Finite Group Databasebeta

← C127×#13068 #44576741

Subgroups — C127×#13068 #44576741

Order 32512.

470 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3407C211yesmaximal, characteristic, complemented
2C2×#472×C127C211yesmaximal, characteristic
3C2×#361×C127C211yesmaximal, characteristic, complemented
4C127×#2140C211yesmaximal, characteristic, complemented
5C127×#1814C211yesmaximal, characteristic, complemented
6C127×#1389C211yesmaximal, characteristic, complemented
7C127×#1385C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C127—21no—
9C22×D8×C127C2211yescharacteristic
10C2×#131×C127C2211yescharacteristic
11#504×C127—22no—
12#504×C127—22no—
13C2×M32×C127—22no—
14C2×M32×C127C2211yescharacteristic
15C2×M32×C127C2211yescharacteristic
16#472×C127—22no—
17#472×C127C414yescomplemented
18C2×#108×C127C2211yescharacteristic
19C2×#108×C127C2211yescharacteristic
20#427×C127C2211yescharacteristic
21#421×C127—21no—
22C2×#100×C127—21no—
23#361×C127—22no—
24#357×C127—22no—
25#344×C127—21no—
26C2×#41×C127D411yescharacteristic
27C22×D4×C127—21no—
28AGL(1,ℤ/8)×C127—24no—
29AGL(1,ℤ/8)×C127—42no—
30C2×SD16×C127—41no—
31C2×D8×C127—21no—
32C2×D8×C127—21no—
33C2×D8×C127—21no—
34C2×D8×C127—21no—
35C2×D8×C127—21no—
36C2×D8×C127—41no—
37C2×D8×C127C2×C412yes—
38#131×C127—21no—
39#131×C127—21no—
40#131×C127C2×C412yes—
41#131×C127D412yes—
42C2×M16×C127—21no—
43C2×M16×C127—21no—
44C2×M16×C127—21no—
45C2×M16×C127D411yescharacteristic
46C22×C8×C127—21no—
47C22×C8×C127C2311yescharacteristic
48#117×C127—21no—
49M32×C127—24no—
50M32×C127—21no—
51M32×C127—21no—
52M32×C127—21no—
53M32×C127D412yes—
54C2×C16×C127—24no—
55C2×C16×C127—21no—
56C2×C16×C127D412yes—
57#108×C127—21no—
58#108×C127—21no—
59#108×C127—21no—
60#108×C127C2×C412yes—
61#107×C127—21no—
62#106×C127—21no—
63#104×C127—42no—
64#100×C127—24no—
65C24×C127—41no—
66#41×C127—21no—
67#41×C127—21no—
68#41×C127C22⋊C412yes—
69#41×C127D812yes—
70C2×Q8×C127SD1611yescharacteristic
71C2×D4×C127—21no—
72C2×D4×C127—22no—
73C2×D4×C127—41no—
74C2×D4×C127—41no—
75C2×D4×C127—41no—
76C2×D4×C127—41no—
77C2×D4×C127—41no—
78C2×D4×C127SD1611yescharacteristic
79C22×C4×C127—21no—
80C22×C4×C127C2×D411yescharacteristic
81SD16×C127—44no—
82D8×C127—22no—
83D8×C127—22no—
84D8×C127—44no—
85D8×C127—41no—
86D8×C127—41no—
87M16×C127—21no—
88M16×C127—22no—
89M16×C127—22no—
90M16×C127—22no—
91M16×C127—41no—
92M16×C127—41no—
93M16×C127—41no—
94M16×C127C22⋊C412yes—
95C2×C8×C127—21no—
96C2×C8×C127—22no—
97C2×C8×C127—22no—
98C2×C8×C127—41no—
99C2×C8×C127—41no—
100C2×C8×C127C22×C411yescharacteristic
101C2×C8×C127C2×D411yescharacteristic
102C2×C8×C127C2×D411yescharacteristic
103C2×C8×C127C2×D411yescharacteristic
104C2×C8×C127#4111yescharacteristic
105C2×C8×C127#4111yescharacteristic
106C4⋊C4×C127—41no—
107C22⋊C4×C127—41no—
108C42×C127—41no—
109C16×C127—22no—
110C16×C127—22no—
111C23×C127—21no—
112C23×C127—42no—
113C23×C127—81no—
114C23×C127—81no—
115C23×C127—162no—
116C23×C127#13511yescharacteristic
117Q8×C127—21no—
118Q8×C127#10212yes—
119D4×C127—22no—
120D4×C127—22no—
121D4×C127—21no—
122D4×C127—42no—
123D4×C127—42no—
124D4×C127—81no—
125D4×C127—81no—
126D4×C127#10212yes—
127C2×C4×C127—21no—
128C2×C4×C127—22no—
129C2×C4×C127—22no—
130C2×C4×C127—81no—
131C2×C4×C127C2×C22⋊C411yescharacteristic
132C2×C4×C127#12111yescharacteristic
133C2×C4×C127#12111yescharacteristic
134C2×C4×C127C2×D811yescharacteristic
135C2×C4×C127C2×SD1611yescharacteristic
136C2×C4×C127#13511yescharacteristic
137C8×C127—22no—
138C8×C127—42no—
139C8×C127—81no—
140C8×C127C4×D411yescharacteristic
141C8×C127C4×D411yescharacteristic
142C8×C127#12011yescharacteristic
143C8×C127#12311yescharacteristic
144C22×C127—22no—
145C22×C127—41no—
146C22×C127—81no—
147C22×C127—82no—
148C22×C127—162no—
149C22×C127—162no—
150C22×C127—162no—
151C22×C127#41611yescharacteristic
152C22×C127#46511yescharacteristic
153C22×C127#46611yescharacteristic
154C4×C127—22no—
155C4×C127—161no—
156C4×C127#38611yescharacteristic
157C4×C127C2×#10211yescharacteristic
158C4×C127#46511yescharacteristic
159C4×C127#46611yescharacteristic
160#13068C12711yesmaximal, characteristic, direct product factor, Sylow
161C2×C127—21no—
162C2×C127—21no—
163C2×C127—21no—
164C2×C127—42no—
165C2×C127—162no—
166C2×C127#189311yescharacteristic
167#3407C2×C12711yescharacteristic, complemented
168C2×#472C2×C12711yescharacteristic
169C2×#361C2×C12711yescharacteristic, complemented
170#2140C2×C12711yescharacteristic, complemented
171#1814C2×C12711yescharacteristic, complemented
172#1389C2×C12711yescharacteristic, complemented
173#1385C2×C12711yescharacteristic, complemented
174C127#1306811yescharacteristic, direct product factor, Sylow
175C2×AGL(1,ℤ/8)—21no—
176C22×D8C22×C12711yescharacteristic
177C2×#131C22×C12711yescharacteristic
178#504—22no—
179#504—22no—
180C2×M32—22no—
181C2×M32C22×C12711yescharacteristic
182C2×M32C22×C12711yescharacteristic
183#472—22no—
184#472C4×C12714yescomplemented
185C2×#108C22×C12711yescharacteristic
186C2×#108C22×C12711yescharacteristic
187#427C22×C12711yescharacteristic
188#421—21no—
189C2×#100—21no—
190#361—22no—
191#357—22no—
192#344—21no—
193C2×#41D4×C12711yescharacteristic
194C22×D4—21no—
195AGL(1,ℤ/8)—24no—
196AGL(1,ℤ/8)—42no—
197C2×SD16—41no—
198C2×D8—21no—
199C2×D8—21no—
200C2×D8—21no—
201C2×D8—21no—
202C2×D8—21no—
203C2×D8—41no—
204C2×D8C2×C4×C12712yes—
205#131—21no—
206#131—21no—
207#131C2×C4×C12712yes—
208#131D4×C12712yes—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16D4×C12711yescharacteristic
213C22×C8—21no—
214C22×C8C23×C12711yescharacteristic
215#117—21no—
216M32—24no—
217M32—21no—
218M32—21no—
219M32—21no—
220M32D4×C12712yes—
221C2×C16—24no—
222C2×C16—21no—
223C2×C16D4×C12712yes—
224#108—21no—
225#108—21no—
226#108—21no—
227#108C2×C4×C12712yes—
228#107—21no—
229#106—21no—
230#104—42no—
231#100—24no—
232C24—41no—
233#41—21no—
234#41—21no—
235#41C22⋊C4×C12712yes—
236#41D8×C12712yes—
237C2×Q8SD16×C12711yescharacteristic
238C2×D4—21no—
239C2×D4—22no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4SD16×C12711yescharacteristic
246C22×C4—21no—
247C22×C4C2×D4×C12711yescharacteristic
248SD16—44no—
249D8—22no—
250D8—22no—
251D8—44no—
252D8—41no—
253D8—41no—
254M16—21no—
255M16—22no—
256M16—22no—
257M16—22no—
258M16—41no—
259M16—41no—
260M16—41no—
261M16C22⋊C4×C12712yes—
262C2×C8—21no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8C22×C4×C12711yescharacteristic
268C2×C8C2×D4×C12711yescharacteristic
269C2×C8C2×D4×C12711yescharacteristic
270C2×C8C2×D4×C12711yescharacteristic
271C2×C8#41×C12711yescharacteristic
272C2×C8#41×C12711yescharacteristic
273C4⋊C4—41no—
274C22⋊C4—41no—
275C42—41no—
276C16—22no—
277C16—22no—
278C23—21no—
279C23—42no—
280C23—81no—
281C23—81no—
282C23—162no—
283C23#135×C12711yescharacteristic
284Q8—21no—
285Q8#102×C12712yes—
286D4—22no—
287D4—22no—
288D4—21no—
289D4—42no—
290D4—42no—
291D4—81no—
292D4—81no—
293D4#102×C12712yes—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—81no—
298C2×C4C2×C22⋊C4×C12711yescharacteristic
299C2×C4#121×C12711yescharacteristic
300C2×C4#121×C12711yescharacteristic
301C2×C4C2×D8×C12711yescharacteristic
302C2×C4C2×SD16×C12711yescharacteristic
303C2×C4#135×C12711yescharacteristic
304C8—22no—
305C8—42no—
306C8—81no—
307C8C4×D4×C12711yescharacteristic
308C8C4×D4×C12711yescharacteristic
309C8#120×C12711yescharacteristic
310C8#123×C12711yescharacteristic
311C22—22no—
312C22—41no—
313C22—81no—
314C22—82no—
315C22—162no—
316C22—162no—
317C22—162no—
318C22#416×C12711yescharacteristic
319C22#465×C12711yescharacteristic
320C22#466×C12711yescharacteristic
321C4—22no—
322C4—161no—
323C4#386×C12711yescharacteristic
324C4C2×#102×C12711yescharacteristic
325C4#465×C12711yescharacteristic
326C4#466×C12711yescharacteristic
327C2—21no—
328C2—21no—
329C2—21no—
330C2—42no—
331C2—162no—
332C2C127×#189311yescharacteristic
subgroupG/H
center, socleC2×C127#1893
commutatorC2×C8C22×C4×C127
Frattini, cosocleC22×C8C23×C127
prime radicalC2×#472×C127C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C22×C4×C127 ◁ C2×#131×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3407C2semidirect
3. C2×#361×C127C2semidirect
4. C127×#2140C2semidirect
5. C127×#1814C2semidirect
6. C127×#1389C2semidirect
7. C127×#1385C2semidirect
17. #472×C127C4semidirect
160. #13068C127direct (N × H)
167. #3407C2×C127semidirect
169. C2×#361C2×C127semidirect
170. #2140C2×C127semidirect
171. #1814C2×C127semidirect
172. #1389C2×C127semidirect
173. #1385C2×C127semidirect
174. C127#13068direct (N × H)
184. #472C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34072
C2×#472×C1272
C2×#361×C1272
C127×#21402
C127×#18142
C127×#13892
C127×#13852
#13068127