Finite Group Databasebeta

← C127×#17074 #44580747

Subgroups — C127×#17074 #44580747

Order 32512.

1186 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#400×C127C212yesmaximal
2C2×#389×C127C211yesmaximal, characteristic
3C2×#389×C127C211yesmaximal, characteristic
4C127×#1445C212yesmaximal
5C127×#1442C212yesmaximal
6C127×#1441C212yesmaximal
7C127×#1441C214yesmaximal
8C127×C4⋊#95C211yesmaximal, characteristic
9C22×C4⋊C4×C127C2212yes—
10C22×C4⋊C4×C127C2212yes—
11C22×C4⋊C4×C127C2211yescharacteristic
12C22×C4⋊C4×C127C2211yescharacteristic
13C22×C4⋊C4×C127C2212yes—
14C22×C4⋊C4×C127C2211yescharacteristic
15C22×C4⋊C4×C127C2211yescharacteristic
16C22×C42×C127C2211yescharacteristic
17#400×C127—28no—
18#389×C127—24no—
19#389×C127—24no—
20C2×#95×C127C2212yes—
21C2×#95×C127C2214yes—
22C2×#95×C127C2214yes—
23C2×#95×C127C2212yes—
24C2×#95×C127C2212yes—
25C2×#95×C127C2212yes—
26C2×#95×C127C2214yes—
27C2×#95×C127C2214yes—
28C23×C4×C127C2312yes—
29C23×C4×C127C2312yes—
30C23×C4×C127C2314yes—
31C23×C4×C127C2312yes—
32C23×C4×C127C2312yes—
33C23×C4×C127C2311yescharacteristic
34C23×C4×C127C2311yescharacteristic
35C23×C4×C127C2311yescharacteristic
36C2×C4⋊C4×C127—22no—
37C2×C4⋊C4×C127—22no—
38C2×C4⋊C4×C127—22no—
39C2×C4⋊C4×C127—22no—
40C2×C4⋊C4×C127—22no—
41C2×C4⋊C4×C127—22no—
42C2×C4⋊C4×C127—24no—
43C2×C4⋊C4×C127—24no—
44C2×C4⋊C4×C127—24no—
45C2×C4⋊C4×C127—24no—
46C2×C4⋊C4×C127—24no—
47C2×C4⋊C4×C127—28no—
48C2×C4⋊C4×C127—42no—
49C2×C4⋊C4×C127—42no—
50C2×C4⋊C4×C127—42no—
51C2×C4⋊C4×C127—42no—
52C2×C4⋊C4×C127D414yes—
53C2×C4⋊C4×C127Q814yes—
54C2×C42×C127—42no—
55C2×C42×C127D414yes—
56#95×C127—24no—
57#95×C127—24no—
58#95×C127—28no—
59#95×C127—24no—
60#95×C127—28no—
61#95×C127—24no—
62#95×C127—44no—
63#95×C127—44no—
64C24×C127C2411yescharacteristic
65C22×C4×C127—22no—
66C22×C4×C127—22no—
67C22×C4×C127—22no—
68C22×C4×C127—21no—
69C22×C4×C127—22no—
70C22×C4×C127—22no—
71C22×C4×C127—21no—
72C22×C4×C127—22no—
73C22×C4×C127—22no—
74C22×C4×C127—22no—
75C22×C4×C127—22no—
76C22×C4×C127—22no—
77C22×C4×C127—24no—
78C22×C4×C127—22no—
79C22×C4×C127—22no—
80C22×C4×C127—24no—
81C22×C4×C127—22no—
82C22×C4×C127—24no—
83C22×C4×C127—22no—
84C22×C4×C127—22no—
85C22×C4×C127—22no—
86C22×C4×C127—24no—
87C22×C4×C127—22no—
88C22×C4×C127—24no—
89C22×C4×C127—24no—
90C22×C4×C127—22no—
91C22×C4×C127—24no—
92C22×C4×C127—24no—
93C22×C4×C127—24no—
94C22×C4×C127—24no—
95C22×C4×C127—24no—
96C22×C4×C127—22no—
97C22×C4×C127—24no—
98C22×C4×C127—24no—
99C22×C4×C127C2×D414yes—
100C22×C4×C127C2×D412yes—
101C22×C4×C127C2×D412yes—
102C22×C4×C127C2×D412yes—
103C22×C4×C127C2×D412yes—
104C22×C4×C127C2×Q814yes—
105C22×C4×C127C2×Q812yes—
106C22×C4×C127#4112yes—
107C22×C4×C127#4112yes—
108C22×C4×C127#4114yes—
109C4⋊C4×C127—28no—
110C4⋊C4×C127—28no—
111C4⋊C4×C127—44no—
112C4⋊C4×C127—44no—
113C4⋊C4×C127—48no—
114C4⋊C4×C127—48no—
115C4⋊C4×C127—48no—
116C42×C127—44no—
117C23×C127C22×D411yescharacteristic
118C23×C127C22×D411yescharacteristic
119C23×C127C22×Q811yescharacteristic
120C23×C127C2×#4111yescharacteristic
121C23×C127C2×#4111yescharacteristic
122C23×C127C2×#4112yes—
123C23×C127H(5,𝔽2)12yes—
124C23×C127H(5,𝔽2)11yescharacteristic
125C23×C1272−1+412yes—
126C23×C1272−1+412yes—
127C23×C1272−1+411yescharacteristic
128C2×C4×C127—22no—
129C2×C4×C127—22no—
130C2×C4×C127—22no—
131C2×C4×C127—22no—
132C2×C4×C127—22no—
133C2×C4×C127—24no—
134C2×C4×C127—22no—
135C2×C4×C127—24no—
136C2×C4×C127—24no—
137C2×C4×C127—24no—
138C2×C4×C127—24no—
139C2×C4×C127—24no—
140C2×C4×C127—24no—
141C2×C4×C127—24no—
142C2×C4×C127—28no—
143C2×C4×C127—24no—
144C2×C4×C127—24no—
145C2×C4×C127—42no—
146C2×C4×C127—42no—
147C2×C4×C127—42no—
148C2×C4×C127—42no—
149C2×C4×C127—42no—
150C2×C4×C127—42no—
151C2×C4×C127—44no—
152C2×C4×C127—42no—
153C2×C4×C127—42no—
154C2×C4×C127—42no—
155C2×C4×C127—42no—
156C2×C4×C127—42no—
157C2×C4×C127—44no—
158C2×C4×C127—42no—
159C2×C4×C127—44no—
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—
167C2×C4×C127—44no—
168C2×C4×C127—44no—
169C2×C4×C127#12114yes—
170C2×C4×C127C22⋊Q814yes—
171C2×C4×C127#12414yes—
172C22×C127C2×#12111yescharacteristic
173C22×C127C2×C22⋊Q811yescharacteristic
174C22×C127C2×#12411yescharacteristic
175C22×C127#53411yescharacteristic
176C22×C127#53611yescharacteristic
177C22×C127#53611yescharacteristic
178C22×C127#53711yescharacteristic
179C22×C127#53712yes—
180C22×C127#53812yes—
181C22×C127#53912yes—
182C22×C127#53912yes—
183C22×C127#53912yes—
184C22×C127#54412yes—
185C22×C127#54712yes—
186C22×C127D4×Q811yescharacteristic
187C22×C127#55011yescharacteristic
188C22×C127#55411yescharacteristic
189C22×C127#55612yes—
190C22×C127#55611yescharacteristic
191C22×C127#55612yes—
192C22×C127#55712yes—
193C22×C127#56312yes—
194C22×C127#56412yes—
195C4×C127—24no—
196C4×C127—24no—
197C4×C127—24no—
198C4×C127—44no—
199C4×C127—44no—
200C4×C127—82no—
201C4×C127—82no—
202C4×C127—84no—
203#17074C12711yesmaximal, characteristic, direct product factor, Sylow
204C2×C127#253811yescharacteristic
205C2×C127#254711yescharacteristic
206C2×C127#256811yescharacteristic
207C2×C127#261811yescharacteristic
208C2×C127#262711yescharacteristic
209C2×C127#263911yescharacteristic
210C2×C127#272612yes—
211C2×C127#272712yes—
212C2×C127#280512yes—
213C2×C127#281012yes—
214C2×C127#282211yescharacteristic
215C2×#400C2×C12712yes—
216C2×#389C2×C12711yescharacteristic
217C2×#389C2×C12711yescharacteristic
218#1445C2×C12712yes—
219#1442C2×C12712yes—
220#1441C2×C12712yes—
221#1441C2×C12714yes—
222C4⋊#95C2×C12711yescharacteristic
223C127#1707411yescharacteristic, direct product factor, Sylow
224C22×C4⋊C4C22×C12712yes—
225C22×C4⋊C4C22×C12712yes—
226C22×C4⋊C4C22×C12711yescharacteristic
227C22×C4⋊C4C22×C12711yescharacteristic
228C22×C4⋊C4C22×C12712yes—
229C22×C4⋊C4C22×C12711yescharacteristic
230C22×C4⋊C4C22×C12711yescharacteristic
231C22×C42C22×C12711yescharacteristic
232#400—28no—
233#389—24no—
234#389—24no—
235C2×#95C22×C12712yes—
236C2×#95C22×C12714yes—
237C2×#95C22×C12714yes—
238C2×#95C22×C12712yes—
239C2×#95C22×C12712yes—
240C2×#95C22×C12712yes—
241C2×#95C22×C12714yes—
242C2×#95C22×C12714yes—
243C23×C4C23×C12712yes—
244C23×C4C23×C12712yes—
245C23×C4C23×C12714yes—
246C23×C4C23×C12712yes—
247C23×C4C23×C12712yes—
248C23×C4C23×C12711yescharacteristic
249C23×C4C23×C12711yescharacteristic
250C23×C4C23×C12711yescharacteristic
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4D4×C12714yes—
268C2×C4⋊C4Q8×C12714yes—
269C2×C42—42no—
270C2×C42D4×C12714yes—
271#95—24no—
272#95—24no—
273#95—28no—
274#95—24no—
275#95—28no—
276#95—24no—
277#95—44no—
278#95—44no—
279C24C24×C12711yescharacteristic
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4C2×D4×C12714yes—
315C22×C4C2×D4×C12712yes—
316C22×C4C2×D4×C12712yes—
317C22×C4C2×D4×C12712yes—
318C22×C4C2×D4×C12712yes—
319C22×C4C2×Q8×C12714yes—
320C22×C4C2×Q8×C12712yes—
321C22×C4#41×C12712yes—
322C22×C4#41×C12712yes—
323C22×C4#41×C12714yes—
324C4⋊C4—28no—
325C4⋊C4—28no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—48no—
329C4⋊C4—48no—
330C4⋊C4—48no—
331C42—44no—
332C23C22×D4×C12711yescharacteristic
333C23C22×D4×C12711yescharacteristic
334C23C22×Q8×C12711yescharacteristic
335C23C2×#41×C12711yescharacteristic
336C23C2×#41×C12711yescharacteristic
337C23C2×#41×C12712yes—
338C23H(5,𝔽2)×C12712yes—
339C23H(5,𝔽2)×C12711yescharacteristic
340C232−1+4×C12712yes—
341C232−1+4×C12712yes—
342C232−1+4×C12711yescharacteristic
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—28no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4#121×C12714yes—
385C2×C4C22⋊Q8×C12714yes—
386C2×C4#124×C12714yes—
387C22C2×#121×C12711yescharacteristic
388C22C2×C22⋊Q8×C12711yescharacteristic
389C22C2×#124×C12711yescharacteristic
390C22#534×C12711yescharacteristic
391C22#536×C12711yescharacteristic
392C22#536×C12711yescharacteristic
393C22#537×C12712yes—
394C22#537×C12711yescharacteristic
395C22#538×C12712yes—
396C22#539×C12712yes—
397C22#539×C12712yes—
398C22#539×C12712yes—
399C22#544×C12712yes—
400C22#547×C12712yes—
401C22D4×Q8×C12711yescharacteristic
402C22#550×C12711yescharacteristic
403C22#554×C12711yescharacteristic
404C22#556×C12712yes—
405C22#556×C12711yescharacteristic
406C22#556×C12712yes—
407C22#557×C12712yes—
408C22#563×C12712yes—
409C22#564×C12712yes—
410C4—24no—
411C4—24no—
412C4—24no—
413C4—44no—
414C4—44no—
415C4—82no—
416C4—82no—
417C4—84no—
418C2C127×#253811yescharacteristic
419C2C127×#254711yescharacteristic
420C2C127×#256811yescharacteristic
421C2C127×#261811yescharacteristic
422C2C127×#262711yescharacteristic
423C2C127×#263911yescharacteristic
424C2C127×#272612yes—
425C2C127×#272712yes—
426C2C127×#280512yes—
427C2C127×#281012yes—
428C2C127×#282211yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C127C24
commutator, Frattini, cosocleC24C24×C127
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
203. #17074C127direct (N × H)
223. C127#17074direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#400×C1272
C2×#389×C1272
C127×#14452
C127×#14422
C127×#14412
C127×C4⋊#952
#17074127