Finite Group Databasebeta

← C127×#20705 #44584378

Subgroups — C127×#20705 #44584378

Order 32512.

1470 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#117×C127C211yesmaximal, characteristic
2C2×#407×C127C211yesmaximal, characteristic
3C127×#2280C211yesmaximal, characteristic
4C127×#1826C218yesmaximal, complemented
5C127×#1733C214yesmaximal, complemented
6C24×C4×C127C2211yescharacteristic
7C22×M16×C127C2212yes—
8C2×#117×C127—22no—
9C2×#117×C127—21no—
10C2×#117×C127—22no—
11C2×#117×C127—21no—
12C2×#117×C127C414yescomplemented
13C2×#117×C127C412yescomplemented
14C2×#117×C127C2214yes—
15C2×#117×C127C2212yes—
16C2×#117×C127C2212yes—
17C22×C4⋊C4×C127C412yescomplemented
18C22×C22⋊C4×C127—21no—
19C22×C42×C127—21no—
20#407×C127—21no—
21#407×C127—21no—
22#407×C127C2214yes—
23C2×C22⋊C8×C127—22no—
24#381×C127—21no—
25#380×C127—22no—
26#379×C127—21no—
27C4×C22⋊C4×C127C2214yes—
28#339×C127C22116yescomplemented
29C25×C127D411yescharacteristic
30C23×C4×C127—28no—
31C23×C4×C127—22no—
32C23×C4×C127C2×C411yescharacteristic
33C23×C4×C127C2×C412yes—
34C23×C4×C127C2×C411yescharacteristic
35C23×C4×C127D412yes—
36C23×C4×C127D411yescharacteristic
37C23×C4×C127D411yescharacteristic
38C23×C4×C127D412yes—
39C23×C4×C127D411yescharacteristic
40C23×C4×C127C2312yes—
41C23×C4×C127C2311yescharacteristic
42C2×M16×C127—22no—
43C2×M16×C127—28no—
44C2×M16×C127C2314yes—
45C22×C8×C127—22no—
46#117×C127—24no—
47#117×C127—24no—
48#117×C127—24no—
49#117×C127—24no—
50#117×C127—22no—
51#117×C127—24no—
52#117×C127—22no—
53#117×C127—28no—
54#117×C127C2×C418yes—
55#117×C127C2×C414yes—
56#117×C127C2314yes—
57C2×C4⋊C4×C127—22no—
58C2×C4⋊C4×C127—22no—
59C2×C4⋊C4×C127—22no—
60C2×C4⋊C4×C127—22no—
61C2×C4⋊C4×C127C2×C414yes—
62C2×C4⋊C4×C127C2×C414yes—
63C2×C4⋊C4×C127C2×C414yes—
64C2×C22⋊C4×C127—24no—
65C2×C22⋊C4×C127—22no—
66C2×C22⋊C4×C127—24no—
67C2×C22⋊C4×C127—22no—
68C2×C22⋊C4×C127—22no—
69C2×C22⋊C4×C127—24no—
70C2×C42×C127—22no—
71C2×C42×C127—24no—
72C2×C42×C127—24no—
73C2×C42×C127—22no—
74C2×C42×C127C2314yes—
75C22⋊C8×C127—24no—
76C22⋊C8×C127—24no—
77#95×C127—24no—
78C24×C127—24no—
79C24×C127—44no—
80C24×C127C22⋊C411yescharacteristic
81C24×C127C22⋊C412yes—
82C24×C127C22⋊C411yescharacteristic
83C24×C127C2×D412yes—
84C24×C127C2×D411yescharacteristic
85C22×C4×C127—24no—
86C22×C4×C127—24no—
87C22×C4×C127—24no—
88C22×C4×C127—28no—
89C22×C4×C127—28no—
90C22×C4×C127—28no—
91C22×C4×C127—24no—
92C22×C4×C127—24no—
93C22×C4×C127—28no—
94C22×C4×C127—24no—
95C22×C4×C127—24no—
96C22×C4×C127—24no—
97C22×C4×C127—24no—
98C22×C4×C127—48no—
99C22×C4×C127C22⋊C411yescharacteristic
100C22×C4×C127C22⋊C412yes—
101C22×C4×C127C22⋊C411yescharacteristic
102C22×C4×C127C22⋊C412yes—
103C22×C4×C127C22⋊C412yes—
104C22×C4×C127C22⋊C412yes—
105C22×C4×C127C22⋊C412yes—
106C22×C4×C127C22×C412yes—
107C22×C4×C127C22×C412yes—
108C22×C4×C127C22×C411yescharacteristic
109C22×C4×C127C22×C411yescharacteristic
110C22×C4×C127C2×D412yes—
111C22×C4×C127C2×D412yes—
112C22×C4×C127C2×D412yes—
113C22×C4×C127C2×D411yescharacteristic
114C22×C4×C127C2×D411yescharacteristic
115C22×C4×C127C2×D411yescharacteristic
116C22×C4×C127#4118yes—
117C22×C4×C127C2411yescharacteristic
118M16×C127—28no—
119M16×C127—44no—
120C2×C8×C127—24no—
121C2×C8×C127—44no—
122C4⋊C4×C127—24no—
123C4⋊C4×C127—24no—
124C4⋊C4×C127—24no—
125C4⋊C4×C127—44no—
126C4⋊C4×C127C22×C418yes—
127C22⋊C4×C127—28no—
128C22⋊C4×C127—24no—
129C22⋊C4×C127—24no—
130C22⋊C4×C127—48no—
131C22⋊C4×C127—44no—
132C42×C127—28no—
133C42×C127—24no—
134C42×C127—24no—
135C42×C127—28no—
136C23×C127—22no—
137C23×C127—24no—
138C23×C127—24no—
139C23×C127—24no—
140C23×C127—22no—
141C23×C127—24no—
142C23×C127—24no—
143C23×C127—42no—
144C23×C127—42no—
145C23×C127—42no—
146C23×C127—42no—
147C23×C127—416no—
148C23×C127C2×C22⋊C412yes—
149C23×C127C2×C22⋊C411yescharacteristic
150C23×C127C2×C22⋊C411yescharacteristic
151C23×C127C2×C22⋊C412yes—
152C23×C127#12314yes—
153C23×C127C22×D411yescharacteristic
154C2×C4×C127—24no—
155C2×C4×C127—22no—
156C2×C4×C127—22no—
157C2×C4×C127—24no—
158C2×C4×C127—24no—
159C2×C4×C127—24no—
160C2×C4×C127—24no—
161C2×C4×C127—24no—
162C2×C4×C127—24no—
163C2×C4×C127—24no—
164C2×C4×C127—24no—
165C2×C4×C127—24no—
166C2×C4×C127—28no—
167C2×C4×C127—48no—
168C2×C4×C127—48no—
169C2×C4×C127—48no—
170C2×C4×C127—48no—
171C2×C4×C127—44no—
172C2×C4×C127C2×C22⋊C412yes—
173C2×C4×C127C2×C22⋊C412yes—
174C2×C4×C127C2×C22⋊C411yescharacteristic
175C2×C4×C127C2×C22⋊C411yescharacteristic
176C2×C4×C127#11714yes—
177C2×C4×C127#11714yes—
178C2×C4×C127#12314yes—
179C2×C4×C127#12314yes—
180C2×C4×C127#12314yes—
181C2×C4×C127C23×C411yescharacteristic
182C2×C4×C127C22×D411yescharacteristic
183C2×C4×C127C2×#4114yes—
184C8×C127—44no—
185C22×C127—28no—
186C22×C127—22no—
187C22×C127—22no—
188C22×C127—22no—
189C22×C127—22no—
190C22×C127—22no—
191C22×C127—22no—
192C22×C127—22no—
193C22×C127—22no—
194C22×C127—44no—
195C22×C127—48no—
196C22×C127—48no—
197C22×C127—44no—
198C22×C127#38112yes—
199C22×C127#38112yes—
200C22×C127#42114yes—
201C22×C127C22×C22⋊C411yescharacteristic
202C22×C127C2×#12312yes—
203C4×C127—22no—
204C4×C127—22no—
205C4×C127—48no—
206C4×C127—84no—
207C4×C127#38112yes—
208C4×C127#38112yes—
209C4×C127C2×#11712yes—
210C4×C127C2×#12312yes—
211#20705C12711yesmaximal, characteristic, direct product factor, Sylow
212C2×C127—22no—
213C2×C127—22no—
214C2×C127—24no—
215C2×C127—22no—
216C2×C127—42no—
217C2×C127C2×#38111yescharacteristic
218C2×C127C2×#42112yes—
219C22×#117C2×C12711yescharacteristic
220C2×#407C2×C12711yescharacteristic
221#2280C2×C12711yescharacteristic
222#1826C2×C12718yescomplemented
223#1733C2×C12714yescomplemented
224C127#2070511yescharacteristic, direct product factor, Sylow
225C24×C4C22×C12711yescharacteristic
226C22×M16C22×C12712yes—
227C2×#117—22no—
228C2×#117—21no—
229C2×#117—22no—
230C2×#117—21no—
231C2×#117C4×C12714yescomplemented
232C2×#117C4×C12712yescomplemented
233C2×#117C22×C12714yes—
234C2×#117C22×C12712yes—
235C2×#117C22×C12712yes—
236C22×C4⋊C4C4×C12712yescomplemented
237C22×C22⋊C4—21no—
238C22×C42—21no—
239#407—21no—
240#407—21no—
241#407C22×C12714yes—
242C2×C22⋊C8—22no—
243#381—21no—
244#380—22no—
245#379—21no—
246C4×C22⋊C4C22×C12714yes—
247#339C22×C127116yescomplemented
248C25D4×C12711yescharacteristic
249C23×C4—28no—
250C23×C4—22no—
251C23×C4C2×C4×C12711yescharacteristic
252C23×C4C2×C4×C12712yes—
253C23×C4C2×C4×C12711yescharacteristic
254C23×C4D4×C12712yes—
255C23×C4D4×C12711yescharacteristic
256C23×C4D4×C12712yes—
257C23×C4D4×C12711yescharacteristic
258C23×C4D4×C12711yescharacteristic
259C23×C4C23×C12712yes—
260C23×C4C23×C12711yescharacteristic
261C2×M16—22no—
262C2×M16—28no—
263C2×M16C23×C12714yes—
264C22×C8—22no—
265#117—24no—
266#117—24no—
267#117—24no—
268#117—24no—
269#117—22no—
270#117—24no—
271#117—22no—
272#117—28no—
273#117C2×C4×C12718yes—
274#117C2×C4×C12714yes—
275#117C23×C12714yes—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4C2×C4×C12714yes—
281C2×C4⋊C4C2×C4×C12714yes—
282C2×C4⋊C4C2×C4×C12714yes—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42C23×C12714yes—
294C22⋊C8—24no—
295C22⋊C8—24no—
296#95—24no—
297C24—24no—
298C24—44no—
299C24C22⋊C4×C12711yescharacteristic
300C24C22⋊C4×C12712yes—
301C24C22⋊C4×C12711yescharacteristic
302C24C2×D4×C12712yes—
303C24C2×D4×C12711yescharacteristic
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—28no—
308C22×C4—28no—
309C22×C4—28no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—28no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—48no—
318C22×C4C22⋊C4×C12711yescharacteristic
319C22×C4C22⋊C4×C12712yes—
320C22×C4C22⋊C4×C12711yescharacteristic
321C22×C4C22⋊C4×C12712yes—
322C22×C4C22⋊C4×C12712yes—
323C22×C4C22⋊C4×C12712yes—
324C22×C4C22⋊C4×C12712yes—
325C22×C4C22×C4×C12712yes—
326C22×C4C22×C4×C12712yes—
327C22×C4C22×C4×C12711yescharacteristic
328C22×C4C22×C4×C12711yescharacteristic
329C22×C4C2×D4×C12712yes—
330C22×C4C2×D4×C12712yes—
331C22×C4C2×D4×C12712yes—
332C22×C4C2×D4×C12711yescharacteristic
333C22×C4C2×D4×C12711yescharacteristic
334C22×C4C2×D4×C12711yescharacteristic
335C22×C4#41×C12718yes—
336C22×C4C24×C12711yescharacteristic
337M16—28no—
338M16—44no—
339C2×C8—24no—
340C2×C8—44no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—44no—
345C4⋊C4C22×C4×C12718yes—
346C22⋊C4—28no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—48no—
350C22⋊C4—44no—
351C42—28no—
352C42—24no—
353C42—24no—
354C42—28no—
355C23—22no—
356C23—24no—
357C23—24no—
358C23—24no—
359C23—22no—
360C23—24no—
361C23—24no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—416no—
367C23C2×C22⋊C4×C12712yes—
368C23C2×C22⋊C4×C12711yescharacteristic
369C23C2×C22⋊C4×C12711yescharacteristic
370C23C2×C22⋊C4×C12712yes—
371C23#123×C12714yes—
372C23C22×D4×C12711yescharacteristic
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—28no—
386C2×C4—48no—
387C2×C4—48no—
388C2×C4—48no—
389C2×C4—48no—
390C2×C4—44no—
391C2×C4C2×C22⋊C4×C12712yes—
392C2×C4C2×C22⋊C4×C12712yes—
393C2×C4C2×C22⋊C4×C12711yescharacteristic
394C2×C4C2×C22⋊C4×C12711yescharacteristic
395C2×C4#117×C12714yes—
396C2×C4#117×C12714yes—
397C2×C4#123×C12714yes—
398C2×C4#123×C12714yes—
399C2×C4#123×C12714yes—
400C2×C4C23×C4×C12711yescharacteristic
401C2×C4C22×D4×C12711yescharacteristic
402C2×C4C2×#41×C12714yes—
403C8—44no—
404C22—28no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—44no—
414C22—48no—
415C22—48no—
416C22—44no—
417C22#381×C12712yes—
418C22#381×C12712yes—
419C22#421×C12714yes—
420C22C22×C22⋊C4×C12711yescharacteristic
421C22C2×#123×C12712yes—
422C4—22no—
423C4—22no—
424C4—48no—
425C4—84no—
426C4#381×C12712yes—
427C4#381×C12712yes—
428C4C2×#117×C12712yes—
429C4C2×#123×C12712yes—
430C2—22no—
431C2—22no—
432C2—24no—
433C2—22no—
434C2—42no—
435C2C2×#381×C12711yescharacteristic
436C2C2×#421×C12712yes—
subgroupG/H
centerC2×C4×C127C22×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
socleC22×C127C22×C22⋊C4
prime radicalC25×C127D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C24×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1826C2semidirect
5. C127×#1733C2semidirect
12. C2×#117×C127C4semidirect
13. C2×#117×C127C4semidirect
17. C22×C4⋊C4×C127C4semidirect
28. #339×C127C22semidirect
211. #20705C127direct (N × H)
222. #1826C2×C127semidirect
223. #1733C2×C127semidirect
224. C127#20705direct (N × H)
231. C2×#117C4×C127semidirect
232. C2×#117C4×C127semidirect
236. C22×C4⋊C4C4×C127semidirect
247. #339C22×C127semidirect

Maximal subgroups

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

subgroupindex
C22×#117×C1272
C2×#407×C1272
C127×#22802
C127×#18262
C127×#17332
#20705127