Finite Group Databasebeta

← C127×#18327 #44582000

Subgroups — C127×#18327 #44582000

Order 32512.

950 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×#544×C127C211yesmaximal, characteristic
2C127×#3195C212yesmaximal
3C127×#2583C211yesmaximal, characteristic
4C127×#1611C214yesmaximal, complemented
5C127×#1605C214yesmaximal, complemented
6C127×#1508C212yesmaximal
7C127×#1463C211yesmaximal, characteristic
8#544×C127—24no—
9#544×C127—22no—
10#544×C127C2214yes—
11C2×C4⋊Q8×C127—22no—
12C2×#125×C127—22no—
13C2×C22⋊Q8×C127—21no—
14C2×C22⋊Q8×C127C2212yes—
15C2×C22⋊Q8×C127C2211yescharacteristic
16C2×#117×C127—21no—
17C22×C4⋊C4×C127C2211yescharacteristic
18#483×C127—22no—
19#483×C127C2214yes—
20#470×C127—24no—
21#451×C127—22no—
22#451×C127C2214yes—
23#407×C127C2212yes—
24#398×C127C2212yes—
25#395×C127C2212yes—
26#389×C127—22no—
27#385×C127—21no—
28#385×C127—21no—
29C4×C22⋊C4×C127C2211yescharacteristic
30#328×C127C2218yes—
31#324×C127C2214yes—
32C22×Q8×C127—21no—
33C22×Q8×C127D412yes—
34C22×Q8×C127C2311yescharacteristic
35C23×C4×C127D411yescharacteristic
36C23×C4×C127D411yescharacteristic
37C23×C4×C127C2311yescharacteristic
38C2×Q16×C127—24no—
39C2×M16×C127—22no—
40C4⋊Q8×C127—28no—
41C4⋊Q8×C127—28no—
42#125×C127—28no—
43#125×C127—28no—
44C22⋊Q8×C127—24no—
45C22⋊Q8×C127—24no—
46C22⋊Q8×C127—22no—
47C22⋊Q8×C127—22no—
48C22⋊Q8×C127—22no—
49C22⋊Q8×C127—22no—
50C22⋊Q8×C127D414yes—
51C22⋊Q8×C127C2314yes—
52#117×C127—24no—
53#117×C127—24no—
54C2×C4⋊C4×C127—21no—
55C2×C4⋊C4×C127—21no—
56C2×C4⋊C4×C127—21no—
57C2×C4⋊C4×C127—22no—
58C2×C4⋊C4×C127—22no—
59C2×C4⋊C4×C127—22no—
60C2×C4⋊C4×C127—22no—
61C2×C4⋊C4×C127—21no—
62C2×C4⋊C4×C127—22no—
63C2×C4⋊C4×C127—22no—
64C2×C4⋊C4×C127D412yes—
65C2×C4⋊C4×C127D412yes—
66C2×C4⋊C4×C127C2311yescharacteristic
67C2×C4⋊C4×C127C2312yes—
68C2×C22⋊C4×C127—22no—
69C2×C22⋊C4×C127—21no—
70C2×C22⋊C4×C127—21no—
71C2×C42×C127—22no—
72C2×C42×C127—21no—
73#106×C127—24no—
74#103×C127—24no—
75#103×C127—24no—
76C22⋊C8×C127—22no—
77C22⋊C8×C127C2314yes—
78#95×C127—22no—
79#95×C127—22no—
80#95×C127C2312yes—
81C24×C127C2×D411yescharacteristic
82C2×Q8×C127—24no—
83C2×Q8×C127—22no—
84C2×Q8×C127—22no—
85C2×Q8×C127—81no—
86C2×Q8×C127—82no—
87C2×Q8×C127—82no—
88C2×Q8×C127C2×D414yes—
89C22×C4×C127—21no—
90C22×C4×C127—21no—
91C22×C4×C127—24no—
92C22×C4×C127—22no—
93C22×C4×C127—22no—
94C22×C4×C127—21no—
95C22×C4×C127—22no—
96C22×C4×C127—21no—
97C22×C4×C127—21no—
98C22×C4×C127—21no—
99C22×C4×C127—81no—
100C22×C4×C127—81no—
101C22×C4×C127C2×D412yes—
102C22×C4×C127C2×D412yes—
103C22×C4×C127C2×D411yescharacteristic
104C22×C4×C127C2×D412yes—
105C22×C4×C127C2×D412yes—
106C22×C4×C127C2411yescharacteristic
107Q16×C127—84no—
108M16×C127—82no—
109C2×C8×C127—24no—
110C4⋊C4×C127—24no—
111C4⋊C4×C127—24no—
112C4⋊C4×C127—24no—
113C4⋊C4×C127—28no—
114C4⋊C4×C127—24no—
115C4⋊C4×C127—24no—
116C4⋊C4×C127—22no—
117C4⋊C4×C127—24no—
118C4⋊C4×C127—24no—
119C4⋊C4×C127—22no—
120C4⋊C4×C127—42no—
121C4⋊C4×C127—44no—
122C4⋊C4×C127—82no—
123C4⋊C4×C127C2×D414yes—
124C22⋊C4×C127—24no—
125C22⋊C4×C127—24no—
126C22⋊C4×C127—24no—
127C22⋊C4×C127—41no—
128C22⋊C4×C127—41no—
129C42×C127—28no—
130C42×C127—81no—
131C23×C127—22no—
132C23×C127—22no—
133C23×C127—41no—
134C23×C127#12011yescharacteristic
135C23×C127#12011yescharacteristic
136C23×C127C22×D411yescharacteristic
137Q8×C127—44no—
138Q8×C127—44no—
139Q8×C127—82no—
140Q8×C127—84no—
141C2×C4×C127—24no—
142C2×C4×C127—24no—
143C2×C4×C127—22no—
144C2×C4×C127—22no—
145C2×C4×C127—24no—
146C2×C4×C127—24no—
147C2×C4×C127—24no—
148C2×C4×C127—22no—
149C2×C4×C127—41no—
150C2×C4×C127—42no—
151C2×C4×C127—42no—
152C2×C4×C127—41no—
153C2×C4×C127—42no—
154C2×C4×C127—42no—
155C2×C4×C127—44no—
156C2×C4×C127—42no—
157C2×C4×C127—44no—
158C2×C4×C127—42no—
159C2×C4×C127—42no—
160C2×C4×C127—41no—
161C2×C4×C127—41no—
162C2×C4×C127—42no—
163C2×C4×C127—81no—
164C2×C4×C127#12012yes—
165C2×C4×C127#13714yes—
166C2×C4×C127C22×D412yes—
167C8×C127—44no—
168C22×C127—21no—
169C22×C127—21no—
170C22×C127—21no—
171C22×C127—21no—
172C22×C127—21no—
173C22×C127—21no—
174C22×C127—21no—
175C22×C127—44no—
176C22×C127#45612yes—
177C22×C127C2×#12011yescharacteristic
178C22×C127C2×#13712yes—
179C4×C127—24no—
180C4×C127—22no—
181C4×C127—22no—
182C4×C127—44no—
183C4×C127—44no—
184C4×C127—42no—
185C4×C127—44no—
186C4×C127—81no—
187C4×C127—81no—
188#18327C12711yesmaximal, characteristic, direct product factor, Sylow
189C2×C127—22no—
190C2×C127—22no—
191C2×C127—41no—
192C2×C127#300411yescharacteristic
193C2×C127C2×#45611yescharacteristic
194C2×C127#302611yescharacteristic
195C2×#544C2×C12711yescharacteristic
196#3195C2×C12712yes—
197#2583C2×C12711yescharacteristic
198#1611C2×C12714yescomplemented
199#1605C2×C12714yescomplemented
200#1508C2×C12712yes—
201#1463C2×C12711yescharacteristic
202C127#1832711yescharacteristic, direct product factor, Sylow
203#544—24no—
204#544—22no—
205#544C22×C12714yes—
206C2×C4⋊Q8—22no—
207C2×#125—22no—
208C2×C22⋊Q8—21no—
209C2×C22⋊Q8C22×C12712yes—
210C2×C22⋊Q8C22×C12711yescharacteristic
211C2×#117—21no—
212C22×C4⋊C4C22×C12711yescharacteristic
213#483—22no—
214#483C22×C12714yes—
215#470—24no—
216#451—22no—
217#451C22×C12714yes—
218#407C22×C12712yes—
219#398C22×C12712yes—
220#395C22×C12712yes—
221#389—22no—
222#385—21no—
223#385—21no—
224C4×C22⋊C4C22×C12711yescharacteristic
225#328C22×C12718yes—
226#324C22×C12714yes—
227C22×Q8—21no—
228C22×Q8D4×C12712yes—
229C22×Q8C23×C12711yescharacteristic
230C23×C4D4×C12711yescharacteristic
231C23×C4D4×C12711yescharacteristic
232C23×C4C23×C12711yescharacteristic
233C2×Q16—24no—
234C2×M16—22no—
235C4⋊Q8—28no—
236C4⋊Q8—28no—
237#125—28no—
238#125—28no—
239C22⋊Q8—24no—
240C22⋊Q8—24no—
241C22⋊Q8—22no—
242C22⋊Q8—22no—
243C22⋊Q8—22no—
244C22⋊Q8—22no—
245C22⋊Q8D4×C12714yes—
246C22⋊Q8C23×C12714yes—
247#117—24no—
248#117—24no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4D4×C12712yes—
260C2×C4⋊C4D4×C12712yes—
261C2×C4⋊C4C23×C12711yescharacteristic
262C2×C4⋊C4C23×C12712yes—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C42—22no—
267C2×C42—21no—
268#106—24no—
269#103—24no—
270#103—24no—
271C22⋊C8—22no—
272C22⋊C8C23×C12714yes—
273#95—22no—
274#95—22no—
275#95C23×C12712yes—
276C24C2×D4×C12711yescharacteristic
277C2×Q8—24no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—81no—
281C2×Q8—82no—
282C2×Q8—82no—
283C2×Q8C2×D4×C12714yes—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4C2×D4×C12712yes—
297C22×C4C2×D4×C12712yes—
298C22×C4C2×D4×C12711yescharacteristic
299C22×C4C2×D4×C12712yes—
300C22×C4C2×D4×C12712yes—
301C22×C4C24×C12711yescharacteristic
302Q16—84no—
303M16—82no—
304C2×C8—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—28no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—82no—
318C4⋊C4C2×D4×C12714yes—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—24no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C42—28no—
325C42—81no—
326C23—22no—
327C23—22no—
328C23—41no—
329C23#120×C12711yescharacteristic
330C23#120×C12711yescharacteristic
331C23C22×D4×C12711yescharacteristic
332Q8—44no—
333Q8—44no—
334Q8—82no—
335Q8—84no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—81no—
359C2×C4#120×C12712yes—
360C2×C4#137×C12714yes—
361C2×C4C22×D4×C12712yes—
362C8—44no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—44no—
371C22#456×C12712yes—
372C22C2×#120×C12711yescharacteristic
373C22C2×#137×C12712yes—
374C4—24no—
375C4—22no—
376C4—22no—
377C4—44no—
378C4—44no—
379C4—42no—
380C4—44no—
381C4—81no—
382C4—81no—
383C2—22no—
384C2—22no—
385C2—41no—
386C2C127×#300411yescharacteristic
387C2C2×#456×C12711yescharacteristic
388C2C127×#302611yescharacteristic
subgroupG/H
center, socleC22×C127C2×#120
commutator, Frattini, cosocleC22×C4C24×C127
prime radicalC24×C127C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1611C2semidirect
5. C127×#1605C2semidirect
188. #18327C127direct (N × H)
198. #1611C2×C127semidirect
199. #1605C2×C127semidirect
202. C127#18327direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#544×C1272
C127×#31952
C127×#25832
C127×#16112
C127×#16052
C127×#15082
C127×#14632
#18327127