Finite Group Databasebeta

← C127×#8386 #44572059

Subgroups — C127×#8386 #44572059

Order 32512.

802 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#559×C127C211yesmaximal, characteristic
2C127×#1801C211yesmaximal, characteristic, complemented
3C127×#1545C212yesmaximal
4C127×#1487C212yesmaximal, complemented
5C127×#1461C211yesmaximal, characteristic, complemented
6#559×C127—24no—
7#559×C127—24no—
8C2×#127×C127C2211yescharacteristic
9C2×#124×C127—21no—
10C2×#124×C127C412yes—
11C2×#123×C127—22no—
12C2×C4×Q8×C127C2211yescharacteristic
13C2×C4×Q8×C127C2211yescharacteristic
14C2×C4×D4×C127—21no—
15C2×C4×D4×C127C412yes—
16#431×C127C2212yes—
17#423×C127C2212yes—
18C2×#101×C127—21no—
19#343×C127—21no—
20#331×C127—24no—
21Q8⋊C8×C127—24no—
22#324×C127—22no—
23C22×Q8×C127C2×C411yescharacteristic
24C22×Q8×C127D411yescharacteristic
25C22×D4×C127—21no—
26C22×D4×C127—21no—
27C22×D4×C127C2×C412yes—
28C23×C4×C127—22no—
29C2×M16×C127—22no—
30#127×C127—22no—
31#127×C127D414yes—
32#124×C127—28no—
33#124×C127—24no—
34#124×C127—24no—
35#123×C127—216no—
36C4×Q8×C127—22no—
37C4×Q8×C127—22no—
38C4×Q8×C127—22no—
39C4×Q8×C127D414yes—
40C4×D4×C127—28no—
41C4×D4×C127—24no—
42C4×D4×C127—24no—
43C2×C4⋊C4×C127—21no—
44C2×C4⋊C4×C127C2×C411yescharacteristic
45C2×C4⋊C4×C127D411yescharacteristic
46C2×C4⋊C4×C127D412yes—
47C2×C22⋊C4×C127—22no—
48C2×C22⋊C4×C127—22no—
49C2×C22⋊C4×C127—22no—
50C2×C42×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—22no—
60C2×Q8×C127—21no—
61C2×Q8×C127—22no—
62C2×Q8×C127—22no—
63C2×Q8×C127—21no—
64C2×Q8×C127—21no—
65C2×Q8×C127—42no—
66C2×Q8×C127C22⋊C412yes—
67C2×D4×C127—22no—
68C2×D4×C127—24no—
69C2×D4×C127—24no—
70C2×D4×C127—44no—
71C2×D4×C127—42no—
72C2×D4×C127—41no—
73C2×D4×C127—41no—
74C2×D4×C127—82no—
75C2×D4×C127C22⋊C414yes—
76C22×C4×C127—21no—
77C22×C4×C127—21no—
78C22×C4×C127—28no—
79C22×C4×C127—84no—
80C22×C4×C127C22⋊C412yes—
81C22×C4×C127C22⋊C412yes—
82C22×C4×C127C22×C411yescharacteristic
83C22×C4×C127C2×D411yescharacteristic
84C22×C4×C127C2×D411yescharacteristic
85M16×C127—42no—
86M16×C127—42no—
87C2×C8×C127—42no—
88C2×C8×C127—42no—
89C4⋊C4×C127—21no—
90C4⋊C4×C127—24no—
91C4⋊C4×C127—22no—
92C4⋊C4×C127—24no—
93C4⋊C4×C127C22⋊C412yes—
94C22⋊C4×C127—28no—
95C22⋊C4×C127—28no—
96C22⋊C4×C127—28no—
97C42×C127—22no—
98C42×C127—22no—
99C42×C127—24no—
100C42×C127C2×D412yes—
101C42×C127C2×D412yes—
102C23×C127—24no—
103C23×C127—44no—
104C23×C127—44no—
105C23×C127—42no—
106C23×C127—42no—
107C23×C127C2×C22⋊C411yescharacteristic
108Q8×C127—24no—
109Q8×C127—42no—
110Q8×C127—42no—
111Q8×C127—42no—
112D4×C127—44no—
113D4×C127—44no—
114D4×C127—44no—
115D4×C127—44no—
116D4×C127—84no—
117C2×C4×C127—22no—
118C2×C4×C127—22no—
119C2×C4×C127—22no—
120C2×C4×C127—22no—
121C2×C4×C127—22no—
122C2×C4×C127—22no—
123C2×C4×C127—22no—
124C2×C4×C127—41no—
125C2×C4×C127—42no—
126C2×C4×C127—42no—
127C2×C4×C127—42no—
128C2×C4×C127—88no—
129C2×C4×C127C2×C22⋊C411yescharacteristic
130C2×C4×C127C2×C22⋊C411yescharacteristic
131C2×C4×C127#12012yes—
132C2×C4×C127#12012yes—
133C2×C4×C127#13512yes—
134C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C127#13711yescharacteristic
136C8×C127—82no—
137C8×C127—82no—
138C22×C127—21no—
139C22×C127—21no—
140C22×C127—44no—
141C22×C127—44no—
142C22×C127—44no—
143C22×C127—84no—
144C22×C127—84no—
145C22×C127#37911yescharacteristic
146C22×C127#41611yescharacteristic
147C22×C127#41711yescharacteristic
148C4×C127—22no—
149C4×C127—21no—
150C4×C127—42no—
151C4×C127—42no—
152C4×C127—42no—
153C4×C127—42no—
154C4×C127#41312yes—
155C4×C127#44912yes—
156C4×C127#45212yes—
157#8386C12711yesmaximal, characteristic, direct product factor, Sylow
158C2×C127—21no—
159C2×C127—21no—
160C2×C127—84no—
161C2×C127#178511yescharacteristic
162C2×C127#178811yescharacteristic
163C2×C127#179511yescharacteristic
164C2×#559C2×C12711yescharacteristic
165#1801C2×C12711yescharacteristic, complemented
166#1545C2×C12712yes—
167#1487C2×C12712yescomplemented
168#1461C2×C12711yescharacteristic, complemented
169C127#838611yescharacteristic, direct product factor, Sylow
170#559—24no—
171#559—24no—
172C2×#127C22×C12711yescharacteristic
173C2×#124—21no—
174C2×#124C4×C12712yes—
175C2×#123—22no—
176C2×C4×Q8C22×C12711yescharacteristic
177C2×C4×Q8C22×C12711yescharacteristic
178C2×C4×D4—21no—
179C2×C4×D4C4×C12712yes—
180#431C22×C12712yes—
181#423C22×C12712yes—
182C2×#101—21no—
183#343—21no—
184#331—24no—
185Q8⋊C8—24no—
186#324—22no—
187C22×Q8C2×C4×C12711yescharacteristic
188C22×Q8D4×C12711yescharacteristic
189C22×D4—21no—
190C22×D4—21no—
191C22×D4C2×C4×C12712yes—
192C23×C4—22no—
193C2×M16—22no—
194#127—22no—
195#127D4×C12714yes—
196#124—28no—
197#124—24no—
198#124—24no—
199#123—216no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8D4×C12714yes—
204C4×D4—28no—
205C4×D4—24no—
206C4×D4—24no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C2×C4×C12711yescharacteristic
209C2×C4⋊C4D4×C12711yescharacteristic
210C2×C4⋊C4D4×C12712yes—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C42—21no—
215C2×C42C2×C4×C12712yes—
216C2×C42C23×C12711yescharacteristic
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220#101—42no—
221C22⋊C8—22no—
222C4×C8—22no—
223C24—22no—
224C2×Q8—21no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—42no—
230C2×Q8C22⋊C4×C12712yes—
231C2×D4—22no—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—44no—
235C2×D4—42no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—82no—
239C2×D4C22⋊C4×C12714yes—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—28no—
243C22×C4—84no—
244C22×C4C22⋊C4×C12712yes—
245C22×C4C22⋊C4×C12712yes—
246C22×C4C22×C4×C12711yescharacteristic
247C22×C4C2×D4×C12711yescharacteristic
248C22×C4C2×D4×C12711yescharacteristic
249M16—42no—
250M16—42no—
251C2×C8—42no—
252C2×C8—42no—
253C4⋊C4—21no—
254C4⋊C4—24no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4C22⋊C4×C12712yes—
258C22⋊C4—28no—
259C22⋊C4—28no—
260C22⋊C4—28no—
261C42—22no—
262C42—22no—
263C42—24no—
264C42C2×D4×C12712yes—
265C42C2×D4×C12712yes—
266C23—24no—
267C23—44no—
268C23—44no—
269C23—42no—
270C23—42no—
271C23C2×C22⋊C4×C12711yescharacteristic
272Q8—24no—
273Q8—42no—
274Q8—42no—
275Q8—42no—
276D4—44no—
277D4—44no—
278D4—44no—
279D4—44no—
280D4—84no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—88no—
293C2×C4C2×C22⋊C4×C12711yescharacteristic
294C2×C4C2×C22⋊C4×C12711yescharacteristic
295C2×C4#120×C12712yes—
296C2×C4#120×C12712yes—
297C2×C4#135×C12712yes—
298C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
299C2×C4#137×C12711yescharacteristic
300C8—82no—
301C8—82no—
302C22—21no—
303C22—21no—
304C22—44no—
305C22—44no—
306C22—44no—
307C22—84no—
308C22—84no—
309C22#379×C12711yescharacteristic
310C22#416×C12711yescharacteristic
311C22#417×C12711yescharacteristic
312C4—21no—
313C4—22no—
314C4—42no—
315C4—42no—
316C4—42no—
317C4—42no—
318C4#413×C12712yes—
319C4#449×C12712yes—
320C4#452×C12712yes—
321C2—21no—
322C2—21no—
323C2—84no—
324C2C127×#178511yescharacteristic
325C2C127×#178811yescharacteristic
326C2C127×#179511yescharacteristic
subgroupG/H
center, socleC22×C127#379
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC2×#127×C127C22
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
4. C127×#1487C2semidirect
5. C127×#1461C2semidirect
157. #8386C127direct (N × H)
165. #1801C2×C127semidirect
167. #1487C2×C127semidirect
168. #1461C2×C127semidirect
169. C127#8386direct (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
C127×#18012
C127×#15452
C127×#14872
C127×#14612
#8386127