Finite Group Databasebeta

← C127×#11831 #44575504

Subgroups — C127×#11831 #44575504

Order 32512.

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

normal subgroups only

This group has 384 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×#2399C211yesmaximal, characteristic
2C127×#1801C211yesmaximal, characteristic
3C127×#1467C211yesmaximal, characteristic
4C127×#1354C212yesmaximal
5C127×#1346C212yesmaximal
6C2×C4×Q8×C127C411yescharacteristic
7C2×C4×Q8×C127C2211yescharacteristic
8#432×C127—21no—
9#423×C127—21no—
10C2×#101×C127—21no—
11#398×C127—21no—
12#398×C127C411yescharacteristic
13#398×C127C2211yescharacteristic
14#395×C127—21no—
15#395×C127C411yescharacteristic
16#395×C127C2211yescharacteristic
17C4×C4⋊C4×C127—21no—
18C4×C4⋊C4×C127—21no—
19#376×C127C411yescharacteristic
20#343×C127C2212yes—
21#324×C127—21no—
22#324×C127C2212yes—
23C22×Q8×C127C2×C411yescharacteristic
24C22×Q8×C127D411yescharacteristic
25C2×M16×C127—22no—
26C4×Q8×C127—41no—
27C4×Q8×C127—41no—
28C4×Q8×C127—41no—
29C4×Q8×C127—41no—
30C2×C4⋊C4×C127—21no—
31C2×C4⋊C4×C127—21no—
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×C42×C127—21no—
40C2×C42×C127—21no—
41C2×C42×C127C2×C411yescharacteristic
42C2×C42×C127D411yescharacteristic
43C2×C42×C127D411yescharacteristic
44C4⋊C8×C127—41no—
45C4⋊C8×C127—41no—
46C4⋊C8×C127—41no—
47#101×C127—42no—
48C22⋊C8×C127—22no—
49#97×C127—41no—
50#95×C127—21no—
51#95×C127—21no—
52#95×C127—21no—
53#95×C127—21no—
54#95×C127—21no—
55#95×C127C2×C411yescharacteristic
56#95×C127C2×C411yescharacteristic
57C2×Q8×C127—21no—
58C2×Q8×C127—21no—
59C2×Q8×C127—41no—
60C2×Q8×C127—41no—
61C2×Q8×C127—42no—
62C2×Q8×C127—41no—
63C22×C4×C127—21no—
64C22×C4×C127—21no—
65C22×C4×C127—21no—
66C22×C4×C127—21no—
67C22×C4×C127C22⋊C411yescharacteristic
68C22×C4×C127C22⋊C411yescharacteristic
69C22×C4×C127C22⋊C411yescharacteristic
70C22×C4×C127C22⋊C411yescharacteristic
71C22×C4×C127C22×C411yescharacteristic
72C22×C4×C127C2×D411yescharacteristic
73C22×C4×C127C2×D411yescharacteristic
74M16×C127—42no—
75M16×C127—42no—
76C2×C8×C127—42no—
77C2×C8×C127—42no—
78C4⋊C4×C127—41no—
79C4⋊C4×C127—41no—
80C4⋊C4×C127—41no—
81C4⋊C4×C127—41no—
82C4⋊C4×C127—42no—
83C4⋊C4×C127—42no—
84C4⋊C4×C127—42no—
85C4⋊C4×C127—42no—
86C4⋊C4×C127—41no—
87C4⋊C4×C127—41no—
88C42×C127—24no—
89C42×C127—21no—
90C42×C127—21no—
91C42×C127—21no—
92C42×C127—21no—
93C42×C127—21no—
94C42×C127—21no—
95C42×C127—41no—
96C42×C127—41no—
97C23×C127C2×C22⋊C411yescharacteristic
98Q8×C127—42no—
99Q8×C127—42no—
100Q8×C127—42no—
101Q8×C127—42no—
102C2×C4×C127—21no—
103C2×C4×C127—21no—
104C2×C4×C127—22no—
105C2×C4×C127—22no—
106C2×C4×C127—22no—
107C2×C4×C127—22no—
108C2×C4×C127—21no—
109C2×C4×C127—21no—
110C2×C4×C127—21no—
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—41no—
118C2×C4×C127—41no—
119C2×C4×C127—41no—
120C2×C4×C127—41no—
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#9911yescharacteristic
130C2×C4×C127#9911yescharacteristic
131C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C127#13711yescharacteristic
133C8×C127—82no—
134C8×C127—82no—
135C22×C127—21no—
136C22×C127—21no—
137C22×C127C2×#9911yescharacteristic
138C22×C127#41711yescharacteristic
139C22×C127#42111yescharacteristic
140C4×C127—21no—
141C4×C127—21no—
142C4×C127—21no—
143C4×C127—42no—
144C4×C127—42no—
145C4×C127—41no—
146C4×C127—41no—
147C4×C127—42no—
148C4×C127—41no—
149C4×C127—41no—
150C4×C127—41no—
151C4×C127—81no—
152C4×C127#35612yes—
153#11831C12711yesmaximal, characteristic, direct product factor, Sylow
154C2×C127—21no—
155C2×C127—21no—
156C2×C127#151311yescharacteristic
157C2×C127#212011yescharacteristic
158C2×C127C2×#35611yescharacteristic
159#2399C2×C12711yescharacteristic
160#1801C2×C12711yescharacteristic
161#1467C2×C12711yescharacteristic
162#1354C2×C12712yes—
163#1346C2×C12712yes—
164C127#1183111yescharacteristic, direct product factor, Sylow
165C2×C4×Q8C4×C12711yescharacteristic
166C2×C4×Q8C22×C12711yescharacteristic
167#432—21no—
168#423—21no—
169C2×#101—21no—
170#398—21no—
171#398C4×C12711yescharacteristic
172#398C22×C12711yescharacteristic
173#395—21no—
174#395C4×C12711yescharacteristic
175#395C22×C12711yescharacteristic
176C4×C4⋊C4—21no—
177C4×C4⋊C4—21no—
178#376C4×C12711yescharacteristic
179#343C22×C12712yes—
180#324—21no—
181#324C22×C12712yes—
182C22×Q8C2×C4×C12711yescharacteristic
183C22×Q8D4×C12711yescharacteristic
184C2×M16—22no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×Q8—41no—
188C4×Q8—41no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C2×C4×C12711yescharacteristic
195C2×C4⋊C4C2×C4×C12711yescharacteristic
196C2×C4⋊C4D4×C12711yescharacteristic
197C2×C4⋊C4C23×C12711yescharacteristic
198C2×C42—21no—
199C2×C42—21no—
200C2×C42C2×C4×C12711yescharacteristic
201C2×C42D4×C12711yescharacteristic
202C2×C42D4×C12711yescharacteristic
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8—41no—
206#101—42no—
207C22⋊C8—22no—
208#97—41no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95C2×C4×C12711yescharacteristic
215#95C2×C4×C12711yescharacteristic
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—42no—
221C2×Q8—41no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4C22⋊C4×C12711yescharacteristic
227C22×C4C22⋊C4×C12711yescharacteristic
228C22×C4C22⋊C4×C12711yescharacteristic
229C22×C4C22⋊C4×C12711yescharacteristic
230C22×C4C22×C4×C12711yescharacteristic
231C22×C4C2×D4×C12711yescharacteristic
232C22×C4C2×D4×C12711yescharacteristic
233M16—42no—
234M16—42no—
235C2×C8—42no—
236C2×C8—42no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C42—24no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—41no—
255C42—41no—
256C23C2×C22⋊C4×C12711yescharacteristic
257Q8—42no—
258Q8—42no—
259Q8—42no—
260Q8—42no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4#99×C12711yescharacteristic
289C2×C4#99×C12711yescharacteristic
290C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
291C2×C4#137×C12711yescharacteristic
292C8—82no—
293C8—82no—
294C22—21no—
295C22—21no—
296C22C2×#99×C12711yescharacteristic
297C22#417×C12711yescharacteristic
298C22#421×C12711yescharacteristic
299C4—21no—
300C4—21no—
301C4—21no—
302C4—42no—
303C4—42no—
304C4—41no—
305C4—42no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—81no—
311C4#356×C12712yes—
312C2—21no—
313C2—21no—
314C2C127×#151311yescharacteristic
315C2C127×#212011yescharacteristic
316C2C2×#356×C12711yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C4⋊C4C23×C127
prime radicalC23×C127C2×C22⋊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
153. #11831C127direct (N × H)
164. C127#11831direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#23992
C127×#18012
C127×#14672
C127×#13542
C127×#13462
#11831127