Finite Group Databasebeta

← C127×#12734 #44576407

Subgroups — C127×#12734 #44576407

Order 32512.

932 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#135×C127C211yesmaximal, characteristic
2C2×#359×C127C214yesmaximal, complemented
3C2×C22⋊C16×C127C211yesmaximal, characteristic, complemented
4C127×#1810C211yesmaximal, characteristic, complemented
5C22×#41×C127—21no—
6C2×#135×C127—28no—
7C2×#135×C127—21no—
8C2×#135×C127—21no—
9C2×#135×C127C412yes—
10C2×#135×C127C2212yes—
11C22×Q16×C127C411yescharacteristic
12C22×SD16×C127—21no—
13C22×D8×C127C411yescharacteristic
14C23×C8×C127C2211yescharacteristic
15C22×C16×C127C2212yescomplemented
16C2×#108×C127C2212yescomplemented
17#407×C127—21no—
18#359×C127—28no—
19C22⋊C16×C127—21no—
20C22⋊C16×C127—21no—
21#341×C127—21no—
22C2×#41×C127—28no—
23C2×#41×C127—22no—
24C2×#41×C127—22no—
25C2×#41×C127—44no—
26C2×#41×C127—41no—
27C2×#41×C127—41no—
28C22×Q8×C127—21no—
29C22×D4×C127—21no—
30C22×D4×C127—41no—
31C23×C4×C127—41no—
32C23×C4×C127D411yescharacteristic
33#135×C127—28no—
34#135×C127—28no—
35#135×C127—24no—
36#135×C127—24no—
37#135×C127—24no—
38#135×C127D418yes—
39C2×Q16×C127—24no—
40C2×Q16×C127—21no—
41C2×Q16×C127C2×C412yes—
42C2×SD16×C127—28no—
43C2×SD16×C127—22no—
44C2×SD16×C127—22no—
45C2×D8×C127—24no—
46C2×D8×C127—21no—
47C2×D8×C127C2×C412yes—
48C2×M16×C127—22no—
49C22×C8×C127—24no—
50C22×C8×C127C2×C411yescharacteristic
51C22×C8×C127C2×C411yescharacteristic
52C22×C8×C127D411yescharacteristic
53C22×C8×C127D411yescharacteristic
54C22×C8×C127D411yescharacteristic
55C22×C8×C127C2311yescharacteristic
56C2×C16×C127—22no—
57C2×C16×C127—24no—
58#108×C127—24no—
59C22⋊C8×C127—41no—
60C22⋊C8×C127—41no—
61C24×C127—41no—
62C24×C127SD1611yescharacteristic
63#41×C127—28no—
64#41×C127—28no—
65#41×C127—48no—
66#41×C127—44no—
67#41×C127—48no—
68#41×C127—44no—
69C2×Q8×C127—24no—
70C2×Q8×C127—22no—
71C2×Q8×C127—42no—
72C2×Q8×C127—41no—
73C2×D4×C127—24no—
74C2×D4×C127—22no—
75C2×D4×C127—42no—
76C2×D4×C127—48no—
77C2×D4×C127—42no—
78C2×D4×C127—42no—
79C2×D4×C127—41no—
80C22×C4×C127—22no—
81C22×C4×C127—22no—
82C22×C4×C127—48no—
83C22×C4×C127—41no—
84C22×C4×C127—42no—
85C22×C4×C127—42no—
86C22×C4×C127C22⋊C411yescharacteristic
87C22×C4×C127C22⋊C411yescharacteristic
88C22×C4×C127D811yescharacteristic
89C22×C4×C127D811yescharacteristic
90C22×C4×C127SD1611yescharacteristic
91C22×C4×C127C2×D411yescharacteristic
92Q16×C127—24no—
93Q16×C127—22no—
94Q16×C127C22⋊C414yes—
95SD16×C127—28no—
96SD16×C127—28no—
97D8×C127—24no—
98D8×C127—22no—
99D8×C127C22⋊C414yes—
100M16×C127—42no—
101M16×C127—42no—
102C2×C8×C127—22no—
103C2×C8×C127—22no—
104C2×C8×C127—22no—
105C2×C8×C127—22no—
106C2×C8×C127—22no—
107C2×C8×C127—42no—
108C2×C8×C127C22⋊C411yescharacteristic
109C2×C8×C127C22⋊C411yescharacteristic
110C2×C8×C127C22×C411yescharacteristic
111C2×C8×C127C2×D414yes—
112C2×C8×C127C2×D411yescharacteristic
113C16×C127—42no—
114C23×C127—22no—
115C23×C127—22no—
116C23×C127—41no—
117C23×C127—44no—
118C23×C127—42no—
119C23×C127—82no—
120C23×C127—82no—
121C23×C127#10211yescharacteristic
122C23×C127#10211yescharacteristic
123C23×C127C2×SD1611yescharacteristic
124Q8×C127—24no—
125Q8×C127—44no—
126Q8×C127—42no—
127D4×C127—24no—
128D4×C127—44no—
129D4×C127—48no—
130D4×C127—48no—
131D4×C127—42no—
132C2×C4×C127—22no—
133C2×C4×C127—21no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—22no—
138C2×C4×C127—22no—
139C2×C4×C127—48no—
140C2×C4×C127—44no—
141C2×C4×C127—42no—
142C2×C4×C127—48no—
143C2×C4×C127#10211yescharacteristic
144C2×C4×C127#10211yescharacteristic
145C2×C4×C127C2×C22⋊C411yescharacteristic
146C2×C4×C127#12012yes—
147C2×C4×C127C2×D811yescharacteristic
148C2×C4×C127AGL(1,ℤ/8)12yes—
149C8×C127—21no—
150C8×C127—21no—
151C8×C127—82no—
152C8×C127C2×C22⋊C412yes—
153C8×C127#12012yes—
154C22×C127—21no—
155C22×C127—21no—
156C22×C127—21no—
157C22×C127—21no—
158C22×C127—21no—
159C22×C127—21no—
160C22×C127—21no—
161C22×C127—44no—
162C22×C127—44no—
163C22×C127—88no—
164C22×C127—82no—
165C22×C127—82no—
166C22×C127#35912yescomplemented
167C22×C127C2×#10211yescharacteristic
168C22×C127#45012yes—
169C4×C127—21no—
170C4×C127—21no—
171C4×C127—44no—
172C4×C127#37911yescharacteristic
173C4×C127#41811yescharacteristic
174C4×C127#44712yes—
175#12734C12711yesmaximal, characteristic, direct product factor, Sylow
176C2×C127—22no—
177C2×C127—22no—
178C2×C127—41no—
179C2×C127—84no—
180C2×C127#178611yescharacteristic
181C2×C127C2×#35911yescharacteristic
182C2×C127#214311yescharacteristic
183C22×#135C2×C12711yescharacteristic
184C2×#359C2×C12714yescomplemented
185C2×C22⋊C16C2×C12711yescharacteristic, complemented
186#1810C2×C12711yescharacteristic, complemented
187C127#1273411yescharacteristic, direct product factor, Sylow
188C22×#41—21no—
189C2×#135—28no—
190C2×#135—21no—
191C2×#135—21no—
192C2×#135C4×C12712yes—
193C2×#135C22×C12712yes—
194C22×Q16C4×C12711yescharacteristic
195C22×SD16—21no—
196C22×D8C4×C12711yescharacteristic
197C23×C8C22×C12711yescharacteristic
198C22×C16C22×C12712yescomplemented
199C2×#108C22×C12712yescomplemented
200#407—21no—
201#359—28no—
202C22⋊C16—21no—
203C22⋊C16—21no—
204#341—21no—
205C2×#41—28no—
206C2×#41—22no—
207C2×#41—22no—
208C2×#41—44no—
209C2×#41—41no—
210C2×#41—41no—
211C22×Q8—21no—
212C22×D4—21no—
213C22×D4—41no—
214C23×C4—41no—
215C23×C4D4×C12711yescharacteristic
216#135—28no—
217#135—28no—
218#135—24no—
219#135—24no—
220#135—24no—
221#135D4×C12718yes—
222C2×Q16—24no—
223C2×Q16—21no—
224C2×Q16C2×C4×C12712yes—
225C2×SD16—28no—
226C2×SD16—22no—
227C2×SD16—22no—
228C2×D8—24no—
229C2×D8—21no—
230C2×D8C2×C4×C12712yes—
231C2×M16—22no—
232C22×C8—24no—
233C22×C8C2×C4×C12711yescharacteristic
234C22×C8C2×C4×C12711yescharacteristic
235C22×C8D4×C12711yescharacteristic
236C22×C8D4×C12711yescharacteristic
237C22×C8D4×C12711yescharacteristic
238C22×C8C23×C12711yescharacteristic
239C2×C16—22no—
240C2×C16—24no—
241#108—24no—
242C22⋊C8—41no—
243C22⋊C8—41no—
244C24—41no—
245C24SD16×C12711yescharacteristic
246#41—28no—
247#41—28no—
248#41—48no—
249#41—44no—
250#41—48no—
251#41—44no—
252C2×Q8—24no—
253C2×Q8—22no—
254C2×Q8—42no—
255C2×Q8—41no—
256C2×D4—24no—
257C2×D4—22no—
258C2×D4—42no—
259C2×D4—48no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—41no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—48no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4C22⋊C4×C12711yescharacteristic
270C22×C4C22⋊C4×C12711yescharacteristic
271C22×C4D8×C12711yescharacteristic
272C22×C4D8×C12711yescharacteristic
273C22×C4SD16×C12711yescharacteristic
274C22×C4C2×D4×C12711yescharacteristic
275Q16—24no—
276Q16—22no—
277Q16C22⋊C4×C12714yes—
278SD16—28no—
279SD16—28no—
280D8—24no—
281D8—22no—
282D8C22⋊C4×C12714yes—
283M16—42no—
284M16—42no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—42no—
291C2×C8C22⋊C4×C12711yescharacteristic
292C2×C8C22⋊C4×C12711yescharacteristic
293C2×C8C22×C4×C12711yescharacteristic
294C2×C8C2×D4×C12714yes—
295C2×C8C2×D4×C12711yescharacteristic
296C16—42no—
297C23—22no—
298C23—22no—
299C23—41no—
300C23—44no—
301C23—42no—
302C23—82no—
303C23—82no—
304C23#102×C12711yescharacteristic
305C23#102×C12711yescharacteristic
306C23C2×SD16×C12711yescharacteristic
307Q8—24no—
308Q8—44no—
309Q8—42no—
310D4—24no—
311D4—44no—
312D4—48no—
313D4—48no—
314D4—42no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—48no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—48no—
326C2×C4#102×C12711yescharacteristic
327C2×C4#102×C12711yescharacteristic
328C2×C4C2×C22⋊C4×C12711yescharacteristic
329C2×C4#120×C12712yes—
330C2×C4C2×D8×C12711yescharacteristic
331C2×C4AGL(1,ℤ/8)×C12712yes—
332C8—21no—
333C8—21no—
334C8—82no—
335C8C2×C22⋊C4×C12712yes—
336C8#120×C12712yes—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—44no—
345C22—44no—
346C22—88no—
347C22—82no—
348C22—82no—
349C22#359×C12712yescomplemented
350C22C2×#102×C12711yescharacteristic
351C22#450×C12712yes—
352C4—21no—
353C4—21no—
354C4—44no—
355C4#379×C12711yescharacteristic
356C4#418×C12711yescharacteristic
357C4#447×C12712yes—
358C2—22no—
359C2—22no—
360C2—41no—
361C2—84no—
362C2C127×#178611yescharacteristic
363C2C2×#359×C12711yescharacteristic
364C2C127×#214311yescharacteristic
subgroupG/H
centerC2×C4×C127C2×D8
commutatorC2×C8C22×C4×C127
Frattini, cosocleC22×C8C23×C127
socleC22×C127C2×#102
prime radicalC22×#135×C127C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C23×C4×C127 ◁ C23×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#359×C127C2semidirect
3. C2×C22⋊C16×C127C2semidirect
4. C127×#1810C2semidirect
15. C22×C16×C127C22semidirect
16. C2×#108×C127C22semidirect
166. C22×C127#359semidirect
175. #12734C127direct (N × H)
184. C2×#359C2×C127semidirect
185. C2×C22⋊C16C2×C127semidirect
186. #1810C2×C127semidirect
187. C127#12734direct (N × H)
198. C22×C16C22×C127semidirect
199. C2×#108C22×C127semidirect
349. C22#359×C127semidirect

Maximal subgroups

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

subgroupindex
C22×#135×C1272
C2×#359×C1272
C2×C22⋊C16×C1272
C127×#18102
#12734127