Finite Group Databasebeta

← C127×#11733 #44575406

Subgroups — C127×#11733 #44575406

Order 32512.

632 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2388C211yesmaximal, characteristic
2C4×#100×C127C211yesmaximal, characteristic, complemented
3C127×#1459C211yesmaximal, characteristic, complemented
4C127×#1354C212yesmaximal, complemented
5C127×#1345C212yesmaximal, complemented
6C22×#41×C127C411yescharacteristic
7C2×C4×D4×C127—21no—
8C2×C4×D4×C127C2211yescharacteristic
9C2×#100×C127—21no—
10C2×C22⋊C8×C127—21no—
11C4×M16×C127—21no—
12#395×C127C2211yescharacteristic
13#394×C127—21no—
14#394×C127C2211yescharacteristic
15#393×C127C411yescharacteristic
16#392×C127—21no—
17#392×C127C411yescharacteristic
18C4×C22⋊C4×C127—21no—
19C4×C22⋊C4×C127C411yescharacteristic
20#343×C127C2212yes—
21#324×C127C2212yes—
22#323×C127—21no—
23C2×#41×C127—44no—
24C2×#41×C127—41no—
25C2×#41×C127—41no—
26C22×Q8×C127C2×C411yescharacteristic
27C22×D4×C127—21no—
28C22×D4×C127—21no—
29C22×D4×C127C2×C411yescharacteristic
30C22×D4×C127D411yescharacteristic
31C23×C4×C127—21no—
32C23×C4×C127C2×C411yescharacteristic
33C23×C4×C127D411yescharacteristic
34C2×M16×C127—22no—
35C22×C8×C127—41no—
36C4×D4×C127—41no—
37C4×D4×C127—41no—
38C4×D4×C127—44no—
39C2×C4⋊C4×C127—21no—
40C2×C4⋊C4×C127C2311yescharacteristic
41C2×C22⋊C4×C127—21no—
42C2×C22⋊C4×C127—21no—
43C2×C22⋊C4×C127—21no—
44C2×C22⋊C4×C127—21no—
45C2×C22⋊C4×C127—21no—
46C2×C22⋊C4×C127C2×C411yescharacteristic
47C2×C22⋊C4×C127D411yescharacteristic
48C2×C42×C127—21no—
49C2×C42×C127D411yescharacteristic
50#100×C127—24no—
51C22⋊C8×C127—22no—
52C22⋊C8×C127—41no—
53#97×C127—41no—
54C4×C8×C127—41no—
55#95×C127—21no—
56#95×C127—21no—
57#95×C127C2×C411yescharacteristic
58#95×C127C2×C411yescharacteristic
59C24×C127—21no—
60C24×C127C22⋊C411yescharacteristic
61C24×C127C22⋊C411yescharacteristic
62#41×C127—44no—
63#41×C127—44no—
64#41×C127—44no—
65#41×C127—44no—
66C2×Q8×C127—42no—
67C2×Q8×C127—41no—
68C2×D4×C127—22no—
69C2×D4×C127—22no—
70C2×D4×C127—41no—
71C2×D4×C127—41no—
72C2×D4×C127—42no—
73C2×D4×C127—42no—
74C2×D4×C127—42no—
75C2×D4×C127—42no—
76C2×D4×C127—42no—
77C2×D4×C127—42no—
78C2×D4×C127—42no—
79C22×C4×C127—24no—
80C22×C4×C127—21no—
81C22×C4×C127—21no—
82C22×C4×C127—21no—
83C22×C4×C127—21no—
84C22×C4×C127—21no—
85C22×C4×C127—21no—
86C22×C4×C127—21no—
87C22×C4×C127—42no—
88C22×C4×C127—42no—
89C22×C4×C127—42no—
90C22×C4×C127—41no—
91C22×C4×C127—41no—
92C22×C4×C127C22⋊C411yescharacteristic
93C22×C4×C127C22⋊C411yescharacteristic
94C22×C4×C127C22×C411yescharacteristic
95C22×C4×C127C2×D411yescharacteristic
96C22×C4×C127C2×D411yescharacteristic
97M16×C127—24no—
98M16×C127—42no—
99C2×C8×C127—42no—
100C2×C8×C127—42no—
101C2×C8×C127—82no—
102C4⋊C4×C127—41no—
103C4⋊C4×C127—42no—
104C22⋊C4×C127—41no—
105C22⋊C4×C127—41no—
106C22⋊C4×C127—42no—
107C22⋊C4×C127—42no—
108C22⋊C4×C127—42no—
109C22⋊C4×C127—42no—
110C22⋊C4×C127—42no—
111C42×C127—21no—
112C42×C127—21no—
113C42×C127—42no—
114C23×C127—21no—
115C23×C127—21no—
116C23×C127—22no—
117C23×C127—21no—
118C23×C127—21no—
119C23×C127—21no—
120C23×C127—41no—
121C23×C127—41no—
122C23×C127—41no—
123C23×C127—41no—
124C23×C127—41no—
125C23×C127—41no—
126C23×C127—41no—
127C23×C127—41no—
128C23×C127—41no—
129C23×C127—41no—
130C23×C127#10412yes—
131C23×C127C2×C22⋊C411yescharacteristic
132Q8×C127—42no—
133Q8×C127—42no—
134D4×C127—44no—
135D4×C127—42no—
136D4×C127—48no—
137D4×C127—48no—
138D4×C127—42no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—22no—
142C2×C4×C127—22no—
143C2×C4×C127—22no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127—44no—
148C2×C4×C127—42no—
149C2×C4×C127—41no—
150C2×C4×C127—41no—
151C2×C4×C127—41no—
152C2×C4×C127—41no—
153C2×C4×C127—42no—
154C2×C4×C127—44no—
155C2×C4×C127—42no—
156C2×C4×C127—41no—
157C2×C4×C127—41no—
158C2×C4×C127—41no—
159C2×C4×C127—41no—
160C2×C4×C127—41no—
161C2×C4×C127—41no—
162C2×C4×C127—81no—
163C2×C4×C127#9911yescharacteristic
164C2×C4×C127#9911yescharacteristic
165C2×C4×C127#13512yes—
166C8×C127—44no—
167C8×C127—82no—
168C22×C127—21no—
169C22×C127—21no—
170C22×C127—22no—
171C22×C127—42no—
172C22×C127—42no—
173C22×C127—42no—
174C22×C127—41no—
175C22×C127—41no—
176C22×C127—41no—
177C22×C127—42no—
178C22×C127—41no—
179C22×C127—41no—
180C22×C127—81no—
181C22×C127—81no—
182C22×C127—81no—
183C22×C127—81no—
184C22×C127—81no—
185C22×C127—81no—
186C22×C127—81no—
187C22×C127C2×#9911yescharacteristic
188C22×C127#41611yescharacteristic
189C22×C127C2×#10411yescharacteristic
190C4×C127—21no—
191C4×C127—22no—
192C4×C127—42no—
193C4×C127—41no—
194C4×C127—42no—
195C4×C127—41no—
196C4×C127—41no—
197C4×C127—81no—
198C4×C127AΣL(1,𝔽16)12yes—
199#11733C12711yesmaximal, characteristic, direct product factor, Sylow
200C2×C127—21no—
201C2×C127—21no—
202C2×C127—42no—
203C2×C127—81no—
204C2×C127—82no—
205C2×C127#150111yescharacteristic
206C2×C127C2×AΣL(1,𝔽16)11yescharacteristic
207C2×C127#213211yescharacteristic
208#2388C2×C12711yescharacteristic
209C4×#100C2×C12711yescharacteristic, complemented
210#1459C2×C12711yescharacteristic, complemented
211#1354C2×C12712yescomplemented
212#1345C2×C12712yescomplemented
213C127#1173311yescharacteristic, direct product factor, Sylow
214C22×#41C4×C12711yescharacteristic
215C2×C4×D4—21no—
216C2×C4×D4C22×C12711yescharacteristic
217C2×#100—21no—
218C2×C22⋊C8—21no—
219C4×M16—21no—
220#395C22×C12711yescharacteristic
221#394—21no—
222#394C22×C12711yescharacteristic
223#393C4×C12711yescharacteristic
224#392—21no—
225#392C4×C12711yescharacteristic
226C4×C22⋊C4—21no—
227C4×C22⋊C4C4×C12711yescharacteristic
228#343C22×C12712yes—
229#324C22×C12712yes—
230#323—21no—
231C2×#41—44no—
232C2×#41—41no—
233C2×#41—41no—
234C22×Q8C2×C4×C12711yescharacteristic
235C22×D4—21no—
236C22×D4—21no—
237C22×D4C2×C4×C12711yescharacteristic
238C22×D4D4×C12711yescharacteristic
239C23×C4—21no—
240C23×C4C2×C4×C12711yescharacteristic
241C23×C4D4×C12711yescharacteristic
242C2×M16—22no—
243C22×C8—41no—
244C4×D4—41no—
245C4×D4—41no—
246C4×D4—44no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4C23×C12711yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4C2×C4×C12711yescharacteristic
255C2×C22⋊C4D4×C12711yescharacteristic
256C2×C42—21no—
257C2×C42D4×C12711yescharacteristic
258#100—24no—
259C22⋊C8—22no—
260C22⋊C8—41no—
261#97—41no—
262C4×C8—41no—
263#95—21no—
264#95—21no—
265#95C2×C4×C12711yescharacteristic
266#95C2×C4×C12711yescharacteristic
267C24—21no—
268C24C22⋊C4×C12711yescharacteristic
269C24C22⋊C4×C12711yescharacteristic
270#41—44no—
271#41—44no—
272#41—44no—
273#41—44no—
274C2×Q8—42no—
275C2×Q8—41no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C22×C4—24no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4C22⋊C4×C12711yescharacteristic
301C22×C4C22⋊C4×C12711yescharacteristic
302C22×C4C22×C4×C12711yescharacteristic
303C22×C4C2×D4×C12711yescharacteristic
304C22×C4C2×D4×C12711yescharacteristic
305M16—24no—
306M16—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—82no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C42—21no—
320C42—21no—
321C42—42no—
322C23—21no—
323C23—21no—
324C23—22no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23#104×C12712yes—
339C23C2×C22⋊C4×C12711yescharacteristic
340Q8—42no—
341Q8—42no—
342D4—44no—
343D4—42no—
344D4—48no—
345D4—42no—
346D4—48no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4#99×C12711yescharacteristic
372C2×C4#99×C12711yescharacteristic
373C2×C4#135×C12712yes—
374C8—44no—
375C8—82no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—42no—
386C22—41no—
387C22—41no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22C2×#99×C12711yescharacteristic
396C22#416×C12711yescharacteristic
397C22C2×#104×C12711yescharacteristic
398C4—21no—
399C4—22no—
400C4—42no—
401C4—41no—
402C4—42no—
403C4—41no—
404C4—41no—
405C4—81no—
406C4AΣL(1,𝔽16)×C12712yes—
407C2—21no—
408C2—21no—
409C2—42no—
410C2—82no—
411C2—81no—
412C2C127×#150111yescharacteristic
413C2C2×AΣL(1,𝔽16)×C12711yescharacteristic
414C2C127×#213211yescharacteristic
subgroupG/H
centerC2×C4×C127#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C4⋊C4C23×C127
socleC22×C127C2×#99
prime radicalC127×#2388C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C22×C4×C127 ◁ C2×C4×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×#100×C127C2semidirect
3. C127×#1459C2semidirect
4. C127×#1354C2semidirect
5. C127×#1345C2semidirect
199. #11733C127direct (N × H)
209. C4×#100C2×C127semidirect
210. #1459C2×C127semidirect
211. #1354C2×C127semidirect
212. #1345C2×C127semidirect
213. C127#11733direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#23882
C4×#100×C1272
C127×#14592
C127×#13542
C127×#13452
#11733127