Finite Group Databasebeta

← C127×#9480 #44573153

Subgroups — C127×#9480 #44573153

Order 32512.

726 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2406C211yesmaximal, characteristic
2C127×#2313C211yesmaximal, characteristic, complemented
3C127×#1789C212yesmaximal
4C127×#1463C211yesmaximal, characteristic, complemented
5C127×#1284C212yesmaximal, complemented
6C22×#41×C127C2211yescharacteristic
7C2×#123×C127—21no—
8C22×C4⋊C4×C127C2211yescharacteristic
9C2×#104×C127—24no—
10#407×C127C2212yes—
11#395×C127C412yescomplemented
12#394×C127C412yescomplemented
13#387×C127—24no—
14#386×C127—21no—
15#385×C127—22no—
16#381×C127—21no—
17#381×C127C2211yescharacteristic
18#380×C127—21no—
19C4×C22⋊C4×C127C2212yes—
20#324×C127—22no—
21C2×#41×C127—22no—
22C2×#41×C127—22no—
23C2×#41×C127—44no—
24C22×Q8×C127C2×C411yescharacteristic
25C22×D4×C127—21no—
26C22×D4×C127C2×C411yescharacteristic
27C23×C4×C127—21no—
28C23×C4×C127D411yescharacteristic
29C23×C4×C127D411yescharacteristic
30C23×C4×C127C2311yescharacteristic
31C2×M16×C127—22no—
32#123×C127—44no—
33C2×C4⋊C4×C127—22no—
34C2×C4⋊C4×C127—22no—
35C2×C4⋊C4×C127—22no—
36C2×C4⋊C4×C127—22no—
37C2×C4⋊C4×C127C2×C412yes—
38C2×C4⋊C4×C127D412yes—
39C2×C22⋊C4×C127—21no—
40C2×C22⋊C4×C127—22no—
41C2×C22⋊C4×C127—22no—
42C2×C22⋊C4×C127—22no—
43C2×C42×C127D414yes—
44#104×C127—48no—
45C22⋊C8×C127—22no—
46C22⋊C8×C127—22no—
47#95×C127—22no—
48#95×C127—24no—
49#95×C127—22no—
50#95×C127—22no—
51#95×C127—21no—
52#95×C127C2×C412yes—
53C24×C127—21no—
54C24×C127C2×D411yescharacteristic
55#41×C127—28no—
56#41×C127—44no—
57#41×C127—44no—
58#41×C127—44no—
59C2×Q8×C127—22no—
60C2×Q8×C127—42no—
61C2×D4×C127—24no—
62C2×D4×C127—22no—
63C2×D4×C127—42no—
64C2×D4×C127—42no—
65C2×D4×C127—41no—
66C2×D4×C127—41no—
67C22×C4×C127—21no—
68C22×C4×C127—21no—
69C22×C4×C127—21no—
70C22×C4×C127—21no—
71C22×C4×C127—21no—
72C22×C4×C127—22no—
73C22×C4×C127—24no—
74C22×C4×C127—42no—
75C22×C4×C127—42no—
76C22×C4×C127—42no—
77C22×C4×C127—42no—
78C22×C4×C127—42no—
79C22×C4×C127—42no—
80C22×C4×C127C22⋊C412yes—
81C22×C4×C127C22⋊C412yes—
82C22×C4×C127C22×C411yescharacteristic
83C22×C4×C127C2×D412yes—
84C22×C4×C127C2×D411yescharacteristic
85C22×C4×C127#4112yes—
86M16×C127—44no—
87C2×C8×C127—42no—
88C4⋊C4×C127—24no—
89C4⋊C4×C127—24no—
90C4⋊C4×C127—44no—
91C4⋊C4×C127—82no—
92C22⋊C4×C127—44no—
93C22⋊C4×C127—44no—
94C22⋊C4×C127—44no—
95C22⋊C4×C127—42no—
96C42×C127—44no—
97C23×C127—22no—
98C23×C127—22no—
99C23×C127—22no—
100C23×C127—41no—
101C23×C127—41no—
102C23×C127—42no—
103C23×C127—41no—
104C23×C127—42no—
105C23×C127C2×C22⋊C411yescharacteristic
106C23×C127#12011yescharacteristic
107C23×C127#12311yescharacteristic
108Q8×C127—24no—
109Q8×C127—42no—
110D4×C127—24no—
111D4×C127—42no—
112D4×C127—44no—
113D4×C127—44no—
114C2×C4×C127—24no—
115C2×C4×C127—22no—
116C2×C4×C127—22no—
117C2×C4×C127—22no—
118C2×C4×C127—22no—
119C2×C4×C127—22no—
120C2×C4×C127—41no—
121C2×C4×C127—41no—
122C2×C4×C127—42no—
123C2×C4×C127—48no—
124C2×C4×C127—42no—
125C2×C4×C127—42no—
126C2×C4×C127—41no—
127C2×C4×C127—41no—
128C2×C4×C127—42no—
129C2×C4×C127—42no—
130C2×C4×C127—42no—
131C2×C4×C127—44no—
132C2×C4×C127—44no—
133C2×C4×C127—42no—
134C2×C4×C127—42no—
135C2×C4×C127—44no—
136C2×C4×C127—81no—
137C2×C4×C127C4×D412yes—
138C2×C4×C127#12112yes—
139C8×C127—82no—
140C22×C127—21no—
141C22×C127—21no—
142C22×C127—21no—
143C22×C127—21no—
144C22×C127—21no—
145C22×C127—21no—
146C22×C127—21no—
147C22×C127—44no—
148C22×C127—44no—
149C22×C127—42no—
150C22×C127—82no—
151C22×C127—81no—
152C22×C127—81no—
153C22×C127#38611yescharacteristic
154C22×C127#42112yes—
155C22×C127B(4,𝔽2)12yes—
156C4×C127—22no—
157C4×C127—22no—
158C4×C127—42no—
159C4×C127—42no—
160C4×C127—42no—
161C4×C127—84no—
162C4×C127—82no—
163#9480C12711yesmaximal, characteristic, direct product factor, Sylow
164C2×C127—22no—
165C2×C127—22no—
166C2×C127—41no—
167C2×C127—82no—
168C2×C127#188911yescharacteristic
169C2×C127#189711yescharacteristic
170C2×C127#191211yescharacteristic
171#2406C2×C12711yescharacteristic
172#2313C2×C12711yescharacteristic, complemented
173#1789C2×C12712yes—
174#1463C2×C12711yescharacteristic, complemented
175#1284C2×C12712yescomplemented
176C127#948011yescharacteristic, direct product factor, Sylow
177C22×#41C22×C12711yescharacteristic
178C2×#123—21no—
179C22×C4⋊C4C22×C12711yescharacteristic
180C2×#104—24no—
181#407C22×C12712yes—
182#395C4×C12712yescomplemented
183#394C4×C12712yescomplemented
184#387—24no—
185#386—21no—
186#385—22no—
187#381—21no—
188#381C22×C12711yescharacteristic
189#380—21no—
190C4×C22⋊C4C22×C12712yes—
191#324—22no—
192C2×#41—22no—
193C2×#41—22no—
194C2×#41—44no—
195C22×Q8C2×C4×C12711yescharacteristic
196C22×D4—21no—
197C22×D4C2×C4×C12711yescharacteristic
198C23×C4—21no—
199C23×C4D4×C12711yescharacteristic
200C23×C4D4×C12711yescharacteristic
201C23×C4C23×C12711yescharacteristic
202C2×M16—22no—
203#123—44no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4C2×C4×C12712yes—
209C2×C4⋊C4D4×C12712yes—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C42D4×C12714yes—
215#104—48no—
216C22⋊C8—22no—
217C22⋊C8—22no—
218#95—22no—
219#95—24no—
220#95—22no—
221#95—22no—
222#95—21no—
223#95C2×C4×C12712yes—
224C24—21no—
225C24C2×D4×C12711yescharacteristic
226#41—28no—
227#41—44no—
228#41—44no—
229#41—44no—
230C2×Q8—22no—
231C2×Q8—42no—
232C2×D4—24no—
233C2×D4—22no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—41no—
237C2×D4—41no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4C22⋊C4×C12712yes—
252C22×C4C22⋊C4×C12712yes—
253C22×C4C22×C4×C12711yescharacteristic
254C22×C4C2×D4×C12712yes—
255C22×C4C2×D4×C12711yescharacteristic
256C22×C4#41×C12712yes—
257M16—44no—
258C2×C8—42no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—44no—
262C4⋊C4—82no—
263C22⋊C4—44no—
264C22⋊C4—44no—
265C22⋊C4—44no—
266C22⋊C4—42no—
267C42—44no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—41no—
272C23—41no—
273C23—42no—
274C23—41no—
275C23—42no—
276C23C2×C22⋊C4×C12711yescharacteristic
277C23#120×C12711yescharacteristic
278C23#123×C12711yescharacteristic
279Q8—24no—
280Q8—42no—
281D4—24no—
282D4—42no—
283D4—44no—
284D4—44no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—48no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—81no—
308C2×C4C4×D4×C12712yes—
309C2×C4#121×C12712yes—
310C8—82no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—44no—
319C22—44no—
320C22—42no—
321C22—82no—
322C22—81no—
323C22—81no—
324C22#386×C12711yescharacteristic
325C22#421×C12712yes—
326C22B(4,𝔽2)×C12712yes—
327C4—22no—
328C4—22no—
329C4—42no—
330C4—42no—
331C4—42no—
332C4—84no—
333C4—82no—
334C2—22no—
335C2—22no—
336C2—41no—
337C2—82no—
338C2C127×#188911yescharacteristic
339C2C127×#189711yescharacteristic
340C2C127×#191211yescharacteristic
subgroupG/H
center, socleC22×C127#386
commutatorC22×C4C22×C4×C127
Frattini, cosocleC23×C4C23×C127
prime radicalC22×#41×C127C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2313C2semidirect
4. C127×#1463C2semidirect
5. C127×#1284C2semidirect
11. #395×C127C4semidirect
12. #394×C127C4semidirect
163. #9480C127direct (N × H)
172. #2313C2×C127semidirect
174. #1463C2×C127semidirect
175. #1284C2×C127semidirect
176. C127#9480direct (N × H)
182. #395C4×C127semidirect
183. #394C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#24062
C127×#23132
C127×#17892
C127×#14632
C127×#12842
#9480127