Finite Group Databasebeta

← C127×#8398 #44572071

Subgroups — C127×#8398 #44572071

Order 32512.

782 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#554×C127C211yesmaximal, characteristic
2C127×#1801C211yesmaximal, characteristic, complemented
3C127×#1549C212yesmaximal, complemented
4C127×#1486C212yesmaximal
5C127×#1461C211yesmaximal, characteristic, complemented
6#554×C127—24no—
7#554×C127—24no—
8C2×C4⋊Q8×C127C2211yescharacteristic
9C2×#125×C127C412yes—
10C2×C22⋊Q8×C127—21no—
11C2×C22⋊Q8×C127—22no—
12C2×C4×Q8×C127C2211yescharacteristic
13C2×C4×D4×C127C412yes—
14C2×C4×D4×C127C2211yescharacteristic
15C22×C4⋊C4×C127—21no—
16#431×C127C2212yes—
17#423×C127C2212yes—
18C2×#101×C127—21no—
19#343×C127—21no—
20#332×C127—24no—
21D4⋊C8×C127—24no—
22#324×C127—22no—
23C22×Q8×C127—21no—
24C22×Q8×C127D411yescharacteristic
25C22×D4×C127C2×C411yescharacteristic
26C23×C4×C127—21no—
27C23×C4×C127—22no—
28C2×M16×C127—22no—
29C4⋊Q8×C127—22no—
30C4⋊Q8×C127D414yes—
31#125×C127—24no—
32#125×C127—24no—
33C22⋊Q8×C127—28no—
34C22⋊Q8×C127—216no—
35C4×Q8×C127—22no—
36C4×Q8×C127—22no—
37C4×D4×C127—24no—
38C4×D4×C127—24no—
39C4×D4×C127—22no—
40C4×D4×C127D414yes—
41C2×C4⋊C4×C127—28no—
42C2×C4⋊C4×C127—22no—
43C2×C4⋊C4×C127—22no—
44C2×C4⋊C4×C127—21no—
45C2×C4⋊C4×C127C2×C412yes—
46C2×C4⋊C4×C127C2×C411yescharacteristic
47C2×C4⋊C4×C127D411yescharacteristic
48C2×C4⋊C4×C127D412yes—
49C2×C22⋊C4×C127—22no—
50C2×C22⋊C4×C127—21no—
51C2×C42×C127C2×C412yes—
52C2×C42×C127C2311yescharacteristic
53C4⋊C8×C127—22no—
54C4⋊C8×C127—22no—
55C4⋊C8×C127—22no—
56#101×C127—42no—
57C22⋊C8×C127—22no—
58C4×C8×C127—22no—
59C24×C127—21no—
60C2×Q8×C127—22no—
61C2×Q8×C127—24no—
62C2×Q8×C127—22no—
63C2×Q8×C127—21no—
64C2×Q8×C127—21no—
65C2×Q8×C127—82no—
66C2×D4×C127—24no—
67C2×D4×C127—21no—
68C2×D4×C127C22⋊C412yes—
69C22×C4×C127—21no—
70C22×C4×C127—22no—
71C22×C4×C127—28no—
72C22×C4×C127—24no—
73C22×C4×C127—42no—
74C22×C4×C127—41no—
75C22×C4×C127—41no—
76C22×C4×C127—84no—
77C22×C4×C127C22⋊C412yes—
78C22×C4×C127C22⋊C412yes—
79C22×C4×C127C22×C411yescharacteristic
80C22×C4×C127C2×D411yescharacteristic
81C22×C4×C127C2×D411yescharacteristic
82M16×C127—42no—
83M16×C127—42no—
84C2×C8×C127—42no—
85C2×C8×C127—42no—
86C4⋊C4×C127—28no—
87C4⋊C4×C127—28no—
88C4⋊C4×C127—21no—
89C4⋊C4×C127—22no—
90C4⋊C4×C127—22no—
91C4⋊C4×C127—24no—
92C4⋊C4×C127—24no—
93C4⋊C4×C127C22⋊C414yes—
94C4⋊C4×C127C22⋊C412yes—
95C22⋊C4×C127—28no—
96C22⋊C4×C127—24no—
97C42×C127—22no—
98C42×C127—22no—
99C42×C127C2×D412yes—
100C42×C127C2×D412yes—
101C23×C127—24no—
102C23×C127—22no—
103C23×C127—42no—
104C23×C127—41no—
105C23×C127—41no—
106C23×C127C2×C22⋊C411yescharacteristic
107Q8×C127—42no—
108Q8×C127—42no—
109Q8×C127—84no—
110D4×C127—24no—
111D4×C127—22no—
112D4×C127#10114yescomplemented
113C2×C4×C127—22no—
114C2×C4×C127—22no—
115C2×C4×C127—22no—
116C2×C4×C127—22no—
117C2×C4×C127—24no—
118C2×C4×C127—22no—
119C2×C4×C127—22no—
120C2×C4×C127—41no—
121C2×C4×C127—44no—
122C2×C4×C127—41no—
123C2×C4×C127—41no—
124C2×C4×C127—44no—
125C2×C4×C127—44no—
126C2×C4×C127—42no—
127C2×C4×C127—88no—
128C2×C4×C127C2×C22⋊C411yescharacteristic
129C2×C4×C127C2×C22⋊C411yescharacteristic
130C2×C4×C127#12012yes—
131C2×C4×C127#12012yes—
132C2×C4×C127#13512yes—
133C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C127#13711yescharacteristic
135C8×C127—82no—
136C8×C127—82no—
137C22×C127—21no—
138C22×C127—21no—
139C22×C127—24no—
140C22×C127—44no—
141C22×C127—44no—
142C22×C127—42no—
143C22×C127—42no—
144C22×C127#37911yescharacteristic
145C22×C127#41611yescharacteristic
146C22×C127#41711yescharacteristic
147C4×C127—22no—
148C4×C127—21no—
149C4×C127—44no—
150C4×C127—42no—
151C4×C127—42no—
152C4×C127—42no—
153C4×C127C2×#10112yes—
154C4×C127#44912yes—
155C4×C127#45212yes—
156#8398C12711yesmaximal, characteristic, direct product factor, Sylow
157C2×C127—21no—
158C2×C127—21no—
159C2×C127—44no—
160C2×C127#178411yescharacteristic
161C2×C127#178811yescharacteristic
162C2×C127#179511yescharacteristic
163C2×#554C2×C12711yescharacteristic
164#1801C2×C12711yescharacteristic, complemented
165#1549C2×C12712yescomplemented
166#1486C2×C12712yes—
167#1461C2×C12711yescharacteristic, complemented
168C127#839811yescharacteristic, direct product factor, Sylow
169#554—24no—
170#554—24no—
171C2×C4⋊Q8C22×C12711yescharacteristic
172C2×#125C4×C12712yes—
173C2×C22⋊Q8—21no—
174C2×C22⋊Q8—22no—
175C2×C4×Q8C22×C12711yescharacteristic
176C2×C4×D4C4×C12712yes—
177C2×C4×D4C22×C12711yescharacteristic
178C22×C4⋊C4—21no—
179#431C22×C12712yes—
180#423C22×C12712yes—
181C2×#101—21no—
182#343—21no—
183#332—24no—
184D4⋊C8—24no—
185#324—22no—
186C22×Q8—21no—
187C22×Q8D4×C12711yescharacteristic
188C22×D4C2×C4×C12711yescharacteristic
189C23×C4—21no—
190C23×C4—22no—
191C2×M16—22no—
192C4⋊Q8—22no—
193C4⋊Q8D4×C12714yes—
194#125—24no—
195#125—24no—
196C22⋊Q8—28no—
197C22⋊Q8—216no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×D4—24no—
201C4×D4—24no—
202C4×D4—22no—
203C4×D4D4×C12714yes—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C2×C4×C12712yes—
209C2×C4⋊C4C2×C4×C12711yescharacteristic
210C2×C4⋊C4D4×C12711yescharacteristic
211C2×C4⋊C4D4×C12712yes—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—21no—
214C2×C42C2×C4×C12712yes—
215C2×C42C23×C12711yescharacteristic
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219#101—42no—
220C22⋊C8—22no—
221C4×C8—22no—
222C24—21no—
223C2×Q8—22no—
224C2×Q8—24no—
225C2×Q8—22no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—82no—
229C2×D4—24no—
230C2×D4—21no—
231C2×D4C22⋊C4×C12712yes—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—28no—
235C22×C4—24no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—84no—
240C22×C4C22⋊C4×C12712yes—
241C22×C4C22⋊C4×C12712yes—
242C22×C4C22×C4×C12711yescharacteristic
243C22×C4C2×D4×C12711yescharacteristic
244C22×C4C2×D4×C12711yescharacteristic
245M16—42no—
246M16—42no—
247C2×C8—42no—
248C2×C8—42no—
249C4⋊C4—28no—
250C4⋊C4—28no—
251C4⋊C4—21no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4C22⋊C4×C12714yes—
257C4⋊C4C22⋊C4×C12712yes—
258C22⋊C4—28no—
259C22⋊C4—24no—
260C42—22no—
261C42—22no—
262C42C2×D4×C12712yes—
263C42C2×D4×C12712yes—
264C23—24no—
265C23—22no—
266C23—42no—
267C23—41no—
268C23—41no—
269C23C2×C22⋊C4×C12711yescharacteristic
270Q8—42no—
271Q8—42no—
272Q8—84no—
273D4—24no—
274D4—22no—
275D4#101×C12714yescomplemented
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—88no—
291C2×C4C2×C22⋊C4×C12711yescharacteristic
292C2×C4C2×C22⋊C4×C12711yescharacteristic
293C2×C4#120×C12712yes—
294C2×C4#120×C12712yes—
295C2×C4#135×C12712yes—
296C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
297C2×C4#137×C12711yescharacteristic
298C8—82no—
299C8—82no—
300C22—21no—
301C22—21no—
302C22—24no—
303C22—44no—
304C22—44no—
305C22—42no—
306C22—42no—
307C22#379×C12711yescharacteristic
308C22#416×C12711yescharacteristic
309C22#417×C12711yescharacteristic
310C4—21no—
311C4—22no—
312C4—42no—
313C4—42no—
314C4—44no—
315C4—42no—
316C4C2×#101×C12712yes—
317C4#449×C12712yes—
318C4#452×C12712yes—
319C2—21no—
320C2—21no—
321C2—44no—
322C2C127×#178411yescharacteristic
323C2C127×#178811yescharacteristic
324C2C127×#179511yescharacteristic
subgroupG/H
center, socleC22×C127#379
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC22×D4×C127C2×C4
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×#1549C2semidirect
5. C127×#1461C2semidirect
112. D4×C127#101semidirect
156. #8398C127direct (N × H)
164. #1801C2×C127semidirect
165. #1549C2×C127semidirect
167. #1461C2×C127semidirect
168. C127#8398direct (N × H)
275. D4#101×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#554×C1272
C127×#18012
C127×#15492
C127×#14862
C127×#14612
#8398127