Finite Group Databasebeta

← C127×#8520 #44572193

Subgroups — C127×#8520 #44572193

Order 32512.

976 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 150 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×#536×C127C211yesmaximal, characteristic
2C127×#1785C211yesmaximal, characteristic
3C127×#1782C211yesmaximal, characteristic
4C127×#1511C214yesmaximal, complemented
5C22×#41×C127C2211yescharacteristic
6C23×Q8×C127C411yescharacteristic, complemented
7#536×C127—28no—
8C2×C4⋊Q8×C127—21no—
9C2×#124×C127—21no—
10C2×#123×C127—21no—
11C2×#123×C127—21no—
12C2×C22⋊Q8×C127C412yescomplemented
13C2×C22⋊Q8×C127C2212yes—
14C2×#117×C127C411yescharacteristic, complemented
15C2×#101×C127—21no—
16C2×#100×C127—21no—
17C2×#99×C127—21no—
18C2×#99×C127—21no—
19#407×C127C2212yes—
20#381×C127C2212yes—
21#328×C127—28no—
22C2×#41×C127—24no—
23C2×#41×C127—42no—
24C2×#41×C127—42no—
25C22×Q8×C127—24no—
26C22×Q8×C127—21no—
27C22×Q8×C127—42no—
28C22×Q8×C127—42no—
29C22×Q8×C127C2×C412yes—
30C22×Q8×C127D411yescharacteristic
31C22×D4×C127D411yescharacteristic
32C22×D4×C127D411yescharacteristic
33C22×D4×C127D411yescharacteristic
34C23×C4×C127—21no—
35C23×C4×C127C2×C411yescharacteristic
36C23×C4×C127C2×C411yescharacteristic
37C23×C4×C127C2311yescharacteristic
38C2×M16×C127—22no—
39C4⋊Q8×C127—28no—
40#124×C127—28no—
41#123×C127—28no—
42#123×C127—28no—
43C22⋊Q8×C127—24no—
44C22⋊Q8×C127—24no—
45C22⋊Q8×C127—24no—
46C22⋊Q8×C127D418yes—
47#117×C127—24no—
48C2×C4⋊C4×C127—22no—
49C2×C4⋊C4×C127—22no—
50C2×C4⋊C4×C127C2×C412yes—
51C2×C22⋊C4×C127—21no—
52C2×C22⋊C4×C127—22no—
53C2×C22⋊C4×C127—22no—
54C2×C22⋊C4×C127—22no—
55C2×C42×C127—21no—
56#101×C127—42no—
57#100×C127—42no—
58#99×C127—42no—
59#99×C127—42no—
60C22⋊C8×C127—24no—
61#95×C127—24no—
62C24×C127C22⋊C411yescharacteristic
63C24×C127C22⋊C411yescharacteristic
64C24×C127C2×D411yescharacteristic
65#41×C127—416no—
66C2×Q8×C127—24no—
67C2×Q8×C127—22no—
68C2×Q8×C127—22no—
69C2×Q8×C127—22no—
70C2×Q8×C127—41no—
71C2×Q8×C127—416no—
72C2×Q8×C127—42no—
73C2×Q8×C127—42no—
74C2×Q8×C127—42no—
75C2×Q8×C127—42no—
76C2×Q8×C127C22⋊C414yes—
77C2×D4×C127—22no—
78C2×D4×C127—22no—
79C2×D4×C127—22no—
80C2×D4×C127—22no—
81C2×D4×C127—22no—
82C2×D4×C127—22no—
83C2×D4×C127—41no—
84C2×D4×C127—41no—
85C2×D4×C127—41no—
86C22×C4×C127—22no—
87C22×C4×C127—22no—
88C22×C4×C127—22no—
89C22×C4×C127—24no—
90C22×C4×C127—22no—
91C22×C4×C127—22no—
92C22×C4×C127—22no—
93C22×C4×C127—41no—
94C22×C4×C127—41no—
95C22×C4×C127—41no—
96C22×C4×C127—41no—
97C22×C4×C127—42no—
98C22×C4×C127—42no—
99C22×C4×C127—42no—
100C22×C4×C127C22⋊C411yescharacteristic
101C22×C4×C127C22⋊C411yescharacteristic
102C22×C4×C127C22×C411yescharacteristic
103C22×C4×C127C2×D411yescharacteristic
104C22×C4×C127C2×D414yes—
105M16×C127—42no—
106M16×C127—42no—
107C2×C8×C127—42no—
108C4⋊C4×C127—28no—
109C4⋊C4×C127—22no—
110C4⋊C4×C127—28no—
111C4⋊C4×C127C22⋊C414yes—
112C22⋊C4×C127—24no—
113C22⋊C4×C127—28no—
114C22⋊C4×C127—28no—
115C22⋊C4×C127—42no—
116C22⋊C4×C127—42no—
117C42×C127—24no—
118C23×C127—21no—
119C23×C127—21no—
120C23×C127—21no—
121C23×C127—21no—
122C23×C127—21no—
123C23×C127—21no—
124C23×C127—21no—
125C23×C127—42no—
126C23×C127—42no—
127C23×C127—42no—
128C23×C127#9912yes—
129C23×C127C2×C22⋊C411yescharacteristic
130C23×C127#12012yes—
131Q8×C127—44no—
132Q8×C127—44no—
133Q8×C127—44no—
134Q8×C127—48no—
135D4×C127—44no—
136D4×C127—44no—
137D4×C127—44no—
138C2×C4×C127—28no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—24no—
146C2×C4×C127—24no—
147C2×C4×C127—21no—
148C2×C4×C127—21no—
149C2×C4×C127—21no—
150C2×C4×C127—44no—
151C2×C4×C127—44no—
152C2×C4×C127—44no—
153C2×C4×C127—48no—
154C2×C4×C127—42no—
155C2×C4×C127—42no—
156C2×C4×C127—82no—
157C2×C4×C127—84no—
158C2×C4×C127#9912yes—
159C2×C4×C127C2×C22⋊C412yes—
160C2×C4×C127#12012yes—
161C8×C127—82no—
162C22×C127—22no—
163C22×C127—24no—
164C22×C127—22no—
165C22×C127—22no—
166C22×C127—42no—
167C22×C127—42no—
168C22×C127—42no—
169C22×C127—42no—
170C22×C127—41no—
171C22×C127—81no—
172C22×C127—81no—
173C22×C127—81no—
174C22×C127—81no—
175C22×C127#37911yescharacteristic
176C22×C127C2×#9911yescharacteristic
177C22×C127C2×#9911yescharacteristic
178C4×C127—22no—
179C4×C127—22no—
180C4×C127—22no—
181C4×C127—44no—
182C4×C127—41no—
183C4×C127—44no—
184C4×C127—162no—
185#8520C12711yesmaximal, characteristic, direct product factor, Sylow
186C2×C127—21no—
187C2×C127—21no—
188C2×C127—42no—
189C2×C127—42no—
190C2×C127—81no—
191C2×C127#178111yescharacteristic
192C2×C127#179612yes—
193C2×#536C2×C12711yescharacteristic
194#1785C2×C12711yescharacteristic
195#1782C2×C12711yescharacteristic
196#1511C2×C12714yescomplemented
197C127#852011yescharacteristic, direct product factor, Sylow
198C22×#41C22×C12711yescharacteristic
199C23×Q8C4×C12711yescharacteristic, complemented
200#536—28no—
201C2×C4⋊Q8—21no—
202C2×#124—21no—
203C2×#123—21no—
204C2×#123—21no—
205C2×C22⋊Q8C4×C12712yescomplemented
206C2×C22⋊Q8C22×C12712yes—
207C2×#117C4×C12711yescharacteristic, complemented
208C2×#101—21no—
209C2×#100—21no—
210C2×#99—21no—
211C2×#99—21no—
212#407C22×C12712yes—
213#381C22×C12712yes—
214#328—28no—
215C2×#41—24no—
216C2×#41—42no—
217C2×#41—42no—
218C22×Q8—24no—
219C22×Q8—21no—
220C22×Q8—42no—
221C22×Q8—42no—
222C22×Q8C2×C4×C12712yes—
223C22×Q8D4×C12711yescharacteristic
224C22×D4D4×C12711yescharacteristic
225C22×D4D4×C12711yescharacteristic
226C22×D4D4×C12711yescharacteristic
227C23×C4—21no—
228C23×C4C2×C4×C12711yescharacteristic
229C23×C4C2×C4×C12711yescharacteristic
230C23×C4C23×C12711yescharacteristic
231C2×M16—22no—
232C4⋊Q8—28no—
233#124—28no—
234#123—28no—
235#123—28no—
236C22⋊Q8—24no—
237C22⋊Q8—24no—
238C22⋊Q8—24no—
239C22⋊Q8D4×C12718yes—
240#117—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4C2×C4×C12712yes—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C42—21no—
249#101—42no—
250#100—42no—
251#99—42no—
252#99—42no—
253C22⋊C8—24no—
254#95—24no—
255C24C22⋊C4×C12711yescharacteristic
256C24C22⋊C4×C12711yescharacteristic
257C24C2×D4×C12711yescharacteristic
258#41—416no—
259C2×Q8—24no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—416no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8C22⋊C4×C12714yes—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4C22⋊C4×C12711yescharacteristic
294C22×C4C22⋊C4×C12711yescharacteristic
295C22×C4C22×C4×C12711yescharacteristic
296C22×C4C2×D4×C12711yescharacteristic
297C22×C4C2×D4×C12714yes—
298M16—42no—
299M16—42no—
300C2×C8—42no—
301C4⋊C4—28no—
302C4⋊C4—22no—
303C4⋊C4—28no—
304C4⋊C4C22⋊C4×C12714yes—
305C22⋊C4—24no—
306C22⋊C4—28no—
307C22⋊C4—28no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C42—24no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23#99×C12712yes—
322C23C2×C22⋊C4×C12711yescharacteristic
323C23#120×C12712yes—
324Q8—44no—
325Q8—44no—
326Q8—44no—
327Q8—48no—
328D4—44no—
329D4—44no—
330D4—44no—
331C2×C4—28no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—48no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—82no—
350C2×C4—84no—
351C2×C4#99×C12712yes—
352C2×C4C2×C22⋊C4×C12712yes—
353C2×C4#120×C12712yes—
354C8—82no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—24no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—41no—
363C22—42no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22#379×C12711yescharacteristic
369C22C2×#99×C12711yescharacteristic
370C22C2×#99×C12711yescharacteristic
371C4—22no—
372C4—22no—
373C4—22no—
374C4—41no—
375C4—44no—
376C4—44no—
377C4—162no—
378C2—21no—
379C2—21no—
380C2—42no—
381C2—42no—
382C2—81no—
383C2C127×#178111yescharacteristic
384C2C127×#179612yes—
subgroupG/H
center, socleC22×C127#379
commutatorC22×C4C22×C4×C127
Frattini, cosocleC23×C4C23×C127
prime radicalC22×#41×C127C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1511C2semidirect
6. C23×Q8×C127C4semidirect
12. C2×C22⋊Q8×C127C4semidirect
14. C2×#117×C127C4semidirect
185. #8520C127direct (N × H)
196. #1511C2×C127semidirect
197. C127#8520direct (N × H)
199. C23×Q8C4×C127semidirect
205. C2×C22⋊Q8C4×C127semidirect
207. C2×#117C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#536×C1272
C127×#17852
C127×#17822
C127×#15112
#8520127