Finite Group Databasebeta

← C127×#13336 #44577009

Subgroups — C127×#13336 #44577009

Order 32512.

586 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#509×C127C211yesmaximal, characteristic, complemented
2C22×D16×C127C211yesmaximal, characteristic, complemented
3C127×#3155C211yesmaximal, characteristic, complemented
4C2×#361×C127C211yesmaximal, characteristic, complemented
5C2×#359×C127C211yesmaximal, characteristic, complemented
6C127×#1978C211yesmaximal, characteristic, complemented
7C127×#1379C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C127—21no—
9C2×#135×C127C2211yescharacteristic
10C22×D8×C127C2211yescharacteristic
11C22×D8×C127C2211yescharacteristic
12#535×C127—21no—
13#509×C127—22no—
14#509×C127—22no—
15C2×SD32×C127—21no—
16C2×D16×C127—22no—
17C2×D16×C127—21no—
18C2×D16×C127—22no—
19C2×D16×C127—22no—
20C2×M32×C127C2211yescharacteristic
21C22×C16×C127C2211yescharacteristic
22#496×C127—22no—
23#495×C127—21no—
24#493×C127—21no—
25C2×#108×C127C2211yescharacteristic
26#418×C127—21no—
27C2×#100×C127—21no—
28#405×C127C2211yescharacteristic
29#361×C127—22no—
30#359×C127—22no—
31C2×#41×C127—21no—
32C22×D4×C127—21no—
33C22×D4×C127—21no—
34AGL(1,ℤ/8)×C127—42no—
35AGL(1,ℤ/8)×C127—42no—
36#135×C127—21no—
37#135×C127—21no—
38#135×C127—21no—
39#135×C127—21no—
40C2×Q16×C127D411yescharacteristic
41C2×SD16×C127—22no—
42C2×SD16×C127—21no—
43C2×D8×C127—22no—
44C2×D8×C127—21no—
45C2×D8×C127—21no—
46C2×D8×C127—21no—
47C2×D8×C127—21no—
48C2×D8×C127—21no—
49C2×D8×C127—21no—
50C2×D8×C127—21no—
51C2×D8×C127—21no—
52C2×D8×C127—21no—
53C2×D8×C127—21no—
54C2×D8×C127D412yes—
55C2×D8×C127D411yescharacteristic
56C2×D8×C127D412yescomplemented
57C2×M16×C127—21no—
58C2×M16×C127D411yescharacteristic
59C22×C8×C127C2311yescharacteristic
60#127×C127—22no—
61#124×C127—22no—
62#121×C127—42no—
63#120×C127—42no—
64#117×C127D411yescharacteristic
65SD32×C127—24no—
66D16×C127—28no—
67D16×C127—24no—
68M32×C127—21no—
69M32×C127D412yescomplemented
70C2×C16×C127—21no—
71C2×C16×C127—21no—
72C2×C16×C127—21no—
73C2×C16×C127D412yescomplemented
74#108×C127—21no—
75#108×C127—21no—
76#102×C127—42no—
77#100×C127—42no—
78#97×C127—21no—
79C4×C8×C127—21no—
80C24×C127—41no—
81C24×C127—41no—
82#41×C127—41no—
83#41×C127—41no—
84#41×C127—41no—
85#41×C127—41no—
86C2×Q8×C127—21no—
87C2×D4×C127—22no—
88C2×D4×C127—21no—
89C2×D4×C127—22no—
90C2×D4×C127—44no—
91C2×D4×C127—41no—
92C2×D4×C127—41no—
93C2×D4×C127—41no—
94C2×D4×C127—41no—
95C2×D4×C127—41no—
96C2×D4×C127—41no—
97C2×D4×C127—41no—
98C2×D4×C127—41no—
99C2×D4×C127—41no—
100C2×D4×C127—41no—
101C2×D4×C127—41no—
102C22×C4×C127—41no—
103C22×C4×C127C2×D411yescharacteristic
104Q16×C127—21no—
105Q16×C127—21no—
106SD16×C127—44no—
107SD16×C127—41no—
108SD16×C127—41no—
109D8×C127—22no—
110D8×C127—22no—
111D8×C127—21no—
112D8×C127—22no—
113D8×C127—22no—
114D8×C127—21no—
115D8×C127—44no—
116D8×C127—41no—
117D8×C127—41no—
118D8×C127—41no—
119D8×C127—41no—
120M16×C127—21no—
121M16×C127—41no—
122M16×C127—41no—
123M16×C127D812yes—
124C2×C8×C127—21no—
125C2×C8×C127—41no—
126C2×C8×C127C2×D411yescharacteristic
127C2×C8×C127C2×D411yescharacteristic
128C2×C8×C127C2×D411yescharacteristic
129C2×C8×C127C2×D411yescharacteristic
130C2×C8×C127C2×D411yescharacteristic
131C2×C8×C127#4111yescharacteristic
132C4⋊C4×C127D812yes—
133C22⋊C4×C127—21no—
134C22⋊C4×C127—44no—
135C42×C127—21no—
136C16×C127—22no—
137C16×C127—22no—
138C23×C127—42no—
139C23×C127—41no—
140C23×C127—42no—
141C23×C127—81no—
142C23×C127—81no—
143C23×C127—81no—
144C23×C127—81no—
145C23×C127—162no—
146C23×C127—162no—
147C23×C127AGL(1,ℤ/8)11yescharacteristic
148Q8×C127—41no—
149Q8×C127—41no—
150D4×C127—42no—
151D4×C127—42no—
152D4×C127—41no—
153D4×C127—41no—
154D4×C127—42no—
155D4×C127—42no—
156D4×C127—81no—
157D4×C127—81no—
158D4×C127—81no—
159D4×C127—81no—
160D4×C127—81no—
161D4×C127—81no—
162D4×C127—164no—
163C2×C4×C127—22no—
164C2×C4×C127—41no—
165C2×C4×C127—81no—
166C2×C4×C127—81no—
167C2×C4×C127#12011yescharacteristic
168C2×C4×C127#12111yescharacteristic
169C2×C4×C127#12711yescharacteristic
170C2×C4×C127C2×D811yescharacteristic
171C2×C4×C127C2×D811yescharacteristic
172C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
173C8×C127—22no—
174C8×C127—81no—
175C8×C127#12011yescharacteristic
176C8×C127#12011yescharacteristic
177C8×C127#12111yescharacteristic
178C8×C127#12111yescharacteristic
179C22×C127—22no—
180C22×C127—82no—
181C22×C127—84no—
182C22×C127—81no—
183C22×C127—82no—
184C22×C127—82no—
185C22×C127—162no—
186C22×C127—162no—
187C22×C127—162no—
188C22×C127—162no—
189C22×C127—162no—
190C22×C127#44711yescharacteristic
191C22×C127#45911yescharacteristic
192C22×C127#49611yescharacteristic
193C4×C127—42no—
194C4×C127—81no—
195C4×C127#39211yescharacteristic
196C4×C127#44711yescharacteristic
197C4×C127#45911yescharacteristic
198C4×C127#49311yescharacteristic
199#13336C12711yesmaximal, characteristic, direct product factor, Sylow
200C2×C127—21no—
201C2×C127—21no—
202C2×C127—161no—
203C2×C127—162no—
204C2×C127—162no—
205C2×C127#201111yescharacteristic
206C2×C127#219212yes—
207C2×#509C2×C12711yescharacteristic, complemented
208C22×D16C2×C12711yescharacteristic, complemented
209#3155C2×C12711yescharacteristic, complemented
210C2×#361C2×C12711yescharacteristic, complemented
211C2×#359C2×C12711yescharacteristic, complemented
212#1978C2×C12711yescharacteristic, complemented
213#1379C2×C12711yescharacteristic, complemented
214C127#1333611yescharacteristic, direct product factor, Sylow
215C2×AGL(1,ℤ/8)—21no—
216C2×#135C22×C12711yescharacteristic
217C22×D8C22×C12711yescharacteristic
218C22×D8C22×C12711yescharacteristic
219#535—21no—
220#509—22no—
221#509—22no—
222C2×SD32—21no—
223C2×D16—22no—
224C2×D16—21no—
225C2×D16—22no—
226C2×D16—22no—
227C2×M32C22×C12711yescharacteristic
228C22×C16C22×C12711yescharacteristic
229#496—22no—
230#495—21no—
231#493—21no—
232C2×#108C22×C12711yescharacteristic
233#418—21no—
234C2×#100—21no—
235#405C22×C12711yescharacteristic
236#361—22no—
237#359—22no—
238C2×#41—21no—
239C22×D4—21no—
240C22×D4—21no—
241AGL(1,ℤ/8)—42no—
242AGL(1,ℤ/8)—42no—
243#135—21no—
244#135—21no—
245#135—21no—
246#135—21no—
247C2×Q16D4×C12711yescharacteristic
248C2×SD16—22no—
249C2×SD16—21no—
250C2×D8—22no—
251C2×D8—21no—
252C2×D8—21no—
253C2×D8—21no—
254C2×D8—21no—
255C2×D8—21no—
256C2×D8—21no—
257C2×D8—21no—
258C2×D8—21no—
259C2×D8—21no—
260C2×D8—21no—
261C2×D8D4×C12712yes—
262C2×D8D4×C12711yescharacteristic
263C2×D8D4×C12712yescomplemented
264C2×M16—21no—
265C2×M16D4×C12711yescharacteristic
266C22×C8C23×C12711yescharacteristic
267#127—22no—
268#124—22no—
269#121—42no—
270#120—42no—
271#117D4×C12711yescharacteristic
272SD32—24no—
273D16—28no—
274D16—24no—
275M32—21no—
276M32D4×C12712yescomplemented
277C2×C16—21no—
278C2×C16—21no—
279C2×C16—21no—
280C2×C16D4×C12712yescomplemented
281#108—21no—
282#108—21no—
283#102—42no—
284#100—42no—
285#97—21no—
286C4×C8—21no—
287C24—41no—
288C24—41no—
289#41—41no—
290#41—41no—
291#41—41no—
292#41—41no—
293C2×Q8—21no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—22no—
297C2×D4—44no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C22×C4—41no—
310C22×C4C2×D4×C12711yescharacteristic
311Q16—21no—
312Q16—21no—
313SD16—44no—
314SD16—41no—
315SD16—41no—
316D8—22no—
317D8—22no—
318D8—21no—
319D8—22no—
320D8—22no—
321D8—21no—
322D8—44no—
323D8—41no—
324D8—41no—
325D8—41no—
326D8—41no—
327M16—21no—
328M16—41no—
329M16—41no—
330M16D8×C12712yes—
331C2×C8—21no—
332C2×C8—41no—
333C2×C8C2×D4×C12711yescharacteristic
334C2×C8C2×D4×C12711yescharacteristic
335C2×C8C2×D4×C12711yescharacteristic
336C2×C8C2×D4×C12711yescharacteristic
337C2×C8C2×D4×C12711yescharacteristic
338C2×C8#41×C12711yescharacteristic
339C4⋊C4D8×C12712yes—
340C22⋊C4—21no—
341C22⋊C4—44no—
342C42—21no—
343C16—22no—
344C16—22no—
345C23—42no—
346C23—41no—
347C23—42no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—162no—
353C23—162no—
354C23AGL(1,ℤ/8)×C12711yescharacteristic
355Q8—41no—
356Q8—41no—
357D4—42no—
358D4—42no—
359D4—41no—
360D4—41no—
361D4—42no—
362D4—42no—
363D4—81no—
364D4—81no—
365D4—81no—
366D4—81no—
367D4—81no—
368D4—81no—
369D4—164no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4#120×C12711yescharacteristic
375C2×C4#121×C12711yescharacteristic
376C2×C4#127×C12711yescharacteristic
377C2×C4C2×D8×C12711yescharacteristic
378C2×C4C2×D8×C12711yescharacteristic
379C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
380C8—22no—
381C8—81no—
382C8#120×C12711yescharacteristic
383C8#120×C12711yescharacteristic
384C8#121×C12711yescharacteristic
385C8#121×C12711yescharacteristic
386C22—22no—
387C22—82no—
388C22—84no—
389C22—81no—
390C22—82no—
391C22—82no—
392C22—162no—
393C22—162no—
394C22—162no—
395C22—162no—
396C22—162no—
397C22#447×C12711yescharacteristic
398C22#459×C12711yescharacteristic
399C22#496×C12711yescharacteristic
400C4—42no—
401C4—81no—
402C4#392×C12711yescharacteristic
403C4#447×C12711yescharacteristic
404C4#459×C12711yescharacteristic
405C4#493×C12711yescharacteristic
406C2—21no—
407C2—21no—
408C2—162no—
409C2—161no—
410C2—162no—
411C2C127×#201111yescharacteristic
412C2C127×#219212yes—
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC22×C8C23×C127
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ #405×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#509×C127C2semidirect
2. C22×D16×C127C2semidirect
3. C127×#3155C2semidirect
4. C2×#361×C127C2semidirect
5. C2×#359×C127C2semidirect
6. C127×#1978C2semidirect
7. C127×#1379C2semidirect
56. C2×D8×C127D4semidirect
69. M32×C127D4semidirect
73. C2×C16×C127D4semidirect
199. #13336C127direct (N × H)
207. C2×#509C2×C127semidirect
208. C22×D16C2×C127semidirect
209. #3155C2×C127semidirect
210. C2×#361C2×C127semidirect
211. C2×#359C2×C127semidirect
212. #1978C2×C127semidirect
213. #1379C2×C127semidirect
214. C127#13336direct (N × H)
263. C2×D8D4×C127semidirect
276. M32D4×C127semidirect
280. C2×C16D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#509×C1272
C22×D16×C1272
C127×#31552
C2×#361×C1272
C2×#359×C1272
C127×#19782
C127×#13792
#13336127