Finite Group Databasebeta

← C127×#15986 #44579659

Subgroups — C127×#15986 #44579659

Order 32512.

2938 conjugacy classes of subgroups; only the 374 normal subgroups are shown.

normal subgroups only

This group has 2938 conjugacy classes of subgroups — too many to draw. Only its 374 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
1C22×#126×C127C211yesmaximal, characteristic, complemented
2C2×#379×C127C213yesmaximal
3C127×#1440C213yesmaximal, complemented
4C127×#1439C216yesmaximal, complemented
5C127×#1437C212yesmaximal, complemented
6C26×C127C2211yescharacteristic
7C2×#126×C127—212no—
8C22×C4⋊C4×C127C2213yescomplemented
9C22×C22⋊C4×C127C2216yes—
10C22×C22⋊C4×C127C2216yes—
11C22×C22⋊C4×C127C2213yes—
12C22×C22⋊C4×C127C2213yes—
13C22×C42×C127C2211yescharacteristic, complemented
14#379×C127—212no—
15C2×#95×C127C2216yescomplemented
16C2×#95×C127C2216yescomplemented
17C25×C127—26no—
18C25×C127—212no—
19C25×C127—46no—
20C25×C127C2311yescharacteristic
21C25×C127C2312yes—
22C23×C4×C127C2316yes—
23C23×C4×C127C2313yes—
24C23×C4×C127C2313yes—
25#126×C127—232no—
26C2×C4⋊C4×C127—26no—
27C2×C4⋊C4×C127—26no—
28C2×C4⋊C4×C127—26no—
29C2×C22⋊C4×C127—26no—
30C2×C22⋊C4×C127—26no—
31C2×C22⋊C4×C127—26no—
32C2×C22⋊C4×C127—212no—
33C2×C22⋊C4×C127—212no—
34C2×C22⋊C4×C127—212no—
35C2×C22⋊C4×C127—212no—
36C2×C22⋊C4×C127—46no—
37C2×C22⋊C4×C127—46no—
38C2×C22⋊C4×C127—46no—
39C2×C22⋊C4×C127D4112yes—
40C2×C42×C127—26no—
41#95×C127—46no—
42#95×C127—46no—
43C24×C127—26no—
44C24×C127—26no—
45C24×C127—26no—
46C24×C127—26no—
47C24×C127—26no—
48C24×C127—26no—
49C24×C127—224no—
50C24×C127—424no—
51C24×C127—424no—
52C24×C127—424no—
53C24×C127—412no—
54C24×C127—424no—
55C24×C127—48no—
56C24×C127—44no—
57C24×C127—48no—
58C24×C127C2×D416yes—
59C24×C127#41112yes—
60C24×C127C2411yescharacteristic
61C22×C4×C127—23no—
62C22×C4×C127—23no—
63C22×C4×C127—23no—
64C22×C4×C127—23no—
65C22×C4×C127—26no—
66C22×C4×C127—26no—
67C22×C4×C127—26no—
68C22×C4×C127—26no—
69C22×C4×C127—26no—
70C22×C4×C127—26no—
71C22×C4×C127—26no—
72C22×C4×C127—26no—
73C22×C4×C127—26no—
74C22×C4×C127—26no—
75C22×C4×C127—26no—
76C22×C4×C127C2×D416yes—
77C22×C4×C127C2×D416yes—
78C4⋊C4×C127—224no—
79C22⋊C4×C127—224no—
80C22⋊C4×C127—424no—
81C22⋊C4×C127—424no—
82C22⋊C4×C127—412no—
83C42×C127—28no—
84C23×C127—26no—
85C23×C127—212no—
86C23×C127—26no—
87C23×C127—212no—
88C23×C127—212no—
89C23×C127—212no—
90C23×C127—212no—
91C23×C127—212no—
92C23×C127—212no—
93C23×C127—212no—
94C23×C127—448no—
95C23×C127—496no—
96C23×C127—496no—
97C23×C127—42no—
98C23×C127—42no—
99C23×C127—42no—
100C23×C127—46no—
101C23×C127—46no—
102C23×C127—46no—
103C23×C127—46no—
104C23×C127—46no—
105C23×C127—46no—
106C23×C127—42no—
107C23×C127—42no—
108C23×C127—42no—
109C23×C127#12014yes—
110C23×C127#12618yescomplemented
111C23×C127C22×D413yes—
112C23×C127C2×#4116yes—
113C23×C127H(5,𝔽2)16yes—
114C2×C4×C127—26no—
115C2×C4×C127—26no—
116C2×C4×C127—212no—
117C2×C4×C127—212no—
118C2×C4×C127—43no—
119C2×C4×C127—43no—
120C2×C4×C127—43no—
121C2×C4×C127—43no—
122C2×C4×C127—46no—
123C2×C4×C127—46no—
124C2×C4×C127—46no—
125C2×C4×C127—46no—
126C2×C4×C127—46no—
127C2×C4×C127—46no—
128C2×C4×C127—46no—
129C2×C4×C127—46no—
130C2×C4×C127—46no—
131C22×C127—212no—
132C22×C127—224no—
133C22×C127—464no—
134C22×C127—412no—
135C22×C127—412no—
136C22×C127—412no—
137C22×C127—412no—
138C22×C127—412no—
139C22×C127—412no—
140C22×C127C2×#12011yescharacteristic
141C22×C127#52513yes—
142C22×C127C2×#12612yes—
143C22×C127#53513yes—
144C22×C127#53816yes—
145C22×C127#54616yes—
146C22×C127#54616yes—
147C22×C127#55116yes—
148C22×C127#56012yes—
149C4×C127—46no—
150C4×C127—46no—
151C4×C127—86no—
152#15986C12711yesmaximal, characteristic, direct product factor, Sylow
153C2×C127—44no—
154C2×C127—48no—
155C2×C127#240413yes—
156C2×C127#248016yes—
157C2×C127#268516yes—
158C22×#126C2×C12711yescharacteristic, complemented
159C2×#379C2×C12713yes—
160#1440C2×C12713yescomplemented
161#1439C2×C12716yescomplemented
162#1437C2×C12712yescomplemented
163C127#1598611yescharacteristic, direct product factor, Sylow
164C26C22×C12711yescharacteristic
165C2×#126—212no—
166C22×C4⋊C4C22×C12713yescomplemented
167C22×C22⋊C4C22×C12716yes—
168C22×C22⋊C4C22×C12716yes—
169C22×C22⋊C4C22×C12713yes—
170C22×C22⋊C4C22×C12713yes—
171C22×C42C22×C12711yescharacteristic, complemented
172#379—212no—
173C2×#95C22×C12716yescomplemented
174C2×#95C22×C12716yescomplemented
175C25—26no—
176C25—212no—
177C25—46no—
178C25C23×C12711yescharacteristic
179C25C23×C12712yes—
180C23×C4C23×C12716yes—
181C23×C4C23×C12713yes—
182C23×C4C23×C12713yes—
183#126—232no—
184C2×C4⋊C4—26no—
185C2×C4⋊C4—26no—
186C2×C4⋊C4—26no—
187C2×C22⋊C4—26no—
188C2×C22⋊C4—26no—
189C2×C22⋊C4—26no—
190C2×C22⋊C4—212no—
191C2×C22⋊C4—212no—
192C2×C22⋊C4—212no—
193C2×C22⋊C4—212no—
194C2×C22⋊C4—46no—
195C2×C22⋊C4—46no—
196C2×C22⋊C4—46no—
197C2×C22⋊C4D4×C127112yes—
198C2×C42—26no—
199#95—46no—
200#95—46no—
201C24—26no—
202C24—26no—
203C24—26no—
204C24—26no—
205C24—26no—
206C24—26no—
207C24—224no—
208C24—424no—
209C24—424no—
210C24—424no—
211C24—412no—
212C24—424no—
213C24—48no—
214C24—44no—
215C24—48no—
216C24C2×D4×C12716yes—
217C24#41×C127112yes—
218C24C24×C12711yescharacteristic
219C22×C4—23no—
220C22×C4—23no—
221C22×C4—23no—
222C22×C4—23no—
223C22×C4—26no—
224C22×C4—26no—
225C22×C4—26no—
226C22×C4—26no—
227C22×C4—26no—
228C22×C4—26no—
229C22×C4—26no—
230C22×C4—26no—
231C22×C4—26no—
232C22×C4—26no—
233C22×C4—26no—
234C22×C4C2×D4×C12716yes—
235C22×C4C2×D4×C12716yes—
236C4⋊C4—224no—
237C22⋊C4—224no—
238C22⋊C4—424no—
239C22⋊C4—424no—
240C22⋊C4—412no—
241C42—28no—
242C23—26no—
243C23—212no—
244C23—26no—
245C23—212no—
246C23—212no—
247C23—212no—
248C23—212no—
249C23—212no—
250C23—212no—
251C23—212no—
252C23—448no—
253C23—496no—
254C23—496no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—46no—
259C23—46no—
260C23—46no—
261C23—46no—
262C23—46no—
263C23—46no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23#120×C12714yes—
268C23#126×C12718yescomplemented
269C23C22×D4×C12713yes—
270C23C2×#41×C12716yes—
271C23H(5,𝔽2)×C12716yes—
272C2×C4—26no—
273C2×C4—26no—
274C2×C4—212no—
275C2×C4—212no—
276C2×C4—43no—
277C2×C4—43no—
278C2×C4—43no—
279C2×C4—43no—
280C2×C4—46no—
281C2×C4—46no—
282C2×C4—46no—
283C2×C4—46no—
284C2×C4—46no—
285C2×C4—46no—
286C2×C4—46no—
287C2×C4—46no—
288C2×C4—46no—
289C22—212no—
290C22—224no—
291C22—464no—
292C22—412no—
293C22—412no—
294C22—412no—
295C22—412no—
296C22—412no—
297C22—412no—
298C22C2×#120×C12711yescharacteristic
299C22#525×C12713yes—
300C22C2×#126×C12712yes—
301C22#535×C12713yes—
302C22#538×C12716yes—
303C22#546×C12716yes—
304C22#546×C12716yes—
305C22#551×C12716yes—
306C22#560×C12712yes—
307C4—46no—
308C4—46no—
309C4—86no—
310C2—44no—
311C2—48no—
312C2C127×#240413yes—
313C2C127×#248016yes—
314C2C127×#268516yes—
subgroupG/H
center, socleC24×C127C24
commutator, Frattini, cosocleC24C24×C127
prime radicalC26×C127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×#126×C127C2semidirect
3. C127×#1440C2semidirect
4. C127×#1439C2semidirect
5. C127×#1437C2semidirect
8. C22×C4⋊C4×C127C22semidirect
13. C22×C42×C127C22semidirect
15. C2×#95×C127C22semidirect
16. C2×#95×C127C22semidirect
110. C23×C127#126semidirect
152. #15986C127direct (N × H)
158. C22×#126C2×C127semidirect
160. #1440C2×C127semidirect
161. #1439C2×C127semidirect
162. #1437C2×C127semidirect
163. C127#15986direct (N × H)
166. C22×C4⋊C4C22×C127semidirect
171. C22×C42C22×C127semidirect
173. C2×#95C22×C127semidirect
174. C2×#95C22×C127semidirect
268. C23#126×C127semidirect

Maximal subgroups

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

subgroupindex
C22×#126×C1272
C2×#379×C1272
C127×#14402
C127×#14392
C127×#14372
#15986127