Finite Group Databasebeta

← C127×#13218 #44576891

Subgroups — C127×#13218 #44576891

Order 32512.

840 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 126 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
1C127×#2408C211yesmaximal, characteristic
2C2×AΣL(1,𝔽16)×C127C214yesmaximal, complemented
3C127×#1799C211yesmaximal, characteristic, complemented
4C4×#100×C127C211yesmaximal, characteristic, complemented
5C23×D4×C127C411yescharacteristic, complemented
6C2×#121×C127—21no—
7C2×#120×C127C2212yes—
8C2×C4×D4×C127C2211yescharacteristic
9C2×#100×C127C2212yes—
10C2×#99×C127C2212yes—
11C4×M16×C127—21no—
12#394×C127—21no—
13#392×C127—21no—
14#390×C127—21no—
15#380×C127—21no—
16#380×C127C411yescharacteristic, complemented
17#379×C127C412yescomplemented
18AΣL(1,𝔽16)×C127—28no—
19#343×C127—21no—
20#342×C127—21no—
21C25×C127C2×C412yes—
22C22×D4×C127—22no—
23C22×D4×C127—21no—
24C22×D4×C127—21no—
25C22×D4×C127—41no—
26C22×D4×C127—41no—
27C22×D4×C127—44no—
28C22×D4×C127C2311yescharacteristic
29C23×C4×C127C2×C411yescharacteristic
30C23×C4×C127D411yescharacteristic
31C2×M16×C127—22no—
32#121×C127—44no—
33#120×C127—24no—
34#120×C127—24no—
35C4×D4×C127—41no—
36C4×D4×C127—41no—
37C2×C4⋊C4×C127—21no—
38C2×C4⋊C4×C127—41no—
39C2×C4⋊C4×C127D411yescharacteristic
40C2×C22⋊C4×C127—22no—
41C2×C22⋊C4×C127—22no—
42C2×C22⋊C4×C127—22no—
43C2×C22⋊C4×C127—22no—
44C2×C22⋊C4×C127C2×C412yes—
45C2×C22⋊C4×C127C2×C411yescharacteristic
46C2×C22⋊C4×C127D411yescharacteristic
47C2×C42×C127D411yescharacteristic
48#100×C127—22no—
49#100×C127D414yescomplemented
50#99×C127—24no—
51#97×C127—41no—
52C4×C8×C127—41no—
53#95×C127—21no—
54#95×C127—41no—
55C24×C127—22no—
56C24×C127—24no—
57C24×C127—42no—
58C24×C127—44no—
59C24×C127—42no—
60C24×C127—82no—
61C24×C127C22×C411yescharacteristic
62C24×C127C2×D411yescharacteristic
63C2×D4×C127—24no—
64C2×D4×C127—41no—
65C2×D4×C127—41no—
66C2×D4×C127—42no—
67C2×D4×C127—42no—
68C2×D4×C127—44no—
69C2×D4×C127—48no—
70C2×D4×C127—44no—
71C2×D4×C127—44no—
72C2×D4×C127—44no—
73C2×D4×C127—42no—
74C2×D4×C127—42no—
75C2×D4×C127—41no—
76C2×D4×C127—41no—
77C2×D4×C127—82no—
78C2×D4×C127C2×D412yes—
79C2×D4×C127#4112yes—
80C22×C4×C127—21no—
81C22×C4×C127—21no—
82C22×C4×C127—21no—
83C22×C4×C127—21no—
84C22×C4×C127—22no—
85C22×C4×C127—42no—
86C22×C4×C127—42no—
87C22×C4×C127—42no—
88C22×C4×C127C22⋊C411yescharacteristic
89C22×C4×C127C22⋊C411yescharacteristic
90C22×C4×C127C22⋊C411yescharacteristic
91C22×C4×C127C22⋊C411yescharacteristic
92C22×C4×C127C2×D411yescharacteristic
93M16×C127—24no—
94M16×C127—42no—
95C2×C8×C127—42no—
96C4⋊C4×C127—41no—
97C4⋊C4×C127—42no—
98C4⋊C4×C127—82no—
99C22⋊C4×C127—24no—
100C22⋊C4×C127—41no—
101C22⋊C4×C127—41no—
102C22⋊C4×C127—44no—
103C22⋊C4×C127—44no—
104C22⋊C4×C127—42no—
105C22⋊C4×C127—42no—
106C22⋊C4×C127—82no—
107C42×C127—21no—
108C42×C127—21no—
109C23×C127—21no—
110C23×C127—21no—
111C23×C127—21no—
112C23×C127—21no—
113C23×C127—24no—
114C23×C127—41no—
115C23×C127—41no—
116C23×C127—41no—
117C23×C127—41no—
118C23×C127—42no—
119C23×C127—42no—
120C23×C127—42no—
121C23×C127—44no—
122C23×C127—44no—
123C23×C127—42no—
124C23×C127—42no—
125C23×C127—42no—
126C23×C127—84no—
127C23×C127—88no—
128C23×C127—88no—
129C23×C127—84no—
130C23×C127C2×C22⋊C411yescharacteristic
131C23×C127C4×D412yes—
132C23×C127#12411yescharacteristic
133C23×C127#12711yescharacteristic
134D4×C127—42no—
135D4×C127—42no—
136D4×C127—44no—
137D4×C127—48no—
138D4×C127—44no—
139D4×C127—84no—
140D4×C127—84no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—22no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127—42no—
148C2×C4×C127—42no—
149C2×C4×C127—42no—
150C2×C4×C127—41no—
151C2×C4×C127—41no—
152C2×C4×C127—41no—
153C2×C4×C127—42no—
154C2×C4×C127—42no—
155C2×C4×C127—42no—
156C2×C4×C127—81no—
157C2×C4×C127—82no—
158C2×C4×C127—82no—
159C2×C4×C127—82no—
160C2×C4×C127#9911yescharacteristic
161C2×C4×C127#9911yescharacteristic
162C2×C4×C127#12112yes—
163C8×C127—44no—
164C22×C127—21no—
165C22×C127—21no—
166C22×C127—21no—
167C22×C127—22no—
168C22×C127—21no—
169C22×C127—41no—
170C22×C127—41no—
171C22×C127—42no—
172C22×C127—42no—
173C22×C127—44no—
174C22×C127—44no—
175C22×C127—41no—
176C22×C127—41no—
177C22×C127—41no—
178C22×C127—41no—
179C22×C127—44no—
180C22×C127—81no—
181C22×C127—82no—
182C22×C127—82no—
183C22×C127—88no—
184C22×C127—88no—
185C22×C127—82no—
186C22×C127—82no—
187C22×C127#39011yescharacteristic
188C22×C127C2×#9911yescharacteristic
189C22×C127#41011yescharacteristic
190C4×C127—21no—
191C4×C127—22no—
192C4×C127—41no—
193C4×C127—41no—
194C4×C127—41no—
195C4×C127—82no—
196C4×C127—162no—
197C4×C127AΣL(1,𝔽16)12yescomplemented
198#13218C12711yesmaximal, characteristic, direct product factor, Sylow
199C2×C127—21no—
200C2×C127—21no—
201C2×C127—41no—
202C2×C127—42no—
203C2×C127—41no—
204C2×C127—84no—
205C2×C127#196411yescharacteristic
206C2×C127C2×AΣL(1,𝔽16)11yescharacteristic
207C2×C127#212211yescharacteristic
208#2408C2×C12711yescharacteristic
209C2×AΣL(1,𝔽16)C2×C12714yescomplemented
210#1799C2×C12711yescharacteristic, complemented
211C4×#100C2×C12711yescharacteristic, complemented
212C127#1321811yescharacteristic, direct product factor, Sylow
213C23×D4C4×C12711yescharacteristic, complemented
214C2×#121—21no—
215C2×#120C22×C12712yes—
216C2×C4×D4C22×C12711yescharacteristic
217C2×#100C22×C12712yes—
218C2×#99C22×C12712yes—
219C4×M16—21no—
220#394—21no—
221#392—21no—
222#390—21no—
223#380—21no—
224#380C4×C12711yescharacteristic, complemented
225#379C4×C12712yescomplemented
226AΣL(1,𝔽16)—28no—
227#343—21no—
228#342—21no—
229C25C2×C4×C12712yes—
230C22×D4—22no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—41no—
234C22×D4—41no—
235C22×D4—44no—
236C22×D4C23×C12711yescharacteristic
237C23×C4C2×C4×C12711yescharacteristic
238C23×C4D4×C12711yescharacteristic
239C2×M16—22no—
240#121—44no—
241#120—24no—
242#120—24no—
243C4×D4—41no—
244C4×D4—41no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4D4×C12711yescharacteristic
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4C2×C4×C12712yes—
253C2×C22⋊C4C2×C4×C12711yescharacteristic
254C2×C22⋊C4D4×C12711yescharacteristic
255C2×C42D4×C12711yescharacteristic
256#100—22no—
257#100D4×C12714yescomplemented
258#99—24no—
259#97—41no—
260C4×C8—41no—
261#95—21no—
262#95—41no—
263C24—22no—
264C24—24no—
265C24—42no—
266C24—44no—
267C24—42no—
268C24—82no—
269C24C22×C4×C12711yescharacteristic
270C24C2×D4×C12711yescharacteristic
271C2×D4—24no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—44no—
277C2×D4—48no—
278C2×D4—44no—
279C2×D4—44no—
280C2×D4—44no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—82no—
286C2×D4C2×D4×C12712yes—
287C2×D4#41×C12712yes—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4C22⋊C4×C12711yescharacteristic
297C22×C4C22⋊C4×C12711yescharacteristic
298C22×C4C22⋊C4×C12711yescharacteristic
299C22×C4C22⋊C4×C12711yescharacteristic
300C22×C4C2×D4×C12711yescharacteristic
301M16—24no—
302M16—42no—
303C2×C8—42no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—82no—
307C22⋊C4—24no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—82no—
315C42—21no—
316C42—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—24no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—44no—
330C23—44no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—84no—
335C23—88no—
336C23—88no—
337C23—84no—
338C23C2×C22⋊C4×C12711yescharacteristic
339C23C4×D4×C12712yes—
340C23#124×C12711yescharacteristic
341C23#127×C12711yescharacteristic
342D4—42no—
343D4—42no—
344D4—44no—
345D4—48no—
346D4—44no—
347D4—84no—
348D4—84no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—81no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4#99×C12711yescharacteristic
369C2×C4#99×C12711yescharacteristic
370C2×C4#121×C12712yes—
371C8—44no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—22no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—42no—
380C22—42no—
381C22—44no—
382C22—44no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—44no—
388C22—81no—
389C22—82no—
390C22—82no—
391C22—88no—
392C22—88no—
393C22—82no—
394C22—82no—
395C22#390×C12711yescharacteristic
396C22C2×#99×C12711yescharacteristic
397C22#410×C12711yescharacteristic
398C4—21no—
399C4—22no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—82no—
404C4—162no—
405C4AΣL(1,𝔽16)×C12712yescomplemented
406C2—21no—
407C2—21no—
408C2—41no—
409C2—42no—
410C2—41no—
411C2—84no—
412C2C127×#196411yescharacteristic
413C2C2×AΣL(1,𝔽16)×C12711yescharacteristic
414C2C127×#212211yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC24C22×C4×C127
Frattini, cosocleC22×D4C23×C127
prime radicalC127×#2408C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C4×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×AΣL(1,𝔽16)×C127C2semidirect
3. C127×#1799C2semidirect
4. C4×#100×C127C2semidirect
5. C23×D4×C127C4semidirect
16. #380×C127C4semidirect
17. #379×C127C4semidirect
49. #100×C127D4semidirect
197. C4×C127AΣL(1,𝔽16)semidirect
198. #13218C127direct (N × H)
209. C2×AΣL(1,𝔽16)C2×C127semidirect
210. #1799C2×C127semidirect
211. C4×#100C2×C127semidirect
212. C127#13218direct (N × H)
213. C23×D4C4×C127semidirect
224. #380C4×C127semidirect
225. #379C4×C127semidirect
257. #100D4×C127semidirect
405. C4AΣL(1,𝔽16)×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#24082
C2×AΣL(1,𝔽16)×C1272
C127×#17992
C4×#100×C1272
#13218127