Finite Group Databasebeta

← C127×#17204 #44580877

Subgroups — C127×#17204 #44580877

Order 32512.

1510 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#399×C127C211yesmaximal, characteristic, complemented
2C2×#396×C127C211yesmaximal, characteristic, complemented
3C127×#1443C211yesmaximal, characteristic, complemented
4C127×#1441C213yesmaximal, complemented
5C127×#1439C213yesmaximal, complemented
6C127×#1439C213yesmaximal, complemented
7C127×#1438C213yesmaximal
8C23×D4×C127C2211yescharacteristic
9C22×C4⋊C4×C127C2213yes—
10C22×C4⋊C4×C127C2213yes—
11C22×C22⋊C4×C127C2213yes—
12C22×C22⋊C4×C127C2213yes—
13C22×C22⋊C4×C127C2213yes—
14C22×C22⋊C4×C127C2213yes—
15#399×C127—24no—
16#396×C127—24no—
17C2×#95×C127C2213yes—
18C2×#95×C127C2213yes—
19C2×#95×C127C2213yes—
20C2×#95×C127C2213yes—
21C2×#95×C127C2213yes—
22C2×#95×C127C2211yescharacteristic
23C25×C127C2311yescharacteristic
24C25×C127C2311yescharacteristic
25C22×D4×C127—43no—
26C22×D4×C127—43no—
27C22×D4×C127Q814yes—
28C23×C4×C127C2313yes—
29C23×C4×C127C2313yes—
30C23×C4×C127C2313yes—
31C23×C4×C127C2311yescharacteristic
32C23×C4×C127C2313yes—
33C2×C4⋊C4×C127—26no—
34C2×C4⋊C4×C127—26no—
35C2×C4⋊C4×C127—26no—
36C2×C4⋊C4×C127—43no—
37C2×C4⋊C4×C127—43no—
38C2×C4⋊C4×C127—43no—
39C2×C22⋊C4×C127—26no—
40C2×C22⋊C4×C127—26no—
41C2×C22⋊C4×C127—26no—
42C2×C22⋊C4×C127—26no—
43C2×C22⋊C4×C127—43no—
44C2×C22⋊C4×C127—43no—
45C2×C22⋊C4×C127—43no—
46C2×C22⋊C4×C127—43no—
47C2×C22⋊C4×C127—43no—
48C2×C22⋊C4×C127—43no—
49C2×C22⋊C4×C127—43no—
50C2×C22⋊C4×C127—43no—
51#95×C127—43no—
52#95×C127—43no—
53#95×C127—43no—
54#95×C127—43no—
55#95×C127—43no—
56#95×C127D414yescomplemented
57C24×C127—21no—
58C24×C127—21no—
59C24×C127—23no—
60C24×C127—23no—
61C24×C127—23no—
62C24×C127—23no—
63C24×C127—23no—
64C24×C127—23no—
65C24×C127—23no—
66C24×C127—23no—
67C24×C127—21no—
68C24×C127—21no—
69C24×C127C2×Q812yes—
70C24×C127C2×Q812yes—
71C24×C127C2411yescharacteristic
72C2×D4×C127—412no—
73C2×D4×C127—44no—
74C2×D4×C127—412no—
75C22×C4×C127—23no—
76C22×C4×C127—23no—
77C22×C4×C127—23no—
78C22×C4×C127—23no—
79C22×C4×C127—23no—
80C22×C4×C127—23no—
81C22×C4×C127—23no—
82C22×C4×C127—23no—
83C22×C4×C127—23no—
84C22×C4×C127—23no—
85C22×C4×C127—23no—
86C22×C4×C127—23no—
87C22×C4×C127—23no—
88C22×C4×C127—23no—
89C22×C4×C127—23no—
90C22×C4×C127—23no—
91C22×C4×C127—23no—
92C22×C4×C127—23no—
93C22×C4×C127—23no—
94C22×C4×C127—23no—
95C22×C4×C127—23no—
96C22×C4×C127—23no—
97C22×C4×C127—23no—
98C22×C4×C127—23no—
99C22×C4×C127—23no—
100C22×C4×C127—23no—
101C22×C4×C127—23no—
102C22×C4×C127—23no—
103C22×C4×C127C2×D416yes—
104C22×C4×C127C2×Q812yes—
105C22×C4×C127#4116yes—
106C4⋊C4×C127—412no—
107C4⋊C4×C127—412no—
108C22⋊C4×C127—412no—
109C22⋊C4×C127—412no—
110C22⋊C4×C127—412no—
111C22⋊C4×C127—412no—
112C23×C127—26no—
113C23×C127—26no—
114C23×C127—22no—
115C23×C127—22no—
116C23×C127—26no—
117C23×C127—26no—
118C23×C127—43no—
119C23×C127—43no—
120C23×C127—43no—
121C23×C127—43no—
122C23×C127—43no—
123C23×C127—43no—
124C23×C127—43no—
125C23×C127—43no—
126C23×C127—43no—
127C23×C127—43no—
128C23×C127—43no—
129C23×C127—43no—
130C23×C127—43no—
131C23×C127—41no—
132C23×C127—41no—
133C23×C127—43no—
134C23×C127—43no—
135C23×C127—43no—
136C23×C127—43no—
137C23×C127—43no—
138C23×C127C22×D411yescharacteristic
139C23×C127C22×Q811yescharacteristic
140C23×C127C2×#4113yes—
141C23×C127H(5,𝔽2)13yes—
142C23×C127H(5,𝔽2)13yes—
143C23×C127H(5,𝔽2)13yes—
144C23×C1272−1+411yescharacteristic
145D4×C127—416no—
146C2×C4×C127—26no—
147C2×C4×C127—26no—
148C2×C4×C127—26no—
149C2×C4×C127—26no—
150C2×C4×C127—26no—
151C2×C4×C127—26no—
152C2×C4×C127—26no—
153C2×C4×C127—43no—
154C2×C4×C127—43no—
155C2×C4×C127—43no—
156C2×C4×C127—43no—
157C2×C4×C127—43no—
158C2×C4×C127—43no—
159C2×C4×C127—43no—
160C2×C4×C127—43no—
161C2×C4×C127—41no—
162C2×C4×C127—43no—
163C2×C4×C127—43no—
164C2×C4×C127—43no—
165C2×C4×C127—43no—
166C2×C4×C127—43no—
167C2×C4×C127—43no—
168C2×C4×C127—43no—
169C2×C4×C127—43no—
170C2×C4×C127—43no—
171C2×C4×C127—43no—
172C2×C4×C127—43no—
173C2×C4×C127—43no—
174C2×C4×C127—43no—
175C2×C4×C127—43no—
176C2×C4×C127—43no—
177C22×C127—42no—
178C22×C127—42no—
179C22×C127—46no—
180C22×C127—46no—
181C22×C127—46no—
182C22×C127—46no—
183C22×C127—81no—
184C22×C127—81no—
185C22×C127—83no—
186C22×C127—83no—
187C22×C127—83no—
188C22×C127—83no—
189C22×C127—81no—
190C22×C127—81no—
191C22×C127#53513yes—
192C22×C127#53813yes—
193C22×C127#54013yes—
194C22×C127#54213yes—
195C22×C127#54313yes—
196C22×C127#54613yes—
197C22×C127D4×Q811yescharacteristic
198C22×C127#55313yes—
199C22×C127#55413yes—
200C22×C127#56013yes—
201C22×C127#56011yescharacteristic
202C22×C127#56013yes—
203C22×C127#56213yes—
204C4×C127—42no—
205C4×C127—46no—
206C4×C127—46no—
207C4×C127—83no—
208C4×C127—83no—
209#17204C12711yesmaximal, characteristic, direct product factor, Sylow
210C2×C127—82no—
211C2×C127—82no—
212C2×C127#269213yes—
213C2×C127#270313yes—
214C2×C127#278013yes—
215C2×C127#280111yescharacteristic
216C2×C127#281811yescharacteristic
217C2×C127#282913yes—
218C2×C127#284211yescharacteristic
219C2×#399C2×C12711yescharacteristic, complemented
220C2×#396C2×C12711yescharacteristic, complemented
221#1443C2×C12711yescharacteristic, complemented
222#1441C2×C12713yescomplemented
223#1439C2×C12713yescomplemented
224#1439C2×C12713yescomplemented
225#1438C2×C12713yes—
226C127#1720411yescharacteristic, direct product factor, Sylow
227C23×D4C22×C12711yescharacteristic
228C22×C4⋊C4C22×C12713yes—
229C22×C4⋊C4C22×C12713yes—
230C22×C22⋊C4C22×C12713yes—
231C22×C22⋊C4C22×C12713yes—
232C22×C22⋊C4C22×C12713yes—
233C22×C22⋊C4C22×C12713yes—
234#399—24no—
235#396—24no—
236C2×#95C22×C12713yes—
237C2×#95C22×C12713yes—
238C2×#95C22×C12713yes—
239C2×#95C22×C12713yes—
240C2×#95C22×C12713yes—
241C2×#95C22×C12711yescharacteristic
242C25C23×C12711yescharacteristic
243C25C23×C12711yescharacteristic
244C22×D4—43no—
245C22×D4—43no—
246C22×D4Q8×C12714yes—
247C23×C4C23×C12713yes—
248C23×C4C23×C12713yes—
249C23×C4C23×C12713yes—
250C23×C4C23×C12711yescharacteristic
251C23×C4C23×C12713yes—
252C2×C4⋊C4—26no—
253C2×C4⋊C4—26no—
254C2×C4⋊C4—26no—
255C2×C4⋊C4—43no—
256C2×C4⋊C4—43no—
257C2×C4⋊C4—43no—
258C2×C22⋊C4—26no—
259C2×C22⋊C4—26no—
260C2×C22⋊C4—26no—
261C2×C22⋊C4—26no—
262C2×C22⋊C4—43no—
263C2×C22⋊C4—43no—
264C2×C22⋊C4—43no—
265C2×C22⋊C4—43no—
266C2×C22⋊C4—43no—
267C2×C22⋊C4—43no—
268C2×C22⋊C4—43no—
269C2×C22⋊C4—43no—
270#95—43no—
271#95—43no—
272#95—43no—
273#95—43no—
274#95—43no—
275#95D4×C12714yescomplemented
276C24—21no—
277C24—21no—
278C24—23no—
279C24—23no—
280C24—23no—
281C24—23no—
282C24—23no—
283C24—23no—
284C24—23no—
285C24—23no—
286C24—21no—
287C24—21no—
288C24C2×Q8×C12712yes—
289C24C2×Q8×C12712yes—
290C24C24×C12711yescharacteristic
291C2×D4—412no—
292C2×D4—412no—
293C2×D4—44no—
294C22×C4—23no—
295C22×C4—23no—
296C22×C4—23no—
297C22×C4—23no—
298C22×C4—23no—
299C22×C4—23no—
300C22×C4—23no—
301C22×C4—23no—
302C22×C4—23no—
303C22×C4—23no—
304C22×C4—23no—
305C22×C4—23no—
306C22×C4—23no—
307C22×C4—23no—
308C22×C4—23no—
309C22×C4—23no—
310C22×C4—23no—
311C22×C4—23no—
312C22×C4—23no—
313C22×C4—23no—
314C22×C4—23no—
315C22×C4—23no—
316C22×C4—23no—
317C22×C4—23no—
318C22×C4—23no—
319C22×C4—23no—
320C22×C4—23no—
321C22×C4—23no—
322C22×C4C2×D4×C12716yes—
323C22×C4C2×Q8×C12712yes—
324C22×C4#41×C12716yes—
325C4⋊C4—412no—
326C4⋊C4—412no—
327C22⋊C4—412no—
328C22⋊C4—412no—
329C22⋊C4—412no—
330C22⋊C4—412no—
331C23—26no—
332C23—26no—
333C23—22no—
334C23—22no—
335C23—26no—
336C23—26no—
337C23—43no—
338C23—43no—
339C23—43no—
340C23—43no—
341C23—43no—
342C23—43no—
343C23—43no—
344C23—43no—
345C23—43no—
346C23—43no—
347C23—43no—
348C23—43no—
349C23—43no—
350C23—43no—
351C23—43no—
352C23—41no—
353C23—41no—
354C23—43no—
355C23—43no—
356C23—43no—
357C23C22×D4×C12711yescharacteristic
358C23C22×Q8×C12711yescharacteristic
359C23C2×#41×C12713yes—
360C23H(5,𝔽2)×C12713yes—
361C23H(5,𝔽2)×C12713yes—
362C23H(5,𝔽2)×C12713yes—
363C232−1+4×C12711yescharacteristic
364D4—416no—
365C2×C4—26no—
366C2×C4—26no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—26no—
370C2×C4—26no—
371C2×C4—26no—
372C2×C4—43no—
373C2×C4—43no—
374C2×C4—43no—
375C2×C4—43no—
376C2×C4—43no—
377C2×C4—43no—
378C2×C4—43no—
379C2×C4—41no—
380C2×C4—43no—
381C2×C4—43no—
382C2×C4—43no—
383C2×C4—43no—
384C2×C4—43no—
385C2×C4—43no—
386C2×C4—43no—
387C2×C4—43no—
388C2×C4—43no—
389C2×C4—43no—
390C2×C4—43no—
391C2×C4—43no—
392C2×C4—43no—
393C2×C4—43no—
394C2×C4—43no—
395C2×C4—43no—
396C22—42no—
397C22—42no—
398C22—46no—
399C22—46no—
400C22—46no—
401C22—46no—
402C22—83no—
403C22—83no—
404C22—83no—
405C22—83no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22#535×C12713yes—
411C22#538×C12713yes—
412C22#540×C12713yes—
413C22#542×C12713yes—
414C22#543×C12713yes—
415C22#546×C12713yes—
416C22D4×Q8×C12711yescharacteristic
417C22#553×C12713yes—
418C22#554×C12713yes—
419C22#560×C12713yes—
420C22#560×C12713yes—
421C22#560×C12711yescharacteristic
422C22#562×C12713yes—
423C4—42no—
424C4—46no—
425C4—46no—
426C4—83no—
427C4—83no—
428C2—82no—
429C2—82no—
430C2C127×#269213yes—
431C2C127×#270313yes—
432C2C127×#278013yes—
433C2C127×#280111yescharacteristic
434C2C127×#281811yescharacteristic
435C2C127×#282913yes—
436C2C127×#284211yescharacteristic
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×#399×C127C2semidirect
2. C2×#396×C127C2semidirect
3. C127×#1443C2semidirect
4. C127×#1441C2semidirect
5. C127×#1439C2semidirect
6. C127×#1439C2semidirect
56. #95×C127D4semidirect
209. #17204C127direct (N × H)
219. C2×#399C2×C127semidirect
220. C2×#396C2×C127semidirect
221. #1443C2×C127semidirect
222. #1441C2×C127semidirect
223. #1439C2×C127semidirect
224. #1439C2×C127semidirect
226. C127#17204direct (N × H)
275. #95D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#399×C1272
C2×#396×C1272
C127×#14432
C127×#14412
C127×#14392
C127×#14382
#17204127