Finite Group Databasebeta

← C127×#13606 #44577279

Subgroups — C127×#13606 #44577279

Order 32512.

320 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3221C211yesmaximal, characteristic
2C127×#2133C211yesmaximal, characteristic, complemented
3C127×#2129C211yesmaximal, characteristic, complemented
4C127×#1414C212yesmaximal, complemented
5C127×#1406C212yesmaximal, complemented
6#540×C127C411yescharacteristic, complemented
7#536×C127C2211yescharacteristic
8#496×C127C2211yescharacteristic
9#492×C127C411yescharacteristic, complemented
10#491×C127C411yescharacteristic, complemented
11#488×C127C2211yescharacteristic
12#463×C127—21no—
13#461×C127—21no—
14#449×C127—21no—
15#448×C127—21no—
16#433×C127C411yescharacteristic, complemented
17C2×#101×C127—21no—
18#410×C127—21no—
19#356×C127—21no—
20#355×C127C2212yes—
21#354×C127—21no—
22#354×C127C2212yes—
23C2×#41×C127D411yescharacteristic
24C22×Q8×C127D411yescharacteristic
25C2×SD16×C127—21no—
26C2×SD16×C127—21no—
27C2×SD16×C127—21no—
28C2×D8×C127—21no—
29C2×M16×C127—41no—
30C4⋊Q8×C127C2×C411yescharacteristic
31C4⋊Q8×C127D411yescharacteristic
32#127×C127C2×C411yescharacteristic
33#125×C127C2×C411yescharacteristic
34#124×C127D411yescharacteristic
35#124×C127C2311yescharacteristic
36#123×C127—41no—
37#123×C127—41no—
38C22⋊Q8×C127—41no—
39#121×C127—21no—
40#121×C127—41no—
41C4×D4×C127—21no—
42#117×C127—41no—
43#117×C127C2×C411yescharacteristic
44C4⋊C8×C127—21no—
45#103×C127—21no—
46#103×C127—21no—
47#103×C127—21no—
48#102×C127—21no—
49#101×C127—22no—
50#101×C127—41no—
51#99×C127—22no—
52#99×C127—41no—
53C22⋊C8×C127—21no—
54#97×C127C2×C411yescharacteristic
55C4×C8×C127C2×C411yescharacteristic
56#41×C127—81no—
57C2×Q8×C127—21no—
58C2×Q8×C127—81no—
59C2×Q8×C127C22⋊C411yescharacteristic
60C2×Q8×C127C22⋊C411yescharacteristic
61C2×Q8×C127C2×D411yescharacteristic
62C2×D4×C127—21no—
63C2×D4×C127—21no—
64C2×D4×C127—41no—
65C2×D4×C127—41no—
66C2×D4×C127C2×D411yescharacteristic
67C22×C4×C127—21no—
68C22×C4×C127—21no—
69C22×C4×C127—41no—
70C22×C4×C127C22⋊C411yescharacteristic
71SD16×C127—81no—
72SD16×C127—81no—
73SD16×C127—81no—
74D8×C127—81no—
75M16×C127—42no—
76M16×C127—81no—
77C2×C8×C127—21no—
78C2×C8×C127—21no—
79C2×C8×C127—81no—
80C4⋊C4×C127—21no—
81C4⋊C4×C127—21no—
82C4⋊C4×C127—41no—
83C4⋊C4×C127—41no—
84C4⋊C4×C127—41no—
85C4⋊C4×C127—81no—
86C22⋊C4×C127—21no—
87C22⋊C4×C127—41no—
88C22⋊C4×C127—41no—
89C22⋊C4×C127—41no—
90C22⋊C4×C127—41no—
91C22⋊C4×C127—42no—
92C42×C127—81no—
93C42×C127C22⋊C411yescharacteristic
94C42×C127C22×C411yescharacteristic
95C23×C127—21no—
96C23×C127—41no—
97C23×C127#9911yescharacteristic
98Q8×C127—41no—
99Q8×C127—41no—
100Q8×C127—41no—
101Q8×C127—81no—
102D4×C127—41no—
103D4×C127—41no—
104D4×C127—41no—
105D4×C127—81no—
106D4×C127—161no—
107D4×C127—161no—
108C2×C4×C127—21no—
109C2×C4×C127—21no—
110C2×C4×C127—21no—
111C2×C4×C127—21no—
112C2×C4×C127—21no—
113C2×C4×C127—41no—
114C2×C4×C127—41no—
115C2×C4×C127—41no—
116C2×C4×C127—41no—
117C2×C4×C127—41no—
118C2×C4×C127—41no—
119C2×C4×C127—82no—
120C2×C4×C127—161no—
121C2×C4×C127#9911yescharacteristic
122C2×C4×C127C2×C22⋊C411yescharacteristic
123C8×C127—41no—
124C8×C127—41no—
125C8×C127—82no—
126C22×C127—21no—
127C22×C127—41no—
128C22×C127—41no—
129C22×C127—81no—
130C22×C127—81no—
131C22×C127—81no—
132C22×C127—81no—
133C22×C127C2×#9911yescharacteristic
134C4×C127—21no—
135C4×C127—21no—
136C4×C127—41no—
137C4×C127—41no—
138C4×C127—41no—
139C4×C127—41no—
140C4×C127—41no—
141C4×C127—81no—
142C4×C127—162no—
143#13606C12711yesmaximal, characteristic, direct product factor, Sylow
144C2×C127—21no—
145C2×C127—41no—
146C2×C127—81no—
147C2×C127—161no—
148C2×C127#212711yescharacteristic
149#3221C2×C12711yescharacteristic
150#2133C2×C12711yescharacteristic, complemented
151#2129C2×C12711yescharacteristic, complemented
152#1414C2×C12712yescomplemented
153#1406C2×C12712yescomplemented
154C127#1360611yescharacteristic, direct product factor, Sylow
155#540C4×C12711yescharacteristic, complemented
156#536C22×C12711yescharacteristic
157#496C22×C12711yescharacteristic
158#492C4×C12711yescharacteristic, complemented
159#491C4×C12711yescharacteristic, complemented
160#488C22×C12711yescharacteristic
161#463—21no—
162#461—21no—
163#449—21no—
164#448—21no—
165#433C4×C12711yescharacteristic, complemented
166C2×#101—21no—
167#410—21no—
168#356—21no—
169#355C22×C12712yes—
170#354—21no—
171#354C22×C12712yes—
172C2×#41D4×C12711yescharacteristic
173C22×Q8D4×C12711yescharacteristic
174C2×SD16—21no—
175C2×SD16—21no—
176C2×SD16—21no—
177C2×D8—21no—
178C2×M16—41no—
179C4⋊Q8C2×C4×C12711yescharacteristic
180C4⋊Q8D4×C12711yescharacteristic
181#127C2×C4×C12711yescharacteristic
182#125C2×C4×C12711yescharacteristic
183#124D4×C12711yescharacteristic
184#124C23×C12711yescharacteristic
185#123—41no—
186#123—41no—
187C22⋊Q8—41no—
188#121—21no—
189#121—41no—
190C4×D4—21no—
191#117—41no—
192#117C2×C4×C12711yescharacteristic
193C4⋊C8—21no—
194#103—21no—
195#103—21no—
196#103—21no—
197#102—21no—
198#101—22no—
199#101—41no—
200#99—22no—
201#99—41no—
202C22⋊C8—21no—
203#97C2×C4×C12711yescharacteristic
204C4×C8C2×C4×C12711yescharacteristic
205#41—81no—
206C2×Q8—21no—
207C2×Q8—81no—
208C2×Q8C22⋊C4×C12711yescharacteristic
209C2×Q8C22⋊C4×C12711yescharacteristic
210C2×Q8C2×D4×C12711yescharacteristic
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4C2×D4×C12711yescharacteristic
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—41no—
219C22×C4C22⋊C4×C12711yescharacteristic
220SD16—81no—
221SD16—81no—
222SD16—81no—
223D8—81no—
224M16—42no—
225M16—81no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—81no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—81no—
235C22⋊C4—21no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—42no—
241C42—81no—
242C42C22⋊C4×C12711yescharacteristic
243C42C22×C4×C12711yescharacteristic
244C23—21no—
245C23—41no—
246C23#99×C12711yescharacteristic
247Q8—41no—
248Q8—41no—
249Q8—41no—
250Q8—81no—
251D4—41no—
252D4—41no—
253D4—41no—
254D4—81no—
255D4—161no—
256D4—161no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—82no—
269C2×C4—161no—
270C2×C4#99×C12711yescharacteristic
271C2×C4C2×C22⋊C4×C12711yescharacteristic
272C8—41no—
273C8—41no—
274C8—82no—
275C22—21no—
276C22—41no—
277C22—41no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22C2×#99×C12711yescharacteristic
283C4—21no—
284C4—21no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—81no—
291C4—162no—
292C2—21no—
293C2—41no—
294C2—81no—
295C2—161no—
296C2C127×#212711yescharacteristic
subgroupG/H
center, socleC2×C127#2127
commutatorC42C22×C4×C127
Frattini, cosocle#124C23×C127
prime radicalC127×#3221C2
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C22×C127 ◁ C22×C4×C127 ◁ #536×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2133C2semidirect
3. C127×#2129C2semidirect
4. C127×#1414C2semidirect
5. C127×#1406C2semidirect
6. #540×C127C4semidirect
9. #492×C127C4semidirect
10. #491×C127C4semidirect
16. #433×C127C4semidirect
143. #13606C127direct (N × H)
150. #2133C2×C127semidirect
151. #2129C2×C127semidirect
152. #1414C2×C127semidirect
153. #1406C2×C127semidirect
154. C127#13606direct (N × H)
155. #540C4×C127semidirect
158. #492C4×C127semidirect
159. #491C4×C127semidirect
165. #433C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#32212
C127×#21332
C127×#21292
C127×#14142
C127×#14062
#13606127