Finite Group Databasebeta

← C4×#352×C127 #44576293

Subgroups — C4×#352×C127 #44576293

Order 32512.

576 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#123×C127C211yesmaximal, characteristic
2C2×#352×C127C212yesmaximal
3C4×#100×C127C211yesmaximal, characteristic
4C4×#99×C127C211yesmaximal, characteristic
5C127×#1390C212yesmaximal
6C2×#123×C127C2211yescharacteristic
7C2×C4×D4×C127C2211yescharacteristic
8C22×C42×C127C411yescharacteristic, complemented
9C2×#100×C127C2212yes—
10C2×#99×C127C2212yes—
11C4×M16×C127—21no—
12#388×C127—21no—
13#387×C127—21no—
14#386×C127C2211yescharacteristic
15#385×C127C411yescharacteristic, complemented
16#381×C127C411yescharacteristic, complemented
17C4×C4⋊C4×C127—21no—
18C4×C22⋊C4×C127—21no—
19C4×C22⋊C4×C127—21no—
20C4×C22⋊C4×C127C411yescharacteristic, complemented
21#352×C127C418yesdirect product factor
22#343×C127—21no—
23#342×C127—21no—
24C22×D4×C127C2311yescharacteristic
25C23×C4×C127C2×C411yescharacteristic
26C23×C4×C127C2×C411yescharacteristic
27C23×C4×C127C2×C411yescharacteristic
28C23×C4×C127D411yescharacteristic
29C2×M16×C127—22no—
30#123×C127—22no—
31#123×C127C2×C414yes—
32C4×D4×C127—41no—
33C4×D4×C127—41no—
34C2×C4⋊C4×C127—21no—
35C2×C4⋊C4×C127—21no—
36C2×C4⋊C4×C127D411yescharacteristic
37C2×C22⋊C4×C127—21no—
38C2×C22⋊C4×C127—22no—
39C2×C22⋊C4×C127—21no—
40C2×C22⋊C4×C127C2×C411yescharacteristic
41C2×C22⋊C4×C127C2×C411yescharacteristic
42C2×C22⋊C4×C127C2×C411yescharacteristic
43C2×C22⋊C4×C127D411yescharacteristic
44C2×C42×C127—21no—
45C2×C42×C127—21no—
46C2×C42×C127—21no—
47C2×C42×C127—41no—
48C2×C42×C127—41no—
49C2×C42×C127—41no—
50C2×C42×C127D411yescharacteristic
51#100×C127—22no—
52#100×C127C2×C414yes—
53#99×C127—22no—
54#99×C127C2×C414yes—
55#97×C127—41no—
56C4×C8×C127—41no—
57#95×C127—21no—
58#95×C127—21no—
59#95×C127—21no—
60#95×C127—21no—
61#95×C127—21no—
62#95×C127—41no—
63C24×C127C22×C411yescharacteristic
64C24×C127C2×D411yescharacteristic
65C2×D4×C127—41no—
66C2×D4×C127—41no—
67C2×D4×C127C22×C412yes—
68C2×D4×C127#4112yes—
69C22×C4×C127—21no—
70C22×C4×C127—21no—
71C22×C4×C127—21no—
72C22×C4×C127—21no—
73C22×C4×C127—21no—
74C22×C4×C127—21no—
75C22×C4×C127—21no—
76C22×C4×C127—21no—
77C22×C4×C127—21no—
78C22×C4×C127—42no—
79C22×C4×C127—42no—
80C22×C4×C127—42no—
81C22×C4×C127—41no—
82C22×C4×C127—41no—
83C22×C4×C127—41no—
84C22×C4×C127—42no—
85C22×C4×C127C4212yescomplemented
86C22×C4×C127C22⋊C411yescharacteristic
87C22×C4×C127C22⋊C411yescharacteristic
88C22×C4×C127C22⋊C411yescharacteristic
89C22×C4×C127C22⋊C411yescharacteristic
90C22×C4×C127C2×D411yescharacteristic
91M16×C127—24no—
92M16×C127—42no—
93C2×C8×C127—42no—
94C4⋊C4×C127—24no—
95C4⋊C4×C127—41no—
96C4⋊C4×C127—42no—
97C22⋊C4×C127—21no—
98C22⋊C4×C127—24no—
99C22⋊C4×C127—24no—
100C22⋊C4×C127—41no—
101C22⋊C4×C127—41no—
102C22⋊C4×C127—42no—
103C22⋊C4×C127—42no—
104C22⋊C4×C127—41no—
105C22⋊C4×C127C4212yescomplemented
106C42×C127—21no—
107C42×C127—21no—
108C42×C127—42no—
109C42×C127—42no—
110C42×C127—41no—
111C42×C127—41no—
112C42×C127—82no—
113C23×C127—21no—
114C23×C127—21no—
115C23×C127—21no—
116C23×C127—21no—
117C23×C127—41no—
118C23×C127—41no—
119C23×C127—41no—
120C23×C127—41no—
121C23×C127C2×C4211yescharacteristic
122C23×C127C2×C22⋊C411yescharacteristic
123C23×C127#11711yescharacteristic
124C23×C127C4×D412yes—
125D4×C127—42no—
126D4×C127—42no—
127C2×C4×C127—21no—
128C2×C4×C127—21no—
129C2×C4×C127—22no—
130C2×C4×C127—22no—
131C2×C4×C127—22no—
132C2×C4×C127—22no—
133C2×C4×C127—21no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—42no—
137C2×C4×C127—41no—
138C2×C4×C127—41no—
139C2×C4×C127—41no—
140C2×C4×C127—42no—
141C2×C4×C127—42no—
142C2×C4×C127—41no—
143C2×C4×C127—41no—
144C2×C4×C127—41no—
145C2×C4×C127—42no—
146C2×C4×C127—42no—
147C2×C4×C127—41no—
148C2×C4×C127—42no—
149C2×C4×C127—41no—
150C2×C4×C127—41no—
151C2×C4×C127—42no—
152C2×C4×C127—41no—
153C2×C4×C127—44no—
154C2×C4×C127—41no—
155C2×C4×C127—41no—
156C2×C4×C127—81no—
157C2×C4×C127—82no—
158C2×C4×C127—82no—
159C2×C4×C127#9911yescharacteristic
160C2×C4×C127#9911yescharacteristic
161C2×C4×C127C4×D412yes—
162C8×C127—44no—
163C22×C127—21no—
164C22×C127—21no—
165C22×C127—21no—
166C22×C127—21no—
167C22×C127—22no—
168C22×C127—41no—
169C22×C127—41no—
170C22×C127—42no—
171C22×C127—42no—
172C22×C127—41no—
173C22×C127—41no—
174C22×C127—41no—
175C22×C127—41no—
176C22×C127—81no—
177C22×C127C4×C22⋊C411yescharacteristic
178C22×C127C2×#9911yescharacteristic
179C22×C127#41011yescharacteristic
180C4×C127—21no—
181C4×C127—22no—
182C4×C127—42no—
183C4×C127—41no—
184C4×C127—41no—
185C4×C127—41no—
186C4×C127—42no—
187C4×C127—42no—
188C4×C127—81no—
189C4×C127—84no—
190C4×C127#35212yesdirect product factor
191C4×#352C12711yesmaximal, characteristic, direct product factor, Sylow
192C2×C127—21no—
193C2×C127—21no—
194C2×C127—42no—
195C2×C127—41no—
196C2×C127—41no—
197C2×C127C4×#9911yescharacteristic
198C2×C127C2×#35211yescharacteristic
199C2×C127#212011yescharacteristic
200C4×#123C2×C12711yescharacteristic
201C2×#352C2×C12712yes—
202C4×#100C2×C12711yescharacteristic
203C4×#99C2×C12711yescharacteristic
204#1390C2×C12712yes—
205C127C4×#35211yescharacteristic, direct product factor, Sylow
206C2×#123C22×C12711yescharacteristic
207C2×C4×D4C22×C12711yescharacteristic
208C22×C42C4×C12711yescharacteristic, complemented
209C2×#100C22×C12712yes—
210C2×#99C22×C12712yes—
211C4×M16—21no—
212#388—21no—
213#387—21no—
214#386C22×C12711yescharacteristic
215#385C4×C12711yescharacteristic, complemented
216#381C4×C12711yescharacteristic, complemented
217C4×C4⋊C4—21no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4C4×C12711yescharacteristic, complemented
221#352C4×C12718yesdirect product factor
222#343—21no—
223#342—21no—
224C22×D4C23×C12711yescharacteristic
225C23×C4C2×C4×C12711yescharacteristic
226C23×C4C2×C4×C12711yescharacteristic
227C23×C4C2×C4×C12711yescharacteristic
228C23×C4D4×C12711yescharacteristic
229C2×M16—22no—
230#123—22no—
231#123C2×C4×C12714yes—
232C4×D4—41no—
233C4×D4—41no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4D4×C12711yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4C2×C4×C12711yescharacteristic
241C2×C22⋊C4C2×C4×C12711yescharacteristic
242C2×C22⋊C4C2×C4×C12711yescharacteristic
243C2×C22⋊C4D4×C12711yescharacteristic
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—41no—
248C2×C42—41no—
249C2×C42—41no—
250C2×C42D4×C12711yescharacteristic
251#100—22no—
252#100C2×C4×C12714yes—
253#99—22no—
254#99C2×C4×C12714yes—
255#97—41no—
256C4×C8—41no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—41no—
263C24C22×C4×C12711yescharacteristic
264C24C2×D4×C12711yescharacteristic
265C2×D4—41no—
266C2×D4—41no—
267C2×D4C22×C4×C12712yes—
268C2×D4#41×C12712yes—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4C42×C12712yescomplemented
286C22×C4C22⋊C4×C12711yescharacteristic
287C22×C4C22⋊C4×C12711yescharacteristic
288C22×C4C22⋊C4×C12711yescharacteristic
289C22×C4C22⋊C4×C12711yescharacteristic
290C22×C4C2×D4×C12711yescharacteristic
291M16—24no—
292M16—42no—
293C2×C8—42no—
294C4⋊C4—24no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C22⋊C4—21no—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—41no—
305C22⋊C4C42×C12712yescomplemented
306C42—21no—
307C42—21no—
308C42—42no—
309C42—42no—
310C42—41no—
311C42—41no—
312C42—82no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23C2×C42×C12711yescharacteristic
322C23C2×C22⋊C4×C12711yescharacteristic
323C23#117×C12711yescharacteristic
324C23C4×D4×C12712yes—
325D4—42no—
326D4—42no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—44no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—81no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4#99×C12711yescharacteristic
360C2×C4#99×C12711yescharacteristic
361C2×C4C4×D4×C12712yes—
362C8—44no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—22no—
368C22—41no—
369C22—41no—
370C22—42no—
371C22—42no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—81no—
377C22C4×C22⋊C4×C12711yescharacteristic
378C22C2×#99×C12711yescharacteristic
379C22#410×C12711yescharacteristic
380C4—21no—
381C4—22no—
382C4—42no—
383C4—41no—
384C4—41no—
385C4—42no—
386C4—41no—
387C4—42no—
388C4—81no—
389C4—84no—
390C4#352×C12712yesdirect product factor
391C2—21no—
392C2—21no—
393C2—42no—
394C2—41no—
395C2—41no—
396C2C4×#99×C12711yescharacteristic
397C2C2×#352×C12711yescharacteristic
398C2C127×#212011yescharacteristic
subgroupG/H
centerC2×C4×C127#99
commutatorC23C2×C42×C127
Frattini, cosocleC22×D4C23×C127
socleC22×C127C2×#99
prime radicalC22×D4×C127C23
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C22×C4×C127 ◁ C2×C4×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
8. C22×C42×C127C4semidirect
15. #385×C127C4semidirect
16. #381×C127C4semidirect
20. C4×C22⋊C4×C127C4semidirect
21. #352×C127C4direct (N × H)
85. C22×C4×C127C42semidirect
105. C22⋊C4×C127C42semidirect
190. C4×C127#352direct (N × H)
191. C4×#352C127direct (N × H)
205. C127C4×#352direct (N × H)
208. C22×C42C4×C127semidirect
215. #385C4×C127semidirect
216. #381C4×C127semidirect
220. C4×C22⋊C4C4×C127semidirect
221. #352C4×C127direct (N × H)
285. C22×C4C42×C127semidirect
305. C22⋊C4C42×C127semidirect
390. C4#352×C127direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#123×C1272
C2×#352×C1272
C4×#100×C1272
C4×#99×C1272
C127×#13902
C4×#352127