Finite Group Databasebeta

← C127×#17137 #44580810

Subgroups — C127×#17137 #44580810

Order 32512.

1722 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#393×C127C214yesmaximal, complemented
2C127×#1446C212yesmaximal, complemented
3C127×#1442C212yesmaximal, complemented
4C127×#1439C214yesmaximal, complemented
5C127×#1438C212yesmaximal
6C127×#1438C211yesmaximal, characteristic
7C23×Q8×C127C2212yes—
8C23×D4×C127C2211yescharacteristic
9C22×C4⋊C4×C127C2212yes—
10C22×C22⋊C4×C127C2212yes—
11C22×C22⋊C4×C127C2214yes—
12C22×C22⋊C4×C127C2214yes—
13C22×C22⋊C4×C127C2212yes—
14#393×C127—216no—
15C2×#95×C127C2214yes—
16C2×#95×C127C2214yes—
17C2×#95×C127C2211yescharacteristic
18C2×#95×C127C2214yes—
19C2×#95×C127C2211yescharacteristic
20C2×#95×C127C2212yes—
21C2×#95×C127C2212yes—
22C25×C127C2312yes—
23C22×Q8×C127—28no—
24C22×Q8×C127—24no—
25C22×Q8×C127—42no—
26C22×Q8×C127—44no—
27C22×Q8×C127—42no—
28C22×D4×C127—44no—
29C22×D4×C127—41no—
30C22×D4×C127—41no—
31C22×D4×C127Q814yescomplemented
32C23×C4×C127C2314yes—
33C23×C4×C127C2312yes—
34C23×C4×C127C2314yes—
35C23×C4×C127C2311yescharacteristic
36C23×C4×C127C2312yes—
37C2×C4⋊C4×C127—42no—
38C2×C4⋊C4×C127—44no—
39C2×C22⋊C4×C127—28no—
40C2×C22⋊C4×C127—28no—
41C2×C22⋊C4×C127—28no—
42C2×C22⋊C4×C127—28no—
43C2×C22⋊C4×C127—44no—
44C2×C22⋊C4×C127—42no—
45C2×C22⋊C4×C127—44no—
46C2×C22⋊C4×C127—44no—
47C2×C22⋊C4×C127—44no—
48C2×C22⋊C4×C127—42no—
49#95×C127—28no—
50#95×C127—28no—
51#95×C127—41no—
52#95×C127—41no—
53#95×C127—44no—
54#95×C127—42no—
55#95×C127D418yescomplemented
56C24×C127—24no—
57C24×C127—24no—
58C24×C127—24no—
59C24×C127—24no—
60C24×C127—24no—
61C24×C127—22no—
62C24×C127—22no—
63C24×C127—24no—
64C24×C127C2×Q814yes—
65C24×C127C2411yescharacteristic
66C2×Q8×C127—48no—
67C2×Q8×C127—416no—
68C2×Q8×C127—416no—
69C2×Q8×C127—48no—
70C2×Q8×C127—48no—
71C2×D4×C127—44no—
72C2×D4×C127—48no—
73C2×D4×C127—48no—
74C2×D4×C127—48no—
75C22×C4×C127—24no—
76C22×C4×C127—24no—
77C22×C4×C127—24no—
78C22×C4×C127—24no—
79C22×C4×C127—22no—
80C22×C4×C127—24no—
81C22×C4×C127—21no—
82C22×C4×C127—21no—
83C22×C4×C127—22no—
84C22×C4×C127—22no—
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—22no—
92C22×C4×C127—24no—
93C22×C4×C127—24no—
94C22×C4×C127—24no—
95C22×C4×C127—24no—
96C22×C4×C127—24no—
97C22×C4×C127—24no—
98C22×C4×C127—24no—
99C22×C4×C127C2×D414yes—
100C22×C4×C127C2×D418yes—
101C22×C4×C127C2×Q812yes—
102C22×C4×C127#4114yes—
103C4⋊C4×C127—48no—
104C22⋊C4×C127—48no—
105C22⋊C4×C127—416no—
106C22⋊C4×C127—416no—
107C22⋊C4×C127—48no—
108C23×C127—24no—
109C23×C127—24no—
110C23×C127—24no—
111C23×C127—28no—
112C23×C127—28no—
113C23×C127—44no—
114C23×C127—44no—
115C23×C127—44no—
116C23×C127—44no—
117C23×C127—44no—
118C23×C127—44no—
119C23×C127—42no—
120C23×C127—44no—
121C23×C127—44no—
122C23×C127—42no—
123C23×C127—42no—
124C23×C127—44no—
125C23×C127—44no—
126C23×C127—44no—
127C23×C127—44no—
128C23×C127—42no—
129C23×C127C22×D412yes—
130C23×C127C22×Q811yescharacteristic
131C23×C127C2×#4112yes—
132C23×C127H(5,𝔽2)12yes—
133C23×C127H(5,𝔽2)14yes—
134C23×C127H(5,𝔽2)11yescharacteristic
135C23×C127H(5,𝔽2)11yescharacteristic
136C23×C1272−1+412yes—
137Q8×C127—432no—
138D4×C127—416no—
139C2×C4×C127—28no—
140C2×C4×C127—24no—
141C2×C4×C127—24no—
142C2×C4×C127—22no—
143C2×C4×C127—24no—
144C2×C4×C127—28no—
145C2×C4×C127—28no—
146C2×C4×C127—28no—
147C2×C4×C127—28no—
148C2×C4×C127—44no—
149C2×C4×C127—44no—
150C2×C4×C127—44no—
151C2×C4×C127—44no—
152C2×C4×C127—42no—
153C2×C4×C127—44no—
154C2×C4×C127—42no—
155C2×C4×C127—42no—
156C2×C4×C127—42no—
157C2×C4×C127—42no—
158C2×C4×C127—44no—
159C2×C4×C127—42no—
160C2×C4×C127—44no—
161C2×C4×C127—44no—
162C2×C4×C127—44no—
163C2×C4×C127—44no—
164C2×C4×C127—44no—
165C2×C4×C127—44no—
166C2×C4×C127—44no—
167C22×C127—44no—
168C22×C127—44no—
169C22×C127—48no—
170C22×C127—48no—
171C22×C127—44no—
172C22×C127—84no—
173C22×C127—84no—
174C22×C127—84no—
175C22×C127—84no—
176C22×C127#53514yes—
177C22×C127#53712yes—
178C22×C127#53812yes—
179C22×C127#54214yes—
180C22×C127#54311yescharacteristic
181C22×C127#54312yes—
182C22×C127D4211yescharacteristic
183C22×C127#54614yes—
184C22×C127D4×Q812yes—
185C22×C127#55412yes—
186C22×C127#55911yescharacteristic
187C22×C127H(3,𝔽4)11yescharacteristic
188C22×C127H(3,𝔽4)12yes—
189C22×C127H(3,𝔽4)12yes—
190C22×C127#56214yes—
191C22×C127#56311yescharacteristic
192C4×C127—42no—
193C4×C127—48no—
194C4×C127—44no—
195C4×C127—44no—
196C4×C127—84no—
197#17137C12711yesmaximal, characteristic, direct product factor, Sylow
198C2×C127—84no—
199C2×C127#267012yes—
200C2×C127#268311yescharacteristic
201C2×C127#269214yes—
202C2×C127#271212yes—
203C2×C127#280411yescharacteristic
204C2×C127#281912yes—
205C2×C127#283212yes—
206C2×C127#284011yescharacteristic
207C2×#393C2×C12714yescomplemented
208#1446C2×C12712yescomplemented
209#1442C2×C12712yescomplemented
210#1439C2×C12714yescomplemented
211#1438C2×C12712yes—
212#1438C2×C12711yescharacteristic
213C127#1713711yescharacteristic, direct product factor, Sylow
214C23×Q8C22×C12712yes—
215C23×D4C22×C12711yescharacteristic
216C22×C4⋊C4C22×C12712yes—
217C22×C22⋊C4C22×C12712yes—
218C22×C22⋊C4C22×C12714yes—
219C22×C22⋊C4C22×C12714yes—
220C22×C22⋊C4C22×C12712yes—
221#393—216no—
222C2×#95C22×C12714yes—
223C2×#95C22×C12714yes—
224C2×#95C22×C12711yescharacteristic
225C2×#95C22×C12714yes—
226C2×#95C22×C12711yescharacteristic
227C2×#95C22×C12712yes—
228C2×#95C22×C12712yes—
229C25C23×C12712yes—
230C22×Q8—28no—
231C22×Q8—24no—
232C22×Q8—42no—
233C22×Q8—44no—
234C22×Q8—42no—
235C22×D4—44no—
236C22×D4—41no—
237C22×D4—41no—
238C22×D4Q8×C12714yescomplemented
239C23×C4C23×C12714yes—
240C23×C4C23×C12712yes—
241C23×C4C23×C12714yes—
242C23×C4C23×C12711yescharacteristic
243C23×C4C23×C12712yes—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—42no—
256#95—28no—
257#95—28no—
258#95—41no—
259#95—41no—
260#95—44no—
261#95—42no—
262#95D4×C12718yescomplemented
263C24—24no—
264C24—24no—
265C24—24no—
266C24—24no—
267C24—24no—
268C24—22no—
269C24—22no—
270C24—24no—
271C24C2×Q8×C12714yes—
272C24C24×C12711yescharacteristic
273C2×Q8—48no—
274C2×Q8—416no—
275C2×Q8—416no—
276C2×Q8—48no—
277C2×Q8—48no—
278C2×D4—44no—
279C2×D4—48no—
280C2×D4—48no—
281C2×D4—48no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4C2×D4×C12714yes—
307C22×C4C2×D4×C12718yes—
308C22×C4C2×Q8×C12712yes—
309C22×C4#41×C12714yes—
310C4⋊C4—48no—
311C22⋊C4—48no—
312C22⋊C4—416no—
313C22⋊C4—416no—
314C22⋊C4—48no—
315C23—24no—
316C23—24no—
317C23—24no—
318C23—28no—
319C23—28no—
320C23—44no—
321C23—44no—
322C23—44no—
323C23—44no—
324C23—44no—
325C23—44no—
326C23—42no—
327C23—44no—
328C23—44no—
329C23—42no—
330C23—42no—
331C23—44no—
332C23—44no—
333C23—44no—
334C23—44no—
335C23—42no—
336C23C22×D4×C12712yes—
337C23C22×Q8×C12711yescharacteristic
338C23C2×#41×C12712yes—
339C23H(5,𝔽2)×C12712yes—
340C23H(5,𝔽2)×C12714yes—
341C23H(5,𝔽2)×C12711yescharacteristic
342C23H(5,𝔽2)×C12711yescharacteristic
343C232−1+4×C12712yes—
344Q8—432no—
345D4—416no—
346C2×C4—28no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—28no—
352C2×C4—28no—
353C2×C4—28no—
354C2×C4—28no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C22—44no—
375C22—44no—
376C22—48no—
377C22—48no—
378C22—44no—
379C22—84no—
380C22—84no—
381C22—84no—
382C22—84no—
383C22#535×C12714yes—
384C22#537×C12712yes—
385C22#538×C12712yes—
386C22#542×C12714yes—
387C22#543×C12711yescharacteristic
388C22#543×C12712yes—
389C22D42×C12711yescharacteristic
390C22#546×C12714yes—
391C22D4×Q8×C12712yes—
392C22#554×C12712yes—
393C22#559×C12711yescharacteristic
394C22H(3,𝔽4)×C12712yes—
395C22H(3,𝔽4)×C12712yes—
396C22H(3,𝔽4)×C12711yescharacteristic
397C22#562×C12714yes—
398C22#563×C12711yescharacteristic
399C4—42no—
400C4—48no—
401C4—44no—
402C4—44no—
403C4—84no—
404C2—84no—
405C2C127×#267012yes—
406C2C127×#268311yescharacteristic
407C2C127×#269214yes—
408C2C127×#271212yes—
409C2C127×#280411yescharacteristic
410C2C127×#281912yes—
411C2C127×#283212yes—
412C2C127×#284011yescharacteristic
subgroupG/H
center, socleC24×C127C24
commutator, Frattini, cosocleC24C24×C127
prime radicalC23×D4×C127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#393×C127C2semidirect
2. C127×#1446C2semidirect
3. C127×#1442C2semidirect
4. C127×#1439C2semidirect
31. C22×D4×C127Q8semidirect
55. #95×C127D4semidirect
197. #17137C127direct (N × H)
207. C2×#393C2×C127semidirect
208. #1446C2×C127semidirect
209. #1442C2×C127semidirect
210. #1439C2×C127semidirect
213. C127#17137direct (N × H)
238. C22×D4Q8×C127semidirect
262. #95D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#393×C1272
C127×#14462
C127×#14422
C127×#14392
C127×#14382
#17137127