Finite Group Databasebeta

← C127×#11778 #44575451

Subgroups — C127×#11778 #44575451

Order 32512.

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

normal subgroups only

This group has 446 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×#2536C211yesmaximal, characteristic
2C4×#101×C127C211yesmaximal, characteristic, complemented
3C127×#1467C211yesmaximal, characteristic, complemented
4C127×#1346C212yesmaximal, complemented
5C127×#1345C212yesmaximal
6C2×C22⋊Q8×C127—21no—
7C2×C4×Q8×C127C2211yescharacteristic
8C2×C4×D4×C127C411yescharacteristic
9#432×C127—21no—
10C2×#101×C127—21no—
11C4×M16×C127—21no—
12#398×C127C411yescharacteristic
13#398×C127C2211yescharacteristic
14#394×C127C411yescharacteristic
15#394×C127C2211yescharacteristic
16#388×C127—21no—
17#388×C127—21no—
18#382×C127C411yescharacteristic
19#380×C127—21no—
20#343×C127C2212yes—
21#324×C127—21no—
22#324×C127C2212yes—
23C22×Q8×C127D411yescharacteristic
24C22×D4×C127C2×C411yescharacteristic
25C23×C4×C127—21no—
26C2×M16×C127—22no—
27C22⋊Q8×C127—44no—
28C4×Q8×C127—41no—
29C4×Q8×C127—41no—
30C4×D4×C127—41no—
31C4×D4×C127—41no—
32C2×C4⋊C4×C127—21no—
33C2×C4⋊C4×C127—21no—
34C2×C4⋊C4×C127—21no—
35C2×C4⋊C4×C127C2×C411yescharacteristic
36C2×C4⋊C4×C127C2×C411yescharacteristic
37C2×C4⋊C4×C127D411yescharacteristic
38C2×C4⋊C4×C127C2311yescharacteristic
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×C127D411yescharacteristic
46C2×C42×C127D411yescharacteristic
47C4⋊C8×C127—41no—
48#101×C127—24no—
49C22⋊C8×C127—22no—
50#97×C127—41no—
51#97×C127—41no—
52C4×C8×C127—41no—
53#95×C127—21no—
54#95×C127—21no—
55#95×C127—21no—
56#95×C127C2×C411yescharacteristic
57#95×C127C2×C411yescharacteristic
58C24×C127—21no—
59C2×Q8×C127—22no—
60C2×Q8×C127—22no—
61C2×Q8×C127—41no—
62C2×D4×C127—41no—
63C2×D4×C127—42no—
64C22×C4×C127—21no—
65C22×C4×C127—21no—
66C22×C4×C127—21no—
67C22×C4×C127—42no—
68C22×C4×C127—42no—
69C22×C4×C127—42no—
70C22×C4×C127C22⋊C411yescharacteristic
71C22×C4×C127C22⋊C411yescharacteristic
72C22×C4×C127C22⋊C411yescharacteristic
73C22×C4×C127C22⋊C411yescharacteristic
74C22×C4×C127C22×C411yescharacteristic
75C22×C4×C127C2×D411yescharacteristic
76C22×C4×C127C2×D411yescharacteristic
77M16×C127—24no—
78M16×C127—42no—
79C2×C8×C127—42no—
80C2×C8×C127—42no—
81C4⋊C4×C127—41no—
82C4⋊C4×C127—41no—
83C4⋊C4×C127—41no—
84C4⋊C4×C127—41no—
85C4⋊C4×C127—42no—
86C4⋊C4×C127—42no—
87C4⋊C4×C127—42no—
88C22⋊C4×C127—42no—
89C22⋊C4×C127—42no—
90C22⋊C4×C127—42no—
91C22⋊C4×C127—41no—
92C22⋊C4×C127—41no—
93C22⋊C4×C127—42no—
94C42×C127—21no—
95C42×C127—21no—
96C42×C127—21no—
97C42×C127—21no—
98C42×C127—21no—
99C42×C127—21no—
100C23×C127—22no—
101C23×C127—41no—
102C23×C127—41no—
103C23×C127—41no—
104C23×C127—41no—
105C23×C127—41no—
106C23×C127—41no—
107C23×C127C2×C22⋊C411yescharacteristic
108Q8×C127—44no—
109D4×C127—42no—
110D4×C127—42no—
111C2×C4×C127—21no—
112C2×C4×C127—21no—
113C2×C4×C127—21no—
114C2×C4×C127—21no—
115C2×C4×C127—21no—
116C2×C4×C127—21no—
117C2×C4×C127—21no—
118C2×C4×C127—21no—
119C2×C4×C127—21no—
120C2×C4×C127—21no—
121C2×C4×C127—41no—
122C2×C4×C127—41no—
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—84no—
135C2×C4×C127#9911yescharacteristic
136C2×C4×C127#9911yescharacteristic
137C2×C4×C127#10412yes—
138C2×C4×C127AGL(1,ℤ/8)12yes—
139C8×C127—44no—
140C8×C127—82no—
141C22×C127—21no—
142C22×C127—21no—
143C22×C127—42no—
144C22×C127—42no—
145C22×C127—42no—
146C22×C127—81no—
147C22×C127—81no—
148C22×C127—81no—
149C22×C127—81no—
150C22×C127C2×#9911yescharacteristic
151C22×C127#41811yescharacteristic
152C22×C127C2×#10411yescharacteristic
153C4×C127—21no—
154C4×C127—21no—
155C4×C127—22no—
156C4×C127—22no—
157C4×C127—41no—
158C4×C127—41no—
159C4×C127—41no—
160C4×C127—41no—
161C4×C127—81no—
162C4×C127—81no—
163C4×C127—81no—
164#11778C12711yesmaximal, characteristic, direct product factor, Sylow
165C2×C127—21no—
166C2×C127—21no—
167C2×C127—82no—
168C2×C127#150711yescharacteristic
169C2×C127#212011yescharacteristic
170C2×C127#213311yescharacteristic
171#2536C2×C12711yescharacteristic
172C4×#101C2×C12711yescharacteristic, complemented
173#1467C2×C12711yescharacteristic, complemented
174#1346C2×C12712yescomplemented
175#1345C2×C12712yes—
176C127#1177811yescharacteristic, direct product factor, Sylow
177C2×C22⋊Q8—21no—
178C2×C4×Q8C22×C12711yescharacteristic
179C2×C4×D4C4×C12711yescharacteristic
180#432—21no—
181C2×#101—21no—
182C4×M16—21no—
183#398C4×C12711yescharacteristic
184#398C22×C12711yescharacteristic
185#394C4×C12711yescharacteristic
186#394C22×C12711yescharacteristic
187#388—21no—
188#388—21no—
189#382C4×C12711yescharacteristic
190#380—21no—
191#343C22×C12712yes—
192#324—21no—
193#324C22×C12712yes—
194C22×Q8D4×C12711yescharacteristic
195C22×D4C2×C4×C12711yescharacteristic
196C23×C4—21no—
197C2×M16—22no—
198C22⋊Q8—44no—
199C4×Q8—41no—
200C4×Q8—41no—
201C4×D4—41no—
202C4×D4—41no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4C2×C4×C12711yescharacteristic
207C2×C4⋊C4C2×C4×C12711yescharacteristic
208C2×C4⋊C4D4×C12711yescharacteristic
209C2×C4⋊C4C23×C12711yescharacteristic
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×C42D4×C12711yescharacteristic
217C2×C42D4×C12711yescharacteristic
218C4⋊C8—41no—
219#101—24no—
220C22⋊C8—22no—
221#97—41no—
222#97—41no—
223C4×C8—41no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95C2×C4×C12711yescharacteristic
228#95C2×C4×C12711yescharacteristic
229C24—21no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—41no—
233C2×D4—41no—
234C2×D4—42no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4C22⋊C4×C12711yescharacteristic
242C22×C4C22⋊C4×C12711yescharacteristic
243C22×C4C22⋊C4×C12711yescharacteristic
244C22×C4C22⋊C4×C12711yescharacteristic
245C22×C4C22×C4×C12711yescharacteristic
246C22×C4C2×D4×C12711yescharacteristic
247C22×C4C2×D4×C12711yescharacteristic
248M16—24no—
249M16—42no—
250C2×C8—42no—
251C2×C8—42no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—42no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C23—22no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23C2×C22⋊C4×C12711yescharacteristic
279Q8—44no—
280D4—42no—
281D4—42no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—41no—
293C2×C4—41no—
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—84no—
306C2×C4#99×C12711yescharacteristic
307C2×C4#99×C12711yescharacteristic
308C2×C4#104×C12712yes—
309C2×C4AGL(1,ℤ/8)×C12712yes—
310C8—44no—
311C8—82no—
312C22—21no—
313C22—21no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22C2×#99×C12711yescharacteristic
322C22#418×C12711yescharacteristic
323C22C2×#104×C12711yescharacteristic
324C4—21no—
325C4—21no—
326C4—22no—
327C4—22no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—81no—
333C4—81no—
334C4—81no—
335C2—21no—
336C2—21no—
337C2—82no—
338C2C127×#150711yescharacteristic
339C2C127×#212011yescharacteristic
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 ▷ C23 ▷ 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. C4×#101×C127C2semidirect
3. C127×#1467C2semidirect
4. C127×#1346C2semidirect
164. #11778C127direct (N × H)
172. C4×#101C2×C127semidirect
173. #1467C2×C127semidirect
174. #1346C2×C127semidirect
176. C127#11778direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#25362
C4×#101×C1272
C127×#14672
C127×#13462
C127×#13452
#11778127