Finite Group Databasebeta

← C127×#9508 #44573181

Subgroups — C127×#9508 #44573181

Order 32512.

542 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2975C211yesmaximal, characteristic
2C127×#2486C211yesmaximal, characteristic
3C127×#1821C212yesmaximal, complemented
4C127×#1465C211yesmaximal, characteristic
5C127×#1305C212yesmaximal, complemented
6C2×#123×C127C412yes—
7C2×C4×D4×C127C2211yescharacteristic
8C22×C4⋊C4×C127C2211yescharacteristic
9#436×C127—24no—
10#431×C127—21no—
11#407×C127—21no—
12#407×C127C2212yes—
13#407×C127C2212yes—
14#398×C127—21no—
15#397×C127—21no—
16#389×C127—21no—
17#381×C127C412yes—
18#380×C127—21no—
19#380×C127C2211yescharacteristic
20#343×C127—24no—
21#324×C127—21no—
22#323×C127—21no—
23C22×D4×C127C2×C411yescharacteristic
24C23×C4×C127C2×C412yes—
25C23×C4×C127D411yescharacteristic
26C23×C4×C127C2311yescharacteristic
27C2×M16×C127—22no—
28C2×M16×C127—22no—
29C2×M16×C127D414yes—
30#123×C127—44no—
31C4×D4×C127—22no—
32C4×D4×C127—22no—
33C2×C4⋊C4×C127—22no—
34C2×C4⋊C4×C127—22no—
35C2×C4⋊C4×C127—21no—
36C2×C4⋊C4×C127—21no—
37C2×C4⋊C4×C127—21no—
38C2×C4⋊C4×C127—22no—
39C2×C4⋊C4×C127D411yescharacteristic
40C2×C22⋊C4×C127—22no—
41C2×C22⋊C4×C127C2×C411yescharacteristic
42C2×C22⋊C4×C127C2×C412yes—
43C2×C22⋊C4×C127D411yescharacteristic
44C2×C42×C127D411yescharacteristic
45C4⋊C8×C127—22no—
46C22⋊C8×C127—22no—
47C22⋊C8×C127—22no—
48C22⋊C8×C127—22no—
49C22⋊C8×C127—22no—
50C22⋊C8×C127—22no—
51C4×C8×C127—22no—
52#95×C127—22no—
53#95×C127—22no—
54#95×C127—21no—
55C24×C127C22⋊C411yescharacteristic
56C24×C127C22×C411yescharacteristic
57C2×D4×C127—22no—
58C2×D4×C127—22no—
59C2×D4×C127—41no—
60C22×C4×C127—21no—
61C22×C4×C127—21no—
62C22×C4×C127—22no—
63C22×C4×C127—22no—
64C22×C4×C127—24no—
65C22×C4×C127—42no—
66C22×C4×C127—42no—
67C22×C4×C127—42no—
68C22×C4×C127—42no—
69C22×C4×C127C22⋊C411yescharacteristic
70C22×C4×C127C22⋊C411yescharacteristic
71C22×C4×C127C22⋊C411yescharacteristic
72C22×C4×C127C2×D412yes—
73C22×C4×C127C2×D411yescharacteristic
74C22×C4×C127C2×D411yescharacteristic
75C22×C4×C127#4112yes—
76M16×C127—44no—
77M16×C127—44no—
78M16×C127—82no—
79C2×C8×C127—24no—
80C2×C8×C127—42no—
81C4⋊C4×C127—22no—
82C4⋊C4×C127—42no—
83C4⋊C4×C127—44no—
84C4⋊C4×C127—41no—
85C4⋊C4×C127—42no—
86C4⋊C4×C127—41no—
87C22⋊C4×C127—22no—
88C22⋊C4×C127—22no—
89C22⋊C4×C127—44no—
90C22⋊C4×C127—42no—
91C42×C127—21no—
92C42×C127—21no—
93C23×C127—21no—
94C23×C127—21no—
95C23×C127—21no—
96C23×C127—21no—
97C23×C127—21no—
98C23×C127—41no—
99C23×C127—41no—
100C23×C127—41no—
101C23×C127—41no—
102C23×C127C2×C22⋊C411yescharacteristic
103C23×C127C4×D412yes—
104D4×C127—44no—
105C2×C4×C127—21no—
106C2×C4×C127—21no—
107C2×C4×C127—22no—
108C2×C4×C127—24no—
109C2×C4×C127—21no—
110C2×C4×C127—21no—
111C2×C4×C127—41no—
112C2×C4×C127—42no—
113C2×C4×C127—42no—
114C2×C4×C127—41no—
115C2×C4×C127—41no—
116C2×C4×C127—42no—
117C2×C4×C127—42no—
118C2×C4×C127—42no—
119C2×C4×C127—42no—
120C2×C4×C127—42no—
121C2×C4×C127—42no—
122C2×C4×C127—81no—
123C2×C4×C127—84no—
124C2×C4×C127#10012yes—
125C2×C4×C127#12011yescharacteristic
126C2×C4×C127#12112yes—
127C2×C4×C127#12311yescharacteristic
128C8×C127—44no—
129C8×C127—82no—
130C22×C127—21no—
131C22×C127—21no—
132C22×C127—22no—
133C22×C127—22no—
134C22×C127—22no—
135C22×C127—42no—
136C22×C127—42no—
137C22×C127—42no—
138C22×C127—41no—
139C22×C127—41no—
140C22×C127—41no—
141C22×C127—81no—
142C22×C127#38611yescharacteristic
143C22×C127C2×#10011yescharacteristic
144C22×C127#41311yescharacteristic
145C4×C127—22no—
146C4×C127—21no—
147C4×C127—22no—
148C4×C127—41no—
149C4×C127—41no—
150C4×C127—82no—
151C4×C127—82no—
152C4×C127#45812yes—
153#9508C12711yesmaximal, characteristic, direct product factor, Sylow
154C2×C127—21no—
155C2×C127—21no—
156C2×C127—42no—
157C2×C127—81no—
158C2×C127#189011yescharacteristic
159C2×C127#190011yescharacteristic
160C2×C127#191511yescharacteristic
161#2975C2×C12711yescharacteristic
162#2486C2×C12711yescharacteristic
163#1821C2×C12712yescomplemented
164#1465C2×C12711yescharacteristic
165#1305C2×C12712yescomplemented
166C127#950811yescharacteristic, direct product factor, Sylow
167C2×#123C4×C12712yes—
168C2×C4×D4C22×C12711yescharacteristic
169C22×C4⋊C4C22×C12711yescharacteristic
170#436—24no—
171#431—21no—
172#407—21no—
173#407C22×C12712yes—
174#407C22×C12712yes—
175#398—21no—
176#397—21no—
177#389—21no—
178#381C4×C12712yes—
179#380—21no—
180#380C22×C12711yescharacteristic
181#343—24no—
182#324—21no—
183#323—21no—
184C22×D4C2×C4×C12711yescharacteristic
185C23×C4C2×C4×C12712yes—
186C23×C4D4×C12711yescharacteristic
187C23×C4C23×C12711yescharacteristic
188C2×M16—22no—
189C2×M16—22no—
190C2×M16D4×C12714yes—
191#123—44no—
192C4×D4—22no—
193C4×D4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4D4×C12711yescharacteristic
201C2×C22⋊C4—22no—
202C2×C22⋊C4C2×C4×C12711yescharacteristic
203C2×C22⋊C4C2×C4×C12712yes—
204C2×C22⋊C4D4×C12711yescharacteristic
205C2×C42D4×C12711yescharacteristic
206C4⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C4×C8—22no—
213#95—22no—
214#95—22no—
215#95—21no—
216C24C22⋊C4×C12711yescharacteristic
217C24C22×C4×C12711yescharacteristic
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—41no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4C22⋊C4×C12711yescharacteristic
231C22×C4C22⋊C4×C12711yescharacteristic
232C22×C4C22⋊C4×C12711yescharacteristic
233C22×C4C2×D4×C12712yes—
234C22×C4C2×D4×C12711yescharacteristic
235C22×C4C2×D4×C12711yescharacteristic
236C22×C4#41×C12712yes—
237M16—44no—
238M16—44no—
239M16—82no—
240C2×C8—24no—
241C2×C8—42no—
242C4⋊C4—22no—
243C4⋊C4—42no—
244C4⋊C4—44no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4—44no—
251C22⋊C4—42no—
252C42—21no—
253C42—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23C2×C22⋊C4×C12711yescharacteristic
264C23C4×D4×C12712yes—
265D4—44no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—24no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—81no—
284C2×C4—84no—
285C2×C4#100×C12712yes—
286C2×C4#120×C12711yescharacteristic
287C2×C4#121×C12712yes—
288C2×C4#123×C12711yescharacteristic
289C8—44no—
290C8—82no—
291C22—21no—
292C22—21no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22#386×C12711yescharacteristic
304C22C2×#100×C12711yescharacteristic
305C22#413×C12711yescharacteristic
306C4—22no—
307C4—21no—
308C4—22no—
309C4—41no—
310C4—41no—
311C4—82no—
312C4—82no—
313C4#458×C12712yes—
314C2—21no—
315C2—21no—
316C2—42no—
317C2—81no—
318C2C127×#189011yescharacteristic
319C2C127×#190011yescharacteristic
320C2C127×#191511yescharacteristic
subgroupG/H
center, socleC22×C127#386
commutatorC24C22×C4×C127
Frattini, cosocleC23×C4C23×C127
prime radicalC22×D4×C127C2×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#1821C2semidirect
5. C127×#1305C2semidirect
153. #9508C127direct (N × H)
163. #1821C2×C127semidirect
165. #1305C2×C127semidirect
166. C127#9508direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#29752
C127×#24862
C127×#18212
C127×#14652
C127×#13052
#9508127