Finite Group Databasebeta

← C127×#11576 #44575249

Subgroups — C127×#11576 #44575249

Order 32512.

426 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2487C211yesmaximal, characteristic
2C127×#1466C211yesmaximal, characteristic
3C127×#1459C211yesmaximal, characteristic
4C127×#1326C214yesmaximal, complemented
5C2×C4×D4×C127C2211yescharacteristic
6C22×C4⋊C4×C127C411yescharacteristic
7#432×C127—21no—
8C2×C22⋊C8×C127—21no—
9#400×C127—21no—
10#399×C127—21no—
11#389×C127—21no—
12#386×C127C412yes—
13#385×C127C2212yes—
14#380×C127—21no—
15#380×C127C411yescharacteristic
16#324×C127—21no—
17#323×C127—21no—
18#323×C127C2212yes—
19#323×C127C2212yes—
20C22×D4×C127C2×C411yescharacteristic
21C23×C4×C127C2×C411yescharacteristic
22C23×C4×C127C2×C412yes—
23C23×C4×C127D411yescharacteristic
24C22×C8×C127—41no—
25C4×D4×C127—41no—
26C4×D4×C127—41no—
27C2×C4⋊C4×C127—21no—
28C2×C4⋊C4×C127—21no—
29C2×C4⋊C4×C127—21no—
30C2×C4⋊C4×C127—22no—
31C2×C4⋊C4×C127—41no—
32C2×C4⋊C4×C127—41no—
33C2×C4⋊C4×C127D411yescharacteristic
34C2×C22⋊C4×C127—22no—
35C2×C22⋊C4×C127C2×C412yes—
36C2×C22⋊C4×C127D411yescharacteristic
37C2×C22⋊C4×C127C2311yescharacteristic
38C2×C42×C127D411yescharacteristic
39C4⋊C8×C127—41no—
40C22⋊C8×C127—22no—
41C22⋊C8×C127—22no—
42C22⋊C8×C127—41no—
43#97×C127—41no—
44#95×C127—21no—
45#95×C127—22no—
46#95×C127—22no—
47C24×C127C22⋊C411yescharacteristic
48C24×C127C22×C411yescharacteristic
49C2×D4×C127—22no—
50C2×D4×C127—22no—
51C2×D4×C127—41no—
52C22×C4×C127—21no—
53C22×C4×C127—21no—
54C22×C4×C127—21no—
55C22×C4×C127—21no—
56C22×C4×C127—22no—
57C22×C4×C127—22no—
58C22×C4×C127—42no—
59C22×C4×C127—42no—
60C22×C4×C127—42no—
61C22×C4×C127—82no—
62C22×C4×C127C22⋊C411yescharacteristic
63C22×C4×C127C22⋊C411yescharacteristic
64C22×C4×C127C22⋊C411yescharacteristic
65C22×C4×C127C2×D411yescharacteristic
66C22×C4×C127C2×D411yescharacteristic
67C2×C8×C127—42no—
68C2×C8×C127—42no—
69C2×C8×C127—82no—
70C4⋊C4×C127—41no—
71C4⋊C4×C127—42no—
72C4⋊C4×C127—42no—
73C4⋊C4×C127—41no—
74C4⋊C4×C127—41no—
75C4⋊C4×C127—82no—
76C22⋊C4×C127—24no—
77C22⋊C4×C127—41no—
78C22⋊C4×C127—41no—
79C22⋊C4×C127—82no—
80C42×C127—21no—
81C42×C127—21no—
82C23×C127—21no—
83C23×C127—21no—
84C23×C127—21no—
85C23×C127—21no—
86C23×C127—41no—
87C23×C127—41no—
88C23×C127—41no—
89C23×C127—41no—
90C23×C127#10112yes—
91C23×C127C2×C22⋊C411yescharacteristic
92C23×C127#13112yes—
93D4×C127—44no—
94C2×C4×C127—22no—
95C2×C4×C127—21no—
96C2×C4×C127—21no—
97C2×C4×C127—21no—
98C2×C4×C127—21no—
99C2×C4×C127—21no—
100C2×C4×C127—21no—
101C2×C4×C127—42no—
102C2×C4×C127—42no—
103C2×C4×C127—42no—
104C2×C4×C127—41no—
105C2×C4×C127—41no—
106C2×C4×C127—41no—
107C2×C4×C127—42no—
108C2×C4×C127—42no—
109C2×C4×C127—42no—
110C2×C4×C127—42no—
111C2×C4×C127—42no—
112C2×C4×C127—42no—
113C2×C4×C127—81no—
114C2×C4×C127—84no—
115C2×C4×C127#9911yescharacteristic
116C2×C4×C127#9911yescharacteristic
117C8×C127—82no—
118C8×C127—82no—
119C22×C127—21no—
120C22×C127—21no—
121C22×C127—22no—
122C22×C127—22no—
123C22×C127—42no—
124C22×C127—42no—
125C22×C127—42no—
126C22×C127—41no—
127C22×C127—41no—
128C22×C127—41no—
129C22×C127—41no—
130C22×C127—81no—
131C22×C127#40811yescharacteristic
132C22×C127C2×#9911yescharacteristic
133C22×C127C2×#10111yescharacteristic
134C4×C127—21no—
135C4×C127—41no—
136C4×C127—41no—
137C4×C127—41no—
138C4×C127—41no—
139C4×C127—82no—
140C4×C127—82no—
141C4×C127#35212yes—
142#11576C12711yesmaximal, characteristic, direct product factor, Sylow
143C2×C127—21no—
144C2×C127—21no—
145C2×C127—42no—
146C2×C127—42no—
147C2×C127#147211yescharacteristic
148C2×C127C2×#35211yescharacteristic
149C2×C127#212211yescharacteristic
150#2487C2×C12711yescharacteristic
151#1466C2×C12711yescharacteristic
152#1459C2×C12711yescharacteristic
153#1326C2×C12714yescomplemented
154C127#1157611yescharacteristic, direct product factor, Sylow
155C2×C4×D4C22×C12711yescharacteristic
156C22×C4⋊C4C4×C12711yescharacteristic
157#432—21no—
158C2×C22⋊C8—21no—
159#400—21no—
160#399—21no—
161#389—21no—
162#386C4×C12712yes—
163#385C22×C12712yes—
164#380—21no—
165#380C4×C12711yescharacteristic
166#324—21no—
167#323—21no—
168#323C22×C12712yes—
169#323C22×C12712yes—
170C22×D4C2×C4×C12711yescharacteristic
171C23×C4C2×C4×C12711yescharacteristic
172C23×C4C2×C4×C12712yes—
173C23×C4D4×C12711yescharacteristic
174C22×C8—41no—
175C4×D4—41no—
176C4×D4—41no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4D4×C12711yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4C2×C4×C12712yes—
186C2×C22⋊C4D4×C12711yescharacteristic
187C2×C22⋊C4C23×C12711yescharacteristic
188C2×C42D4×C12711yescharacteristic
189C4⋊C8—41no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—41no—
193#97—41no—
194#95—21no—
195#95—22no—
196#95—22no—
197C24C22⋊C4×C12711yescharacteristic
198C24C22×C4×C12711yescharacteristic
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—41no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—82no—
212C22×C4C22⋊C4×C12711yescharacteristic
213C22×C4C22⋊C4×C12711yescharacteristic
214C22×C4C22⋊C4×C12711yescharacteristic
215C22×C4C2×D4×C12711yescharacteristic
216C22×C4C2×D4×C12711yescharacteristic
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—82no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—82no—
226C22⋊C4—24no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—82no—
230C42—21no—
231C42—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23#101×C12712yes—
241C23C2×C22⋊C4×C12711yescharacteristic
242C23#131×C12712yes—
243D4—44no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—81no—
264C2×C4—84no—
265C2×C4#99×C12711yescharacteristic
266C2×C4#99×C12711yescharacteristic
267C8—82no—
268C8—82no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—81no—
281C22#408×C12711yescharacteristic
282C22C2×#99×C12711yescharacteristic
283C22C2×#101×C12711yescharacteristic
284C4—21no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—82no—
290C4—82no—
291C4#352×C12712yes—
292C2—21no—
293C2—21no—
294C2—42no—
295C2—42no—
296C2C127×#147211yescharacteristic
297C2C2×#352×C12711yescharacteristic
298C2C127×#212211yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC24C22×C4×C127
Frattini, cosocleC2×C22⋊C4C23×C127
prime radicalC22×D4×C127C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C4×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1326C2semidirect
142. #11576C127direct (N × H)
153. #1326C2×C127semidirect
154. C127#11576direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#24872
C127×#14662
C127×#14592
C127×#13262
#11576127