Finite Group Databasebeta

← C127×#14297 #44577970

Subgroups — C127×#14297 #44577970

Order 32512.

1566 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#381×C127C211yesmaximal, characteristic
2C2×#380×C127C211yesmaximal, characteristic
3C2×C4×C22⋊C4×C127C211yesmaximal, characteristic
4C127×#1441C214yesmaximal, complemented
5C127×#1439C214yesmaximal
6C127×#1436C214yesmaximal, complemented
7C24×C4×C127C2211yescharacteristic
8C22×C4⋊C4×C127C2214yes—
9C22×C4⋊C4×C127C2212yes—
10C22×C22⋊C4×C127C2212yes—
11C22×C22⋊C4×C127C2212yes—
12C22×C22⋊C4×C127C2212yes—
13C22×C42×C127C2212yes—
14#381×C127—24no—
15#380×C127C418yescomplemented
16C4×C22⋊C4×C127—24no—
17C2×#95×C127C2212yes—
18C2×#95×C127C2214yes—
19C2×#95×C127C2212yes—
20C2×#95×C127C2212yes—
21C2×#95×C127C2214yes—
22C2×#95×C127C2214yes—
23C2×#95×C127C2212yes—
24C25×C127C2311yescharacteristic
25C23×C4×C127—42no—
26C23×C4×C127—42no—
27C23×C4×C127C2×C414yes—
28C23×C4×C127D414yes—
29C23×C4×C127Q814yes—
30C23×C4×C127C2314yes—
31C23×C4×C127C2314yes—
32C23×C4×C127C2314yes—
33C23×C4×C127C2311yescharacteristic
34C23×C4×C127C2311yescharacteristic
35C2×C4⋊C4×C127—28no—
36C2×C4⋊C4×C127—42no—
37C2×C4⋊C4×C127—42no—
38C2×C4⋊C4×C127—44no—
39C2×C4⋊C4×C127—44no—
40C2×C4⋊C4×C127C2×C418yes—
41C2×C22⋊C4×C127—24no—
42C2×C22⋊C4×C127—28no—
43C2×C22⋊C4×C127—24no—
44C2×C22⋊C4×C127—42no—
45C2×C22⋊C4×C127—42no—
46C2×C22⋊C4×C127—42no—
47C2×C22⋊C4×C127—42no—
48C2×C22⋊C4×C127C2×C418yes—
49C2×C42×C127—24no—
50C2×C42×C127—42no—
51C2×C42×C127—42no—
52#95×C127—24no—
53#95×C127—24no—
54#95×C127—24no—
55#95×C127—28no—
56#95×C127—44no—
57#95×C127—44no—
58#95×C127C2×C418yes—
59C24×C127—24no—
60C24×C127—22no—
61C24×C127—22no—
62C24×C127—22no—
63C24×C127—22no—
64C24×C127C22×C412yes—
65C24×C127C2×D412yes—
66C24×C127C2×Q812yes—
67C24×C127C2411yescharacteristic
68C22×C4×C127—22no—
69C22×C4×C127—22no—
70C22×C4×C127—22no—
71C22×C4×C127—22no—
72C22×C4×C127—24no—
73C22×C4×C127—24no—
74C22×C4×C127—24no—
75C22×C4×C127—24no—
76C22×C4×C127—24no—
77C22×C4×C127—24no—
78C22×C4×C127—24no—
79C22×C4×C127—24no—
80C22×C4×C127—24no—
81C22×C4×C127—24no—
82C22×C4×C127—24no—
83C22×C4×C127—24no—
84C22×C4×C127—24no—
85C22×C4×C127—24no—
86C22×C4×C127—24no—
87C22×C4×C127—24no—
88C22×C4×C127—24no—
89C22×C4×C127—24no—
90C22×C4×C127—24no—
91C22×C4×C127—48no—
92C22×C4×C127—48no—
93C22×C4×C127—44no—
94C22×C4×C127—44no—
95C22×C4×C127C4⋊C4116yescomplemented
96C22×C4×C127C22×C418yes—
97C22×C4×C127C22×C412yes—
98C22×C4×C127C22×C412yes—
99C22×C4×C127C2×D412yes—
100C22×C4×C127C2×D412yes—
101C22×C4×C127C2×Q812yes—
102C22×C4×C127C2×Q812yes—
103C22×C4×C127#4118yes—
104C4⋊C4×C127—48no—
105C4⋊C4×C127—416no—
106C22⋊C4×C127—216no—
107C22⋊C4×C127—48no—
108C22⋊C4×C127—48no—
109C42×C127—48no—
110C23×C127—24no—
111C23×C127—22no—
112C23×C127—24no—
113C23×C127—22no—
114C23×C127—24no—
115C23×C127—24no—
116C23×C127—24no—
117C23×C127—24no—
118C23×C127—22no—
119C23×C127—22no—
120C23×C127—24no—
121C23×C127—44no—
122C23×C127—44no—
123C23×C127—44no—
124C23×C127—44no—
125C23×C127C2×C4⋊C414yes—
126C23×C127C23×C411yescharacteristic
127C23×C127C22×D411yescharacteristic
128C23×C127C22×Q811yescharacteristic
129C23×C127C2×#4114yes—
130C23×C127H(5,𝔽2)12yes—
131C23×C127H(5,𝔽2)12yes—
132C23×C127H(5,𝔽2)12yes—
133C23×C1272−1+412yes—
134C2×C4×C127—24no—
135C2×C4×C127—24no—
136C2×C4×C127—24no—
137C2×C4×C127—24no—
138C2×C4×C127—22no—
139C2×C4×C127—22no—
140C2×C4×C127—22no—
141C2×C4×C127—22no—
142C2×C4×C127—28no—
143C2×C4×C127—28no—
144C2×C4×C127—28no—
145C2×C4×C127—28no—
146C2×C4×C127—28no—
147C2×C4×C127—28no—
148C2×C4×C127—416no—
149C2×C4×C127—44no—
150C2×C4×C127—44no—
151C2×C4×C127—44no—
152C2×C4×C127—44no—
153C2×C4×C127—44no—
154C2×C4×C127—44no—
155C2×C4×C127—44no—
156C2×C4×C127—44no—
157C2×C4×C127—44no—
158C2×C4×C127—44no—
159C2×C4×C127—44no—
160C2×C4×C127—44no—
161C2×C4×C127—44no—
162C2×C4×C127—44no—
163C2×C4×C127—44no—
164C2×C4×C127C2×C4⋊C414yes—
165C2×C4×C127C2×C4⋊C414yes—
166C22×C127—24no—
167C22×C127—24no—
168C22×C127—24no—
169C22×C127—44no—
170C22×C127—44no—
171C22×C127—48no—
172C22×C127—44no—
173C22×C127—44no—
174C22×C127C22×C4⋊C411yescharacteristic
175C22×C127C4×#4112yes—
176C22×C127#51812yes—
177C22×C127#51811yescharacteristic
178C22×C127#51911yescharacteristic
179C22×C127#52512yes—
180C22×C127#53312yes—
181C22×C127#53412yes—
182C22×C127#53712yes—
183C22×C127#53814yes—
184C22×C127#53914yes—
185C22×C127#54014yes—
186C22×C127#54214yes—
187C22×C127#54311yescharacteristic
188C22×C127#54311yescharacteristic
189C22×C127#54412yes—
190C4×C127—24no—
191C4×C127—24no—
192C4×C127—48no—
193C4×C127—48no—
194C4×C127—84no—
195#14297C12711yesmaximal, characteristic, direct product factor, Sylow
196C2×C127—44no—
197C2×C127#229611yescharacteristic
198C2×C127#231511yescharacteristic
199C2×C127#231611yescharacteristic
200C2×C127#233212yes—
201C2×C127#233312yes—
202C2×C127#263014yes—
203C2×C127#266512yes—
204C2×C127#266712yes—
205C2×#381C2×C12711yescharacteristic
206C2×#380C2×C12711yescharacteristic
207C2×C4×C22⋊C4C2×C12711yescharacteristic
208#1441C2×C12714yescomplemented
209#1439C2×C12714yes—
210#1436C2×C12714yescomplemented
211C127#1429711yescharacteristic, direct product factor, Sylow
212C24×C4C22×C12711yescharacteristic
213C22×C4⋊C4C22×C12714yes—
214C22×C4⋊C4C22×C12712yes—
215C22×C22⋊C4C22×C12712yes—
216C22×C22⋊C4C22×C12712yes—
217C22×C22⋊C4C22×C12712yes—
218C22×C42C22×C12712yes—
219#381—24no—
220#380C4×C12718yescomplemented
221C4×C22⋊C4—24no—
222C2×#95C22×C12712yes—
223C2×#95C22×C12714yes—
224C2×#95C22×C12712yes—
225C2×#95C22×C12712yes—
226C2×#95C22×C12714yes—
227C2×#95C22×C12714yes—
228C2×#95C22×C12712yes—
229C25C23×C12711yescharacteristic
230C23×C4—42no—
231C23×C4—42no—
232C23×C4C2×C4×C12714yes—
233C23×C4D4×C12714yes—
234C23×C4Q8×C12714yes—
235C23×C4C23×C12714yes—
236C23×C4C23×C12714yes—
237C23×C4C23×C12714yes—
238C23×C4C23×C12711yescharacteristic
239C23×C4C23×C12711yescharacteristic
240C2×C4⋊C4—28no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4C2×C4×C12718yes—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4C2×C4×C12718yes—
254C2×C42—24no—
255C2×C42—42no—
256C2×C42—42no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—28no—
261#95—44no—
262#95—44no—
263#95C2×C4×C12718yes—
264C24—24no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24C22×C4×C12712yes—
270C24C2×D4×C12712yes—
271C24C2×Q8×C12712yes—
272C24C24×C12711yescharacteristic
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—48no—
297C22×C4—48no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4C4⋊C4×C127116yescomplemented
301C22×C4C22×C4×C12718yes—
302C22×C4C22×C4×C12712yes—
303C22×C4C22×C4×C12712yes—
304C22×C4C2×D4×C12712yes—
305C22×C4C2×D4×C12712yes—
306C22×C4C2×Q8×C12712yes—
307C22×C4C2×Q8×C12712yes—
308C22×C4#41×C12718yes—
309C4⋊C4—48no—
310C4⋊C4—416no—
311C22⋊C4—216no—
312C22⋊C4—48no—
313C22⋊C4—48no—
314C42—48no—
315C23—24no—
316C23—22no—
317C23—24no—
318C23—22no—
319C23—24no—
320C23—24no—
321C23—24no—
322C23—24no—
323C23—22no—
324C23—22no—
325C23—24no—
326C23—44no—
327C23—44no—
328C23—44no—
329C23—44no—
330C23C2×C4⋊C4×C12714yes—
331C23C23×C4×C12711yescharacteristic
332C23C22×D4×C12711yescharacteristic
333C23C22×Q8×C12711yescharacteristic
334C23C2×#41×C12714yes—
335C23H(5,𝔽2)×C12712yes—
336C23H(5,𝔽2)×C12712yes—
337C23H(5,𝔽2)×C12712yes—
338C232−1+4×C12712yes—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—28no—
348C2×C4—28no—
349C2×C4—28no—
350C2×C4—28no—
351C2×C4—28no—
352C2×C4—28no—
353C2×C4—416no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4C2×C4⋊C4×C12714yes—
370C2×C4C2×C4⋊C4×C12714yes—
371C22—24no—
372C22—24no—
373C22—24no—
374C22—44no—
375C22—44no—
376C22—48no—
377C22—44no—
378C22—44no—
379C22C22×C4⋊C4×C12711yescharacteristic
380C22C4×#41×C12712yes—
381C22#518×C12712yes—
382C22#518×C12711yescharacteristic
383C22#519×C12711yescharacteristic
384C22#525×C12712yes—
385C22#533×C12712yes—
386C22#534×C12712yes—
387C22#537×C12712yes—
388C22#538×C12714yes—
389C22#539×C12714yes—
390C22#540×C12714yes—
391C22#542×C12714yes—
392C22#543×C12711yescharacteristic
393C22#543×C12711yescharacteristic
394C22#544×C12712yes—
395C4—24no—
396C4—24no—
397C4—48no—
398C4—48no—
399C4—84no—
400C2—44no—
401C2C127×#229611yescharacteristic
402C2C127×#231511yescharacteristic
403C2C127×#231611yescharacteristic
404C2C127×#233212yes—
405C2C127×#233312yes—
406C2C127×#263014yes—
407C2C127×#266512yes—
408C2C127×#266712yes—
subgroupG/H
center, socleC24×C127C24
commutatorC23C23×C4×C127
Frattini, cosocleC24C24×C127
prime radicalC25×C127C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1441C2semidirect
6. C127×#1436C2semidirect
15. #380×C127C4semidirect
95. C22×C4×C127C4⋊C4semidirect
195. #14297C127direct (N × H)
208. #1441C2×C127semidirect
210. #1436C2×C127semidirect
211. C127#14297direct (N × H)
220. #380C4×C127semidirect
300. C22×C4C4⋊C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#381×C1272
C2×#380×C1272
C2×C4×C22⋊C4×C1272
C127×#14412
C127×#14392
C127×#14362
#14297127