Finite Group Databasebeta

← C127×#10968 #44574641

Subgroups — C127×#10968 #44574641

Order 32512.

1434 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8×C127C211yesmaximal, characteristic
2C22×#101×C127C212yesmaximal, complemented
3C2×#407×C127C212yesmaximal
4C2×#343×C127C212yesmaximal, complemented
5C23×Q8×C127C2211yescharacteristic
6C24×C4×C127C2211yescharacteristic
7C22×M16×C127C2214yes—
8C2×C22⋊Q8×C127—24no—
9C2×C22⋊Q8×C127—24no—
10C2×C22⋊Q8×C127—24no—
11C22×C4⋊C4×C127C412yes—
12C22×C4⋊C4×C127C2211yescharacteristic
13C22×C22⋊C4×C127C412yes—
14C2×#101×C127—24no—
15C2×#101×C127—28no—
16#407×C127—28no—
17C2×C22⋊C8×C127—24no—
18#343×C127—24no—
19C25×C127C2×C411yescharacteristic
20C22×Q8×C127—22no—
21C22×Q8×C127—22no—
22C22×Q8×C127—24no—
23C22×Q8×C127—24no—
24C22×Q8×C127D414yes—
25C23×C4×C127—22no—
26C23×C4×C127—22no—
27C23×C4×C127—24no—
28C23×C4×C127—24no—
29C23×C4×C127C2×C412yes—
30C23×C4×C127C2×C411yescharacteristic
31C23×C4×C127C2×C412yes—
32C23×C4×C127D414yes—
33C23×C4×C127C2311yescharacteristic
34C2×M16×C127—24no—
35C2×M16×C127—24no—
36C2×M16×C127—24no—
37C2×M16×C127—24no—
38C2×M16×C127—24no—
39C2×M16×C127D418yes—
40C22×C8×C127—24no—
41C22⋊Q8×C127—232no—
42C2×C4⋊C4×C127—24no—
43C2×C4⋊C4×C127—24no—
44C2×C4⋊C4×C127—24no—
45C2×C4⋊C4×C127—22no—
46C2×C4⋊C4×C127—22no—
47C2×C4⋊C4×C127D414yes—
48C2×C22⋊C4×C127—24no—
49C2×C22⋊C4×C127—24no—
50C2×C22⋊C4×C127—24no—
51#101×C127—216no—
52C22⋊C8×C127—216no—
53C24×C127—24no—
54C24×C127—21no—
55C24×C127—21no—
56C24×C127—24no—
57C24×C127—42no—
58C24×C127C22⋊C412yes—
59C24×C127C22×C411yescharacteristic
60C2×Q8×C127—24no—
61C2×Q8×C127—28no—
62C2×Q8×C127—28no—
63C2×Q8×C127—24no—
64C2×Q8×C127—42no—
65C2×Q8×C127—44no—
66C2×Q8×C127—44no—
67C2×Q8×C127—42no—
68C2×Q8×C127—42no—
69C2×Q8×C127—42no—
70C22×C4×C127—22no—
71C22×C4×C127—22no—
72C22×C4×C127—21no—
73C22×C4×C127—21no—
74C22×C4×C127—22no—
75C22×C4×C127—22no—
76C22×C4×C127—24no—
77C22×C4×C127—28no—
78C22×C4×C127—24no—
79C22×C4×C127—28no—
80C22×C4×C127—28no—
81C22×C4×C127—24no—
82C22×C4×C127—42no—
83C22×C4×C127—44no—
84C22×C4×C127—44no—
85C22×C4×C127—42no—
86C22×C4×C127—42no—
87C22×C4×C127—42no—
88C22×C4×C127—44no—
89C22×C4×C127C22⋊C414yes—
90C22×C4×C127C22⋊C412yes—
91C22×C4×C127C22⋊C414yes—
92C22×C4×C127C2×D412yes—
93C22×C4×C127C2×D412yes—
94C22×C4×C127C2×D414yes—
95C22×C4×C127C2×D412yes—
96C22×C4×C127#4114yes—
97M16×C127—28no—
98M16×C127—28no—
99M16×C127—44no—
100M16×C127—44no—
101C2×C8×C127—28no—
102C2×C8×C127—44no—
103C2×C8×C127—44no—
104C4⋊C4×C127—216no—
105C4⋊C4×C127—28no—
106C22⋊C4×C127—216no—
107C23×C127—22no—
108C23×C127—22no—
109C23×C127—24no—
110C23×C127—24no—
111C23×C127—22no—
112C23×C127—24no—
113C23×C127—22no—
114C23×C127—24no—
115C23×C127—416no—
116C23×C127—42no—
117C23×C127—42no—
118C23×C127—42no—
119C23×C127—42no—
120C23×C127#10114yescomplemented
121C23×C127C2×C22⋊C411yescharacteristic
122C23×C127C2×C22⋊C411yescharacteristic
123C23×C127C2×C22⋊C411yescharacteristic
124C23×C127C4×D414yes—
125Q8×C127—48no—
126Q8×C127—48no—
127C2×C4×C127—24no—
128C2×C4×C127—24no—
129C2×C4×C127—22no—
130C2×C4×C127—24no—
131C2×C4×C127—28no—
132C2×C4×C127—44no—
133C2×C4×C127—42no—
134C2×C4×C127—44no—
135C2×C4×C127—42no—
136C2×C4×C127—44no—
137C2×C4×C127—44no—
138C2×C4×C127—48no—
139C2×C4×C127—48no—
140C2×C4×C127—44no—
141C2×C4×C127#12012yes—
142C2×C4×C127#12012yes—
143C2×C4×C127#12014yes—
144C2×C4×C127#12114yes—
145C2×C4×C127#12114yes—
146C2×C4×C127#12114yes—
147C2×C4×C127#12312yes—
148C2×C4×C127#12312yes—
149C2×C4×C127#12412yes—
150C2×C4×C127#12712yes—
151C8×C127—48no—
152C22×C127—28no—
153C22×C127—21no—
154C22×C127—21no—
155C22×C127—21no—
156C22×C127—21no—
157C22×C127—28no—
158C22×C127—24no—
159C22×C127—48no—
160C22×C127—48no—
161C22×C127—44no—
162C22×C127—42no—
163C22×C127—42no—
164C22×C127#37911yescharacteristic
165C22×C127#38612yes—
166C22×C127#38612yes—
167C22×C127#39012yes—
168C22×C127C2×#10112yes—
169C22×C127#41312yes—
170C4×C127—42no—
171C4×C127—44no—
172C4×C127—44no—
173C4×C127#39214yes—
174C4×C127#39414yes—
175#10968C12711yesmaximal, characteristic, direct product factor, Sylow
176C2×C127—22no—
177C2×C127—22no—
178C2×C127—44no—
179C2×C127#143811yescharacteristic
180C2×C127#178411yescharacteristic
181C2×C127#178511yescharacteristic
182C2×C127#190114yes—
183C22×C22⋊Q8C2×C12711yescharacteristic
184C22×#101C2×C12712yescomplemented
185C2×#407C2×C12712yes—
186C2×#343C2×C12712yescomplemented
187C127#1096811yescharacteristic, direct product factor, Sylow
188C23×Q8C22×C12711yescharacteristic
189C24×C4C22×C12711yescharacteristic
190C22×M16C22×C12714yes—
191C2×C22⋊Q8—24no—
192C2×C22⋊Q8—24no—
193C2×C22⋊Q8—24no—
194C22×C4⋊C4C4×C12712yes—
195C22×C4⋊C4C22×C12711yescharacteristic
196C22×C22⋊C4C4×C12712yes—
197C2×#101—24no—
198C2×#101—28no—
199#407—28no—
200C2×C22⋊C8—24no—
201#343—24no—
202C25C2×C4×C12711yescharacteristic
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8D4×C12714yes—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4C2×C4×C12712yes—
213C23×C4C2×C4×C12711yescharacteristic
214C23×C4C2×C4×C12712yes—
215C23×C4D4×C12714yes—
216C23×C4C23×C12711yescharacteristic
217C2×M16—24no—
218C2×M16—24no—
219C2×M16—24no—
220C2×M16—24no—
221C2×M16—24no—
222C2×M16D4×C12718yes—
223C22×C8—24no—
224C22⋊Q8—232no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4D4×C12714yes—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234#101—216no—
235C22⋊C8—216no—
236C24—24no—
237C24—21no—
238C24—21no—
239C24—24no—
240C24—42no—
241C24C22⋊C4×C12712yes—
242C24C22×C4×C12711yescharacteristic
243C2×Q8—24no—
244C2×Q8—28no—
245C2×Q8—28no—
246C2×Q8—24no—
247C2×Q8—42no—
248C2×Q8—44no—
249C2×Q8—44no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—24no—
260C22×C4—28no—
261C22×C4—24no—
262C22×C4—28no—
263C22×C4—28no—
264C22×C4—24no—
265C22×C4—42no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4C22⋊C4×C12714yes—
273C22×C4C22⋊C4×C12712yes—
274C22×C4C22⋊C4×C12714yes—
275C22×C4C2×D4×C12712yes—
276C22×C4C2×D4×C12712yes—
277C22×C4C2×D4×C12714yes—
278C22×C4C2×D4×C12712yes—
279C22×C4#41×C12714yes—
280M16—28no—
281M16—28no—
282M16—44no—
283M16—44no—
284C2×C8—28no—
285C2×C8—44no—
286C2×C8—44no—
287C4⋊C4—216no—
288C4⋊C4—28no—
289C22⋊C4—216no—
290C23—22no—
291C23—22no—
292C23—24no—
293C23—24no—
294C23—22no—
295C23—24no—
296C23—22no—
297C23—24no—
298C23—416no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23#101×C12714yescomplemented
304C23C2×C22⋊C4×C12711yescharacteristic
305C23C2×C22⋊C4×C12711yescharacteristic
306C23C2×C22⋊C4×C12711yescharacteristic
307C23C4×D4×C12714yes—
308Q8—48no—
309Q8—48no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—28no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—48no—
322C2×C4—48no—
323C2×C4—44no—
324C2×C4#120×C12712yes—
325C2×C4#120×C12712yes—
326C2×C4#120×C12714yes—
327C2×C4#121×C12714yes—
328C2×C4#121×C12714yes—
329C2×C4#121×C12714yes—
330C2×C4#123×C12712yes—
331C2×C4#123×C12712yes—
332C2×C4#124×C12712yes—
333C2×C4#127×C12712yes—
334C8—48no—
335C22—28no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—28no—
341C22—24no—
342C22—48no—
343C22—48no—
344C22—44no—
345C22—42no—
346C22—42no—
347C22#379×C12711yescharacteristic
348C22#386×C12712yes—
349C22#386×C12712yes—
350C22#390×C12712yes—
351C22C2×#101×C12712yes—
352C22#413×C12712yes—
353C4—42no—
354C4—44no—
355C4—44no—
356C4#392×C12714yes—
357C4#394×C12714yes—
358C2—22no—
359C2—22no—
360C2—44no—
361C2C127×#143811yescharacteristic
362C2C127×#178411yescharacteristic
363C2C127×#178511yescharacteristic
364C2C127×#190114yes—
subgroupG/H
center, socleC23×C127C2×C22⋊C4
commutatorC24C22×C4×C127
Frattini, cosocleC23×C4C23×C127
prime radicalC25×C127C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C24×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×#101×C127C2semidirect
4. C2×#343×C127C2semidirect
120. C23×C127#101semidirect
175. #10968C127direct (N × H)
184. C22×#101C2×C127semidirect
186. C2×#343C2×C127semidirect
187. C127#10968direct (N × H)
303. C23#101×C127semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q8×C1272
C22×#101×C1272
C2×#407×C1272
C2×#343×C1272
#10968127