22×Q8Q8×C12714yes—
201C22×Q8C23×C12712yes—
202C23×C4D4×C12712yes—
203C23×C4C23×C12711yescharacteristic
204C2×M16—22no—
205C2×M16—24no—
206C2×M16C2×C4×C12718yes—
207C2×M16C23×C12714yes—
208C22×C8—22no—
209C4×Q8—216no—
210C4×Q8—216no—
211#117—28no—
212#117—28no—
213#117—22no—
214#117—22no—
215#117—24no—
216#117—22no—
217#117—22no—
218#117D4×C12714yes—
219#117C23×C12714yes—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4D4×C12714yes—
223C2×C4⋊C4C23×C12712yes—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—21no—
230C2×C42C23×C12712yes—
231#107—28no—
232#106—28no—
233#101C2×C4×C127116yescomplemented
234#95—28no—
235C24C2×D4×C12711yescharacteristic
236C2×Q8—28no—
237C2×Q8—28no—
238C2×Q8—24no—
239C2×Q8—28no—
240C2×Q8—28no—
241C2×Q8—24no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8C4⋊C4×C12718yes—
245C2×Q8C22×C4×C12714yes—
246C2×Q8C2×Q8×C12714yes—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4C4⋊C4×C12718yes—
254C22×C4C22×C4×C12712yes—
255C22×C4C2×D4×C12714yes—
256C22×C4C2×D4×C12711yescharacteristic
257C22×C4C2×D4×C12712yes—
258C22×C4C2×D4×C12712yes—
259C22×C4C2×Q8×C12712yes—
260C22×C4#41×C12714yes—
261C22×C4C24×C12711yescharacteristic
262M16—28no—
263M16—24no—
264M16C22×C4×C12718yes—
265C2×C8—28no—
266C2×C8—24no—
267C4⋊C4—216no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—42no—
271C4⋊C4C2×D4×C12718yes—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—42no—
277C42—216no—
278C42—24no—
279C42—24no—
280C42—24no—
281C42—42no—
282C23—42no—
283C23C2×C4⋊C4×C12712yes—
284C23#120×C12711yescharacteristic
285C23#120×C12711yescharacteristic
286C23#123×C12712yes—
287C23C22×D4×C12711yescharacteristic
288Q8—28no—
289Q8—28no—
290Q8—216no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—28no—
294C2×C4—28no—
295C2×C4—28no—
296C2×C4—48no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4C2×C4⋊C4×C12718yes—
302C2×C4C2×C4⋊C4×C12712yes—
303C2×C4C4×D4×C12714yes—
304C2×C4C4×D4×C12714yes—
305C2×C4#120×C12712yes—
306C2×C4C22⋊Q8×C12714yes—
307C2×C4C22⋊Q8×C12714yes—
308C2×C4#123×C12712yes—
309C2×C4C23×C4×C12711yescharacteristic
310C2×C4C22×D4×C12712yes—
311C2×C4C22×Q8×C12711yescharacteristic
312C2×C4C2×#41×C12712yes—
313C8—28no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—42no—
318C22—42no—
319C22#385×C12712yes—
320C22#385×C12712yes—
321C22C22×C4⋊C4×C12711yescharacteristic
322C22C2×#120×C12711yescharacteristic
323C22C2×#123×C12711yescharacteristic
324C4—28no—
325C4—48no—
326C4—82no—
327C4—82no—
328C4#385×C12714yes—
329C4C2×C4×D4×C12712yes—
330C4C2×C22⋊Q8×C12712yes—
331C2—21no—
332C2—21no—
333C2—22no—
334C2—22no—
335C2C2×#385×C12711yescharacteristic
336C2C127×#302012yes—
subgroupG/H
center, socleC22×C127C2×#120
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC24×C127C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
5. C127×#1857C2semidirect
20. C2×#101×C127C4semidirect
64. #101×C127C2×C4semidirect
162. #20857C127direct (N × H)
173. #1857C2×C127semidirect
175. C127#20857direct (N × H)
189. C2×#101C4×C127semidirect
233. #101C2×C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#519×C1272
C127×#29112
C22×#101×C1272
C127×#23102
C127×#18572
C127×#17332
#20857127