Finite Group Databasebeta

← C127×#11828 #44575501

Subgroups — C127×#11828 #44575501

Order 32512.

612 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 102 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
2C127×#1800C211yesmaximal, characteristic, complemented
3C127×#1465C211yesmaximal, characteristic, complemented
4C127×#1354C212yesmaximal, complemented
5C127×#1345C212yesmaximal, complemented
6C22×#41×C127C411yescharacteristic
7C2×C4×D4×C127—21no—
8C2×C4×D4×C127C2211yescharacteristic
9#423×C127—21no—
10C2×#100×C127—21no—
11#407×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—
35C2×M16×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
50C4⋊C8×C127—41no—
51C4⋊C8×C127—41no—
52#100×C127—42no—
53C22⋊C8×C127—22no—
54C22⋊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—21no—
69C2×D4×C127—21no—
70C2×D4×C127—41no—
71C2×D4×C127—41no—
72C2×D4×C127—41no—
73C2×D4×C127—42no—
74C2×D4×C127—42no—
75C2×D4×C127—42no—
76C2×D4×C127—42no—
77C2×D4×C127—42no—
78C2×D4×C127—42no—
79C2×D4×C127—42no—
80C22×C4×C127—24no—
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—21no—
88C22×C4×C127—42no—
89C22×C4×C127—42no—
90C22×C4×C127—42no—
91C22×C4×C127—41no—
92C22×C4×C127—41no—
93C22×C4×C127C22⋊C411yescharacteristic
94C22×C4×C127C22⋊C411yescharacteristic
95C22×C4×C127C22×C411yescharacteristic
96C22×C4×C127C2×D411yescharacteristic
97C22×C4×C127C2×D411yescharacteristic
98M16×C127—42no—
99M16×C127—42no—
100M16×C127—82no—
101C2×C8×C127—42no—
102C2×C8×C127—42no—
103C4⋊C4×C127—41no—
104C4⋊C4×C127—42no—
105C22⋊C4×C127—41no—
106C22⋊C4×C127—41no—
107C22⋊C4×C127—42no—
108C22⋊C4×C127—42no—
109C22⋊C4×C127—42no—
110C22⋊C4×C127—42no—
111C22⋊C4×C127—41no—
112C22⋊C4×C127—41no—
113C42×C127—21no—
114C42×C127—21no—
115C42×C127—42no—
116C23×C127—22no—
117C23×C127—21no—
118C23×C127—21no—
119C23×C127—21no—
120C23×C127—21no—
121C23×C127—21no—
122C23×C127—21no—
123C23×C127—41no—
124C23×C127—41no—
125C23×C127—41no—
126C23×C127—41no—
127C23×C127—41no—
128C23×C127—41no—
129C23×C127—41no—
130C23×C127—41no—
131C23×C127—41no—
132C23×C127—41no—
133C23×C127C2×C22⋊C411yescharacteristic
134Q8×C127—42no—
135Q8×C127—42no—
136D4×C127—42no—
137D4×C127—42no—
138D4×C127—42no—
139D4×C127—48no—
140D4×C127—48no—
141D4×C127—42no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—22no—
145C2×C4×C127—22no—
146C2×C4×C127—22no—
147C2×C4×C127—21no—
148C2×C4×C127—21no—
149C2×C4×C127—21no—
150C2×C4×C127—44no—
151C2×C4×C127—42no—
152C2×C4×C127—41no—
153C2×C4×C127—41no—
154C2×C4×C127—41no—
155C2×C4×C127—41no—
156C2×C4×C127—42no—
157C2×C4×C127—44no—
158C2×C4×C127—42no—
159C2×C4×C127—41no—
160C2×C4×C127—41no—
161C2×C4×C127—41no—
162C2×C4×C127—41no—
163C2×C4×C127—41no—
164C2×C4×C127—41no—
165C2×C4×C127—81no—
166C2×C4×C127#9911yescharacteristic
167C2×C4×C127#9911yescharacteristic
168C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C127#13711yescharacteristic
170C8×C127—82no—
171C8×C127—82no—
172C22×C127—21no—
173C22×C127—21no—
174C22×C127—42no—
175C22×C127—42no—
176C22×C127—42no—
177C22×C127—41no—
178C22×C127—42no—
179C22×C127—41no—
180C22×C127—41no—
181C22×C127—41no—
182C22×C127—41no—
183C22×C127—41no—
184C22×C127—81no—
185C22×C127—81no—
186C22×C127—81no—
187C22×C127—81no—
188C22×C127—81no—
189C22×C127—81no—
190C22×C127—81no—
191C22×C127C2×#9911yescharacteristic
192C22×C127#41711yescharacteristic
193C22×C127#42111yescharacteristic
194C4×C127—21no—
195C4×C127—21no—
196C4×C127—21no—
197C4×C127—42no—
198C4×C127—41no—
199C4×C127—42no—
200C4×C127—41no—
201C4×C127—41no—
202C4×C127—81no—
203C4×C127AΣL(1,𝔽16)12yes—
204#11828C12711yesmaximal, characteristic, direct product factor, Sylow
205C2×C127—21no—
206C2×C127—21no—
207C2×C127—81no—
208C2×C127—81no—
209C2×C127—82no—
210C2×C127#151311yescharacteristic
211C2×C127C2×AΣL(1,𝔽16)11yescharacteristic
212C2×C127#213211yescharacteristic
213#2388C2×C12711yescharacteristic
214#1800C2×C12711yescharacteristic, complemented
215#1465C2×C12711yescharacteristic, complemented
216#1354C2×C12712yescomplemented
217#1345C2×C12712yescomplemented
218C127#1182811yescharacteristic, direct product factor, Sylow
219C22×#41C4×C12711yescharacteristic
220C2×C4×D4—21no—
221C2×C4×D4C22×C12711yescharacteristic
222#423—21no—
223C2×#100—21no—
224#407—21no—
225#395C22×C12711yescharacteristic
226#394—21no—
227#394C22×C12711yescharacteristic
228#393C4×C12711yescharacteristic
229#392—21no—
230#392C4×C12711yescharacteristic
231C4×C22⋊C4—21no—
232C4×C22⋊C4C4×C12711yescharacteristic
233#343C22×C12712yes—
234#324C22×C12712yes—
235#323—21no—
236C2×#41—44no—
237C2×#41—41no—
238C2×#41—41no—
239C22×Q8C2×C4×C12711yescharacteristic
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2×C4×C12711yescharacteristic
243C22×D4D4×C12711yescharacteristic
244C23×C4—21no—
245C23×C4C2×C4×C12711yescharacteristic
246C23×C4D4×C12711yescharacteristic
247C2×M16—22no—
248C2×M16—41no—
249C4×D4—41no—
250C4×D4—41no—
251C4×D4—44no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4C23×C12711yescharacteristic
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4C2×C4×C12711yescharacteristic
260C2×C22⋊C4D4×C12711yescharacteristic
261C2×C42—21no—
262C2×C42D4×C12711yescharacteristic
263C4⋊C8—41no—
264C4⋊C8—41no—
265#100—42no—
266C22⋊C8—22no—
267C22⋊C8—41no—
268#95—21no—
269#95—21no—
270#95C2×C4×C12711yescharacteristic
271#95C2×C4×C12711yescharacteristic
272C24—21no—
273C24C22⋊C4×C12711yescharacteristic
274C24C22⋊C4×C12711yescharacteristic
275#41—44no—
276#41—44no—
277#41—44no—
278#41—44no—
279C2×Q8—42no—
280C2×Q8—41no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C22×C4—24no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4C22⋊C4×C12711yescharacteristic
307C22×C4C22⋊C4×C12711yescharacteristic
308C22×C4C22×C4×C12711yescharacteristic
309C22×C4C2×D4×C12711yescharacteristic
310C22×C4C2×D4×C12711yescharacteristic
311M16—42no—
312M16—42no—
313M16—82no—
314C2×C8—42no—
315C2×C8—42no—
316C4⋊C4—41no—
317C4⋊C4—42no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C42—21no—
327C42—21no—
328C42—42no—
329C23—22no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23C2×C22⋊C4×C12711yescharacteristic
347Q8—42no—
348Q8—42no—
349D4—42no—
350D4—42no—
351D4—42no—
352D4—48no—
353D4—42no—
354D4—48no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—81no—
379C2×C4#99×C12711yescharacteristic
380C2×C4#99×C12711yescharacteristic
381C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
382C2×C4#137×C12711yescharacteristic
383C8—82no—
384C8—82no—
385C22—21no—
386C22—21no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—41no—
391C22—42no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22C2×#99×C12711yescharacteristic
405C22#417×C12711yescharacteristic
406C22#421×C12711yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—42no—
411C4—41no—
412C4—42no—
413C4—41no—
414C4—41no—
415C4—81no—
416C4AΣL(1,𝔽16)×C12712yes—
417C2—21no—
418C2—21no—
419C2—81no—
420C2—82no—
421C2—81no—
422C2C127×#151311yescharacteristic
423C2C2×AΣL(1,𝔽16)×C12711yescharacteristic
424C2C127×#213211yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C4⋊C4C23×C127
prime radicalC127×#2388C2
LCSG ▷ C22×C4 ▷ C23 ▷ 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. C127×#1800C2semidirect
3. C127×#1465C2semidirect
4. C127×#1354C2semidirect
5. C127×#1345C2semidirect
204. #11828C127direct (N × H)
214. #1800C2×C127semidirect
215. #1465C2×C127semidirect
216. #1354C2×C127semidirect
217. #1345C2×C127semidirect
218. C127#11828direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#23882
C127×#18002
C127×#14652
C127×#13542
C127×#13452
#11828127