Finite Group Databasebeta

← C127×#13319 #44576992

Subgroups — C127×#13319 #44576992

Order 32512.

572 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D16×C127C211yesmaximal, characteristic, complemented
2C2×#472×C127C211yesmaximal, characteristic, complemented
3C127×#3058C211yesmaximal, characteristic, complemented
4C2×#361×C127C211yesmaximal, characteristic, complemented
5C127×#2140C211yesmaximal, characteristic, complemented
6C2×#350×C127C211yesmaximal, characteristic, complemented
7C127×#1382C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C127—21no—
9C22×D8×C127C2211yescharacteristic
10C22×D8×C127C2211yescharacteristic
11C2×#131×C127C2211yescharacteristic
12#535×C127—21no—
13C2×D16×C127—22no—
14C2×D16×C127—22no—
15C2×D16×C127—22no—
16C2×M32×C127C2211yescharacteristic
17C22×C16×C127C2211yescharacteristic
18#472×C127—22no—
19#472×C127—22no—
20#469×C127—22no—
21#466×C127—22no—
22C2×#108×C127C2211yescharacteristic
23#427×C127C2211yescharacteristic
24#418×C127—21no—
25C2×#100×C127—21no—
26#361×C127—22no—
27#357×C127—22no—
28#350×C127—22no—
29C2×#41×C127D411yescharacteristic
30C22×D4×C127—21no—
31C22×D4×C127—21no—
32AGL(1,ℤ/8)×C127—24no—
33AGL(1,ℤ/8)×C127—42no—
34C2×SD16×C127—41no—
35C2×D8×C127—21no—
36C2×D8×C127—21no—
37C2×D8×C127—21no—
38C2×D8×C127—21no—
39C2×D8×C127—21no—
40C2×D8×C127—21no—
41C2×D8×C127—21no—
42C2×D8×C127—21no—
43C2×D8×C127—21no—
44C2×D8×C127—21no—
45C2×D8×C127—41no—
46C2×D8×C127D412yes—
47C2×D8×C127D412yescomplemented
48#131×C127—21no—
49#131×C127—21no—
50#131×C127—21no—
51#131×C127—21no—
52C2×M16×C127—21no—
53C2×M16×C127—21no—
54C2×M16×C127D411yescharacteristic
55C22×C8×C127—21no—
56C22×C8×C127C2311yescharacteristic
57#127×C127—41no—
58#124×C127—41no—
59#121×C127—24no—
60#120×C127—42no—
61#117×C127—21no—
62D16×C127—28no—
63M32×C127—21no—
64M32×C127—21no—
65C2×C16×C127—21no—
66C2×C16×C127—21no—
67C2×C16×C127—21no—
68C2×C16×C127D412yescomplemented
69#108×C127—21no—
70#108×C127D412yescomplemented
71#107×C127D412yes—
72#106×C127—21no—
73#102×C127—24no—
74#100×C127—24no—
75C24×C127—41no—
76C24×C127—41no—
77#41×C127—21no—
78#41×C127—21no—
79#41×C127—21no—
80#41×C127—21no—
81C2×Q8×C127D811yescharacteristic
82C2×D4×C127—21no—
83C2×D4×C127—22no—
84C2×D4×C127—22no—
85C2×D4×C127—44no—
86C2×D4×C127—41no—
87C2×D4×C127—41no—
88C2×D4×C127—41no—
89C2×D4×C127—41no—
90C2×D4×C127—41no—
91C2×D4×C127—41no—
92C2×D4×C127—41no—
93C2×D4×C127—41no—
94C2×D4×C127—41no—
95C2×D4×C127—41no—
96C2×D4×C127D811yescharacteristic
97C22×C4×C127—21no—
98C22×C4×C127C2×D411yescharacteristic
99SD16×C127—44no—
100D8×C127—22no—
101D8×C127—22no—
102D8×C127—22no—
103D8×C127—22no—
104D8×C127—44no—
105D8×C127—41no—
106D8×C127—41no—
107D8×C127—41no—
108D8×C127—41no—
109M16×C127—21no—
110M16×C127—22no—
111M16×C127—21no—
112M16×C127—41no—
113M16×C127—41no—
114M16×C127—41no—
115C2×C8×C127—21no—
116C2×C8×C127—41no—
117C2×C8×C127—41no—
118C2×C8×C127—41no—
119C2×C8×C127C2×D411yescharacteristic
120C2×C8×C127C2×D411yescharacteristic
121C2×C8×C127C2×D411yescharacteristic
122C2×C8×C127C2×D411yescharacteristic
123C2×C8×C127C2×D411yescharacteristic
124C2×C8×C127#4111yescharacteristic
125C4⋊C4×C127—22no—
126C22⋊C4×C127—44no—
127C22⋊C4×C127—41no—
128C42×C127—41no—
129C16×C127—22no—
130C16×C127—41no—
131C23×C127—21no—
132C23×C127—42no—
133C23×C127—42no—
134C23×C127—81no—
135C23×C127—81no—
136C23×C127—81no—
137C23×C127—81no—
138C23×C127—162no—
139C23×C127—162no—
140C23×C127AGL(1,ℤ/8)11yescharacteristic
141Q8×C127—21no—
142Q8×C127—21no—
143D4×C127—21no—
144D4×C127—21no—
145D4×C127—41no—
146D4×C127—41no—
147D4×C127—42no—
148D4×C127—42no—
149D4×C127—42no—
150D4×C127—42no—
151D4×C127—81no—
152D4×C127—81no—
153D4×C127—81no—
154D4×C127—81no—
155D4×C127—164no—
156C2×C4×C127—21no—
157C2×C4×C127—41no—
158C2×C4×C127—41no—
159C2×C4×C127—42no—
160C2×C4×C127#12011yescharacteristic
161C2×C4×C127#12011yescharacteristic
162C2×C4×C127#12111yescharacteristic
163C2×C4×C127C2×D811yescharacteristic
164C2×C4×C127#13511yescharacteristic
165C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
166C8×C127—41no—
167C8×C127—42no—
168C8×C127#12011yescharacteristic
169C8×C127#12111yescharacteristic
170C8×C127#12111yescharacteristic
171C8×C127#12711yescharacteristic
172C22×C127—22no—
173C22×C127—42no—
174C22×C127—41no—
175C22×C127—84no—
176C22×C127—82no—
177C22×C127—82no—
178C22×C127—162no—
179C22×C127—162no—
180C22×C127—162no—
181C22×C127—162no—
182C22×C127—162no—
183C22×C127#44711yescharacteristic
184C22×C127#44911yescharacteristic
185C22×C127#46911yescharacteristic
186C4×C127—41no—
187C4×C127—82no—
188C4×C127#39211yescharacteristic
189C4×C127#44711yescharacteristic
190C4×C127#44911yescharacteristic
191C4×C127#46611yescharacteristic
192#13319C12711yesmaximal, characteristic, direct product factor, Sylow
193C2×C127—21no—
194C2×C127—21no—
195C2×C127—81no—
196C2×C127—162no—
197C2×C127—162no—
198C2×C127#199911yescharacteristic
199C2×C127#219212yes—
200C22×D16C2×C12711yescharacteristic, complemented
201C2×#472C2×C12711yescharacteristic, complemented
202#3058C2×C12711yescharacteristic, complemented
203C2×#361C2×C12711yescharacteristic, complemented
204#2140C2×C12711yescharacteristic, complemented
205C2×#350C2×C12711yescharacteristic, complemented
206#1382C2×C12711yescharacteristic, complemented
207C127#1331911yescharacteristic, direct product factor, Sylow
208C2×AGL(1,ℤ/8)—21no—
209C22×D8C22×C12711yescharacteristic
210C22×D8C22×C12711yescharacteristic
211C2×#131C22×C12711yescharacteristic
212#535—21no—
213C2×D16—22no—
214C2×D16—22no—
215C2×D16—22no—
216C2×M32C22×C12711yescharacteristic
217C22×C16C22×C12711yescharacteristic
218#472—22no—
219#472—22no—
220#469—22no—
221#466—22no—
222C2×#108C22×C12711yescharacteristic
223#427C22×C12711yescharacteristic
224#418—21no—
225C2×#100—21no—
226#361—22no—
227#357—22no—
228#350—22no—
229C2×#41D4×C12711yescharacteristic
230C22×D4—21no—
231C22×D4—21no—
232AGL(1,ℤ/8)—24no—
233AGL(1,ℤ/8)—42no—
234C2×SD16—41no—
235C2×D8—21no—
236C2×D8—21no—
237C2×D8—21no—
238C2×D8—21no—
239C2×D8—21no—
240C2×D8—21no—
241C2×D8—21no—
242C2×D8—21no—
243C2×D8—21no—
244C2×D8—21no—
245C2×D8—41no—
246C2×D8D4×C12712yes—
247C2×D8D4×C12712yescomplemented
248#131—21no—
249#131—21no—
250#131—21no—
251#131—21no—
252C2×M16—21no—
253C2×M16—21no—
254C2×M16D4×C12711yescharacteristic
255C22×C8—21no—
256C22×C8C23×C12711yescharacteristic
257#127—41no—
258#124—41no—
259#121—24no—
260#120—42no—
261#117—21no—
262D16—28no—
263M32—21no—
264M32—21no—
265C2×C16—21no—
266C2×C16—21no—
267C2×C16—21no—
268C2×C16D4×C12712yescomplemented
269#108—21no—
270#108D4×C12712yescomplemented
271#107D4×C12712yes—
272#106—21no—
273#102—24no—
274#100—24no—
275C24—41no—
276C24—41no—
277#41—21no—
278#41—21no—
279#41—21no—
280#41—21no—
281C2×Q8D8×C12711yescharacteristic
282C2×D4—21no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—44no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4D8×C12711yescharacteristic
297C22×C4—21no—
298C22×C4C2×D4×C12711yescharacteristic
299SD16—44no—
300D8—22no—
301D8—22no—
302D8—22no—
303D8—22no—
304D8—44no—
305D8—41no—
306D8—41no—
307D8—41no—
308D8—41no—
309M16—21no—
310M16—22no—
311M16—21no—
312M16—41no—
313M16—41no—
314M16—41no—
315C2×C8—21no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8C2×D4×C12711yescharacteristic
320C2×C8C2×D4×C12711yescharacteristic
321C2×C8C2×D4×C12711yescharacteristic
322C2×C8C2×D4×C12711yescharacteristic
323C2×C8C2×D4×C12711yescharacteristic
324C2×C8#41×C12711yescharacteristic
325C4⋊C4—22no—
326C22⋊C4—44no—
327C22⋊C4—41no—
328C42—41no—
329C16—22no—
330C16—41no—
331C23—21no—
332C23—42no—
333C23—42no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23—162no—
339C23—162no—
340C23AGL(1,ℤ/8)×C12711yescharacteristic
341Q8—21no—
342Q8—21no—
343D4—21no—
344D4—21no—
345D4—41no—
346D4—41no—
347D4—42no—
348D4—42no—
349D4—42no—
350D4—42no—
351D4—81no—
352D4—81no—
353D4—81no—
354D4—81no—
355D4—164no—
356C2×C4—21no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4#120×C12711yescharacteristic
361C2×C4#120×C12711yescharacteristic
362C2×C4#121×C12711yescharacteristic
363C2×C4C2×D8×C12711yescharacteristic
364C2×C4#135×C12711yescharacteristic
365C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
366C8—41no—
367C8—42no—
368C8#120×C12711yescharacteristic
369C8#121×C12711yescharacteristic
370C8#121×C12711yescharacteristic
371C8#127×C12711yescharacteristic
372C22—22no—
373C22—42no—
374C22—41no—
375C22—84no—
376C22—82no—
377C22—82no—
378C22—162no—
379C22—162no—
380C22—162no—
381C22—162no—
382C22—162no—
383C22#447×C12711yescharacteristic
384C22#449×C12711yescharacteristic
385C22#469×C12711yescharacteristic
386C4—41no—
387C4—82no—
388C4#392×C12711yescharacteristic
389C4#447×C12711yescharacteristic
390C4#449×C12711yescharacteristic
391C4#466×C12711yescharacteristic
392C2—21no—
393C2—21no—
394C2—81no—
395C2—162no—
396C2—162no—
397C2C127×#199911yescharacteristic
398C2C127×#219212yes—
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC22×C8C23×C127
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×#131×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×D16×C127C2semidirect
2. C2×#472×C127C2semidirect
3. C127×#3058C2semidirect
4. C2×#361×C127C2semidirect
5. C127×#2140C2semidirect
6. C2×#350×C127C2semidirect
7. C127×#1382C2semidirect
47. C2×D8×C127D4semidirect
68. C2×C16×C127D4semidirect
70. #108×C127D4semidirect
192. #13319C127direct (N × H)
200. C22×D16C2×C127semidirect
201. C2×#472C2×C127semidirect
202. #3058C2×C127semidirect
203. C2×#361C2×C127semidirect
204. #2140C2×C127semidirect
205. C2×#350C2×C127semidirect
206. #1382C2×C127semidirect
207. C127#13319direct (N × H)
247. C2×D8D4×C127semidirect
268. C2×C16D4×C127semidirect
270. #108D4×C127semidirect

Maximal subgroups

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

subgroupindex
C22×D16×C1272
C2×#472×C1272
C127×#30582
C2×#361×C1272
C127×#21402
C2×#350×C1272
C127×#13822
#13319127