Finite Group Databasebeta

← C127×#8370 #44572043

Subgroups — C127×#8370 #44572043

Order 32512.

808 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#559×C127C211yesmaximal, characteristic
2C4×#101×C127C211yesmaximal, characteristic, complemented
3C127×#1527C212yesmaximal, complemented
4C127×#1490C212yesmaximal, complemented
5C127×#1467C211yesmaximal, characteristic, complemented
6#559×C127—24no—
7#559×C127—24no—
8C2×#127×C127C411yescharacteristic
9C2×#124×C127—21no—
10C2×#124×C127C411yescharacteristic
11C2×#124×C127C2211yescharacteristic
12C2×#123×C127—21no—
13C2×#123×C127—21no—
14C2×C4×Q8×C127C411yescharacteristic
15C2×C4×Q8×C127C2211yescharacteristic
16C2×C4×D4×C127—21no—
17C2×C4×D4×C127C411yescharacteristic
18C2×C4×D4×C127C2211yescharacteristic
19#432×C127C2212yes—
20C2×#101×C127—21no—
21C4×M16×C127C2212yes—
22#343×C127—21no—
23#329×C127—24no—
24D4⋊C8×C127—24no—
25#324×C127—21no—
26#324×C127—21no—
27C22×Q8×C127C2×C411yescharacteristic
28C22×Q8×C127D411yescharacteristic
29C22×D4×C127—21no—
30C22×D4×C127—21no—
31C22×D4×C127C2×C411yescharacteristic
32C22×D4×C127C2×C411yescharacteristic
33C23×C4×C127—21no—
34C23×C4×C127—21no—
35C2×M16×C127—22no—
36#127×C127—22no—
37#127×C127—22no—
38#124×C127—28no—
39#124×C127—22no—
40#124×C127—22no—
41#124×C127—22no—
42#124×C127D414yes—
43#123×C127—28no—
44#123×C127—28no—
45C4×Q8×C127—22no—
46C4×Q8×C127—22no—
47C4×Q8×C127—22no—
48C4×Q8×C127—22no—
49C4×D4×C127—28no—
50C4×D4×C127—22no—
51C4×D4×C127—22no—
52C4×D4×C127—22no—
53C4×D4×C127D414yes—
54C2×C4⋊C4×C127—21no—
55C2×C4⋊C4×C127C2×C411yescharacteristic
56C2×C4⋊C4×C127D411yescharacteristic
57C2×C4⋊C4×C127D411yescharacteristic
58C2×C4⋊C4×C127D411yescharacteristic
59C2×C22⋊C4×C127—21no—
60C2×C22⋊C4×C127—21no—
61C2×C22⋊C4×C127—21no—
62C2×C22⋊C4×C127—21no—
63C2×C22⋊C4×C127—21no—
64C2×C22⋊C4×C127—21no—
65C2×C42×C127—21no—
66C2×C42×C127C2×C411yescharacteristic
67C2×C42×C127C2×C411yescharacteristic
68C2×C42×C127C2311yescharacteristic
69C4⋊C8×C127—22no—
70#101×C127—24no—
71C22⋊C8×C127—22no—
72#97×C127—22no—
73#97×C127—22no—
74C4×C8×C127—22no—
75C24×C127—21no—
76C24×C127—21no—
77C2×Q8×C127—22no—
78C2×Q8×C127—21no—
79C2×Q8×C127—22no—
80C2×Q8×C127—41no—
81C2×Q8×C127—42no—
82C2×Q8×C127C22⋊C412yes—
83C2×D4×C127—24no—
84C2×D4×C127—24no—
85C2×D4×C127—21no—
86C2×D4×C127—21no—
87C2×D4×C127—42no—
88C2×D4×C127—42no—
89C2×D4×C127—42no—
90C2×D4×C127—42no—
91C2×D4×C127—82no—
92C2×D4×C127C22⋊C412yes—
93C2×D4×C127C22⋊C412yes—
94C22×C4×C127—21no—
95C22×C4×C127—21no—
96C22×C4×C127—24no—
97C22×C4×C127—24no—
98C22×C4×C127—82no—
99C22×C4×C127—82no—
100C22×C4×C127C22⋊C411yescharacteristic
101C22×C4×C127C22⋊C411yescharacteristic
102C22×C4×C127C22⋊C411yescharacteristic
103C22×C4×C127C22⋊C411yescharacteristic
104C22×C4×C127C22×C411yescharacteristic
105C22×C4×C127C2×D411yescharacteristic
106C22×C4×C127C2×D411yescharacteristic
107M16×C127—24no—
108M16×C127—42no—
109C2×C8×C127—42no—
110C2×C8×C127—42no—
111C4⋊C4×C127—24no—
112C4⋊C4×C127—21no—
113C4⋊C4×C127—22no—
114C4⋊C4×C127—22no—
115C4⋊C4×C127—22no—
116C4⋊C4×C127C22⋊C412yes—
117C22⋊C4×C127—24no—
118C22⋊C4×C127—24no—
119C22⋊C4×C127—24no—
120C22⋊C4×C127—24no—
121C22⋊C4×C127—24no—
122C22⋊C4×C127—24no—
123C42×C127—24no—
124C42×C127—21no—
125C42×C127—21no—
126C42×C127—21no—
127C42×C127—21no—
128C42×C127C2×D412yes—
129C42×C127C2×D412yes—
130C23×C127—22no—
131C23×C127—22no—
132C23×C127—42no—
133C23×C127—42no—
134C23×C127—42no—
135C23×C127—42no—
136C23×C127—41no—
137C23×C127—41no—
138C23×C127—41no—
139C23×C127—41no—
140C23×C127C2×C22⋊C411yescharacteristic
141Q8×C127—44no—
142Q8×C127—42no—
143Q8×C127—42no—
144D4×C127—24no—
145D4×C127—42no—
146D4×C127—42no—
147D4×C127—48no—
148D4×C127—42no—
149D4×C127—84no—
150C2×C4×C127—21no—
151C2×C4×C127—21no—
152C2×C4×C127—21no—
153C2×C4×C127—21no—
154C2×C4×C127—22no—
155C2×C4×C127—22no—
156C2×C4×C127—21no—
157C2×C4×C127—21no—
158C2×C4×C127—22no—
159C2×C4×C127—21no—
160C2×C4×C127—21no—
161C2×C4×C127—21no—
162C2×C4×C127—41no—
163C2×C4×C127—42no—
164C2×C4×C127—42no—
165C2×C4×C127—41no—
166C2×C4×C127—41no—
167C2×C4×C127—84no—
168C2×C4×C127—84no—
169C2×C4×C127#10412yes—
170C2×C4×C127C2×C22⋊C411yescharacteristic
171C2×C4×C127C2×C22⋊C411yescharacteristic
172C2×C4×C127#12012yes—
173C2×C4×C127#12012yes—
174C8×C127—44no—
175C8×C127—82no—
176C22×C127—21no—
177C22×C127—21no—
178C22×C127—42no—
179C22×C127—42no—
180C22×C127—42no—
181C22×C127—42no—
182C22×C127—42no—
183C22×C127—42no—
184C22×C127—82no—
185C22×C127—82no—
186C22×C127—82no—
187C22×C127—82no—
188C22×C127#37911yescharacteristic
189C22×C127C2×#10411yescharacteristic
190C22×C127#42111yescharacteristic
191C4×C127—22no—
192C4×C127—21no—
193C4×C127—22no—
194C4×C127—22no—
195C4×C127—42no—
196C4×C127—41no—
197C4×C127—41no—
198C4×C127—42no—
199C4×C127—41no—
200C4×C127#41312yes—
201#8370C12711yesmaximal, characteristic, direct product factor, Sylow
202C2×C127—21no—
203C2×C127—21no—
204C2×C127—82no—
205C2×C127—82no—
206C2×C127#178511yescharacteristic
207C2×C127#178911yescharacteristic
208C2×C127#179511yescharacteristic
209C2×#559C2×C12711yescharacteristic
210C4×#101C2×C12711yescharacteristic, complemented
211#1527C2×C12712yescomplemented
212#1490C2×C12712yescomplemented
213#1467C2×C12711yescharacteristic, complemented
214C127#837011yescharacteristic, direct product factor, Sylow
215#559—24no—
216#559—24no—
217C2×#127C4×C12711yescharacteristic
218C2×#124—21no—
219C2×#124C4×C12711yescharacteristic
220C2×#124C22×C12711yescharacteristic
221C2×#123—21no—
222C2×#123—21no—
223C2×C4×Q8C4×C12711yescharacteristic
224C2×C4×Q8C22×C12711yescharacteristic
225C2×C4×D4—21no—
226C2×C4×D4C4×C12711yescharacteristic
227C2×C4×D4C22×C12711yescharacteristic
228#432C22×C12712yes—
229C2×#101—21no—
230C4×M16C22×C12712yes—
231#343—21no—
232#329—24no—
233D4⋊C8—24no—
234#324—21no—
235#324—21no—
236C22×Q8C2×C4×C12711yescharacteristic
237C22×Q8D4×C12711yescharacteristic
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2×C4×C12711yescharacteristic
241C22×D4C2×C4×C12711yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C2×M16—22no—
245#127—22no—
246#127—22no—
247#124—28no—
248#124—22no—
249#124—22no—
250#124—22no—
251#124D4×C12714yes—
252#123—28no—
253#123—28no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×Q8—22no—
258C4×D4—28no—
259C4×D4—22no—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4D4×C12714yes—
263C2×C4⋊C4—21no—
264C2×C4⋊C4C2×C4×C12711yescharacteristic
265C2×C4⋊C4D4×C12711yescharacteristic
266C2×C4⋊C4D4×C12711yescharacteristic
267C2×C4⋊C4D4×C12711yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C42—21no—
275C2×C42C2×C4×C12711yescharacteristic
276C2×C42C2×C4×C12711yescharacteristic
277C2×C42C23×C12711yescharacteristic
278C4⋊C8—22no—
279#101—24no—
280C22⋊C8—22no—
281#97—22no—
282#97—22no—
283C4×C8—22no—
284C24—21no—
285C24—21no—
286C2×Q8—22no—
287C2×Q8—21no—
288C2×Q8—22no—
289C2×Q8—41no—
290C2×Q8—42no—
291C2×Q8C22⋊C4×C12712yes—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—82no—
301C2×D4C22⋊C4×C12712yes—
302C2×D4C22⋊C4×C12712yes—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4C22⋊C4×C12711yescharacteristic
310C22×C4C22⋊C4×C12711yescharacteristic
311C22×C4C22⋊C4×C12711yescharacteristic
312C22×C4C22⋊C4×C12711yescharacteristic
313C22×C4C22×C4×C12711yescharacteristic
314C22×C4C2×D4×C12711yescharacteristic
315C22×C4C2×D4×C12711yescharacteristic
316M16—24no—
317M16—42no—
318C2×C8—42no—
319C2×C8—42no—
320C4⋊C4—24no—
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4C22⋊C4×C12712yes—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C42—24no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42C2×D4×C12712yes—
338C42C2×D4×C12712yes—
339C23—22no—
340C23—22no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23C2×C22⋊C4×C12711yescharacteristic
350Q8—44no—
351Q8—42no—
352Q8—42no—
353D4—24no—
354D4—42no—
355D4—42no—
356D4—48no—
357D4—42no—
358D4—84no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4#104×C12712yes—
379C2×C4C2×C22⋊C4×C12711yescharacteristic
380C2×C4C2×C22⋊C4×C12711yescharacteristic
381C2×C4#120×C12712yes—
382C2×C4#120×C12712yes—
383C8—44no—
384C8—82no—
385C22—21no—
386C22—21no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—82no—
394C22—82no—
395C22—82no—
396C22—82no—
397C22#379×C12711yescharacteristic
398C22C2×#104×C12711yescharacteristic
399C22#421×C12711yescharacteristic
400C4—21no—
401C4—22no—
402C4—22no—
403C4—22no—
404C4—42no—
405C4—41no—
406C4—42no—
407C4—41no—
408C4—41no—
409C4#413×C12712yes—
410C2—21no—
411C2—21no—
412C2—82no—
413C2—82no—
414C2C127×#178511yescharacteristic
415C2C127×#178911yescharacteristic
416C2C127×#179511yescharacteristic
subgroupG/H
center, socleC22×C127#379
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC2×#127×C127C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×#101×C127C2semidirect
3. C127×#1527C2semidirect
4. C127×#1490C2semidirect
5. C127×#1467C2semidirect
201. #8370C127direct (N × H)
210. C4×#101C2×C127semidirect
211. #1527C2×C127semidirect
212. #1490C2×C127semidirect
213. #1467C2×C127semidirect
214. C127#8370direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#559×C1272
C4×#101×C1272
C127×#15272
C127×#14902
C127×#14672
#8370127