Finite Group Databasebeta

← C127×#11777 #44575450

Subgroups — C127×#11777 #44575450

Order 32512.

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

normal subgroups only

This group has 480 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×#2547C211yesmaximal, characteristic
2C4×#99×C127C211yesmaximal, characteristic
3C127×#1465C211yesmaximal, characteristic
4C127×#1344C214yesmaximal, complemented
5C2×C4⋊Q8×C127C411yescharacteristic, complemented
6C2×#124×C127—21no—
7C2×C4×D4×C127C2211yescharacteristic
8C2×#99×C127—21no—
9#407×C127—21no—
10#397×C127C412yescomplemented
11#397×C127C2212yes—
12#391×C127—21no—
13#380×C127—21no—
14#380×C127—21no—
15C4×C4⋊C4×C127C411yescharacteristic, complemented
16C4×C22⋊C4×C127—21no—
17#342×C127C2212yes—
18#324×C127—21no—
19#323×C127C2212yes—
20C22×Q8×C127—21no—
21C22×D4×C127D411yescharacteristic
22C23×C4×C127D411yescharacteristic
23C23×C4×C127D411yescharacteristic
24C2×M16×C127—41no—
25C4⋊Q8×C127—41no—
26C4⋊Q8×C127—41no—
27#124×C127—44no—
28C4×D4×C127—41no—
29C4×D4×C127—41no—
30C2×C4⋊C4×C127—21no—
31C2×C4⋊C4×C127—21no—
32C2×C4⋊C4×C127C2×C412yes—
33C2×C4⋊C4×C127D411yescharacteristic
34C2×C22⋊C4×C127—22no—
35C2×C22⋊C4×C127—22no—
36C2×C22⋊C4×C127—22no—
37C2×C22⋊C4×C127C2×C411yescharacteristic
38C2×C22⋊C4×C127C2311yescharacteristic
39C2×C42×C127—21no—
40C2×C42×C127—41no—
41C2×C42×C127C2×C411yescharacteristic
42#99×C127—24no—
43C22⋊C8×C127—22no—
44C22⋊C8×C127—41no—
45#95×C127—22no—
46#95×C127—22no—
47#95×C127—41no—
48#95×C127C2×C412yes—
49C24×C127C22⋊C411yescharacteristic
50C24×C127C2×D411yescharacteristic
51C2×Q8×C127—42no—
52C2×Q8×C127—42no—
53C2×Q8×C127—42no—
54C2×D4×C127—22no—
55C2×D4×C127—22no—
56C2×D4×C127—41no—
57C22×C4×C127—21no—
58C22×C4×C127—21no—
59C22×C4×C127—21no—
60C22×C4×C127—21no—
61C22×C4×C127—22no—
62C22×C4×C127—22no—
63C22×C4×C127—42no—
64C22×C4×C127—42no—
65C22×C4×C127—42no—
66C22×C4×C127C22⋊C411yescharacteristic
67C22×C4×C127C22⋊C411yescharacteristic
68C22×C4×C127C22⋊C411yescharacteristic
69C22×C4×C127C22×C411yescharacteristic
70C22×C4×C127C2×D411yescharacteristic
71M16×C127—82no—
72C2×C8×C127—42no—
73C4⋊C4×C127—41no—
74C4⋊C4×C127—42no—
75C4⋊C4×C127—42no—
76C4⋊C4×C127—41no—
77C4⋊C4×C127—41no—
78C22⋊C4×C127—24no—
79C22⋊C4×C127—41no—
80C22⋊C4×C127—41no—
81C22⋊C4×C127—44no—
82C22⋊C4×C127—42no—
83C22⋊C4×C127—44no—
84C42×C127—21no—
85C42×C127—21no—
86C42×C127—42no—
87C42×C127—82no—
88C23×C127—21no—
89C23×C127—21no—
90C23×C127—21no—
91C23×C127—21no—
92C23×C127—21no—
93C23×C127—41no—
94C23×C127—41no—
95C23×C127—41no—
96C23×C127—41no—
97C23×C127C2×C22⋊C411yescharacteristic
98C23×C127AGL(1,ℤ/8)12yes—
99Q8×C127—84no—
100D4×C127—44no—
101C2×C4×C127—21no—
102C2×C4×C127—21no—
103C2×C4×C127—22no—
104C2×C4×C127—21no—
105C2×C4×C127—21no—
106C2×C4×C127—21no—
107C2×C4×C127—42no—
108C2×C4×C127—42no—
109C2×C4×C127—41no—
110C2×C4×C127—41no—
111C2×C4×C127—41no—
112C2×C4×C127—44no—
113C2×C4×C127—42no—
114C2×C4×C127—42no—
115C2×C4×C127—42no—
116C2×C4×C127—42no—
117C2×C4×C127—42no—
118C2×C4×C127—42no—
119C2×C4×C127—81no—
120C2×C4×C127—81no—
121C2×C4×C127—82no—
122C2×C4×C127—82no—
123C2×C4×C127#9911yescharacteristic
124C2×C4×C127#9911yescharacteristic
125C2×C4×C127#10412yes—
126C8×C127—82no—
127C22×C127—21no—
128C22×C127—21no—
129C22×C127—22no—
130C22×C127—42no—
131C22×C127—42no—
132C22×C127—41no—
133C22×C127—41no—
134C22×C127—41no—
135C22×C127—41no—
136C22×C127—41no—
137C22×C127—81no—
138C22×C127—81no—
139C22×C127C2×#9911yescharacteristic
140C22×C127#41811yescharacteristic
141C22×C127C2×#10411yescharacteristic
142C4×C127—21no—
143C4×C127—22no—
144C4×C127—41no—
145C4×C127—41no—
146C4×C127—41no—
147C4×C127—82no—
148C4×C127—84no—
149C4×C127—82no—
150C4×C127#35412yes—
151#11777C12711yesmaximal, characteristic, direct product factor, Sylow
152C2×C127—21no—
153C2×C127—21no—
154C2×C127—42no—
155C2×C127—81no—
156C2×C127#150711yescharacteristic
157C2×C127#212311yescharacteristic
158C2×C127C2×#35411yescharacteristic
159#2547C2×C12711yescharacteristic
160C4×#99C2×C12711yescharacteristic
161#1465C2×C12711yescharacteristic
162#1344C2×C12714yescomplemented
163C127#1177711yescharacteristic, direct product factor, Sylow
164C2×C4⋊Q8C4×C12711yescharacteristic, complemented
165C2×#124—21no—
166C2×C4×D4C22×C12711yescharacteristic
167C2×#99—21no—
168#407—21no—
169#397C4×C12712yescomplemented
170#397C22×C12712yes—
171#391—21no—
172#380—21no—
173#380—21no—
174C4×C4⋊C4C4×C12711yescharacteristic, complemented
175C4×C22⋊C4—21no—
176#342C22×C12712yes—
177#324—21no—
178#323C22×C12712yes—
179C22×Q8—21no—
180C22×D4D4×C12711yescharacteristic
181C23×C4D4×C12711yescharacteristic
182C23×C4D4×C12711yescharacteristic
183C2×M16—41no—
184C4⋊Q8—41no—
185C4⋊Q8—41no—
186#124—44no—
187C4×D4—41no—
188C4×D4—41no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4C2×C4×C12712yes—
192C2×C4⋊C4D4×C12711yescharacteristic
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4C2×C4×C12711yescharacteristic
197C2×C22⋊C4C23×C12711yescharacteristic
198C2×C42—21no—
199C2×C42—41no—
200C2×C42C2×C4×C12711yescharacteristic
201#99—24no—
202C22⋊C8—22no—
203C22⋊C8—41no—
204#95—22no—
205#95—22no—
206#95—41no—
207#95C2×C4×C12712yes—
208C24C22⋊C4×C12711yescharacteristic
209C24C2×D4×C12711yescharacteristic
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—41no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4C22⋊C4×C12711yescharacteristic
226C22×C4C22⋊C4×C12711yescharacteristic
227C22×C4C22⋊C4×C12711yescharacteristic
228C22×C4C22×C4×C12711yescharacteristic
229C22×C4C2×D4×C12711yescharacteristic
230M16—82no—
231C2×C8—42no—
232C4⋊C4—41no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C22⋊C4—24no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—44no—
241C22⋊C4—42no—
242C22⋊C4—44no—
243C42—21no—
244C42—21no—
245C42—42no—
246C42—82no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23C2×C22⋊C4×C12711yescharacteristic
257C23AGL(1,ℤ/8)×C12712yes—
258Q8—84no—
259D4—44no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—44no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4#99×C12711yescharacteristic
283C2×C4#99×C12711yescharacteristic
284C2×C4#104×C12712yes—
285C8—82no—
286C22—21no—
287C22—21no—
288C22—22no—
289C22—42no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—81no—
297C22—81no—
298C22C2×#99×C12711yescharacteristic
299C22#418×C12711yescharacteristic
300C22C2×#104×C12711yescharacteristic
301C4—21no—
302C4—22no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—82no—
307C4—84no—
308C4—82no—
309C4#354×C12712yes—
310C2—21no—
311C2—21no—
312C2—42no—
313C2—81no—
314C2C127×#150711yescharacteristic
315C2C127×#212311yescharacteristic
316C2C2×#354×C12711yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C22⋊C4C23×C127
prime radicalC22×D4×C127D4
LCSG ▷ C22×C4 ▷ C23 ▷ 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×#1344C2semidirect
5. C2×C4⋊Q8×C127C4semidirect
10. #397×C127C4semidirect
15. C4×C4⋊C4×C127C4semidirect
151. #11777C127direct (N × H)
162. #1344C2×C127semidirect
163. C127#11777direct (N × H)
164. C2×C4⋊Q8C4×C127semidirect
169. #397C4×C127semidirect
174. C4×C4⋊C4C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#25472
C4×#99×C1272
C127×#14652
C127×#13442
#11777127