Finite Group Databasebeta

← C127×#12686 #44576359

Subgroups — C127×#12686 #44576359

Order 32512.

904 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#568×C127C211yesmaximal, characteristic
2C127×#2117C214yesmaximal, complemented
3C2×#349×C127C212yesmaximal, complemented
4C22×#41×C127C411yescharacteristic
5#568×C127—28no—
6C2×#131×C127—22no—
7C2×#131×C127C412yes—
8C2×#131×C127C2212yes—
9C22×M16×C127—22no—
10C22×M16×C127C411yescharacteristic
11C22×M16×C127C2211yescharacteristic
12C2×M32×C127C2214yescomplemented
13#350×C127—28no—
14#349×C127—22no—
15#349×C127—22no—
16C2×#41×C127—24no—
17C2×#41×C127—24no—
18C2×#41×C127—21no—
19C2×#41×C127—21no—
20C2×#41×C127C812yes—
21C2×#41×C127C2×C412yes—
22C22×Q8×C127C811yescharacteristic
23C22×D4×C127—21no—
24C22×D4×C127C811yescharacteristic
25C23×C4×C127—21no—
26C23×C4×C127C2×C411yescharacteristic
27#131×C127—216no—
28#131×C127—24no—
29#131×C127—24no—
30#131×C127—24no—
31#131×C127D418yes—
32C2×M16×C127—216no—
33C2×M16×C127—21no—
34C2×M16×C127—24no—
35C2×M16×C127—22no—
36C2×M16×C127—22no—
37C2×M16×C127—24no—
38C2×M16×C127C2×C412yes—
39C2×M16×C127D412yes—
40C2×M16×C127D412yes—
41C22×C8×C127—21no—
42C22×C8×C127—22no—
43C22×C8×C127—22no—
44C22×C8×C127C2×C411yescharacteristic
45C22×C8×C127C2311yescharacteristic
46M32×C127—24no—
47M32×C127—24no—
48C2×C16×C127—24no—
49C24×C127—21no—
50C24×C127M1611yescharacteristic
51#41×C127—24no—
52#41×C127—28no—
53#41×C127—24no—
54#41×C127—24no—
55#41×C127—24no—
56#41×C127—22no—
57#41×C127—22no—
58#41×C127C22⋊C414yes—
59#41×C127M1614yes—
60C2×Q8×C127—24no—
61C2×Q8×C127—21no—
62C2×Q8×C127C2×C812yes—
63C2×D4×C127—28no—
64C2×D4×C127—22no—
65C2×D4×C127—22no—
66C2×D4×C127—24no—
67C2×D4×C127—21no—
68C2×D4×C127C2×C812yes—
69C22×C4×C127—24no—
70C22×C4×C127—24no—
71C22×C4×C127—21no—
72C22×C4×C127—22no—
73C22×C4×C127—22no—
74C22×C4×C127—22no—
75C22×C4×C127—22no—
76C22×C4×C127C22⋊C411yescharacteristic
77C22×C4×C127C22⋊C411yescharacteristic
78C22×C4×C127C2×C811yescharacteristic
79C22×C4×C127C2×C811yescharacteristic
80C22×C4×C127M1611yescharacteristic
81C22×C4×C127C22×C411yescharacteristic
82M16×C127—24no—
83M16×C127—28no—
84M16×C127—22no—
85M16×C127—24no—
86M16×C127—28no—
87M16×C127—24no—
88M16×C127C22⋊C414yes—
89C2×C8×C127—24no—
90C2×C8×C127—28no—
91C2×C8×C127—22no—
92C2×C8×C127—22no—
93C2×C8×C127—28no—
94C2×C8×C127C22⋊C412yes—
95C2×C8×C127C2×D414yes—
96C2×C8×C127C2×D412yes—
97C16×C127—44no—
98C23×C127—24no—
99C23×C127—22no—
100C23×C127—22no—
101C23×C127—81no—
102C23×C127—82no—
103C23×C127C22⋊C811yescharacteristic
104C23×C127C22⋊C811yescharacteristic
105C23×C127C2×M1611yescharacteristic
106Q8×C127—24no—
107Q8×C127—22no—
108Q8×C127C22⋊C814yes—
109D4×C127—28no—
110D4×C127—24no—
111D4×C127—24no—
112D4×C127—24no—
113D4×C127—22no—
114D4×C127C22⋊C814yes—
115C2×C4×C127—24no—
116C2×C4×C127—28no—
117C2×C4×C127—22no—
118C2×C4×C127—22no—
119C2×C4×C127—24no—
120C2×C4×C127—21no—
121C2×C4×C127—21no—
122C2×C4×C127—22no—
123C2×C4×C127—22no—
124C2×C4×C127—21no—
125C2×C4×C127—24no—
126C2×C4×C127—21no—
127C2×C4×C127C22⋊C811yescharacteristic
128C2×C4×C127C22⋊C811yescharacteristic
129C2×C4×C127C2×C22⋊C412yes—
130C2×C4×C127C2×C22⋊C411yescharacteristic
131C2×C4×C127C22×C811yescharacteristic
132C2×C4×C127C2×M1612yes—
133C8×C127—24no—
134C8×C127—22no—
135C8×C127#12014yes—
136C22×C127—21no—
137C22×C127—21no—
138C22×C127—24no—
139C22×C127—41no—
140C22×C127—41no—
141C22×C127—41no—
142C22×C127—44no—
143C22×C127—82no—
144C22×C127—84no—
145C22×C127C2×C22⋊C811yescharacteristic
146C22×C127#40712yes—
147C4×C127—21no—
148C4×C127—24no—
149C4×C127—21no—
150C4×C127#37911yescharacteristic
151C4×C127C2×C22⋊C812yes—
152C4×C127#40711yescharacteristic
153#12686C12711yesmaximal, characteristic, direct product factor, Sylow
154C2×C127—21no—
155C2×C127—22no—
156C2×C127—44no—
157C2×C127—41no—
158C2×C127—41no—
159C2×C127#177911yescharacteristic
160C2×#568C2×C12711yescharacteristic
161#2117C2×C12714yescomplemented
162C2×#349C2×C12712yescomplemented
163C127#1268611yescharacteristic, direct product factor, Sylow
164C22×#41C4×C12711yescharacteristic
165#568—28no—
166C2×#131—22no—
167C2×#131C4×C12712yes—
168C2×#131C22×C12712yes—
169C22×M16—22no—
170C22×M16C4×C12711yescharacteristic
171C22×M16C22×C12711yescharacteristic
172C2×M32C22×C12714yescomplemented
173#350—28no—
174#349—22no—
175#349—22no—
176C2×#41—24no—
177C2×#41—24no—
178C2×#41—21no—
179C2×#41—21no—
180C2×#41C8×C12712yes—
181C2×#41C2×C4×C12712yes—
182C22×Q8C8×C12711yescharacteristic
183C22×D4—21no—
184C22×D4C8×C12711yescharacteristic
185C23×C4—21no—
186C23×C4C2×C4×C12711yescharacteristic
187#131—216no—
188#131—24no—
189#131—24no—
190#131—24no—
191#131D4×C12718yes—
192C2×M16—216no—
193C2×M16—21no—
194C2×M16—24no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—24no—
198C2×M16C2×C4×C12712yes—
199C2×M16D4×C12712yes—
200C2×M16D4×C12712yes—
201C22×C8—21no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8C2×C4×C12711yescharacteristic
205C22×C8C23×C12711yescharacteristic
206M32—24no—
207M32—24no—
208C2×C16—24no—
209C24—21no—
210C24M16×C12711yescharacteristic
211#41—24no—
212#41—28no—
213#41—24no—
214#41—24no—
215#41—24no—
216#41—22no—
217#41—22no—
218#41C22⋊C4×C12714yes—
219#41M16×C12714yes—
220C2×Q8—24no—
221C2×Q8—21no—
222C2×Q8C2×C8×C12712yes—
223C2×D4—28no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—24no—
227C2×D4—21no—
228C2×D4C2×C8×C12712yes—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4C22⋊C4×C12711yescharacteristic
237C22×C4C22⋊C4×C12711yescharacteristic
238C22×C4C2×C8×C12711yescharacteristic
239C22×C4C2×C8×C12711yescharacteristic
240C22×C4M16×C12711yescharacteristic
241C22×C4C22×C4×C12711yescharacteristic
242M16—24no—
243M16—28no—
244M16—22no—
245M16—24no—
246M16—28no—
247M16—24no—
248M16C22⋊C4×C12714yes—
249C2×C8—24no—
250C2×C8—28no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—28no—
254C2×C8C22⋊C4×C12712yes—
255C2×C8C2×D4×C12714yes—
256C2×C8C2×D4×C12712yes—
257C16—44no—
258C23—24no—
259C23—22no—
260C23—22no—
261C23—81no—
262C23—82no—
263C23C22⋊C8×C12711yescharacteristic
264C23C22⋊C8×C12711yescharacteristic
265C23C2×M16×C12711yescharacteristic
266Q8—24no—
267Q8—22no—
268Q8C22⋊C8×C12714yes—
269D4—28no—
270D4—24no—
271D4—24no—
272D4—24no—
273D4—22no—
274D4C22⋊C8×C12714yes—
275C2×C4—24no—
276C2×C4—28no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—24no—
286C2×C4—21no—
287C2×C4C22⋊C8×C12711yescharacteristic
288C2×C4C22⋊C8×C12711yescharacteristic
289C2×C4C2×C22⋊C4×C12712yes—
290C2×C4C2×C22⋊C4×C12711yescharacteristic
291C2×C4C22×C8×C12711yescharacteristic
292C2×C4C2×M16×C12712yes—
293C8—24no—
294C8—22no—
295C8#120×C12714yes—
296C22—21no—
297C22—21no—
298C22—24no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—44no—
303C22—82no—
304C22—84no—
305C22C2×C22⋊C8×C12711yescharacteristic
306C22#407×C12712yes—
307C4—21no—
308C4—24no—
309C4—21no—
310C4#379×C12711yescharacteristic
311C4C2×C22⋊C8×C12712yes—
312C4#407×C12711yescharacteristic
313C2—21no—
314C2—22no—
315C2—44no—
316C2—41no—
317C2—41no—
318C2C127×#177911yescharacteristic
subgroupG/H
centerC4×C127#379
commutatorC2×C4C22×C8×C127
Frattini, cosocleC22×C8C23×C127
socleC2×C127#1779
prime radicalC22×#41×C127C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C127 ◁ C22×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2117C2semidirect
3. C2×#349×C127C2semidirect
12. C2×M32×C127C22semidirect
153. #12686C127direct (N × H)
161. #2117C2×C127semidirect
162. C2×#349C2×C127semidirect
163. C127#12686direct (N × H)
172. C2×M32C22×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#568×C1272
C127×#21172
C2×#349×C1272
#12686127