Finite Group Databasebeta

← C127×#8384 #44572057

Subgroups — C127×#8384 #44572057

Order 32512.

1050 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#550×C127C211yesmaximal, characteristic
2C127×#1801C211yesmaximal, characteristic, complemented
3C127×#1545C212yesmaximal, complemented
4C127×#1486C212yesmaximal, complemented
5C127×#1461C211yesmaximal, characteristic, complemented
6C22×#41×C127—21no—
7#550×C127—24no—
8#550×C127—24no—
9C2×#127×C127C412yes—
10C2×#127×C127C2211yescharacteristic
11C2×#121×C127—21no—
12C2×#121×C127—22no—
13C2×C4×Q8×C127C2211yescharacteristic
14C2×C4×D4×C127C412yes—
15C2×C4×D4×C127C2211yescharacteristic
16#431×C127C2212yes—
17#423×C127C2212yes—
18C2×#101×C127—21no—
19#343×C127—21no—
20#331×C127—24no—
21D4⋊C8×C127—24no—
22#324×C127—22no—
23C2×#41×C127—28no—
24C2×#41×C127—44no—
25C2×#41×C127—42no—
26C2×#41×C127—42no—
27C22×Q8×C127D411yescharacteristic
28C22×D4×C127—22no—
29C22×D4×C127—22no—
30C22×D4×C127—21no—
31C22×D4×C127—21no—
32C22×D4×C127C2×C412yes—
33C22×D4×C127C2×C411yescharacteristic
34C23×C4×C127—22no—
35C23×C4×C127—21no—
36C2×M16×C127—22no—
37#127×C127—24no—
38#127×C127—24no—
39#127×C127—22no—
40#127×C127D414yes—
41#121×C127—28no—
42#121×C127—216no—
43C4×Q8×C127—22no—
44C4×Q8×C127—22no—
45C4×D4×C127—24no—
46C4×D4×C127—24no—
47C4×D4×C127—22no—
48C4×D4×C127D414yes—
49C2×C4⋊C4×C127C2×C411yescharacteristic
50C2×C4⋊C4×C127D412yes—
51C2×C4⋊C4×C127D411yescharacteristic
52C2×C22⋊C4×C127—22no—
53C2×C22⋊C4×C127—21no—
54C2×C42×C127C2×C412yes—
55C2×C42×C127C2311yescharacteristic
56C4⋊C8×C127—22no—
57C4⋊C8×C127—22no—
58C4⋊C8×C127—22no—
59#101×C127—42no—
60C22⋊C8×C127—22no—
61C4×C8×C127—22no—
62C24×C127—22no—
63C24×C127—21no—
64#41×C127—416no—
65#41×C127—416no—
66C2×Q8×C127—22no—
67C2×Q8×C127—22no—
68C2×Q8×C127D812yes—
69C2×Q8×C127SD1612yes—
70C2×D4×C127—28no—
71C2×D4×C127—28no—
72C2×D4×C127—21no—
73C2×D4×C127—22no—
74C2×D4×C127—24no—
75C2×D4×C127—24no—
76C2×D4×C127—42no—
77C2×D4×C127—44no—
78C2×D4×C127—44no—
79C2×D4×C127—44no—
80C2×D4×C127—41no—
81C2×D4×C127—41no—
82C2×D4×C127—42no—
83C2×D4×C127—82no—
84C2×D4×C127—84no—
85C2×D4×C127C22⋊C412yes—
86C2×D4×C127C22⋊C414yes—
87C22×C4×C127—28no—
88C22×C4×C127—21no—
89C22×C4×C127—24no—
90C22×C4×C127—44no—
91C22×C4×C127—44no—
92C22×C4×C127—41no—
93C22×C4×C127—41no—
94C22×C4×C127—42no—
95C22×C4×C127C22⋊C412yes—
96C22×C4×C127C22⋊C412yes—
97C22×C4×C127C22×C411yescharacteristic
98C22×C4×C127C2×D411yescharacteristic
99C22×C4×C127C2×D411yescharacteristic
100M16×C127—42no—
101M16×C127—42no—
102C2×C8×C127—42no—
103C2×C8×C127—42no—
104C4⋊C4×C127—21no—
105C4⋊C4×C127—24no—
106C4⋊C4×C127—22no—
107C4⋊C4×C127C22⋊C412yes—
108C22⋊C4×C127—28no—
109C22⋊C4×C127—24no—
110C42×C127—22no—
111C42×C127—22no—
112C42×C127C2×D412yes—
113C42×C127C2×D412yes—
114C23×C127—22no—
115C23×C127—24no—
116C23×C127—42no—
117C23×C127—44no—
118C23×C127—44no—
119C23×C127—42no—
120C23×C127—42no—
121C23×C127—41no—
122C23×C127—42no—
123C23×C127—41no—
124C23×C127C2×C22⋊C411yescharacteristic
125Q8×C127—24no—
126Q8×C127—24no—
127D4×C127—24no—
128D4×C127—44no—
129D4×C127—42no—
130D4×C127—416no—
131D4×C127—44no—
132D4×C127—44no—
133D4×C127—44no—
134D4×C127—84no—
135D4×C127—88no—
136C2×C4×C127—22no—
137C2×C4×C127—22no—
138C2×C4×C127—22no—
139C2×C4×C127—22no—
140C2×C4×C127—22no—
141C2×C4×C127—41no—
142C2×C4×C127—416no—
143C2×C4×C127—41no—
144C2×C4×C127—41no—
145C2×C4×C127—44no—
146C2×C4×C127—44no—
147C2×C4×C127—42no—
148C2×C4×C127#10214yes—
149C2×C4×C127C2×C22⋊C411yescharacteristic
150C2×C4×C127C2×C22⋊C411yescharacteristic
151C2×C4×C127#12012yes—
152C2×C4×C127#12012yes—
153C2×C4×C127C2×D811yescharacteristic
154C2×C4×C127C2×SD1611yescharacteristic
155C2×C4×C127AGL(1,ℤ/8)12yes—
156C8×C127—82no—
157C8×C127—82no—
158C22×C127—21no—
159C22×C127—21no—
160C22×C127—44no—
161C22×C127—42no—
162C22×C127—42no—
163C22×C127—44no—
164C22×C127—44no—
165C22×C127—42no—
166C22×C127—82no—
167C22×C127—84no—
168C22×C127—82no—
169C22×C127—84no—
170C22×C127#37911yescharacteristic
171C22×C127C2×#10211yescharacteristic
172C22×C127#41811yescharacteristic
173C4×C127—22no—
174C4×C127—21no—
175C4×C127—24no—
176C4×C127—42no—
177C4×C127—42no—
178C4×C127#41312yes—
179C4×C127#44712yes—
180C4×C127#45012yes—
181#8384C12711yesmaximal, characteristic, direct product factor, Sylow
182C2×C127—21no—
183C2×C127—21no—
184C2×C127—82no—
185C2×C127—84no—
186C2×C127#178511yescharacteristic
187C2×C127#178611yescharacteristic
188C2×C127#179511yescharacteristic
189C2×#550C2×C12711yescharacteristic
190#1801C2×C12711yescharacteristic, complemented
191#1545C2×C12712yescomplemented
192#1486C2×C12712yescomplemented
193#1461C2×C12711yescharacteristic, complemented
194C127#838411yescharacteristic, direct product factor, Sylow
195C22×#41—21no—
196#550—24no—
197#550—24no—
198C2×#127C4×C12712yes—
199C2×#127C22×C12711yescharacteristic
200C2×#121—21no—
201C2×#121—22no—
202C2×C4×Q8C22×C12711yescharacteristic
203C2×C4×D4C4×C12712yes—
204C2×C4×D4C22×C12711yescharacteristic
205#431C22×C12712yes—
206#423C22×C12712yes—
207C2×#101—21no—
208#343—21no—
209#331—24no—
210D4⋊C8—24no—
211#324—22no—
212C2×#41—28no—
213C2×#41—44no—
214C2×#41—42no—
215C2×#41—42no—
216C22×Q8D4×C12711yescharacteristic
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—21no—
220C22×D4—21no—
221C22×D4C2×C4×C12712yes—
222C22×D4C2×C4×C12711yescharacteristic
223C23×C4—22no—
224C23×C4—21no—
225C2×M16—22no—
226#127—24no—
227#127—24no—
228#127—22no—
229#127D4×C12714yes—
230#121—28no—
231#121—216no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×D4—24no—
235C4×D4—24no—
236C4×D4—22no—
237C4×D4D4×C12714yes—
238C2×C4⋊C4C2×C4×C12711yescharacteristic
239C2×C4⋊C4D4×C12712yes—
240C2×C4⋊C4D4×C12711yescharacteristic
241C2×C22⋊C4—22no—
242C2×C22⋊C4—21no—
243C2×C42C2×C4×C12712yes—
244C2×C42C23×C12711yescharacteristic
245C4⋊C8—22no—
246C4⋊C8—22no—
247C4⋊C8—22no—
248#101—42no—
249C22⋊C8—22no—
250C4×C8—22no—
251C24—22no—
252C24—21no—
253#41—416no—
254#41—416no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8D8×C12712yes—
258C2×Q8SD16×C12712yes—
259C2×D4—28no—
260C2×D4—28no—
261C2×D4—21no—
262C2×D4—22no—
263C2×D4—24no—
264C2×D4—24no—
265C2×D4—42no—
266C2×D4—44no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—42no—
272C2×D4—82no—
273C2×D4—84no—
274C2×D4C22⋊C4×C12712yes—
275C2×D4C22⋊C4×C12714yes—
276C22×C4—28no—
277C22×C4—21no—
278C22×C4—24no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4C22⋊C4×C12712yes—
285C22×C4C22⋊C4×C12712yes—
286C22×C4C22×C4×C12711yescharacteristic
287C22×C4C2×D4×C12711yescharacteristic
288C22×C4C2×D4×C12711yescharacteristic
289M16—42no—
290M16—42no—
291C2×C8—42no—
292C2×C8—42no—
293C4⋊C4—21no—
294C4⋊C4—24no—
295C4⋊C4—22no—
296C4⋊C4C22⋊C4×C12712yes—
297C22⋊C4—28no—
298C22⋊C4—24no—
299C42—22no—
300C42—22no—
301C42C2×D4×C12712yes—
302C42C2×D4×C12712yes—
303C23—22no—
304C23—24no—
305C23—42no—
306C23—44no—
307C23—44no—
308C23—42no—
309C23—42no—
310C23—41no—
311C23—42no—
312C23—41no—
313C23C2×C22⋊C4×C12711yescharacteristic
314Q8—24no—
315Q8—24no—
316D4—24no—
317D4—44no—
318D4—416no—
319D4—42no—
320D4—44no—
321D4—44no—
322D4—44no—
323D4—84no—
324D4—88no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—416no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4#102×C12714yes—
338C2×C4C2×C22⋊C4×C12711yescharacteristic
339C2×C4C2×C22⋊C4×C12711yescharacteristic
340C2×C4#120×C12712yes—
341C2×C4#120×C12712yes—
342C2×C4C2×D8×C12711yescharacteristic
343C2×C4C2×SD16×C12711yescharacteristic
344C2×C4AGL(1,ℤ/8)×C12712yes—
345C8—82no—
346C8—82no—
347C22—21no—
348C22—21no—
349C22—42no—
350C22—44no—
351C22—44no—
352C22—42no—
353C22—44no—
354C22—42no—
355C22—82no—
356C22—84no—
357C22—82no—
358C22—84no—
359C22#379×C12711yescharacteristic
360C22C2×#102×C12711yescharacteristic
361C22#418×C12711yescharacteristic
362C4—21no—
363C4—24no—
364C4—22no—
365C4—42no—
366C4—42no—
367C4#413×C12712yes—
368C4#447×C12712yes—
369C4#450×C12712yes—
370C2—21no—
371C2—21no—
372C2—82no—
373C2—84no—
374C2C127×#178511yescharacteristic
375C2C127×#178611yescharacteristic
376C2C127×#179511yescharacteristic
subgroupG/H
center, socleC22×C127#379
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC2×#550×C127C2
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. C127×#1801C2semidirect
3. C127×#1545C2semidirect
4. C127×#1486C2semidirect
5. C127×#1461C2semidirect
181. #8384C127direct (N × H)
190. #1801C2×C127semidirect
191. #1545C2×C127semidirect
192. #1486C2×C127semidirect
193. #1461C2×C127semidirect
194. C127#8384direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#550×C1272
C127×#18012
C127×#15452
C127×#14862
C127×#14612
#8384127