Finite Group Databasebeta

← C127×#20694 #44584367

Subgroups — C127×#20694 #44584367

Order 32512.

1430 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#117×C127C211yesmaximal, characteristic
2C2×#407×C127C212yesmaximal
3C127×#1822C214yesmaximal, complemented
4C127×#1821C214yesmaximal, complemented
5C127×#1742C214yesmaximal, complemented
6C24×C4×C127C2211yescharacteristic
7C22×M16×C127C2214yes—
8C2×#117×C127—24no—
9C2×#117×C127—24no—
10C2×#117×C127C412yes—
11C2×#117×C127C412yes—
12C2×#117×C127C2212yes—
13C2×#117×C127C2212yes—
14C22×C4⋊C4×C127C2212yes—
15C22×C22⋊C4×C127C412yes—
16C22×C42×C127C412yes—
17#407×C127—24no—
18#407×C127C2214yes—
19#407×C127C2214yes—
20C2×C22⋊C8×C127—24no—
21#343×C127C22116yescomplemented
22C25×C127C2×C411yescharacteristic
23C23×C4×C127—28no—
24C23×C4×C127C2×C411yescharacteristic
25C23×C4×C127C2×C411yescharacteristic
26C23×C4×C127C2×C411yescharacteristic
27C23×C4×C127C2×C414yes—
28C23×C4×C127D414yes—
29C23×C4×C127D414yes—
30C23×C4×C127C2311yescharacteristic
31C23×C4×C127C2311yescharacteristic
32C23×C4×C127C2311yescharacteristic
33C2×M16×C127—24no—
34C2×M16×C127—216no—
35C2×M16×C127C2318yes—
36C22×C8×C127—24no—
37#117×C127—28no—
38#117×C127—24no—
39#117×C127—24no—
40#117×C127—28no—
41#117×C127—24no—
42#117×C127—24no—
43C2×C4⋊C4×C127—24no—
44C2×C4⋊C4×C127—24no—
45C2×C4⋊C4×C127C2×C414yes—
46C2×C4⋊C4×C127C2314yes—
47C2×C22⋊C4×C127—28no—
48C2×C22⋊C4×C127C2×C414yes—
49C2×C22⋊C4×C127C2×C414yes—
50C2×C42×C127—24no—
51C2×C42×C127—24no—
52C2×C42×C127C2×C414yes—
53C2×C42×C127C2×C414yes—
54C22⋊C8×C127—28no—
55C22⋊C8×C127—28no—
56C24×C127—24no—
57C24×C127—44no—
58C24×C127C22⋊C414yes—
59C24×C127C22×C411yescharacteristic
60C24×C127C22×C411yescharacteristic
61C24×C127C22×C411yescharacteristic
62C22×C4×C127—24no—
63C22×C4×C127—24no—
64C22×C4×C127—24no—
65C22×C4×C127—216no—
66C22×C4×C127—216no—
67C22×C4×C127—24no—
68C22×C4×C127—24no—
69C22×C4×C127—24no—
70C22×C4×C127—24no—
71C22×C4×C127—48no—
72C22×C4×C127C22⋊C412yes—
73C22×C4×C127C22⋊C414yes—
74C22×C4×C127C22⋊C412yes—
75C22×C4×C127C22⋊C412yes—
76C22×C4×C127C22⋊C412yes—
77C22×C4×C127C22×C411yescharacteristic
78C22×C4×C127C22×C411yescharacteristic
79C22×C4×C127C22×C418yes—
80C22×C4×C127C22×C411yescharacteristic
81C22×C4×C127C2×D414yes—
82C22×C4×C127C2×D414yes—
83C22×C4×C127C2×D412yes—
84C22×C4×C127C2×D412yes—
85C22×C4×C127#4118yes—
86C22×C4×C127C2411yescharacteristic
87M16×C127—216no—
88M16×C127—48no—
89C2×C8×C127—28no—
90C2×C8×C127—48no—
91C4⋊C4×C127—28no—
92C4⋊C4×C127—28no—
93C22⋊C4×C127—28no—
94C22⋊C4×C127—28no—
95C42×C127—28no—
96C42×C127—24no—
97C42×C127—24no—
98C23×C127—22no—
99C23×C127—24no—
100C23×C127—22no—
101C23×C127—28no—
102C23×C127—28no—
103C23×C127—44no—
104C23×C127—42no—
105C23×C127—42no—
106C23×C127—416no—
107C23×C127C2×C22⋊C412yes—
108C23×C127C2×C22⋊C412yes—
109C23×C127C2×C22⋊C412yes—
110C23×C127#11714yes—
111C23×C127C23×C411yescharacteristic
112C2×C4×C127—22no—
113C2×C4×C127—22no—
114C2×C4×C127—28no—
115C2×C4×C127—28no—
116C2×C4×C127—28no—
117C2×C4×C127—24no—
118C2×C4×C127—24no—
119C2×C4×C127—24no—
120C2×C4×C127—24no—
121C2×C4×C127—28no—
122C2×C4×C127—48no—
123C2×C4×C127—48no—
124C2×C4×C127—48no—
125C2×C4×C127C2×C22⋊C412yes—
126C2×C4×C127C2×C22⋊C412yes—
127C2×C4×C127C2×C22⋊C411yescharacteristic
128C2×C4×C127C2×C22⋊C411yescharacteristic
129C2×C4×C127#11714yes—
130C2×C4×C127#12318yes—
131C2×C4×C127#12318yes—
132C2×C4×C127C22×D412yes—
133C2×C4×C127C2×#4114yes—
134C8×C127—48no—
135C22×C127—28no—
136C22×C127—22no—
137C22×C127—22no—
138C22×C127—24no—
139C22×C127—22no—
140C22×C127—24no—
141C22×C127—22no—
142C22×C127—44no—
143C22×C127—416no—
144C22×C127—44no—
145C22×C127#38114yes—
146C22×C127#41314yes—
147C22×C127C22×C22⋊C411yescharacteristic
148C22×C127C2×#11712yes—
149C4×C127—22no—
150C4×C127—22no—
151C4×C127—48no—
152C4×C127#38112yes—
153C4×C127#38112yes—
154C4×C127C2×#12314yes—
155#20694C12711yesmaximal, characteristic, direct product factor, Sylow
156C2×C127—22no—
157C2×C127—24no—
158C2×C127—24no—
159C2×C127—42no—
160C2×C127C2×#38111yescharacteristic
161C2×C127C2×#41312yes—
162C22×#117C2×C12711yescharacteristic
163C2×#407C2×C12712yes—
164#1822C2×C12714yescomplemented
165#1821C2×C12714yescomplemented
166#1742C2×C12714yescomplemented
167C127#2069411yescharacteristic, direct product factor, Sylow
168C24×C4C22×C12711yescharacteristic
169C22×M16C22×C12714yes—
170C2×#117—24no—
171C2×#117—24no—
172C2×#117C4×C12712yes—
173C2×#117C4×C12712yes—
174C2×#117C22×C12712yes—
175C2×#117C22×C12712yes—
176C22×C4⋊C4C22×C12712yes—
177C22×C22⋊C4C4×C12712yes—
178C22×C42C4×C12712yes—
179#407—24no—
180#407C22×C12714yes—
181#407C22×C12714yes—
182C2×C22⋊C8—24no—
183#343C22×C127116yescomplemented
184C25C2×C4×C12711yescharacteristic
185C23×C4—28no—
186C23×C4C2×C4×C12711yescharacteristic
187C23×C4C2×C4×C12711yescharacteristic
188C23×C4C2×C4×C12711yescharacteristic
189C23×C4C2×C4×C12714yes—
190C23×C4D4×C12714yes—
191C23×C4D4×C12714yes—
192C23×C4C23×C12711yescharacteristic
193C23×C4C23×C12711yescharacteristic
194C23×C4C23×C12711yescharacteristic
195C2×M16—24no—
196C2×M16—216no—
197C2×M16C23×C12718yes—
198C22×C8—24no—
199#117—28no—
200#117—24no—
201#117—24no—
202#117—28no—
203#117—24no—
204#117—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4C2×C4×C12714yes—
208C2×C4⋊C4C23×C12714yes—
209C2×C22⋊C4—28no—
210C2×C22⋊C4C2×C4×C12714yes—
211C2×C22⋊C4C2×C4×C12714yes—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42C2×C4×C12714yes—
215C2×C42C2×C4×C12714yes—
216C22⋊C8—28no—
217C22⋊C8—28no—
218C24—24no—
219C24—44no—
220C24C22⋊C4×C12714yes—
221C24C22×C4×C12711yescharacteristic
222C24C22×C4×C12711yescharacteristic
223C24C22×C4×C12711yescharacteristic
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—216no—
228C22×C4—216no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—48no—
234C22×C4C22⋊C4×C12712yes—
235C22×C4C22⋊C4×C12712yes—
236C22×C4C22⋊C4×C12714yes—
237C22×C4C22⋊C4×C12712yes—
238C22×C4C22⋊C4×C12712yes—
239C22×C4C22×C4×C12711yescharacteristic
240C22×C4C22×C4×C12711yescharacteristic
241C22×C4C22×C4×C12718yes—
242C22×C4C22×C4×C12711yescharacteristic
243C22×C4C2×D4×C12714yes—
244C22×C4C2×D4×C12714yes—
245C22×C4C2×D4×C12712yes—
246C22×C4C2×D4×C12712yes—
247C22×C4#41×C12718yes—
248C22×C4C24×C12711yescharacteristic
249M16—216no—
250M16—48no—
251C2×C8—28no—
252C2×C8—48no—
253C4⋊C4—28no—
254C4⋊C4—28no—
255C22⋊C4—28no—
256C22⋊C4—28no—
257C42—28no—
258C42—24no—
259C42—24no—
260C23—22no—
261C23—24no—
262C23—22no—
263C23—28no—
264C23—28no—
265C23—44no—
266C23—42no—
267C23—42no—
268C23—416no—
269C23C2×C22⋊C4×C12712yes—
270C23C2×C22⋊C4×C12712yes—
271C23C2×C22⋊C4×C12712yes—
272C23#117×C12714yes—
273C23C23×C4×C12711yescharacteristic
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—28no—
277C2×C4—28no—
278C2×C4—28no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—28no—
284C2×C4—48no—
285C2×C4—48no—
286C2×C4—48no—
287C2×C4C2×C22⋊C4×C12712yes—
288C2×C4C2×C22⋊C4×C12712yes—
289C2×C4C2×C22⋊C4×C12711yescharacteristic
290C2×C4C2×C22⋊C4×C12711yescharacteristic
291C2×C4#117×C12714yes—
292C2×C4#123×C12718yes—
293C2×C4#123×C12718yes—
294C2×C4C22×D4×C12712yes—
295C2×C4C2×#41×C12714yes—
296C8—48no—
297C22—28no—
298C22—22no—
299C22—22no—
300C22—24no—
301C22—22no—
302C22—24no—
303C22—22no—
304C22—44no—
305C22—416no—
306C22—44no—
307C22#381×C12714yes—
308C22#413×C12714yes—
309C22C22×C22⋊C4×C12711yescharacteristic
310C22C2×#117×C12712yes—
311C4—22no—
312C4—22no—
313C4—48no—
314C4#381×C12712yes—
315C4#381×C12712yes—
316C4C2×#123×C12714yes—
317C2—22no—
318C2—24no—
319C2—24no—
320C2—42no—
321C2C2×#381×C12711yescharacteristic
322C2C2×#413×C12712yes—
subgroupG/H
centerC2×C4×C127C2×C22⋊C4
commutatorC23C23×C4×C127
Frattini, cosocleC22×C4C24×C127
socleC22×C127C22×C22⋊C4
prime radicalC25×C127C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C24×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#1822C2semidirect
4. C127×#1821C2semidirect
5. C127×#1742C2semidirect
21. #343×C127C22semidirect
155. #20694C127direct (N × H)
164. #1822C2×C127semidirect
165. #1821C2×C127semidirect
166. #1742C2×C127semidirect
167. C127#20694direct (N × H)
183. #343C22×C127semidirect

Maximal subgroups

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

subgroupindex
C22×#117×C1272
C2×#407×C1272
C127×#18222
C127×#18212
C127×#17422
#20694127