Finite Group Databasebeta

← C127×#9334 #44573007

Subgroups — C127×#9334 #44573007

Order 32512.

398 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 114 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×#3215C211yesmaximal, characteristic
2C127×#2987C211yesmaximal, characteristic, complemented
3C127×#1554C211yesmaximal, characteristic, complemented
4C127×#1533C211yesmaximal, characteristic
5C127×#1469C211yesmaximal, characteristic
6C127×#1308C211yesmaximal, characteristic, complemented
7C127×#1287C211yesmaximal, characteristic, complemented
8#533×C127C2211yescharacteristic
9C2×#124×C127C2211yescharacteristic
10#497×C127—21no—
11#489×C127—21no—
12#464×C127—22no—
13#451×C127C412yes—
14#450×C127C412yes—
15#446×C127—22no—
16#446×C127—22no—
17#433×C127—22no—
18#432×C127C2211yescharacteristic
19#432×C127C2211yescharacteristic
20#432×C127C2211yescharacteristic
21#432×C127C2211yescharacteristic
22#431×C127C2211yescharacteristic
23#333×C127—21no—
24#332×C127—21no—
25#329×C127—22no—
26#323×C127—22no—
27C22×Q8×C127C2×C411yescharacteristic
28C22×D4×C127C2×C411yescharacteristic
29C2×Q16×C127—22no—
30C2×SD16×C127—22no—
31C4⋊Q8×C127D411yescharacteristic
32C4⋊Q8×C127D411yescharacteristic
33#125×C127—21no—
34#124×C127—21no—
35#124×C127—21no—
36#124×C127—21no—
37#124×C127—21no—
38C22⋊Q8×C127—21no—
39C22⋊Q8×C127C2×C412yes—
40C4×Q8×C127—21no—
41C4×Q8×C127D412yes—
42#117×C127—21no—
43C2×C22⋊C4×C127—21no—
44C2×C22⋊C4×C127—21no—
45C2×C42×C127C2311yescharacteristic
46C4⋊C8×C127—24no—
47C4⋊C8×C127—21no—
48C4⋊C8×C127—21no—
49C4⋊C8×C127—21no—
50C4⋊C8×C127—21no—
51C4⋊C8×C127—21no—
52#103×C127—22no—
53#102×C127—22no—
54C22⋊C8×C127—21no—
55C22⋊C8×C127—21no—
56C22⋊C8×C127—21no—
57C22⋊C8×C127C2×C412yes—
58C22⋊C8×C127D412yes—
59#97×C127—22no—
60#97×C127—21no—
61#97×C127—21no—
62#97×C127—21no—
63#97×C127D412yes—
64C4×C8×C127—22no—
65C4×C8×C127—21no—
66C24×C127—21no—
67C2×Q8×C127—21no—
68C2×Q8×C127—21no—
69C2×Q8×C127—21no—
70C2×Q8×C127—81no—
71C2×Q8×C127C22⋊C412yes—
72C2×D4×C127—21no—
73C2×D4×C127—21no—
74C2×D4×C127—81no—
75C22×C4×C127—21no—
76C22×C4×C127C22×C411yescharacteristic
77C22×C4×C127C2×D411yescharacteristic
78C22×C4×C127#4111yescharacteristic
79Q16×C127—82no—
80SD16×C127—82no—
81C2×C8×C127—22no—
82C2×C8×C127—22no—
83C2×C8×C127—22no—
84C2×C8×C127—41no—
85C2×C8×C127—41no—
86C4⋊C4×C127—22no—
87C4⋊C4×C127—21no—
88C4⋊C4×C127—21no—
89C4⋊C4×C127—21no—
90C4⋊C4×C127—22no—
91C4⋊C4×C127C22⋊C412yes—
92C22⋊C4×C127—22no—
93C22⋊C4×C127—41no—
94C22⋊C4×C127—41no—
95C22⋊C4×C127—41no—
96C22⋊C4×C127—41no—
97C42×C127—22no—
98C42×C127C2×D411yescharacteristic
99C42×C127C2×D411yescharacteristic
100C42×C127C2×D411yescharacteristic
101C42×C127#4111yescharacteristic
102C23×C127—42no—
103C23×C127—41no—
104C23×C127—81no—
105C23×C127—81no—
106C23×C127C4×D411yescharacteristic
107Q8×C127—42no—
108Q8×C127—42no—
109Q8×C127—81no—
110Q8×C127—81no—
111D4×C127—42no—
112D4×C127—81no—
113C2×C4×C127—22no—
114C2×C4×C127—22no—
115C2×C4×C127—41no—
116C2×C4×C127—41no—
117C2×C4×C127—41no—
118C2×C4×C127—41no—
119C2×C4×C127—81no—
120C2×C4×C127C2×C22⋊C411yescharacteristic
121C2×C4×C127C4×D411yescharacteristic
122C2×C4×C127#12011yescharacteristic
123C2×C4×C127#12111yescharacteristic
124C2×C4×C127#12111yescharacteristic
125C2×C4×C127#12311yescharacteristic
126C8×C127—42no—
127C8×C127—42no—
128C8×C127—42no—
129C8×C127—81no—
130C8×C127—81no—
131C22×C127—21no—
132C22×C127—21no—
133C22×C127—21no—
134C22×C127—81no—
135C22×C127—81no—
136C22×C127—81no—
137C22×C127—162no—
138C22×C127#38611yescharacteristic
139C4×C127—21no—
140C4×C127—21no—
141C4×C127—21no—
142C4×C127—21no—
143C4×C127—21no—
144C4×C127—21no—
145C4×C127—42no—
146C4×C127—42no—
147C4×C127—81no—
148#9334C12711yesmaximal, characteristic, direct product factor, Sylow
149C2×C127—41no—
150C2×C127—161no—
151C2×C127#188111yescharacteristic
152C2×C127#190511yescharacteristic
153C2×C127#191411yescharacteristic
154#3215C2×C12711yescharacteristic
155#2987C2×C12711yescharacteristic, complemented
156#1554C2×C12711yescharacteristic, complemented
157#1533C2×C12711yescharacteristic
158#1469C2×C12711yescharacteristic
159#1308C2×C12711yescharacteristic, complemented
160#1287C2×C12711yescharacteristic, complemented
161C127#933411yescharacteristic, direct product factor, Sylow
162#533C22×C12711yescharacteristic
163C2×#124C22×C12711yescharacteristic
164#497—21no—
165#489—21no—
166#464—22no—
167#451C4×C12712yes—
168#450C4×C12712yes—
169#446—22no—
170#446—22no—
171#433—22no—
172#432C22×C12711yescharacteristic
173#432C22×C12711yescharacteristic
174#432C22×C12711yescharacteristic
175#432C22×C12711yescharacteristic
176#431C22×C12711yescharacteristic
177#333—21no—
178#332—21no—
179#329—22no—
180#323—22no—
181C22×Q8C2×C4×C12711yescharacteristic
182C22×D4C2×C4×C12711yescharacteristic
183C2×Q16—22no—
184C2×SD16—22no—
185C4⋊Q8D4×C12711yescharacteristic
186C4⋊Q8D4×C12711yescharacteristic
187#125—21no—
188#124—21no—
189#124—21no—
190#124—21no—
191#124—21no—
192C22⋊Q8—21no—
193C22⋊Q8C2×C4×C12712yes—
194C4×Q8—21no—
195C4×Q8D4×C12712yes—
196#117—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C42C23×C12711yescharacteristic
200C4⋊C8—24no—
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206#103—22no—
207#102—22no—
208C22⋊C8—21no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8C2×C4×C12712yes—
212C22⋊C8D4×C12712yes—
213#97—22no—
214#97—21no—
215#97—21no—
216#97—21no—
217#97D4×C12712yes—
218C4×C8—22no—
219C4×C8—21no—
220C24—21no—
221C2×Q8—21no—
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—81no—
225C2×Q8C22⋊C4×C12712yes—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—81no—
229C22×C4—21no—
230C22×C4C22×C4×C12711yescharacteristic
231C22×C4C2×D4×C12711yescharacteristic
232C22×C4#41×C12711yescharacteristic
233Q16—82no—
234SD16—82no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—41no—
239C2×C8—41no—
240C4⋊C4—22no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—22no—
245C4⋊C4C22⋊C4×C12712yes—
246C22⋊C4—22no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C42—22no—
252C42C2×D4×C12711yescharacteristic
253C42C2×D4×C12711yescharacteristic
254C42C2×D4×C12711yescharacteristic
255C42#41×C12711yescharacteristic
256C23—42no—
257C23—41no—
258C23—81no—
259C23—81no—
260C23C4×D4×C12711yescharacteristic
261Q8—42no—
262Q8—42no—
263Q8—81no—
264Q8—81no—
265D4—42no—
266D4—81no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—81no—
274C2×C4C2×C22⋊C4×C12711yescharacteristic
275C2×C4C4×D4×C12711yescharacteristic
276C2×C4#120×C12711yescharacteristic
277C2×C4#121×C12711yescharacteristic
278C2×C4#121×C12711yescharacteristic
279C2×C4#123×C12711yescharacteristic
280C8—42no—
281C8—42no—
282C8—42no—
283C8—81no—
284C8—81no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22—162no—
292C22#386×C12711yescharacteristic
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—21no—
299C4—42no—
300C4—42no—
301C4—81no—
302C2—41no—
303C2—161no—
304C2C127×#188111yescharacteristic
305C2C127×#190511yescharacteristic
306C2C127×#191411yescharacteristic
subgroupG/H
center, socleC22×C127#386
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC22×D4×C127C2×C4
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×#2987C2semidirect
3. C127×#1554C2semidirect
6. C127×#1308C2semidirect
7. C127×#1287C2semidirect
148. #9334C127direct (N × H)
155. #2987C2×C127semidirect
156. #1554C2×C127semidirect
159. #1308C2×C127semidirect
160. #1287C2×C127semidirect
161. C127#9334direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32152
C127×#29872
C127×#15542
C127×#15332
C127×#14692
C127×#13082
C127×#12872
#9334127