Finite Group Databasebeta

← C127×#12691 #44576364

Subgroups — C127×#12691 #44576364

Order 32512.

1168 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 150 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×#535×C127C211yesmaximal, characteristic
2C2×AΣL(1,𝔽16)×C127C214yesmaximal, complemented
3C127×#1785C211yesmaximal, characteristic, complemented
4C127×#1782C211yesmaximal, characteristic, complemented
5C22×#41×C127C2211yescharacteristic
6C23×D4×C127C411yescharacteristic, complemented
7#535×C127—28no—
8C2×#127×C127—21no—
9C2×#124×C127—21no—
10C2×#121×C127—21no—
11C2×#121×C127—21no—
12C2×#120×C127C412yescomplemented
13C2×#120×C127C2212yes—
14C2×#117×C127C411yescharacteristic, complemented
15C2×#101×C127—21no—
16C2×#100×C127C2212yes—
17C2×#99×C127—21no—
18C2×#99×C127C2212yes—
19#407×C127—21no—
20#381×C127—21no—
21AΣL(1,𝔽16)×C127—28no—
22C25×C127C2×C412yes—
23C2×#41×C127—24no—
24C2×#41×C127—42no—
25C2×#41×C127—42no—
26C22×Q8×C127D411yescharacteristic
27C22×D4×C127—24no—
28C22×D4×C127—22no—
29C22×D4×C127—22no—
30C22×D4×C127—21no—
31C22×D4×C127—21no—
32C22×D4×C127—42no—
33C22×D4×C127—42no—
34C22×D4×C127C2×C411yescharacteristic
35C22×D4×C127D411yescharacteristic
36C22×D4×C127C2311yescharacteristic
37C23×C4×C127C2×C411yescharacteristic
38C23×C4×C127D411yescharacteristic
39C23×C4×C127D411yescharacteristic
40C2×M16×C127—22no—
41#127×C127—28no—
42#124×C127—28no—
43#121×C127—28no—
44#121×C127—28no—
45#120×C127—24no—
46#120×C127—24no—
47#120×C127—24no—
48#120×C127D418yes—
49#117×C127—24no—
50C2×C4⋊C4×C127—21no—
51C2×C22⋊C4×C127—22no—
52C2×C22⋊C4×C127—22no—
53C2×C22⋊C4×C127—22no—
54C2×C22⋊C4×C127C2×C412yes—
55C2×C42×C127—21no—
56#101×C127—42no—
57#100×C127—24no—
58#99×C127—24no—
59#99×C127—42no—
60C22⋊C8×C127—42no—
61#95×C127—42no—
62C24×C127—22no—
63C24×C127—22no—
64C24×C127—42no—
65C24×C127—42no—
66C24×C127—84no—
67C24×C127C22⋊C411yescharacteristic
68C24×C127C22⋊C414yes—
69C24×C127C22×C411yescharacteristic
70C24×C127C2×D411yescharacteristic
71#41×C127—416no—
72C2×Q8×C127—22no—
73C2×Q8×C127—22no—
74C2×Q8×C127—41no—
75C2×D4×C127—24no—
76C2×D4×C127—24no—
77C2×D4×C127—22no—
78C2×D4×C127—22no—
79C2×D4×C127—28no—
80C2×D4×C127—28no—
81C2×D4×C127—22no—
82C2×D4×C127—41no—
83C2×D4×C127—41no—
84C2×D4×C127—41no—
85C2×D4×C127—41no—
86C2×D4×C127—41no—
87C2×D4×C127—416no—
88C2×D4×C127—42no—
89C2×D4×C127—42no—
90C2×D4×C127—42no—
91C2×D4×C127—42no—
92C2×D4×C127—84no—
93C2×D4×C127—84no—
94C2×D4×C127C2×D414yes—
95C22×C4×C127—22no—
96C22×C4×C127—22no—
97C22×C4×C127—22no—
98C22×C4×C127—22no—
99C22×C4×C127—22no—
100C22×C4×C127—22no—
101C22×C4×C127—22no—
102C22×C4×C127—22no—
103C22×C4×C127—41no—
104C22×C4×C127—41no—
105C22×C4×C127—42no—
106C22×C4×C127C22⋊C411yescharacteristic
107C22×C4×C127C22⋊C411yescharacteristic
108C22×C4×C127C22⋊C411yescharacteristic
109C22×C4×C127C2×D411yescharacteristic
110M16×C127—42no—
111M16×C127—42no—
112C2×C8×C127—42no—
113C4⋊C4×C127—24no—
114C22⋊C4×C127—28no—
115C22⋊C4×C127—22no—
116C22⋊C4×C127—28no—
117C22⋊C4×C127—42no—
118C22⋊C4×C127—42no—
119C22⋊C4×C127C22⋊C414yes—
120C42×C127—24no—
121C23×C127—21no—
122C23×C127—21no—
123C23×C127—21no—
124C23×C127—21no—
125C23×C127—24no—
126C23×C127—21no—
127C23×C127—21no—
128C23×C127—21no—
129C23×C127—42no—
130C23×C127—42no—
131C23×C127—42no—
132C23×C127—44no—
133C23×C127—44no—
134C23×C127—44no—
135C23×C127—44no—
136C23×C127—42no—
137C23×C127—42no—
138C23×C127—84no—
139C23×C127—84no—
140C23×C127—88no—
141C23×C127—88no—
142C23×C127C2×C22⋊C411yescharacteristic
143C23×C127C2×C22⋊C412yes—
144C23×C127#12012yes—
145Q8×C127—44no—
146D4×C127—44no—
147D4×C127—44no—
148D4×C127—48no—
149D4×C127—48no—
150D4×C127—44no—
151D4×C127—88no—
152D4×C127—88no—
153C2×C4×C127—28no—
154C2×C4×C127—21no—
155C2×C4×C127—21no—
156C2×C4×C127—21no—
157C2×C4×C127—21no—
158C2×C4×C127—21no—
159C2×C4×C127—21no—
160C2×C4×C127—24no—
161C2×C4×C127—21no—
162C2×C4×C127—21no—
163C2×C4×C127—21no—
164C2×C4×C127—44no—
165C2×C4×C127—44no—
166C2×C4×C127—44no—
167C2×C4×C127—82no—
168C2×C4×C127—84no—
169C2×C4×C127#9912yes—
170C2×C4×C127#9912yes—
171C2×C4×C127#12012yes—
172C8×C127—82no—
173C22×C127—22no—
174C22×C127—22no—
175C22×C127—22no—
176C22×C127—42no—
177C22×C127—42no—
178C22×C127—42no—
179C22×C127—42no—
180C22×C127—44no—
181C22×C127—44no—
182C22×C127—42no—
183C22×C127—42no—
184C22×C127—42no—
185C22×C127—41no—
186C22×C127—44no—
187C22×C127—81no—
188C22×C127—81no—
189C22×C127—84no—
190C22×C127—88no—
191C22×C127—88no—
192C22×C127—84no—
193C22×C127#37911yescharacteristic
194C22×C127C2×#9911yescharacteristic
195C22×C127C2×#9911yescharacteristic
196C4×C127—22no—
197C4×C127—22no—
198C4×C127—22no—
199C4×C127—44no—
200C4×C127—41no—
201C4×C127—162no—
202#12691C12711yesmaximal, characteristic, direct product factor, Sylow
203C2×C127—21no—
204C2×C127—21no—
205C2×C127—42no—
206C2×C127—42no—
207C2×C127—81no—
208C2×C127—84no—
209C2×C127#178111yescharacteristic
210C2×C127#212212yes—
211C2×#535C2×C12711yescharacteristic
212C2×AΣL(1,𝔽16)C2×C12714yescomplemented
213#1785C2×C12711yescharacteristic, complemented
214#1782C2×C12711yescharacteristic, complemented
215C127#1269111yescharacteristic, direct product factor, Sylow
216C22×#41C22×C12711yescharacteristic
217C23×D4C4×C12711yescharacteristic, complemented
218#535—28no—
219C2×#127—21no—
220C2×#124—21no—
221C2×#121—21no—
222C2×#121—21no—
223C2×#120C4×C12712yescomplemented
224C2×#120C22×C12712yes—
225C2×#117C4×C12711yescharacteristic, complemented
226C2×#101—21no—
227C2×#100C22×C12712yes—
228C2×#99—21no—
229C2×#99C22×C12712yes—
230#407—21no—
231#381—21no—
232AΣL(1,𝔽16)—28no—
233C25C2×C4×C12712yes—
234C2×#41—24no—
235C2×#41—42no—
236C2×#41—42no—
237C22×Q8D4×C12711yescharacteristic
238C22×D4—24no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—42no—
244C22×D4—42no—
245C22×D4C2×C4×C12711yescharacteristic
246C22×D4D4×C12711yescharacteristic
247C22×D4C23×C12711yescharacteristic
248C23×C4C2×C4×C12711yescharacteristic
249C23×C4D4×C12711yescharacteristic
250C23×C4D4×C12711yescharacteristic
251C2×M16—22no—
252#127—28no—
253#124—28no—
254#121—28no—
255#121—28no—
256#120—24no—
257#120—24no—
258#120—24no—
259#120D4×C12718yes—
260#117—24no—
261C2×C4⋊C4—21no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4C2×C4×C12712yes—
266C2×C42—21no—
267#101—42no—
268#100—24no—
269#99—24no—
270#99—42no—
271C22⋊C8—42no—
272#95—42no—
273C24—22no—
274C24—22no—
275C24—42no—
276C24—42no—
277C24—84no—
278C24C22⋊C4×C12711yescharacteristic
279C24C22⋊C4×C12714yes—
280C24C22×C4×C12711yescharacteristic
281C24C2×D4×C12711yescharacteristic
282#41—416no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—41no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—28no—
291C2×D4—28no—
292C2×D4—22no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—416no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—84no—
304C2×D4—84no—
305C2×D4C2×D4×C12714yes—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4C22⋊C4×C12711yescharacteristic
318C22×C4C22⋊C4×C12711yescharacteristic
319C22×C4C22⋊C4×C12711yescharacteristic
320C22×C4C2×D4×C12711yescharacteristic
321M16—42no—
322M16—42no—
323C2×C8—42no—
324C4⋊C4—24no—
325C22⋊C4—28no—
326C22⋊C4—22no—
327C22⋊C4—28no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4C22⋊C4×C12714yes—
331C42—24no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—24no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—44no—
344C23—44no—
345C23—44no—
346C23—44no—
347C23—42no—
348C23—42no—
349C23—84no—
350C23—84no—
351C23—88no—
352C23—88no—
353C23C2×C22⋊C4×C12711yescharacteristic
354C23C2×C22⋊C4×C12712yes—
355C23#120×C12712yes—
356Q8—44no—
357D4—44no—
358D4—44no—
359D4—48no—
360D4—48no—
361D4—44no—
362D4—88no—
363D4—88no—
364C2×C4—28no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—24no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—82no—
379C2×C4—84no—
380C2×C4#99×C12712yes—
381C2×C4#99×C12712yes—
382C2×C4#120×C12712yes—
383C8—82no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—44no—
392C22—44no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—41no—
397C22—44no—
398C22—81no—
399C22—81no—
400C22—84no—
401C22—88no—
402C22—88no—
403C22—84no—
404C22#379×C12711yescharacteristic
405C22C2×#99×C12711yescharacteristic
406C22C2×#99×C12711yescharacteristic
407C4—22no—
408C4—22no—
409C4—22no—
410C4—44no—
411C4—41no—
412C4—162no—
413C2—21no—
414C2—21no—
415C2—42no—
416C2—42no—
417C2—81no—
418C2—84no—
419C2C127×#178111yescharacteristic
420C2C127×#212212yes—
subgroupG/H
center, socleC22×C127C2×#99
commutatorC24C22×C4×C127
Frattini, cosocleC22×D4C23×C127
prime radicalC2×#535×C127C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C22×#41×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×AΣL(1,𝔽16)×C127C2semidirect
3. C127×#1785C2semidirect
4. C127×#1782C2semidirect
6. C23×D4×C127C4semidirect
12. C2×#120×C127C4semidirect
14. C2×#117×C127C4semidirect
202. #12691C127direct (N × H)
212. C2×AΣL(1,𝔽16)C2×C127semidirect
213. #1785C2×C127semidirect
214. #1782C2×C127semidirect
215. C127#12691direct (N × H)
217. C23×D4C4×C127semidirect
223. C2×#120C4×C127semidirect
225. C2×#117C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#535×C1272
C2×AΣL(1,𝔽16)×C1272
C127×#17852
C127×#17822
#12691127