Finite Group Databasebeta

← C127×#13221 #44576894

Subgroups — C127×#13221 #44576894

Order 32512.

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

normal subgroups only

This group has 844 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×#1800C211yesmaximal, characteristic, complemented
4C4×#99×C127C211yesmaximal, characteristic, complemented
5C23×D4×C127C411yescharacteristic, complemented
6C2×#121×C127—21no—
7C2×#120×C127C2212yes—
8C2×C4×D4×C127C2211yescharacteristic
9#423×C127—21no—
10C2×#100×C127C2212yes—
11C2×#99×C127C2212yes—
12#394×C127—21no—
13#392×C127—21no—
14#390×C127—21no—
15#380×C127C411yescharacteristic, complemented
16#379×C127C412yescomplemented
17C4×C22⋊C4×C127—21no—
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×C127D411yescharacteristic
39C2×C22⋊C4×C127—22no—
40C2×C22⋊C4×C127—22no—
41C2×C22⋊C4×C127—22no—
42C2×C22⋊C4×C127—22no—
43C2×C22⋊C4×C127C2×C412yes—
44C2×C22⋊C4×C127C2×C411yescharacteristic
45C2×C22⋊C4×C127D411yescharacteristic
46C2×C42×C127—41no—
47C2×C42×C127D411yescharacteristic
48C4⋊C8×C127—41no—
49C4⋊C8×C127—41no—
50#100×C127—24no—
51#99×C127—22no—
52#99×C127D414yescomplemented
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—44no—
74C2×D4×C127—41no—
75C2×D4×C127—41no—
76C2×D4×C127—82no—
77C2×D4×C127C2×D412yes—
78C2×D4×C127#4112yes—
79C22×C4×C127—21no—
80C22×C4×C127—21no—
81C22×C4×C127—21no—
82C22×C4×C127—21no—
83C22×C4×C127—22no—
84C22×C4×C127—42no—
85C22×C4×C127—42no—
86C22×C4×C127—42no—
87C22×C4×C127C22⋊C411yescharacteristic
88C22×C4×C127C22⋊C411yescharacteristic
89C22×C4×C127C22⋊C411yescharacteristic
90C22×C4×C127C22⋊C411yescharacteristic
91C22×C4×C127C2×D411yescharacteristic
92M16×C127—42no—
93M16×C127—42no—
94C2×C8×C127—42no—
95C4⋊C4×C127—41no—
96C4⋊C4×C127—42no—
97C22⋊C4×C127—24no—
98C22⋊C4×C127—24no—
99C22⋊C4×C127—41no—
100C22⋊C4×C127—41no—
101C22⋊C4×C127—42no—
102C22⋊C4×C127—44no—
103C22⋊C4×C127—42no—
104C22⋊C4×C127—42no—
105C22⋊C4×C127—82no—
106C42×C127—21no—
107C42×C127—21no—
108C42×C127—82no—
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#12112yes—
133D4×C127—42no—
134D4×C127—42no—
135D4×C127—44no—
136D4×C127—48no—
137D4×C127—44no—
138D4×C127—88no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—22no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—42no—
146C2×C4×C127—42no—
147C2×C4×C127—42no—
148C2×C4×C127—41no—
149C2×C4×C127—41no—
150C2×C4×C127—41no—
151C2×C4×C127—44no—
152C2×C4×C127—42no—
153C2×C4×C127—42no—
154C2×C4×C127—42no—
155C2×C4×C127—81no—
156C2×C4×C127—82no—
157C2×C4×C127—82no—
158C2×C4×C127#9911yescharacteristic
159C2×C4×C127#9911yescharacteristic
160C2×C4×C127#12411yescharacteristic
161C2×C4×C127#12711yescharacteristic
162C8×C127—82no—
163C22×C127—21no—
164C22×C127—21no—
165C22×C127—22no—
166C22×C127—22no—
167C22×C127—42no—
168C22×C127—42no—
169C22×C127—42no—
170C22×C127—44no—
171C22×C127—44no—
172C22×C127—41no—
173C22×C127—41no—
174C22×C127—41no—
175C22×C127—41no—
176C22×C127—44no—
177C22×C127—81no—
178C22×C127—82no—
179C22×C127—82no—
180C22×C127—88no—
181C22×C127—88no—
182C22×C127—82no—
183C22×C127—82no—
184C22×C127#39011yescharacteristic
185C22×C127C2×#9911yescharacteristic
186C22×C127#41011yescharacteristic
187C4×C127—21no—
188C4×C127—21no—
189C4×C127—21no—
190C4×C127—41no—
191C4×C127—41no—
192C4×C127—41no—
193C4×C127—82no—
194C4×C127—84no—
195C4×C127AΣL(1,𝔽16)12yescomplemented
196#13221C12711yesmaximal, characteristic, direct product factor, Sylow
197C2×C127—21no—
198C2×C127—21no—
199C2×C127—42no—
200C2×C127—42no—
201C2×C127—84no—
202C2×C127#196411yescharacteristic
203C2×C127C2×AΣL(1,𝔽16)11yescharacteristic
204C2×C127#212211yescharacteristic
205#2408C2×C12711yescharacteristic
206C2×AΣL(1,𝔽16)C2×C12714yescomplemented
207#1800C2×C12711yescharacteristic, complemented
208C4×#99C2×C12711yescharacteristic, complemented
209C127#1322111yescharacteristic, direct product factor, Sylow
210C23×D4C4×C12711yescharacteristic, complemented
211C2×#121—21no—
212C2×#120C22×C12712yes—
213C2×C4×D4C22×C12711yescharacteristic
214#423—21no—
215C2×#100C22×C12712yes—
216C2×#99C22×C12712yes—
217#394—21no—
218#392—21no—
219#390—21no—
220#380C4×C12711yescharacteristic, complemented
221#379C4×C12712yescomplemented
222C4×C22⋊C4—21no—
223AΣL(1,𝔽16)—28no—
224#343—21no—
225#342—21no—
226C25C2×C4×C12712yes—
227C22×D4—22no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4—41no—
231C22×D4—41no—
232C22×D4—44no—
233C22×D4C23×C12711yescharacteristic
234C23×C4C2×C4×C12711yescharacteristic
235C23×C4D4×C12711yescharacteristic
236C2×M16—22no—
237#121—44no—
238#120—24no—
239#120—24no—
240C4×D4—41no—
241C4×D4—41no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4D4×C12711yescharacteristic
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4C2×C4×C12712yes—
249C2×C22⋊C4C2×C4×C12711yescharacteristic
250C2×C22⋊C4D4×C12711yescharacteristic
251C2×C42—41no—
252C2×C42D4×C12711yescharacteristic
253C4⋊C8—41no—
254C4⋊C8—41no—
255#100—24no—
256#99—22no—
257#99D4×C12714yescomplemented
258#95—21no—
259#95—41no—
260C24—22no—
261C24—24no—
262C24—42no—
263C24—44no—
264C24—42no—
265C24—82no—
266C24C22×C4×C12711yescharacteristic
267C24C2×D4×C12711yescharacteristic
268C2×D4—24no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—44no—
274C2×D4—48no—
275C2×D4—44no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—82no—
282C2×D4C2×D4×C12712yes—
283C2×D4#41×C12712yes—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4C22⋊C4×C12711yescharacteristic
293C22×C4C22⋊C4×C12711yescharacteristic
294C22×C4C22⋊C4×C12711yescharacteristic
295C22×C4C22⋊C4×C12711yescharacteristic
296C22×C4C2×D4×C12711yescharacteristic
297M16—42no—
298M16—42no—
299C2×C8—42no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—42no—
307C22⋊C4—44no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—82no—
311C42—21no—
312C42—21no—
313C42—82no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—24no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—44no—
327C23—44no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—84no—
332C23—88no—
333C23—88no—
334C23—84no—
335C23C2×C22⋊C4×C12711yescharacteristic
336C23C4×D4×C12712yes—
337C23#121×C12712yes—
338D4—42no—
339D4—42no—
340D4—44no—
341D4—48no—
342D4—44no—
343D4—88no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4#99×C12711yescharacteristic
364C2×C4#99×C12711yescharacteristic
365C2×C4#124×C12711yescharacteristic
366C2×C4#127×C12711yescharacteristic
367C8—82no—
368C22—21no—
369C22—21no—
370C22—22no—
371C22—22no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—44no—
376C22—44no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—44no—
382C22—81no—
383C22—82no—
384C22—82no—
385C22—88no—
386C22—88no—
387C22—82no—
388C22—82no—
389C22#390×C12711yescharacteristic
390C22C2×#99×C12711yescharacteristic
391C22#410×C12711yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—82no—
399C4—84no—
400C4AΣL(1,𝔽16)×C12712yescomplemented
401C2—21no—
402C2—21no—
403C2—42no—
404C2—42no—
405C2—84no—
406C2C127×#196411yescharacteristic
407C2C2×AΣL(1,𝔽16)×C12711yescharacteristic
408C2C127×#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×#1800C2semidirect
4. C4×#99×C127C2semidirect
5. C23×D4×C127C4semidirect
15. #380×C127C4semidirect
16. #379×C127C4semidirect
52. #99×C127D4semidirect
195. C4×C127AΣL(1,𝔽16)semidirect
196. #13221C127direct (N × H)
206. C2×AΣL(1,𝔽16)C2×C127semidirect
207. #1800C2×C127semidirect
208. C4×#99C2×C127semidirect
209. C127#13221direct (N × H)
210. C23×D4C4×C127semidirect
220. #380C4×C127semidirect
221. #379C4×C127semidirect
257. #99D4×C127semidirect
400. 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×#18002
C4×#99×C1272
#13221127