Finite Group Databasebeta

← C127×#21836 #44585509

Subgroups — C127×#21836 #44585509

Order 32512.

1042 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#125×C127C211yesmaximal, characteristic
2C2×#431×C127C211yesmaximal, characteristic
3C127×#2283C211yesmaximal, characteristic
4C127×#1991C214yesmaximal, complemented
5C127×#1989C214yesmaximal, complemented
6C127×#1824C212yesmaximal
7C127×#1823C212yesmaximal
8C2×#125×C127—21no—
9C2×#125×C127—21no—
10C2×#125×C127—24no—
11C2×#125×C127C418yescomplemented
12C2×#125×C127C2212yes—
13C2×#125×C127C2212yes—
14C22×C4⋊C4×C127—21no—
15C22×C4⋊C4×C127—21no—
16C22×C4⋊C4×C127C2211yescharacteristic
17C22×C4⋊C4×C127C2211yescharacteristic
18C22×C42×C127C2211yescharacteristic
19#431×C127—24no—
20C2×C4⋊C8×C127C2212yes—
21C2×C22⋊C8×C127C2212yes—
22C2×C4×C8×C127C2212yes—
23#383×C127—21no—
24#382×C127C2212yes—
25#381×C127—21no—
26#380×C127C2212yes—
27C4×C4⋊C4×C127C2212yes—
28C4×C22⋊C4×C127—21no—
29#376×C127—21no—
30#340×C127C2218yes—
31#340×C127C2218yes—
32C23×C4×C127—21no—
33C23×C4×C127—21no—
34C23×C4×C127D411yescharacteristic
35C23×C4×C127D411yescharacteristic
36C23×C4×C127C2311yescharacteristic
37C22×C8×C127C2314yes—
38#125×C127—28no—
39#125×C127—28no—
40#125×C127—24no—
41#125×C127—24no—
42#125×C127C2×C418yes—
43#125×C127C2×C418yes—
44C2×C4⋊C4×C127—28no—
45C2×C4⋊C4×C127—28no—
46C2×C4⋊C4×C127—22no—
47C2×C4⋊C4×C127—22no—
48C2×C4⋊C4×C127—22no—
49C2×C4⋊C4×C127—22no—
50C2×C4⋊C4×C127—21no—
51C2×C4⋊C4×C127—21no—
52C2×C4⋊C4×C127—22no—
53C2×C4⋊C4×C127—22no—
54C2×C4⋊C4×C127C2×C414yes—
55C2×C4⋊C4×C127C2×C414yes—
56C2×C4⋊C4×C127C2314yes—
57C2×C4⋊C4×C127C2312yes—
58C2×C4⋊C4×C127C2312yes—
59C2×C22⋊C4×C127—22no—
60C2×C42×C127—22no—
61C2×C42×C127C2×C414yes—
62C2×C42×C127D411yescharacteristic
63C2×C42×C127D411yescharacteristic
64C2×C42×C127D414yes—
65C2×C42×C127C2311yescharacteristic
66C2×C42×C127C2311yescharacteristic
67C4⋊C8×C127—24no—
68C22⋊C8×C127—24no—
69C4×C8×C127—24no—
70#95×C127—22no—
71#95×C127—22no—
72#95×C127—22no—
73#95×C127—22no—
74C24×C127C2×D411yescharacteristic
75C22×C4×C127—24no—
76C22×C4×C127—24no—
77C22×C4×C127—24no—
78C22×C4×C127—24no—
79C22×C4×C127—22no—
80C22×C4×C127—22no—
81C22×C4×C127—24no—
82C22×C4×C127—42no—
83C22×C4×C127—44no—
84C22×C4×C127—44no—
85C22×C4×C127C22⋊C414yes—
86C22×C4×C127C22⋊C414yes—
87C22×C4×C127C22×C412yes—
88C22×C4×C127C2×D411yescharacteristic
89C22×C4×C127C2×D411yescharacteristic
90C22×C4×C127C2×D411yescharacteristic
91C22×C4×C127C2×D411yescharacteristic
92C22×C4×C127C2×D411yescharacteristic
93C22×C4×C127C2×D412yes—
94C22×C4×C127C2411yescharacteristic
95C2×C8×C127—24no—
96C2×C8×C127—28no—
97C4⋊C4×C127—28no—
98C4⋊C4×C127—28no—
99C4⋊C4×C127—28no—
100C4⋊C4×C127—28no—
101C4⋊C4×C127—24no—
102C4⋊C4×C127—24no—
103C4⋊C4×C127—22no—
104C4⋊C4×C127—22no—
105C4⋊C4×C127—28no—
106C4⋊C4×C127C22×C414yes—
107C4⋊C4×C127C22×C414yes—
108C22⋊C4×C127—44no—
109C42×C127—44no—
110C42×C127C22⋊C414yes—
111C42×C127C22⋊C414yes—
112C42×C127C22×C412yes—
113C42×C127C22×C412yes—
114C42×C127C2×D412yes—
115C42×C127C2×D412yes—
116C23×C127—22no—
117C23×C127—22no—
118C23×C127C2×C22⋊C412yes—
119C23×C127#13514yes—
120C23×C127C22×D411yescharacteristic
121C2×C4×C127—22no—
122C2×C4×C127—22no—
123C2×C4×C127—22no—
124C2×C4×C127—22no—
125C2×C4×C127—22no—
126C2×C4×C127—28no—
127C2×C4×C127—24no—
128C2×C4×C127—24no—
129C2×C4×C127—24no—
130C2×C4×C127—24no—
131C2×C4×C127—44no—
132C2×C4×C127—48no—
133C2×C4×C127—44no—
134C2×C4×C127—44no—
135C2×C4×C127—48no—
136C2×C4×C127—44no—
137C2×C4×C127—44no—
138C2×C4×C127—44no—
139C2×C4×C127C2×C22⋊C412yes—
140C2×C4×C127C2×C22⋊C412yes—
141C2×C4×C127C2×C22⋊C412yes—
142C2×C4×C127C2×C22⋊C412yes—
143C2×C4×C127C2×C22⋊C412yes—
144C2×C4×C127C23×C411yescharacteristic
145C2×C4×C127C22×D411yescharacteristic
146C2×C4×C1272−1+414yes—
147C8×C127—44no—
148C22×C127—22no—
149C22×C127—22no—
150C22×C127—28no—
151C22×C127#41614yes—
152C22×C127C22×C22⋊C411yescharacteristic
153C22×C127#53612yes—
154C22×C127C2×#13512yes—
155C22×C127#57512yes—
156C4×C127—22no—
157C4×C127—22no—
158C4×C127—22no—
159C4×C127—22no—
160C4×C127—22no—
161C4×C127—44no—
162C4×C127—44no—
163C4×C127—84no—
164C4×C127#51912yes—
165C4×C127#53612yes—
166#21836C12711yesmaximal, characteristic, direct product factor, Sylow
167C2×C127—24no—
168C2×C127#231011yescharacteristic
169C2×C127C2×#41611yescharacteristic
170C2×C127#289611yescharacteristic
171C2×C127#323214yes—
172C22×#125C2×C12711yescharacteristic
173C2×#431C2×C12711yescharacteristic
174#2283C2×C12711yescharacteristic
175#1991C2×C12714yescomplemented
176#1989C2×C12714yescomplemented
177#1824C2×C12712yes—
178#1823C2×C12712yes—
179C127#2183611yescharacteristic, direct product factor, Sylow
180C2×#125—21no—
181C2×#125—21no—
182C2×#125—24no—
183C2×#125C4×C12718yescomplemented
184C2×#125C22×C12712yes—
185C2×#125C22×C12712yes—
186C22×C4⋊C4—21no—
187C22×C4⋊C4—21no—
188C22×C4⋊C4C22×C12711yescharacteristic
189C22×C4⋊C4C22×C12711yescharacteristic
190C22×C42C22×C12711yescharacteristic
191#431—24no—
192C2×C4⋊C8C22×C12712yes—
193C2×C22⋊C8C22×C12712yes—
194C2×C4×C8C22×C12712yes—
195#383—21no—
196#382C22×C12712yes—
197#381—21no—
198#380C22×C12712yes—
199C4×C4⋊C4C22×C12712yes—
200C4×C22⋊C4—21no—
201#376—21no—
202#340C22×C12718yes—
203#340C22×C12718yes—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4D4×C12711yescharacteristic
207C23×C4D4×C12711yescharacteristic
208C23×C4C23×C12711yescharacteristic
209C22×C8C23×C12714yes—
210#125—28no—
211#125—28no—
212#125—24no—
213#125—24no—
214#125C2×C4×C12718yes—
215#125C2×C4×C12718yes—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4C2×C4×C12714yes—
227C2×C4⋊C4C2×C4×C12714yes—
228C2×C4⋊C4C23×C12714yes—
229C2×C4⋊C4C23×C12712yes—
230C2×C4⋊C4C23×C12712yes—
231C2×C22⋊C4—22no—
232C2×C42—22no—
233C2×C42C2×C4×C12714yes—
234C2×C42D4×C12711yescharacteristic
235C2×C42D4×C12711yescharacteristic
236C2×C42D4×C12714yes—
237C2×C42C23×C12711yescharacteristic
238C2×C42C23×C12711yescharacteristic
239C4⋊C8—24no—
240C22⋊C8—24no—
241C4×C8—24no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246C24C2×D4×C12711yescharacteristic
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4C22⋊C4×C12714yes—
258C22×C4C22⋊C4×C12714yes—
259C22×C4C22×C4×C12712yes—
260C22×C4C2×D4×C12711yescharacteristic
261C22×C4C2×D4×C12711yescharacteristic
262C22×C4C2×D4×C12711yescharacteristic
263C22×C4C2×D4×C12711yescharacteristic
264C22×C4C2×D4×C12711yescharacteristic
265C22×C4C2×D4×C12712yes—
266C22×C4C24×C12711yescharacteristic
267C2×C8—24no—
268C2×C8—28no—
269C4⋊C4—28no—
270C4⋊C4—28no—
271C4⋊C4—28no—
272C4⋊C4—28no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—28no—
278C4⋊C4C22×C4×C12714yes—
279C4⋊C4C22×C4×C12714yes—
280C22⋊C4—44no—
281C42—44no—
282C42C22⋊C4×C12714yes—
283C42C22⋊C4×C12714yes—
284C42C22×C4×C12712yes—
285C42C22×C4×C12712yes—
286C42C2×D4×C12712yes—
287C42C2×D4×C12712yes—
288C23—22no—
289C23—22no—
290C23C2×C22⋊C4×C12712yes—
291C23#135×C12714yes—
292C23C22×D4×C12711yescharacteristic
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—28no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—44no—
304C2×C4—48no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—48no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4C2×C22⋊C4×C12712yes—
312C2×C4C2×C22⋊C4×C12712yes—
313C2×C4C2×C22⋊C4×C12712yes—
314C2×C4C2×C22⋊C4×C12712yes—
315C2×C4C2×C22⋊C4×C12712yes—
316C2×C4C23×C4×C12711yescharacteristic
317C2×C4C22×D4×C12711yescharacteristic
318C2×C42−1+4×C12714yes—
319C8—44no—
320C22—22no—
321C22—28no—
322C22—22no—
323C22#416×C12714yes—
324C22C22×C22⋊C4×C12711yescharacteristic
325C22#536×C12712yes—
326C22C2×#135×C12712yes—
327C22#575×C12712yes—
328C4—22no—
329C4—22no—
330C4—22no—
331C4—22no—
332C4—22no—
333C4—44no—
334C4—44no—
335C4—84no—
336C4#519×C12712yes—
337C4#536×C12712yes—
338C2—24no—
339C2C127×#231011yescharacteristic
340C2C2×#416×C12711yescharacteristic
341C2C127×#289611yescharacteristic
342C2C127×#323214yes—
subgroupG/H
center, socleC23×C127C22×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC24×C127C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C22×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1991C2semidirect
5. C127×#1989C2semidirect
11. C2×#125×C127C4semidirect
166. #21836C127direct (N × H)
175. #1991C2×C127semidirect
176. #1989C2×C127semidirect
179. C127#21836direct (N × H)
183. C2×#125C4×C127semidirect

Maximal subgroups

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

subgroupindex
C22×#125×C1272
C2×#431×C1272
C127×#22832
C127×#19912
C127×#19892
C127×#18242
C127×#18232
#21836127