Finite Group Databasebeta

← C127×#11517 #44575190

Subgroups — C127×#11517 #44575190

Order 32512.

750 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 118 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×#385×C127C211yesmaximal, characteristic
2C127×#1462C211yesmaximal, characteristic
3C127×#1462C211yesmaximal, characteristic
4C127×#1326C214yesmaximal, complemented
5C24×C4×C127C411yescharacteristic
6C22×C4⋊C4×C127—21no—
7C22×C22⋊C4×C127C411yescharacteristic
8C22×C22⋊C4×C127C2211yescharacteristic
9#407×C127—21no—
10#407×C127—21no—
11#385×C127—21no—
12#385×C127—21no—
13#385×C127C412yes—
14#385×C127C2212yes—
15C2×#95×C127—21no—
16#323×C127—21no—
17#323×C127—21no—
18#323×C127C2212yes—
19#323×C127C2212yes—
20C25×C127C2×C411yescharacteristic
21C23×C4×C127—21no—
22C23×C4×C127—21no—
23C23×C4×C127—21no—
24C23×C4×C127—44no—
25C23×C4×C127—41no—
26C23×C4×C127—41no—
27C23×C4×C127C2×C412yes—
28C23×C4×C127D411yescharacteristic
29C23×C4×C127D411yescharacteristic
30C2×M16×C127—41no—
31C2×M16×C127—41no—
32C2×C4⋊C4×C127—22no—
33C2×C4⋊C4×C127—22no—
34C2×C4⋊C4×C127—44no—
35C2×C22⋊C4×C127—21no—
36C2×C22⋊C4×C127—24no—
37C2×C22⋊C4×C127—41no—
38C2×C22⋊C4×C127—41no—
39C2×C22⋊C4×C127C2×C412yes—
40C2×C22⋊C4×C127C2×C411yescharacteristic
41C2×C22⋊C4×C127D411yescharacteristic
42C2×C22⋊C4×C127D411yescharacteristic
43C2×C22⋊C4×C127C2311yescharacteristic
44C22⋊C8×C127—22no—
45C22⋊C8×C127—22no—
46C22⋊C8×C127—41no—
47C22⋊C8×C127—41no—
48#95×C127—22no—
49#95×C127—22no—
50C24×C127—21no—
51C24×C127—21no—
52C24×C127—22no—
53C24×C127—22no—
54C24×C127—42no—
55C24×C127—42no—
56C24×C127C22⋊C411yescharacteristic
57C24×C127C22⋊C411yescharacteristic
58C24×C127C22×C411yescharacteristic
59C22×C4×C127—24no—
60C22×C4×C127—24no—
61C22×C4×C127—21no—
62C22×C4×C127—21no—
63C22×C4×C127—22no—
64C22×C4×C127—22no—
65C22×C4×C127—21no—
66C22×C4×C127—21no—
67C22×C4×C127—44no—
68C22×C4×C127—44no—
69C22×C4×C127—42no—
70C22×C4×C127—42no—
71C22×C4×C127—42no—
72C22×C4×C127—42no—
73C22×C4×C127—48no—
74C22×C4×C127—44no—
75C22×C4×C127—41no—
76C22×C4×C127—41no—
77C22×C4×C127—42no—
78C22×C4×C127—44no—
79C22×C4×C127—42no—
80C22×C4×C127—42no—
81C22×C4×C127C22⋊C411yescharacteristic
82C22×C4×C127C22⋊C411yescharacteristic
83C22×C4×C127C2×D411yescharacteristic
84C22×C4×C127C2×D411yescharacteristic
85M16×C127—82no—
86M16×C127—82no—
87C2×C8×C127—42no—
88C2×C8×C127—42no—
89C4⋊C4×C127—44no—
90C4⋊C4×C127—44no—
91C22⋊C4×C127—24no—
92C22⋊C4×C127—22no—
93C22⋊C4×C127—41no—
94C22⋊C4×C127—41no—
95C22⋊C4×C127—41no—
96C22⋊C4×C127—41no—
97C22⋊C4×C127M1614yescomplemented
98C23×C127—22no—
99C23×C127—22no—
100C23×C127—21no—
101C23×C127—21no—
102C23×C127—24no—
103C23×C127—24no—
104C23×C127—21no—
105C23×C127—21no—
106C23×C127—21no—
107C23×C127—21no—
108C23×C127—41no—
109C23×C127—41no—
110C23×C127—41no—
111C23×C127—41no—
112C23×C127—42no—
113C23×C127—48no—
114C23×C127—48no—
115C23×C127—42no—
116C23×C127—42no—
117C23×C127—42no—
118C23×C127#9911yescharacteristic
119C23×C127#9911yescharacteristic
120C23×C127#10112yes—
121C23×C127C2×C22⋊C411yescharacteristic
122C23×C127C2×M1612yes—
123C2×C4×C127—24no—
124C2×C4×C127—24no—
125C2×C4×C127—21no—
126C2×C4×C127—21no—
127C2×C4×C127—21no—
128C2×C4×C127—21no—
129C2×C4×C127—42no—
130C2×C4×C127—48no—
131C2×C4×C127—42no—
132C2×C4×C127—42no—
133C2×C4×C127—44no—
134C2×C4×C127—48no—
135C2×C4×C127—44no—
136C2×C4×C127—42no—
137C2×C4×C127—44no—
138C2×C4×C127—42no—
139C2×C4×C127—42no—
140C2×C4×C127—41no—
141C2×C4×C127—41no—
142C2×C4×C127—41no—
143C2×C4×C127—41no—
144C2×C4×C127—44no—
145C2×C4×C127—81no—
146C2×C4×C127—81no—
147C8×C127—82no—
148C8×C127—82no—
149C22×C127—24no—
150C22×C127—24no—
151C22×C127—21no—
152C22×C127—21no—
153C22×C127—21no—
154C22×C127—21no—
155C22×C127—21no—
156C22×C127—21no—
157C22×C127—22no—
158C22×C127—22no—
159C22×C127—42no—
160C22×C127—42no—
161C22×C127—42no—
162C22×C127—48no—
163C22×C127—48no—
164C22×C127—42no—
165C22×C127—41no—
166C22×C127—41no—
167C22×C127—41no—
168C22×C127—41no—
169C22×C127AΣL(1,𝔽16)12yes—
170C22×C127#35212yes—
171C22×C127#40711yescharacteristic
172C22×C127C2×#9911yescharacteristic
173C22×C127C2×#10111yescharacteristic
174C4×C127—44no—
175C4×C127—41no—
176C4×C127—44no—
177C4×C127—41no—
178C4×C127—41no—
179C4×C127—41no—
180#11517C12711yesmaximal, characteristic, direct product factor, Sylow
181C2×C127—21no—
182C2×C127—22no—
183C2×C127—22no—
184C2×C127—21no—
185C2×C127—42no—
186C2×C127—42no—
187C2×C127#146311yescharacteristic
188C2×C127C2×AΣL(1,𝔽16)11yescharacteristic
189C2×C127C2×#35211yescharacteristic
190C2×#385C2×C12711yescharacteristic
191#1462C2×C12711yescharacteristic
192#1462C2×C12711yescharacteristic
193#1326C2×C12714yescomplemented
194C127#1151711yescharacteristic, direct product factor, Sylow
195C24×C4C4×C12711yescharacteristic
196C22×C4⋊C4—21no—
197C22×C22⋊C4C4×C12711yescharacteristic
198C22×C22⋊C4C22×C12711yescharacteristic
199#407—21no—
200#407—21no—
201#385—21no—
202#385—21no—
203#385C4×C12712yes—
204#385C22×C12712yes—
205C2×#95—21no—
206#323—21no—
207#323—21no—
208#323C22×C12712yes—
209#323C22×C12712yes—
210C25C2×C4×C12711yescharacteristic
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—44no—
215C23×C4—41no—
216C23×C4—41no—
217C23×C4C2×C4×C12712yes—
218C23×C4D4×C12711yescharacteristic
219C23×C4D4×C12711yescharacteristic
220C2×M16—41no—
221C2×M16—41no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—44no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4C2×C4×C12712yes—
230C2×C22⋊C4C2×C4×C12711yescharacteristic
231C2×C22⋊C4D4×C12711yescharacteristic
232C2×C22⋊C4D4×C12711yescharacteristic
233C2×C22⋊C4C23×C12711yescharacteristic
234C22⋊C8—22no—
235C22⋊C8—22no—
236C22⋊C8—41no—
237C22⋊C8—41no—
238#95—22no—
239#95—22no—
240C24—21no—
241C24—21no—
242C24—22no—
243C24—22no—
244C24—42no—
245C24—42no—
246C24C22⋊C4×C12711yescharacteristic
247C24C22⋊C4×C12711yescharacteristic
248C24C22×C4×C12711yescharacteristic
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—48no—
264C22×C4—44no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4C22⋊C4×C12711yescharacteristic
272C22×C4C22⋊C4×C12711yescharacteristic
273C22×C4C2×D4×C12711yescharacteristic
274C22×C4C2×D4×C12711yescharacteristic
275M16—82no—
276M16—82no—
277C2×C8—42no—
278C2×C8—42no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C22⋊C4—24no—
282C22⋊C4—22no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4M16×C12714yescomplemented
288C23—22no—
289C23—22no—
290C23—21no—
291C23—21no—
292C23—24no—
293C23—24no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—42no—
303C23—48no—
304C23—48no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23#99×C12711yescharacteristic
309C23#99×C12711yescharacteristic
310C23#101×C12712yes—
311C23C2×C22⋊C4×C12711yescharacteristic
312C23C2×M16×C12712yes—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—48no—
323C2×C4—42no—
324C2×C4—48no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—44no—
335C2×C4—81no—
336C2×C4—81no—
337C8—82no—
338C8—82no—
339C22—24no—
340C22—24no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—22no—
348C22—22no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—48no—
353C22—48no—
354C22—42no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22AΣL(1,𝔽16)×C12712yes—
360C22#352×C12712yes—
361C22#407×C12711yescharacteristic
362C22C2×#99×C12711yescharacteristic
363C22C2×#101×C12711yescharacteristic
364C4—44no—
365C4—41no—
366C4—44no—
367C4—41no—
368C4—41no—
369C4—41no—
370C2—21no—
371C2—22no—
372C2—22no—
373C2—21no—
374C2—42no—
375C2—42no—
376C2C127×#146311yescharacteristic
377C2C2×AΣL(1,𝔽16)×C12711yescharacteristic
378C2C2×#352×C12711yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC24C22×C4×C127
Frattini, cosocleC2×C22⋊C4C23×C127
prime radicalC25×C127C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C24×C127 ◁ C22×C22⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1326C2semidirect
97. C22⋊C4×C127M16semidirect
180. #11517C127direct (N × H)
193. #1326C2×C127semidirect
194. C127#11517direct (N × H)
287. C22⋊C4M16×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#385×C1272
C127×#14622
C127×#13262
#11517127