Finite Group Databasebeta

← C127×#11565 #44575238

Subgroups — C127×#11565 #44575238

Order 32512.

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

normal subgroups only

This group has 412 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×#2545C211yesmaximal, characteristic
2C127×#1467C211yesmaximal, characteristic, complemented
3C127×#1461C211yesmaximal, characteristic, complemented
4C127×#1319C212yesmaximal, complemented
5C127×#1318C212yesmaximal
6C2×C4⋊Q8×C127C411yescharacteristic
7C2×#124×C127C411yescharacteristic
8C2×#124×C127C411yescharacteristic
9C2×C4×Q8×C127C2211yescharacteristic
10#432×C127—21no—
11#431×C127—21no—
12#396×C127—21no—
13#396×C127—21no—
14#391×C127C2211yescharacteristic
15#390×C127C2211yescharacteristic
16#388×C127—21no—
17#388×C127—21no—
18C4×C4⋊C4×C127C411yescharacteristic
19#324×C127—21no—
20#324×C127—21no—
21#324×C127C2212yes—
22#324×C127C2212yes—
23C22×Q8×C127C2×C411yescharacteristic
24C22×Q8×C127C2×C411yescharacteristic
25C22×D4×C127C2×C411yescharacteristic
26C4⋊Q8×C127—42no—
27#124×C127—41no—
28#124×C127—41no—
29#124×C127—42no—
30C4×Q8×C127—41no—
31C4×Q8×C127—41no—
32C2×C4⋊C4×C127—21no—
33C2×C4⋊C4×C127—21no—
34C2×C4⋊C4×C127—21no—
35C2×C4⋊C4×C127D411yescharacteristic
36C2×C4⋊C4×C127D411yescharacteristic
37C2×C4⋊C4×C127C2311yescharacteristic
38C2×C22⋊C4×C127—21no—
39C2×C22⋊C4×C127—21no—
40C2×C22⋊C4×C127—21no—
41C2×C22⋊C4×C127—21no—
42C2×C22⋊C4×C127—21no—
43C2×C22⋊C4×C127—21no—
44C2×C42×C127C2×C411yescharacteristic
45C2×C42×C127C2×C411yescharacteristic
46C2×C42×C127C2×C411yescharacteristic
47C2×C42×C127D411yescharacteristic
48C2×C42×C127D411yescharacteristic
49C4⋊C8×C127—41no—
50C4⋊C8×C127—41no—
51C22⋊C8×C127—22no—
52C22⋊C8×C127—22no—
53#97×C127—41no—
54C4×C8×C127—41no—
55#95×C127—21no—
56#95×C127—21no—
57#95×C127—21no—
58C24×C127—21no—
59C2×Q8×C127—22no—
60C2×Q8×C127—22no—
61C2×Q8×C127—22no—
62C2×Q8×C127—41no—
63C2×Q8×C127—41no—
64C2×Q8×C127—41no—
65C2×D4×C127—22no—
66C2×D4×C127—41no—
67C2×D4×C127—41no—
68C22×C4×C127—21no—
69C22×C4×C127—21no—
70C22×C4×C127—21no—
71C22×C4×C127C22⋊C411yescharacteristic
72C22×C4×C127C22⋊C411yescharacteristic
73C22×C4×C127C22⋊C411yescharacteristic
74C22×C4×C127C22⋊C411yescharacteristic
75C22×C4×C127C22×C411yescharacteristic
76C22×C4×C127C2×D411yescharacteristic
77C22×C4×C127C2×D411yescharacteristic
78C2×C8×C127—42no—
79C2×C8×C127—42no—
80C4⋊C4×C127—41no—
81C4⋊C4×C127—41no—
82C4⋊C4×C127—41no—
83C4⋊C4×C127—42no—
84C4⋊C4×C127—42no—
85C4⋊C4×C127—42no—
86C22⋊C4×C127—42no—
87C22⋊C4×C127—42no—
88C22⋊C4×C127—42no—
89C22⋊C4×C127—42no—
90C22⋊C4×C127—41no—
91C22⋊C4×C127—41no—
92C22⋊C4×C127—42no—
93C42×C127—21no—
94C42×C127—21no—
95C42×C127—21no—
96C42×C127—21no—
97C42×C127—21no—
98C42×C127—21no—
99C42×C127—41no—
100C42×C127—41no—
101C23×C127—41no—
102C23×C127—41no—
103C23×C127—41no—
104C23×C127—41no—
105C23×C127—41no—
106C23×C127—41no—
107C23×C127—41no—
108C23×C127C2×C22⋊C411yescharacteristic
109Q8×C127—44no—
110Q8×C127—82no—
111D4×C127—82no—
112C2×C4×C127—21no—
113C2×C4×C127—21no—
114C2×C4×C127—21no—
115C2×C4×C127—21no—
116C2×C4×C127—22no—
117C2×C4×C127—21no—
118C2×C4×C127—21no—
119C2×C4×C127—21no—
120C2×C4×C127—21no—
121C2×C4×C127—21no—
122C2×C4×C127—21no—
123C2×C4×C127—41no—
124C2×C4×C127—41no—
125C2×C4×C127—41no—
126C2×C4×C127—41no—
127C2×C4×C127—41no—
128C2×C4×C127—41no—
129C2×C4×C127—41no—
130C2×C4×C127—41no—
131C2×C4×C127—41no—
132C2×C4×C127—41no—
133C2×C4×C127—41no—
134C2×C4×C127—41no—
135C2×C4×C127#9911yescharacteristic
136C2×C4×C127#9911yescharacteristic
137C2×C4×C127#10012yes—
138C2×C4×C127#13112yes—
139C8×C127—82no—
140C8×C127—82no—
141C22×C127—21no—
142C22×C127—21no—
143C22×C127—81no—
144C22×C127—81no—
145C22×C127—81no—
146C22×C127—81no—
147C22×C127—81no—
148C22×C127—81no—
149C22×C127—81no—
150C22×C127#40811yescharacteristic
151C22×C127C2×#9911yescharacteristic
152C22×C127C2×#10011yescharacteristic
153C4×C127—21no—
154C4×C127—21no—
155C4×C127—22no—
156C4×C127—22no—
157C4×C127—42no—
158C4×C127—41no—
159C4×C127—41no—
160C4×C127—41no—
161C4×C127—41no—
162C4×C127—81no—
163C4×C127—81no—
164#11565C12711yesmaximal, characteristic, direct product factor, Sylow
165C2×C127—21no—
166C2×C127—21no—
167C2×C127—161no—
168C2×C127#147111yescharacteristic
169C2×C127#212611yescharacteristic
170C2×C127#213311yescharacteristic
171#2545C2×C12711yescharacteristic
172#1467C2×C12711yescharacteristic, complemented
173#1461C2×C12711yescharacteristic, complemented
174#1319C2×C12712yescomplemented
175#1318C2×C12712yes—
176C127#1156511yescharacteristic, direct product factor, Sylow
177C2×C4⋊Q8C4×C12711yescharacteristic
178C2×#124C4×C12711yescharacteristic
179C2×#124C4×C12711yescharacteristic
180C2×C4×Q8C22×C12711yescharacteristic
181#432—21no—
182#431—21no—
183#396—21no—
184#396—21no—
185#391C22×C12711yescharacteristic
186#390C22×C12711yescharacteristic
187#388—21no—
188#388—21no—
189C4×C4⋊C4C4×C12711yescharacteristic
190#324—21no—
191#324—21no—
192#324C22×C12712yes—
193#324C22×C12712yes—
194C22×Q8C2×C4×C12711yescharacteristic
195C22×Q8C2×C4×C12711yescharacteristic
196C22×D4C2×C4×C12711yescharacteristic
197C4⋊Q8—42no—
198#124—41no—
199#124—41no—
200#124—42no—
201C4×Q8—41no—
202C4×Q8—41no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4D4×C12711yescharacteristic
207C2×C4⋊C4D4×C12711yescharacteristic
208C2×C4⋊C4C23×C12711yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C42C2×C4×C12711yescharacteristic
216C2×C42C2×C4×C12711yescharacteristic
217C2×C42C2×C4×C12711yescharacteristic
218C2×C42D4×C12711yescharacteristic
219C2×C42D4×C12711yescharacteristic
220C4⋊C8—41no—
221C4⋊C8—41no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224#97—41no—
225C4×C8—41no—
226#95—21no—
227#95—21no—
228#95—21no—
229C24—21no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×D4—22no—
237C2×D4—41no—
238C2×D4—41no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4C22⋊C4×C12711yescharacteristic
243C22×C4C22⋊C4×C12711yescharacteristic
244C22×C4C22⋊C4×C12711yescharacteristic
245C22×C4C22⋊C4×C12711yescharacteristic
246C22×C4C22×C4×C12711yescharacteristic
247C22×C4C2×D4×C12711yescharacteristic
248C22×C4C2×D4×C12711yescharacteristic
249C2×C8—42no—
250C2×C8—42no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—42no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—41no—
271C42—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23C2×C22⋊C4×C12711yescharacteristic
280Q8—44no—
281Q8—82no—
282D4—82no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4#99×C12711yescharacteristic
307C2×C4#99×C12711yescharacteristic
308C2×C4#100×C12712yes—
309C2×C4#131×C12712yes—
310C8—82no—
311C8—82no—
312C22—21no—
313C22—21no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22#408×C12711yescharacteristic
322C22C2×#99×C12711yescharacteristic
323C22C2×#100×C12711yescharacteristic
324C4—21no—
325C4—21no—
326C4—22no—
327C4—22no—
328C4—42no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—81no—
334C4—81no—
335C2—21no—
336C2—21no—
337C2—161no—
338C2C127×#147111yescharacteristic
339C2C127×#212611yescharacteristic
340C2C127×#213311yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C4⋊C4C23×C127
prime radicalC22×D4×C127C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C4×Q8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1467C2semidirect
3. C127×#1461C2semidirect
4. C127×#1319C2semidirect
164. #11565C127direct (N × H)
172. #1467C2×C127semidirect
173. #1461C2×C127semidirect
174. #1319C2×C127semidirect
176. C127#11565direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#25452
C127×#14672
C127×#14612
C127×#13192
C127×#13182
#11565127