Finite Group Databasebeta

← C127×#11780 #44575453

Subgroups — C127×#11780 #44575453

Order 32512.

464 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2486C211yesmaximal, characteristic
2C4×#100×C127C211yesmaximal, characteristic
3C127×#1465C211yesmaximal, characteristic
4C127×#1346C214yesmaximal, complemented
5C2×#123×C127—21no—
6C2×C4×D4×C127C2211yescharacteristic
7C22×C4⋊C4×C127C411yescharacteristic
8C2×#100×C127—21no—
9#407×C127—21no—
10C4×M16×C127—21no—
11#398×C127C2212yes—
12#397×C127C412yes—
13#389×C127—21no—
14#381×C127—21no—
15#380×C127—21no—
16#380×C127C411yescharacteristic
17#343×C127C2212yes—
18#324×C127C2212yes—
19#323×C127—21no—
20C22×D4×C127D411yescharacteristic
21C23×C4×C127—21no—
22C23×C4×C127C2×C411yescharacteristic
23C23×C4×C127D411yescharacteristic
24C2×M16×C127—22no—
25C2×M16×C127—41no—
26#123×C127—44no—
27C4×D4×C127—41no—
28C4×D4×C127—41no—
29C2×C4⋊C4×C127—21no—
30C2×C4⋊C4×C127—21no—
31C2×C4⋊C4×C127—22no—
32C2×C4⋊C4×C127—41no—
33C2×C4⋊C4×C127—41no—
34C2×C4⋊C4×C127C2×C412yes—
35C2×C4⋊C4×C127C2311yescharacteristic
36C2×C22⋊C4×C127—22no—
37C2×C22⋊C4×C127—21no—
38C2×C22⋊C4×C127C2×C411yescharacteristic
39C2×C22⋊C4×C127D411yescharacteristic
40C2×C42×C127D411yescharacteristic
41#100×C127—24no—
42C22⋊C8×C127—22no—
43C22⋊C8×C127—41no—
44#97×C127—41no—
45C4×C8×C127—41no—
46#95×C127—22no—
47#95×C127—22no—
48#95×C127C2×C412yes—
49C24×C127C22⋊C411yescharacteristic
50C24×C127C22⋊C411yescharacteristic
51C2×D4×C127—22no—
52C2×D4×C127—22no—
53C2×D4×C127—41no—
54C22×C4×C127—21no—
55C22×C4×C127—21no—
56C22×C4×C127—21no—
57C22×C4×C127—21no—
58C22×C4×C127—22no—
59C22×C4×C127—22no—
60C22×C4×C127—42no—
61C22×C4×C127—42no—
62C22×C4×C127—42no—
63C22×C4×C127—42no—
64C22×C4×C127—42no—
65C22×C4×C127C22⋊C411yescharacteristic
66C22×C4×C127C22⋊C411yescharacteristic
67C22×C4×C127C22×C411yescharacteristic
68C22×C4×C127C2×D411yescharacteristic
69C22×C4×C127C2×D411yescharacteristic
70M16×C127—24no—
71M16×C127—42no—
72M16×C127—82no—
73C2×C8×C127—42no—
74C2×C8×C127—42no—
75C4⋊C4×C127—41no—
76C4⋊C4×C127—42no—
77C4⋊C4×C127—42no—
78C4⋊C4×C127—41no—
79C4⋊C4×C127—41no—
80C4⋊C4×C127—44no—
81C22⋊C4×C127—41no—
82C22⋊C4×C127—41no—
83C22⋊C4×C127—42no—
84C22⋊C4×C127—44no—
85C42×C127—21no—
86C42×C127—21no—
87C23×C127—21no—
88C23×C127—21no—
89C23×C127—21no—
90C23×C127—21no—
91C23×C127—21no—
92C23×C127—41no—
93C23×C127—41no—
94C23×C127—41no—
95C23×C127—41no—
96C23×C127#10412yes—
97C23×C127C2×C22⋊C411yescharacteristic
98D4×C127—44no—
99C2×C4×C127—21no—
100C2×C4×C127—21no—
101C2×C4×C127—22no—
102C2×C4×C127—21no—
103C2×C4×C127—21no—
104C2×C4×C127—21no—
105C2×C4×C127—42no—
106C2×C4×C127—42no—
107C2×C4×C127—42no—
108C2×C4×C127—41no—
109C2×C4×C127—41no—
110C2×C4×C127—41no—
111C2×C4×C127—42no—
112C2×C4×C127—42no—
113C2×C4×C127—42no—
114C2×C4×C127—42no—
115C2×C4×C127—42no—
116C2×C4×C127—42no—
117C2×C4×C127—81no—
118C2×C4×C127—84no—
119C2×C4×C127#9911yescharacteristic
120C2×C4×C127#9911yescharacteristic
121C2×C4×C127AGL(1,ℤ/8)12yes—
122C8×C127—44no—
123C8×C127—82no—
124C22×C127—21no—
125C22×C127—21no—
126C22×C127—22no—
127C22×C127—42no—
128C22×C127—42no—
129C22×C127—42no—
130C22×C127—41no—
131C22×C127—41no—
132C22×C127—41no—
133C22×C127—41no—
134C22×C127—41no—
135C22×C127—81no—
136C22×C127C2×#9911yescharacteristic
137C22×C127#41811yescharacteristic
138C22×C127C2×#10411yescharacteristic
139C4×C127—21no—
140C4×C127—22no—
141C4×C127—41no—
142C4×C127—41no—
143C4×C127—41no—
144C4×C127—82no—
145C4×C127—82no—
146C4×C127#35212yes—
147#11780C12711yesmaximal, characteristic, direct product factor, Sylow
148C2×C127—21no—
149C2×C127—21no—
150C2×C127—42no—
151C2×C127—81no—
152C2×C127#150711yescharacteristic
153C2×C127C2×#35211yescharacteristic
154C2×C127#213511yescharacteristic
155#2486C2×C12711yescharacteristic
156C4×#100C2×C12711yescharacteristic
157#1465C2×C12711yescharacteristic
158#1346C2×C12714yescomplemented
159C127#1178011yescharacteristic, direct product factor, Sylow
160C2×#123—21no—
161C2×C4×D4C22×C12711yescharacteristic
162C22×C4⋊C4C4×C12711yescharacteristic
163C2×#100—21no—
164#407—21no—
165C4×M16—21no—
166#398C22×C12712yes—
167#397C4×C12712yes—
168#389—21no—
169#381—21no—
170#380—21no—
171#380C4×C12711yescharacteristic
172#343C22×C12712yes—
173#324C22×C12712yes—
174#323—21no—
175C22×D4D4×C12711yescharacteristic
176C23×C4—21no—
177C23×C4C2×C4×C12711yescharacteristic
178C23×C4D4×C12711yescharacteristic
179C2×M16—22no—
180C2×M16—41no—
181#123—44no—
182C4×D4—41no—
183C4×D4—41no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4C2×C4×C12712yes—
190C2×C4⋊C4C23×C12711yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4C2×C4×C12711yescharacteristic
194C2×C22⋊C4D4×C12711yescharacteristic
195C2×C42D4×C12711yescharacteristic
196#100—24no—
197C22⋊C8—22no—
198C22⋊C8—41no—
199#97—41no—
200C4×C8—41no—
201#95—22no—
202#95—22no—
203#95C2×C4×C12712yes—
204C24C22⋊C4×C12711yescharacteristic
205C24C22⋊C4×C12711yescharacteristic
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—41no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4C22⋊C4×C12711yescharacteristic
221C22×C4C22⋊C4×C12711yescharacteristic
222C22×C4C22×C4×C12711yescharacteristic
223C22×C4C2×D4×C12711yescharacteristic
224C22×C4C2×D4×C12711yescharacteristic
225M16—24no—
226M16—42no—
227M16—82no—
228C2×C8—42no—
229C2×C8—42no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—44no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—42no—
239C22⋊C4—44no—
240C42—21no—
241C42—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23#104×C12712yes—
252C23C2×C22⋊C4×C12711yescharacteristic
253D4—44no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—81no—
273C2×C4—84no—
274C2×C4#99×C12711yescharacteristic
275C2×C4#99×C12711yescharacteristic
276C2×C4AGL(1,ℤ/8)×C12712yes—
277C8—44no—
278C8—82no—
279C22—21no—
280C22—21no—
281C22—22no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—81no—
291C22C2×#99×C12711yescharacteristic
292C22#418×C12711yescharacteristic
293C22C2×#104×C12711yescharacteristic
294C4—21no—
295C4—22no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—82no—
300C4—82no—
301C4#352×C12712yes—
302C2—21no—
303C2—21no—
304C2—42no—
305C2—81no—
306C2C127×#150711yescharacteristic
307C2C2×#352×C12711yescharacteristic
308C2C127×#213511yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C4⋊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×#1346C2semidirect
147. #11780C127direct (N × H)
158. #1346C2×C127semidirect
159. C127#11780direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#24862
C4×#100×C1272
C127×#14652
C127×#13462
#11780127