Finite Group Databasebeta

← C127×#11898 #44575571

Subgroups — C127×#11898 #44575571

Order 32512.

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

normal subgroups only

This group has 580 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
1C127×#2409C211yesmaximal, characteristic
2C127×#1799C211yesmaximal, characteristic
3C127×#1460C211yesmaximal, characteristic
4C127×#1351C214yesmaximal, complemented
5C23×Q8×C127C411yescharacteristic, complemented
6C2×C22⋊Q8×C127—21no—
7C2×C4×D4×C127C2211yescharacteristic
8#431×C127—21no—
9C2×#99×C127—21no—
10#395×C127C412yescomplemented
11#393×C127C2212yes—
12#391×C127—21no—
13#386×C127—21no—
14#385×C127—21no—
15#380×C127—21no—
16#380×C127C411yescharacteristic, complemented
17#342×C127C2212yes—
18#323×C127—21no—
19#323×C127C2212yes—
20C22×Q8×C127—21no—
21C22×Q8×C127—41no—
22C22×Q8×C127—41no—
23C22×Q8×C127—44no—
24C22×Q8×C127C2×C412yes—
25C22×D4×C127D411yescharacteristic
26C23×C4×C127—21no—
27C23×C4×C127C2×C411yescharacteristic
28C23×C4×C127D411yescharacteristic
29C22⋊Q8×C127—44no—
30C4×D4×C127—41no—
31C4×D4×C127—41no—
32C2×C4⋊C4×C127—22no—
33C2×C4⋊C4×C127—21no—
34C2×C4⋊C4×C127—41no—
35C2×C4⋊C4×C127C2×C411yescharacteristic
36C2×C22⋊C4×C127—21no—
37C2×C22⋊C4×C127—22no—
38C2×C22⋊C4×C127—22no—
39C2×C22⋊C4×C127D411yescharacteristic
40C2×C22⋊C4×C127C2311yescharacteristic
41C2×C42×C127D411yescharacteristic
42C4⋊C8×C127—41no—
43#99×C127—42no—
44C22⋊C8×C127—22no—
45C4×C8×C127—41no—
46#95×C127—22no—
47#95×C127—22no—
48#95×C127—41no—
49#95×C127C2×C412yes—
50C24×C127C22⋊C411yescharacteristic
51C24×C127C2×D411yescharacteristic
52C2×Q8×C127—24no—
53C2×Q8×C127—44no—
54C2×Q8×C127—42no—
55C2×Q8×C127—42no—
56C2×Q8×C127—44no—
57C2×Q8×C127—48no—
58C2×Q8×C127—44no—
59C2×Q8×C127—41no—
60C2×Q8×C127—41no—
61C2×D4×C127—41no—
62C2×D4×C127—41no—
63C2×D4×C127SD1612yes—
64C2×D4×C127Q1612yes—
65C22×C4×C127—22no—
66C22×C4×C127—21no—
67C22×C4×C127—21no—
68C22×C4×C127—21no—
69C22×C4×C127—21no—
70C22×C4×C127—22no—
71C22×C4×C127—44no—
72C22×C4×C127—42no—
73C22×C4×C127—42no—
74C22×C4×C127—42no—
75C22×C4×C127—42no—
76C22×C4×C127C22⋊C411yescharacteristic
77C22×C4×C127C22⋊C411yescharacteristic
78C22×C4×C127C22⋊C411yescharacteristic
79C22×C4×C127C22×C411yescharacteristic
80C22×C4×C127C2×D411yescharacteristic
81C2×C8×C127—42no—
82C4⋊C4×C127—41no—
83C4⋊C4×C127—42no—
84C4⋊C4×C127—44no—
85C4⋊C4×C127—82no—
86C22⋊C4×C127—41no—
87C22⋊C4×C127—41no—
88C22⋊C4×C127—44no—
89C22⋊C4×C127—42no—
90C22⋊C4×C127—42no—
91C22⋊C4×C127—42no—
92C42×C127—21no—
93C42×C127—21no—
94C23×C127—21no—
95C23×C127—21no—
96C23×C127—21no—
97C23×C127—21no—
98C23×C127—41no—
99C23×C127—41no—
100C23×C127—41no—
101C23×C127—41no—
102C23×C127#10312yes—
103C23×C127C2×C22⋊C411yescharacteristic
104C23×C127C2×SD1611yescharacteristic
105C23×C127C2×Q1611yescharacteristic
106Q8×C127—44no—
107Q8×C127—44no—
108Q8×C127—44no—
109Q8×C127—44no—
110D4×C127—42no—
111D4×C127—42no—
112C2×C4×C127—21no—
113C2×C4×C127—21no—
114C2×C4×C127—22no—
115C2×C4×C127—24no—
116C2×C4×C127—21no—
117C2×C4×C127—21no—
118C2×C4×C127—21no—
119C2×C4×C127—42no—
120C2×C4×C127—42no—
121C2×C4×C127—48no—
122C2×C4×C127—42no—
123C2×C4×C127—42no—
124C2×C4×C127—42no—
125C2×C4×C127—41no—
126C2×C4×C127—41no—
127C2×C4×C127—41no—
128C2×C4×C127—42no—
129C2×C4×C127—42no—
130C2×C4×C127—42no—
131C2×C4×C127—81no—
132C2×C4×C127—82no—
133C2×C4×C127—82no—
134C2×C4×C127—82no—
135C2×C4×C127#9911yescharacteristic
136C2×C4×C127#9911yescharacteristic
137C2×C4×C127#10312yes—
138C8×C127—82no—
139C22×C127—21no—
140C22×C127—21no—
141C22×C127—21no—
142C22×C127—22no—
143C22×C127—21no—
144C22×C127—42no—
145C22×C127—41no—
146C22×C127—41no—
147C22×C127—42no—
148C22×C127—41no—
149C22×C127—41no—
150C22×C127—41no—
151C22×C127—41no—
152C22×C127—81no—
153C22×C127C2×#9911yescharacteristic
154C22×C127C2×#10311yescharacteristic
155C22×C127#42111yescharacteristic
156C4×C127—21no—
157C4×C127—22no—
158C4×C127—44no—
159C4×C127—41no—
160C4×C127—41no—
161C4×C127—41no—
162C4×C127—82no—
163C4×C127—162no—
164C4×C127AΣL(1,𝔽16)12yes—
165#11898C12711yesmaximal, characteristic, direct product factor, Sylow
166C2×C127—21no—
167C2×C127—21no—
168C2×C127—41no—
169C2×C127—42no—
170C2×C127—41no—
171C2×C127#152011yescharacteristic
172C2×C127C2×AΣL(1,𝔽16)11yescharacteristic
173C2×C127#212911yescharacteristic
174#2409C2×C12711yescharacteristic
175#1799C2×C12711yescharacteristic
176#1460C2×C12711yescharacteristic
177#1351C2×C12714yescomplemented
178C127#1189811yescharacteristic, direct product factor, Sylow
179C23×Q8C4×C12711yescharacteristic, complemented
180C2×C22⋊Q8—21no—
181C2×C4×D4C22×C12711yescharacteristic
182#431—21no—
183C2×#99—21no—
184#395C4×C12712yescomplemented
185#393C22×C12712yes—
186#391—21no—
187#386—21no—
188#385—21no—
189#380—21no—
190#380C4×C12711yescharacteristic, complemented
191#342C22×C12712yes—
192#323—21no—
193#323C22×C12712yes—
194C22×Q8—21no—
195C22×Q8—41no—
196C22×Q8—41no—
197C22×Q8—44no—
198C22×Q8C2×C4×C12712yes—
199C22×D4D4×C12711yescharacteristic
200C23×C4—21no—
201C23×C4C2×C4×C12711yescharacteristic
202C23×C4D4×C12711yescharacteristic
203C22⋊Q8—44no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4C2×C4×C12711yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4D4×C12711yescharacteristic
214C2×C22⋊C4C23×C12711yescharacteristic
215C2×C42D4×C12711yescharacteristic
216C4⋊C8—41no—
217#99—42no—
218C22⋊C8—22no—
219C4×C8—41no—
220#95—22no—
221#95—22no—
222#95—41no—
223#95C2×C4×C12712yes—
224C24C22⋊C4×C12711yescharacteristic
225C24C2×D4×C12711yescharacteristic
226C2×Q8—24no—
227C2×Q8—44no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—44no—
231C2×Q8—48no—
232C2×Q8—44no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4SD16×C12712yes—
238C2×D4Q16×C12712yes—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4C22⋊C4×C12711yescharacteristic
251C22×C4C22⋊C4×C12711yescharacteristic
252C22×C4C22⋊C4×C12711yescharacteristic
253C22×C4C22×C4×C12711yescharacteristic
254C22×C4C2×D4×C12711yescharacteristic
255C2×C8—42no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—44no—
259C4⋊C4—82no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—44no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C42—21no—
267C42—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23#103×C12712yes—
277C23C2×C22⋊C4×C12711yescharacteristic
278C23C2×SD16×C12711yescharacteristic
279C23C2×Q16×C12711yescharacteristic
280Q8—44no—
281Q8—44no—
282Q8—44no—
283Q8—44no—
284D4—42no—
285D4—42no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—48no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4#99×C12711yescharacteristic
310C2×C4#99×C12711yescharacteristic
311C2×C4#103×C12712yes—
312C8—82no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—42no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22C2×#99×C12711yescharacteristic
328C22C2×#103×C12711yescharacteristic
329C22#421×C12711yescharacteristic
330C4—21no—
331C4—22no—
332C4—44no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—82no—
337C4—162no—
338C4AΣL(1,𝔽16)×C12712yes—
339C2—21no—
340C2—21no—
341C2—41no—
342C2—42no—
343C2—41no—
344C2C127×#152011yescharacteristic
345C2C2×AΣL(1,𝔽16)×C12711yescharacteristic
346C2C127×#212911yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C22⋊C4C23×C127
prime radicalC22×D4×C127D4
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
4. C127×#1351C2semidirect
5. C23×Q8×C127C4semidirect
10. #395×C127C4semidirect
16. #380×C127C4semidirect
165. #11898C127direct (N × H)
177. #1351C2×C127semidirect
178. C127#11898direct (N × H)
179. C23×Q8C4×C127semidirect
184. #395C4×C127semidirect
190. #380C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#24092
C127×#17992
C127×#14602
C127×#13512
#11898127