Finite Group Databasebeta

← C127×#13049 #44576722

Subgroups — C127×#13049 #44576722

Order 32512.

796 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 796 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#453×C127C211yesmaximal, characteristic
2C127×#2380C211yesmaximal, characteristic, complemented
3C2×#353×C127C211yesmaximal, characteristic, complemented
4C127×#1790C212yesmaximal, complemented
5C127×#1390C212yesmaximal, complemented
6C2×H(5,𝔽2)×C127C2211yescharacteristic
7C2×AGL(1,ℤ/8)×C127—21no—
8C2×#127×C127C2211yescharacteristic
9#453×C127—22no—
10#453×C127C414yescomplemented
11C2×#104×C127—21no—
12C2×#100×C127C2211yescharacteristic
13C2×#99×C127—22no—
14C2×#99×C127C2212yes—
15#386×C127—21no—
16#386×C127—22no—
17#386×C127—22no—
18#386×C127—22no—
19#386×C127C2212yes—
20#383×C127—21no—
21#353×C127—22no—
22H(5,𝔽2)×C127—22no—
23H(5,𝔽2)×C127—21no—
24H(5,𝔽2)×C127—42no—
25H(5,𝔽2)×C127C2×C412yescomplemented
26C2×#41×C127—21no—
27C2×#41×C127—41no—
28C22×D4×C127—22no—
29C22×D4×C127—22no—
30C22×D4×C127—41no—
31C22×D4×C127D411yescharacteristic
32C22×D4×C127C2311yescharacteristic
33C23×C4×C127—21no—
34C23×C4×C127D412yes—
35AGL(1,ℤ/8)×C127—24no—
36AGL(1,ℤ/8)×C127—42no—
37C2×SD16×C127—41no—
38C2×D8×C127—41no—
39C2×M16×C127—21no—
40#127×C127—22no—
41#127×C127C2×C412yes—
42#127×C127D412yes—
43C2×C22⋊C4×C127—22no—
44C2×C22⋊C4×C127—21no—
45C2×C22⋊C4×C127—22no—
46C2×C22⋊C4×C127—22no—
47C2×C22⋊C4×C127—42no—
48C2×C22⋊C4×C127D412yes—
49C2×C42×C127D411yescharacteristic
50#104×C127—24no—
51#100×C127—21no—
52#100×C127C2×C412yescomplemented
53#99×C127—22no—
54#99×C127—22no—
55#99×C127—44no—
56#95×C127—21no—
57#95×C127—22no—
58#95×C127—21no—
59#95×C127—22no—
60#95×C127—42no—
61C24×C127—22no—
62C24×C127—21no—
63C24×C127C2×D412yes—
64#41×C127—22no—
65#41×C127—22no—
66#41×C127—42no—
67#41×C127—41no—
68#41×C127—44no—
69#41×C127—42no—
70#41×C127—41no—
71C2×Q8×C127—21no—
72C2×D4×C127—21no—
73C2×D4×C127—24no—
74C2×D4×C127—24no—
75C2×D4×C127—24no—
76C2×D4×C127—24no—
77C2×D4×C127—21no—
78C2×D4×C127—41no—
79C2×D4×C127—41no—
80C2×D4×C127—41no—
81C2×D4×C127—41no—
82C2×D4×C127—42no—
83C2×D4×C127—42no—
84C2×D4×C127—42no—
85C2×D4×C127—42no—
86C2×D4×C127—44no—
87C2×D4×C127—41no—
88C2×D4×C127—42no—
89C2×D4×C127—82no—
90C2×D4×C127—82no—
91C2×D4×C127C22⋊C412yescomplemented
92C2×D4×C127C22×C411yescharacteristic
93C2×D4×C127C2×D411yescharacteristic
94C2×D4×C127#4112yes—
95C22×C4×C127—22no—
96C22×C4×C127—21no—
97C22×C4×C127—22no—
98C22×C4×C127—22no—
99C22×C4×C127—41no—
100C22×C4×C127—42no—
101C22×C4×C127—42no—
102C22×C4×C127—82no—
103C22×C4×C127—82no—
104C22×C4×C127—81no—
105C22×C4×C127C2×D411yescharacteristic
106SD16×C127—44no—
107D8×C127—44no—
108M16×C127—22no—
109M16×C127—41no—
110C2×C8×C127—41no—
111C22⋊C4×C127—42no—
112C22⋊C4×C127—42no—
113C22⋊C4×C127—42no—
114C22⋊C4×C127—82no—
115C22⋊C4×C127—82no—
116C22⋊C4×C127—81no—
117C22⋊C4×C127—84no—
118C42×C127—21no—
119C42×C127C22⋊C412yescomplemented
120C23×C127—22no—
121C23×C127—22no—
122C23×C127—24no—
123C23×C127—22no—
124C23×C127—42no—
125C23×C127—42no—
126C23×C127—42no—
127C23×C127—42no—
128C23×C127—42no—
129C23×C127—41no—
130C23×C127—41no—
131C23×C127—41no—
132C23×C127—82no—
133C23×C127—82no—
134C23×C127C4×D412yes—
135C23×C127#12011yescharacteristic
136C23×C127#12112yes—
137Q8×C127—22no—
138Q8×C127—41no—
139D4×C127—22no—
140D4×C127—41no—
141D4×C127—41no—
142D4×C127—42no—
143D4×C127—42no—
144D4×C127—41no—
145D4×C127—44no—
146D4×C127—44no—
147D4×C127—42no—
148D4×C127—42no—
149D4×C127—81no—
150D4×C127—84no—
151D4×C127—84no—
152C2×C4×C127—21no—
153C2×C4×C127—21no—
154C2×C4×C127—24no—
155C2×C4×C127—22no—
156C2×C4×C127—42no—
157C2×C4×C127—42no—
158C2×C4×C127—42no—
159C2×C4×C127—42no—
160C2×C4×C127—41no—
161C2×C4×C127—42no—
162C2×C4×C127—42no—
163C2×C4×C127—42no—
164C2×C4×C127—42no—
165C2×C4×C127—44no—
166C2×C4×C127—42no—
167C2×C4×C127—42no—
168C2×C4×C127—84no—
169C2×C4×C127—82no—
170C2×C4×C127—82no—
171C2×C4×C127—82no—
172C2×C4×C127—82no—
173C2×C4×C127C2×C22⋊C411yescharacteristic
174C2×C4×C127#12311yescharacteristic
175C8×C127—42no—
176C22×C127—22no—
177C22×C127—22no—
178C22×C127—22no—
179C22×C127—42no—
180C22×C127—42no—
181C22×C127—44no—
182C22×C127—42no—
183C22×C127—42no—
184C22×C127—42no—
185C22×C127—42no—
186C22×C127—82no—
187C22×C127—81no—
188C22×C127—81no—
189C22×C127—84no—
190C22×C127—82no—
191C22×C127—82no—
192C22×C127—162no—
193C22×C127#38611yescharacteristic
194C22×C127B(4,𝔽2)11yescharacteristic
195C22×C127B(4,𝔽2)11yescharacteristic
196C4×C127—21no—
197C4×C127—21no—
198C4×C127—42no—
199C4×C127—42no—
200C4×C127—82no—
201C4×C127—82no—
202C4×C127—162no—
203#13049C12711yesmaximal, characteristic, direct product factor, Sylow
204C2×C127—21no—
205C2×C127—21no—
206C2×C127—42no—
207C2×C127—42no—
208C2×C127—82no—
209C2×C127—82no—
210C2×C127#188911yescharacteristic
211C2×C127#220012yes—
212C2×#453C2×C12711yescharacteristic
213#2380C2×C12711yescharacteristic, complemented
214C2×#353C2×C12711yescharacteristic, complemented
215#1790C2×C12712yescomplemented
216#1390C2×C12712yescomplemented
217C127#1304911yescharacteristic, direct product factor, Sylow
218C2×H(5,𝔽2)C22×C12711yescharacteristic
219C2×AGL(1,ℤ/8)—21no—
220C2×#127C22×C12711yescharacteristic
221#453—22no—
222#453C4×C12714yescomplemented
223C2×#104—21no—
224C2×#100C22×C12711yescharacteristic
225C2×#99—22no—
226C2×#99C22×C12712yes—
227#386—21no—
228#386—22no—
229#386—22no—
230#386—22no—
231#386C22×C12712yes—
232#383—21no—
233#353—22no—
234H(5,𝔽2)—22no—
235H(5,𝔽2)—21no—
236H(5,𝔽2)—42no—
237H(5,𝔽2)C2×C4×C12712yescomplemented
238C2×#41—21no—
239C2×#41—41no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—41no—
243C22×D4D4×C12711yescharacteristic
244C22×D4C23×C12711yescharacteristic
245C23×C4—21no—
246C23×C4D4×C12712yes—
247AGL(1,ℤ/8)—24no—
248AGL(1,ℤ/8)—42no—
249C2×SD16—41no—
250C2×D8—41no—
251C2×M16—21no—
252#127—22no—
253#127C2×C4×C12712yes—
254#127D4×C12712yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4D4×C12712yes—
261C2×C42D4×C12711yescharacteristic
262#104—24no—
263#100—21no—
264#100C2×C4×C12712yescomplemented
265#99—22no—
266#99—22no—
267#99—44no—
268#95—21no—
269#95—22no—
270#95—21no—
271#95—22no—
272#95—42no—
273C24—22no—
274C24—21no—
275C24C2×D4×C12712yes—
276#41—22no—
277#41—22no—
278#41—42no—
279#41—41no—
280#41—44no—
281#41—42no—
282#41—41no—
283C2×Q8—21no—
284C2×D4—21no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—21no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—44no—
299C2×D4—41no—
300C2×D4—42no—
301C2×D4—82no—
302C2×D4—82no—
303C2×D4C22⋊C4×C12712yescomplemented
304C2×D4C22×C4×C12711yescharacteristic
305C2×D4C2×D4×C12711yescharacteristic
306C2×D4#41×C12712yes—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—81no—
317C22×C4C2×D4×C12711yescharacteristic
318SD16—44no—
319D8—44no—
320M16—22no—
321M16—41no—
322C2×C8—41no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C22⋊C4—81no—
329C22⋊C4—84no—
330C42—21no—
331C42C22⋊C4×C12712yescomplemented
332C23—22no—
333C23—22no—
334C23—24no—
335C23—22no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—82no—
345C23—82no—
346C23C4×D4×C12712yes—
347C23#120×C12711yescharacteristic
348C23#121×C12712yes—
349Q8—22no—
350Q8—41no—
351D4—22no—
352D4—41no—
353D4—41no—
354D4—42no—
355D4—42no—
356D4—41no—
357D4—44no—
358D4—44no—
359D4—42no—
360D4—42no—
361D4—81no—
362D4—84no—
363D4—84no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—84no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4C2×C22⋊C4×C12711yescharacteristic
386C2×C4#123×C12711yescharacteristic
387C8—42no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—42no—
392C22—42no—
393C22—44no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—82no—
399C22—81no—
400C22—81no—
401C22—84no—
402C22—82no—
403C22—82no—
404C22—162no—
405C22#386×C12711yescharacteristic
406C22B(4,𝔽2)×C12711yescharacteristic
407C22B(4,𝔽2)×C12711yescharacteristic
408C4—21no—
409C4—21no—
410C4—42no—
411C4—42no—
412C4—82no—
413C4—82no—
414C4—162no—
415C2—21no—
416C2—21no—
417C2—42no—
418C2—42no—
419C2—82no—
420C2—82no—
421C2C127×#188911yescharacteristic
422C2C127×#220012yes—
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutatorC2×D4C22×C4×C127
Frattini, cosocleC22×D4C23×C127
prime radicalC2×#453×C127C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C127 ◁ C22×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2380C2semidirect
3. C2×#353×C127C2semidirect
4. C127×#1790C2semidirect
5. C127×#1390C2semidirect
10. #453×C127C4semidirect
25. H(5,𝔽2)×C127C2×C4semidirect
52. #100×C127C2×C4semidirect
91. C2×D4×C127C22⋊C4semidirect
119. C42×C127C22⋊C4semidirect
203. #13049C127direct (N × H)
213. #2380C2×C127semidirect
214. C2×#353C2×C127semidirect
215. #1790C2×C127semidirect
216. #1390C2×C127semidirect
217. C127#13049direct (N × H)
222. #453C4×C127semidirect
237. H(5,𝔽2)C2×C4×C127semidirect
264. #100C2×C4×C127semidirect
303. C2×D4C22⋊C4×C127semidirect
331. C42C22⋊C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#453×C1272
C127×#23802
C2×#353×C1272
C127×#17902
C127×#13902
#13049127