Finite Group Databasebeta

← C127×#12122 #44575795

Subgroups — C127×#12122 #44575795

Order 32512.

458 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2607C211yesmaximal, characteristic, complemented
2C127×#2022C211yesmaximal, characteristic
3C127×#1613C212yesmaximal
4C127×#1352C212yesmaximal, complemented
5C127×#1317C211yesmaximal, characteristic, complemented
6#534×C127C2211yescharacteristic
7C2×#123×C127—21no—
8#481×C127—22no—
9C2×#99×C127—21no—
10#399×C127C2212yes—
11#393×C127—21no—
12#391×C127—22no—
13#387×C127—22no—
14#381×C127C2211yescharacteristic
15C4×C22⋊C4×C127—21no—
16#342×C127C2211yescharacteristic
17#327×C127—22no—
18#323×C127C2212yes—
19C22×Q8×C127—21no—
20C22×D4×C127—21no—
21C23×C4×C127D411yescharacteristic
22#123×C127—42no—
23#123×C127—41no—
24#123×C127—42no—
25#121×C127—22no—
26#120×C127—41no—
27C2×C4⋊C4×C127—22no—
28C2×C4⋊C4×C127—41no—
29C2×C4⋊C4×C127D412yes—
30C2×C22⋊C4×C127—21no—
31C2×C22⋊C4×C127—22no—
32C2×C22⋊C4×C127—41no—
33C2×C22⋊C4×C127D411yescharacteristic
34C2×C22⋊C4×C127C2311yescharacteristic
35C2×C42×C127—21no—
36C2×C42×C127—21no—
37#106×C127—42no—
38#102×C127—42no—
39#99×C127—42no—
40C22⋊C8×C127—22no—
41#95×C127—22no—
42#95×C127—21no—
43#95×C127—22no—
44#95×C127—22no—
45#95×C127D412yes—
46C24×C127—21no—
47C24×C127C2×D411yescharacteristic
48C2×Q8×C127—44no—
49C2×Q8×C127—81no—
50C2×D4×C127—22no—
51C2×D4×C127—41no—
52C2×D4×C127—41no—
53C2×D4×C127—41no—
54C2×D4×C127—41no—
55C2×D4×C127—41no—
56C2×D4×C127—41no—
57C22×C4×C127—21no—
58C22×C4×C127—22no—
59C22×C4×C127—21no—
60C22×C4×C127—21no—
61C22×C4×C127—21no—
62C22×C4×C127—22no—
63C22×C4×C127—41no—
64C22×C4×C127—41no—
65C22×C4×C127—41no—
66C22×C4×C127SD1612yes—
67C22×C4×C127C2×D412yes—
68C2×C8×C127—42no—
69C4⋊C4×C127—22no—
70C4⋊C4×C127—42no—
71C4⋊C4×C127—44no—
72C4⋊C4×C127—44no—
73C22⋊C4×C127—22no—
74C22⋊C4×C127—41no—
75C22⋊C4×C127—44no—
76C22⋊C4×C127—42no—
77C22⋊C4×C127—44no—
78C22⋊C4×C127—41no—
79C22⋊C4×C127—41no—
80C42×C127—81no—
81C42×C127—81no—
82C23×C127—21no—
83C23×C127—21no—
84C23×C127—42no—
85C23×C127—41no—
86C23×C127—41no—
87C23×C127—41no—
88C23×C127—82no—
89C23×C127#12011yescharacteristic
90C23×C127C2×SD1611yescharacteristic
91C23×C127#13711yescharacteristic
92Q8×C127—84no—
93D4×C127—82no—
94D4×C127—81no—
95D4×C127—81no—
96D4×C127—81no—
97D4×C127—81no—
98C2×C4×C127—24no—
99C2×C4×C127—22no—
100C2×C4×C127—22no—
101C2×C4×C127—42no—
102C2×C4×C127—41no—
103C2×C4×C127—42no—
104C2×C4×C127—41no—
105C2×C4×C127—42no—
106C2×C4×C127—42no—
107C2×C4×C127—42no—
108C2×C4×C127—42no—
109C2×C4×C127—41no—
110C2×C4×C127—42no—
111C2×C4×C127—41no—
112C2×C4×C127—42no—
113C2×C4×C127—41no—
114C2×C4×C127—42no—
115C2×C4×C127—41no—
116C2×C4×C127—81no—
117C2×C4×C127—82no—
118C8×C127—82no—
119C22×C127—22no—
120C22×C127—21no—
121C22×C127—21no—
122C22×C127—41no—
123C22×C127—41no—
124C22×C127—42no—
125C22×C127—81no—
126C22×C127—82no—
127C22×C127—81no—
128C22×C127—82no—
129C22×C127—161no—
130C22×C127#44811yescharacteristic
131C22×C127#45311yescharacteristic
132C22×C127B(4,𝔽2)11yescharacteristic
133C4×C127—44no—
134C4×C127—42no—
135C4×C127—42no—
136C4×C127—81no—
137C4×C127—82no—
138C4×C127—81no—
139C4×C127—82no—
140#12122C12711yesmaximal, characteristic, direct product factor, Sylow
141C2×C127—21no—
142C2×C127—21no—
143C2×C127—41no—
144C2×C127—41no—
145C2×C127—161no—
146C2×C127#159611yescharacteristic
147C2×C127#219812yes—
148#2607C2×C12711yescharacteristic, complemented
149#2022C2×C12711yescharacteristic
150#1613C2×C12712yes—
151#1352C2×C12712yescomplemented
152#1317C2×C12711yescharacteristic, complemented
153C127#1212211yescharacteristic, direct product factor, Sylow
154#534C22×C12711yescharacteristic
155C2×#123—21no—
156#481—22no—
157C2×#99—21no—
158#399C22×C12712yes—
159#393—21no—
160#391—22no—
161#387—22no—
162#381C22×C12711yescharacteristic
163C4×C22⋊C4—21no—
164#342C22×C12711yescharacteristic
165#327—22no—
166#323C22×C12712yes—
167C22×Q8—21no—
168C22×D4—21no—
169C23×C4D4×C12711yescharacteristic
170#123—42no—
171#123—41no—
172#123—42no—
173#121—22no—
174#120—41no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4D4×C12712yes—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4D4×C12711yescharacteristic
182C2×C22⋊C4C23×C12711yescharacteristic
183C2×C42—21no—
184C2×C42—21no—
185#106—42no—
186#102—42no—
187#99—42no—
188C22⋊C8—22no—
189#95—22no—
190#95—21no—
191#95—22no—
192#95—22no—
193#95D4×C12712yes—
194C24—21no—
195C24C2×D4×C12711yescharacteristic
196C2×Q8—44no—
197C2×Q8—81no—
198C2×D4—22no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4SD16×C12712yes—
215C22×C4C2×D4×C12712yes—
216C2×C8—42no—
217C4⋊C4—22no—
218C4⋊C4—42no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C22⋊C4—22no—
222C22⋊C4—41no—
223C22⋊C4—44no—
224C22⋊C4—42no—
225C22⋊C4—44no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C42—81no—
229C42—81no—
230C23—21no—
231C23—21no—
232C23—42no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—82no—
237C23#120×C12711yescharacteristic
238C23C2×SD16×C12711yescharacteristic
239C23#137×C12711yescharacteristic
240Q8—84no—
241D4—82no—
242D4—81no—
243D4—81no—
244D4—81no—
245D4—81no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—81no—
265C2×C4—82no—
266C8—82no—
267C22—22no—
268C22—21no—
269C22—21no—
270C22—41no—
271C22—41no—
272C22—42no—
273C22—81no—
274C22—82no—
275C22—81no—
276C22—82no—
277C22—161no—
278C22#448×C12711yescharacteristic
279C22#453×C12711yescharacteristic
280C22B(4,𝔽2)×C12711yescharacteristic
281C4—44no—
282C4—42no—
283C4—42no—
284C4—81no—
285C4—82no—
286C4—81no—
287C4—82no—
288C2—21no—
289C2—21no—
290C2—41no—
291C2—41no—
292C2—161no—
293C2C127×#159611yescharacteristic
294C2C127×#219812yes—
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C127
prime radical#534×C127C22
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C127 ◁ C2×C22⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#2607C2semidirect
4. C127×#1352C2semidirect
5. C127×#1317C2semidirect
140. #12122C127direct (N × H)
148. #2607C2×C127semidirect
151. #1352C2×C127semidirect
152. #1317C2×C127semidirect
153. C127#12122direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#26072
C127×#20222
C127×#16132
C127×#13522
C127×#13172
#12122127