Finite Group Databasebeta

← C127×#11722 #44575395

Subgroups — C127×#11722 #44575395

Order 32512.

452 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#2627C211yesmaximal, characteristic
2C2×#343×C127C211yesmaximal, characteristic
3C2×#324×C127C211yesmaximal, characteristic
4C127×#1354C212yesmaximal, complemented
5C127×#1353C212yesmaximal, complemented
6C22×M16×C127—21no—
7C2×C22⋊Q8×C127C411yescharacteristic
8C22×C4⋊C4×C127C2211yescharacteristic
9C2×C22⋊C8×C127—21no—
10#400×C127C411yescharacteristic
11#400×C127C2211yescharacteristic
12#398×C127—21no—
13#398×C127—21no—
14#395×C127C411yescharacteristic
15#395×C127C2211yescharacteristic
16#385×C127—21no—
17#381×C127—21no—
18#380×C127C411yescharacteristic
19#343×C127—21no—
20#343×C127C2212yes—
21#324×C127—21no—
22#324×C127C2212yes—
23C22×Q8×C127C2×C411yescharacteristic
24C23×C4×C127C2×C411yescharacteristic
25C23×C4×C127D411yescharacteristic
26C23×C4×C127D411yescharacteristic
27C2×M16×C127—24no—
28C2×M16×C127—22no—
29C2×M16×C127—41no—
30C2×M16×C127—42no—
31C22×C8×C127—41no—
32C22×C8×C127—41no—
33C22⋊Q8×C127—41no—
34C22⋊Q8×C127—41no—
35C2×C4⋊C4×C127—21no—
36C2×C4⋊C4×C127—21no—
37C2×C4⋊C4×C127—21no—
38C2×C4⋊C4×C127—21no—
39C2×C4⋊C4×C127—41no—
40C2×C4⋊C4×C127—41no—
41C2×C4⋊C4×C127C2×C411yescharacteristic
42C2×C4⋊C4×C127C2×C411yescharacteristic
43C2×C4⋊C4×C127D411yescharacteristic
44C2×C4⋊C4×C127D411yescharacteristic
45C2×C4⋊C4×C127C2311yescharacteristic
46C2×C22⋊C4×C127—21no—
47C2×C22⋊C4×C127—21no—
48C22⋊C8×C127—22no—
49C22⋊C8×C127—41no—
50#95×C127—21no—
51#95×C127—21no—
52#95×C127—21no—
53#95×C127—21no—
54#95×C127—21no—
55#95×C127—21no—
56#95×C127C2×C411yescharacteristic
57#95×C127C2×C411yescharacteristic
58C24×C127C22⋊C411yescharacteristic
59C2×Q8×C127—41no—
60C2×Q8×C127—81no—
61C22×C4×C127—21no—
62C22×C4×C127—21no—
63C22×C4×C127—22no—
64C22×C4×C127—21no—
65C22×C4×C127—21no—
66C22×C4×C127—21no—
67C22×C4×C127—21no—
68C22×C4×C127—21no—
69C22×C4×C127—21no—
70C22×C4×C127—21no—
71C22×C4×C127—21no—
72C22×C4×C127—81no—
73C22×C4×C127—81no—
74C22×C4×C127C22⋊C411yescharacteristic
75C22×C4×C127C22⋊C411yescharacteristic
76C22×C4×C127C22⋊C411yescharacteristic
77C22×C4×C127SD1612yes—
78C22×C4×C127Q1612yes—
79C22×C4×C127C22×C411yescharacteristic
80C22×C4×C127C2×D411yescharacteristic
81C22×C4×C127C2×D411yescharacteristic
82M16×C127—24no—
83M16×C127—44no—
84M16×C127—42no—
85C2×C8×C127—44no—
86C2×C8×C127—42no—
87C2×C8×C127—42no—
88C2×C8×C127—82no—
89C4⋊C4×C127—41no—
90C4⋊C4×C127—41no—
91C4⋊C4×C127—41no—
92C4⋊C4×C127—41no—
93C4⋊C4×C127—41no—
94C4⋊C4×C127—42no—
95C4⋊C4×C127—81no—
96C4⋊C4×C127—81no—
97C4⋊C4×C127—81no—
98C22⋊C4×C127—41no—
99C22⋊C4×C127—41no—
100C22⋊C4×C127—42no—
101C23×C127—41no—
102C23×C127—41no—
103C23×C127#9911yescharacteristic
104C23×C127#9911yescharacteristic
105C23×C127#10312yes—
106C23×C127#10412yes—
107C23×C127C2×C22⋊C411yescharacteristic
108Q8×C127—82no—
109C2×C4×C127—22no—
110C2×C4×C127—22no—
111C2×C4×C127—22no—
112C2×C4×C127—22no—
113C2×C4×C127—21no—
114C2×C4×C127—21no—
115C2×C4×C127—21no—
116C2×C4×C127—21no—
117C2×C4×C127—21no—
118C2×C4×C127—21no—
119C2×C4×C127—21no—
120C2×C4×C127—21no—
121C2×C4×C127—42no—
122C2×C4×C127—41no—
123C2×C4×C127—41no—
124C2×C4×C127—41no—
125C2×C4×C127—41no—
126C2×C4×C127—41no—
127C2×C4×C127—41no—
128C2×C4×C127—41no—
129C2×C4×C127—41no—
130C2×C4×C127—41no—
131C2×C4×C127—41no—
132C2×C4×C127—42no—
133C2×C4×C127—41no—
134C2×C4×C127—41no—
135C2×C4×C127—41no—
136C2×C4×C127—41no—
137C2×C4×C127—81no—
138C2×C4×C127—81no—
139C2×C4×C127—81no—
140C2×C4×C127#10312yes—
141C2×C4×C127C2×SD1611yescharacteristic
142C2×C4×C127C2×Q1611yescharacteristic
143C8×C127—44no—
144C8×C127—82no—
145C22×C127—21no—
146C22×C127—21no—
147C22×C127—21no—
148C22×C127—21no—
149C22×C127—21no—
150C22×C127—21no—
151C22×C127—42no—
152C22×C127—42no—
153C22×C127#32812yes—
154C22×C127#32812yes—
155C22×C127C2×#9911yescharacteristic
156C22×C127C2×#10311yescharacteristic
157C22×C127C2×#10411yescharacteristic
158C4×C127—21no—
159C4×C127—22no—
160C4×C127—21no—
161C4×C127—41no—
162C4×C127—41no—
163C4×C127—41no—
164C4×C127—41no—
165C4×C127—81no—
166C4×C127—81no—
167C4×C127—81no—
168C4×C127—81no—
169#11722C12711yesmaximal, characteristic, direct product factor, Sylow
170C2×C127—21no—
171C2×C127—22no—
172C2×C127—22no—
173C2×C127—21no—
174C2×C127C2×#32811yescharacteristic
175C2×C127#212111yescharacteristic
176C2×C127#213311yescharacteristic
177#2627C2×C12711yescharacteristic
178C2×#343C2×C12711yescharacteristic
179C2×#324C2×C12711yescharacteristic
180#1354C2×C12712yescomplemented
181#1353C2×C12712yescomplemented
182C127#1172211yescharacteristic, direct product factor, Sylow
183C22×M16—21no—
184C2×C22⋊Q8C4×C12711yescharacteristic
185C22×C4⋊C4C22×C12711yescharacteristic
186C2×C22⋊C8—21no—
187#400C4×C12711yescharacteristic
188#400C22×C12711yescharacteristic
189#398—21no—
190#398—21no—
191#395C4×C12711yescharacteristic
192#395C22×C12711yescharacteristic
193#385—21no—
194#381—21no—
195#380C4×C12711yescharacteristic
196#343—21no—
197#343C22×C12712yes—
198#324—21no—
199#324C22×C12712yes—
200C22×Q8C2×C4×C12711yescharacteristic
201C23×C4C2×C4×C12711yescharacteristic
202C23×C4D4×C12711yescharacteristic
203C23×C4D4×C12711yescharacteristic
204C2×M16—24no—
205C2×M16—22no—
206C2×M16—41no—
207C2×M16—42no—
208C22×C8—41no—
209C22×C8—41no—
210C22⋊Q8—41no—
211C22⋊Q8—41no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4C2×C4×C12711yescharacteristic
219C2×C4⋊C4C2×C4×C12711yescharacteristic
220C2×C4⋊C4D4×C12711yescharacteristic
221C2×C4⋊C4D4×C12711yescharacteristic
222C2×C4⋊C4C23×C12711yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C22⋊C8—22no—
226C22⋊C8—41no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95C2×C4×C12711yescharacteristic
234#95C2×C4×C12711yescharacteristic
235C24C22⋊C4×C12711yescharacteristic
236C2×Q8—41no—
237C2×Q8—81no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4C22⋊C4×C12711yescharacteristic
252C22×C4C22⋊C4×C12711yescharacteristic
253C22×C4C22⋊C4×C12711yescharacteristic
254C22×C4SD16×C12712yes—
255C22×C4Q16×C12712yes—
256C22×C4C22×C4×C12711yescharacteristic
257C22×C4C2×D4×C12711yescharacteristic
258C22×C4C2×D4×C12711yescharacteristic
259M16—24no—
260M16—44no—
261M16—42no—
262C2×C8—44no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—82no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—81no—
273C4⋊C4—81no—
274C4⋊C4—81no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—42no—
278C23—41no—
279C23—41no—
280C23#99×C12711yescharacteristic
281C23#99×C12711yescharacteristic
282C23#103×C12712yes—
283C23#104×C12712yes—
284C23C2×C22⋊C4×C12711yescharacteristic
285Q8—82no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4#103×C12712yes—
318C2×C4C2×SD16×C12711yescharacteristic
319C2×C4C2×Q16×C12711yescharacteristic
320C8—44no—
321C8—82no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—42no—
329C22—42no—
330C22#328×C12712yes—
331C22#328×C12712yes—
332C22C2×#99×C12711yescharacteristic
333C22C2×#103×C12711yescharacteristic
334C22C2×#104×C12711yescharacteristic
335C4—21no—
336C4—22no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—81no—
346C2—21no—
347C2—22no—
348C2—22no—
349C2—21no—
350C2C2×#328×C12711yescharacteristic
351C2C127×#212111yescharacteristic
352C2C127×#213311yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C4⋊C4C23×C127
prime radicalC24×C127C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C24×C127 ◁ C22×C4⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1354C2semidirect
5. C127×#1353C2semidirect
169. #11722C127direct (N × H)
180. #1354C2×C127semidirect
181. #1353C2×C127semidirect
182. C127#11722direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#26272
C2×#343×C1272
C2×#324×C1272
C127×#13542
C127×#13532
#11722127