Finite Group Databasebeta

← C127×#16823 #44580496

Subgroups — C127×#16823 #44580496

Order 32512.

1562 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×C127C212yesmaximal
2C127×#1446C214yesmaximal, complemented
3C127×#1442C214yesmaximal, complemented
4C127×#1439C212yesmaximal
5C127×C22⋊#95C212yesmaximal
6C127×C22⋊#95C211yesmaximal, characteristic
7C23×Q8×C127C2212yes—
8C24×C4×C127C2211yescharacteristic
9C22×C4⋊C4×C127C2212yes—
10C22×C22⋊C4×C127C2214yes—
11C22×C22⋊C4×C127C2211yescharacteristic
12C22×C22⋊C4×C127C2211yescharacteristic
13#393×C127—28no—
14C2×#95×C127C2212yes—
15C2×#95×C127C2214yes—
16C2×#95×C127C2214yes—
17C2×#95×C127C2212yes—
18C2×#95×C127C2214yes—
19C2×#95×C127C2214yes—
20C2×#95×C127C2212yes—
21C2×#95×C127C2212yes—
22C25×C127C2311yescharacteristic
23C22×Q8×C127—24no—
24C22×Q8×C127—28no—
25C22×Q8×C127—44no—
26C22×Q8×C127—42no—
27C22×Q8×C127—42no—
28C23×C4×C127—41no—
29C23×C4×C127—41no—
30C23×C4×C127—44no—
31C23×C4×C127Q814yescomplemented
32C23×C4×C127C2314yes—
33C23×C4×C127C2312yes—
34C23×C4×C127C2312yes—
35C23×C4×C127C2312yes—
36C23×C4×C127C2314yes—
37C2×C4⋊C4×C127—44no—
38C2×C4⋊C4×C127—42no—
39C2×C22⋊C4×C127—28no—
40C2×C22⋊C4×C127—28no—
41C2×C22⋊C4×C127—41no—
42C2×C22⋊C4×C127—41no—
43C2×C22⋊C4×C127—41no—
44C2×C22⋊C4×C127—41no—
45C2×C22⋊C4×C127—44no—
46C2×C22⋊C4×C127D414yes—
47C2×C22⋊C4×C127Q814yescomplemented
48#95×C127—28no—
49#95×C127—24no—
50#95×C127—28no—
51#95×C127—42no—
52#95×C127—44no—
53#95×C127—42no—
54#95×C127—44no—
55#95×C127—42no—
56C24×C127—24no—
57C24×C127—21no—
58C24×C127—21no—
59C24×C127—21no—
60C24×C127—24no—
61C24×C127—21no—
62C24×C127C2×D412yes—
63C24×C127C2×Q812yes—
64C24×C127C2×Q812yes—
65C24×C127C2411yescharacteristic
66C2×Q8×C127—416no—
67C2×Q8×C127—48no—
68C2×Q8×C127—48no—
69C2×Q8×C127—416no—
70C2×Q8×C127—48no—
71C22×C4×C127—24no—
72C22×C4×C127—24no—
73C22×C4×C127—24no—
74C22×C4×C127—24no—
75C22×C4×C127—22no—
76C22×C4×C127—24no—
77C22×C4×C127—22no—
78C22×C4×C127—22no—
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—22no—
86C22×C4×C127—22no—
87C22×C4×C127—22no—
88C22×C4×C127—24no—
89C22×C4×C127—24no—
90C22×C4×C127—24no—
91C22×C4×C127—24no—
92C22×C4×C127—24no—
93C22×C4×C127—24no—
94C22×C4×C127—24no—
95C22×C4×C127—24no—
96C22×C4×C127—44no—
97C22×C4×C127—48no—
98C22×C4×C127—48no—
99C22×C4×C127—48no—
100C22×C4×C127C2×D414yes—
101C22×C4×C127C2×Q814yes—
102C22×C4×C127C2×Q814yes—
103C22×C4×C127#4118yes—
104C4⋊C4×C127—48no—
105C22⋊C4×C127—44no—
106C22⋊C4×C127—44no—
107C22⋊C4×C127—416no—
108C23×C127—24no—
109C23×C127—22no—
110C23×C127—22no—
111C23×C127—22no—
112C23×C127—22no—
113C23×C127—24no—
114C23×C127—22no—
115C23×C127—22no—
116C23×C127—28no—
117C23×C127—28no—
118C23×C127—44no—
119C23×C127—44no—
120C23×C127—42no—
121C23×C127—42no—
122C23×C127—42no—
123C23×C127—42no—
124C23×C127C4⋊Q814yes—
125C23×C127C22×D411yescharacteristic
126C23×C127C22×Q811yescharacteristic
127C23×C127C22×Q811yescharacteristic
128C23×C127C2×#4114yes—
129C23×C127H(5,𝔽2)11yescharacteristic
130C23×C127H(5,𝔽2)11yescharacteristic
131C23×C127H(5,𝔽2)11yescharacteristic
132C23×C127H(5,𝔽2)14yes—
133C23×C1272−1+411yescharacteristic
134Q8×C127—432no—
135C2×C4×C127—24no—
136C2×C4×C127—28no—
137C2×C4×C127—24no—
138C2×C4×C127—24no—
139C2×C4×C127—24no—
140C2×C4×C127—24no—
141C2×C4×C127—28no—
142C2×C4×C127—28no—
143C2×C4×C127—28no—
144C2×C4×C127—28no—
145C2×C4×C127—416no—
146C2×C4×C127—44no—
147C2×C4×C127—44no—
148C2×C4×C127—44no—
149C2×C4×C127—44no—
150C2×C4×C127—44no—
151C2×C4×C127—42no—
152C2×C4×C127—42no—
153C2×C4×C127—42no—
154C2×C4×C127—44no—
155C2×C4×C127—44no—
156C2×C4×C127—42no—
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×C127—44no—
165C22×C127—24no—
166C22×C127—28no—
167C22×C127—42no—
168C22×C127—42no—
169C22×C127—44no—
170C22×C127—44no—
171C22×C127—44no—
172C22×C127—44no—
173C22×C127—44no—
174C22×C127C2×C4⋊Q811yescharacteristic
175C22×C127#53312yes—
176C22×C127#53411yescharacteristic
177C22×C127#53711yescharacteristic
178C22×C127#53814yes—
179C22×C127#54214yes—
180C22×C127#54311yescharacteristic
181C22×C127#54312yes—
182C22×C127#54311yescharacteristic
183C22×C127#54411yescharacteristic
184C22×C127#54411yescharacteristic
185C22×C127#54614yes—
186C22×C127#55112yes—
187C22×C127#55414yes—
188C22×C127#55912yes—
189C22×C127H(3,𝔽4)12yes—
190C22×C127H(3,𝔽4)12yes—
191C4×C127—44no—
192C4×C127—44no—
193C4×C127—48no—
194C4×C127—44no—
195C4×C127—84no—
196#16823C12711yesmaximal, characteristic, direct product factor, Sylow
197C2×C127—44no—
198C2×C127#254912yes—
199C2×C127#260711yescharacteristic
200C2×C127#265911yescharacteristic
201C2×C127#266512yes—
202C2×C127#268412yes—
203C2×C127#271212yes—
204C2×C127#276312yes—
205C2×C127#280412yes—
206C2×C127#284011yescharacteristic
207C2×#393C2×C12712yes—
208#1446C2×C12714yescomplemented
209#1442C2×C12714yescomplemented
210#1439C2×C12712yes—
211C22⋊#95C2×C12712yes—
212C22⋊#95C2×C12711yescharacteristic
213C127#1682311yescharacteristic, direct product factor, Sylow
214C23×Q8C22×C12712yes—
215C24×C4C22×C12711yescharacteristic
216C22×C4⋊C4C22×C12712yes—
217C22×C22⋊C4C22×C12714yes—
218C22×C22⋊C4C22×C12711yescharacteristic
219C22×C22⋊C4C22×C12711yescharacteristic
220#393—28no—
221C2×#95C22×C12712yes—
222C2×#95C22×C12714yes—
223C2×#95C22×C12714yes—
224C2×#95C22×C12712yes—
225C2×#95C22×C12714yes—
226C2×#95C22×C12714yes—
227C2×#95C22×C12712yes—
228C2×#95C22×C12712yes—
229C25C23×C12711yescharacteristic
230C22×Q8—24no—
231C22×Q8—28no—
232C22×Q8—44no—
233C22×Q8—42no—
234C22×Q8—42no—
235C23×C4—41no—
236C23×C4—41no—
237C23×C4—44no—
238C23×C4Q8×C12714yescomplemented
239C23×C4C23×C12714yes—
240C23×C4C23×C12712yes—
241C23×C4C23×C12712yes—
242C23×C4C23×C12712yes—
243C23×C4C23×C12714yes—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4D4×C12714yes—
254C2×C22⋊C4Q8×C12714yescomplemented
255#95—28no—
256#95—24no—
257#95—28no—
258#95—42no—
259#95—44no—
260#95—42no—
261#95—44no—
262#95—42no—
263C24—24no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—24no—
268C24—21no—
269C24C2×D4×C12712yes—
270C24C2×Q8×C12712yes—
271C24C2×Q8×C12712yes—
272C24C24×C12711yescharacteristic
273C2×Q8—416no—
274C2×Q8—48no—
275C2×Q8—48no—
276C2×Q8—416no—
277C2×Q8—48no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—44no—
304C22×C4—48no—
305C22×C4—48no—
306C22×C4—48no—
307C22×C4C2×D4×C12714yes—
308C22×C4C2×Q8×C12714yes—
309C22×C4C2×Q8×C12714yes—
310C22×C4#41×C12718yes—
311C4⋊C4—48no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—416no—
315C23—24no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—24no—
321C23—22no—
322C23—22no—
323C23—28no—
324C23—28no—
325C23—44no—
326C23—44no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23C4⋊Q8×C12714yes—
332C23C22×D4×C12711yescharacteristic
333C23C22×Q8×C12711yescharacteristic
334C23C22×Q8×C12711yescharacteristic
335C23C2×#41×C12714yes—
336C23H(5,𝔽2)×C12711yescharacteristic
337C23H(5,𝔽2)×C12711yescharacteristic
338C23H(5,𝔽2)×C12711yescharacteristic
339C23H(5,𝔽2)×C12714yes—
340C232−1+4×C12711yescharacteristic
341Q8—432no—
342C2×C4—24no—
343C2×C4—28no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—28no—
349C2×C4—28no—
350C2×C4—28no—
351C2×C4—28no—
352C2×C4—416no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C22—24no—
373C22—28no—
374C22—42no—
375C22—42no—
376C22—44no—
377C22—44no—
378C22—44no—
379C22—44no—
380C22—44no—
381C22C2×C4⋊Q8×C12711yescharacteristic
382C22#533×C12712yes—
383C22#534×C12711yescharacteristic
384C22#537×C12711yescharacteristic
385C22#538×C12714yes—
386C22#542×C12714yes—
387C22#543×C12711yescharacteristic
388C22#543×C12712yes—
389C22#543×C12711yescharacteristic
390C22#544×C12711yescharacteristic
391C22#544×C12711yescharacteristic
392C22#546×C12714yes—
393C22#551×C12712yes—
394C22#554×C12714yes—
395C22#559×C12712yes—
396C22H(3,𝔽4)×C12712yes—
397C22H(3,𝔽4)×C12712yes—
398C4—44no—
399C4—44no—
400C4—48no—
401C4—44no—
402C4—84no—
403C2—44no—
404C2C127×#254912yes—
405C2C127×#260711yescharacteristic
406C2C127×#265911yescharacteristic
407C2C127×#266512yes—
408C2C127×#268412yes—
409C2C127×#271212yes—
410C2C127×#276312yes—
411C2C127×#280412yes—
412C2C127×#284011yescharacteristic
subgroupG/H
center, socleC24×C127C24
commutator, Frattini, cosocleC24C24×C127
prime radicalC25×C127C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1446C2semidirect
3. C127×#1442C2semidirect
31. C23×C4×C127Q8semidirect
47. C2×C22⋊C4×C127Q8semidirect
196. #16823C127direct (N × H)
208. #1446C2×C127semidirect
209. #1442C2×C127semidirect
213. C127#16823direct (N × H)
238. C23×C4Q8×C127semidirect
254. C2×C22⋊C4Q8×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×C22⋊#952
#16823127