Finite Group Databasebeta

← C127×#16824 #44580497

Subgroups — C127×#16824 #44580497

Order 32512.

1434 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#399×C127C212yesmaximal
2C127×#1441C214yesmaximal, complemented
3C127×#1441C214yesmaximal, complemented
4C127×#1439C212yesmaximal
5C127×C22⋊#95C212yesmaximal
6C127×C22⋊#95C211yesmaximal, characteristic
7C24×C4×C127C2211yescharacteristic
8C22×C4⋊C4×C127C2214yes—
9C22×C4⋊C4×C127C2212yes—
10C22×C4⋊C4×C127C2212yes—
11C22×C22⋊C4×C127C2214yes—
12C22×C22⋊C4×C127C2211yescharacteristic
13C22×C22⋊C4×C127C2211yescharacteristic
14#399×C127—28no—
15C2×#95×C127C2212yes—
16C2×#95×C127C2212yes—
17C2×#95×C127C2214yes—
18C2×#95×C127C2214yes—
19C2×#95×C127C2212yes—
20C2×#95×C127C2212yes—
21C2×#95×C127C2214yes—
22C25×C127C2311yescharacteristic
23C23×C4×C127—41no—
24C23×C4×C127—41no—
25C23×C4×C127—44no—
26C23×C4×C127Q814yes—
27C23×C4×C127C2314yes—
28C23×C4×C127C2312yes—
29C23×C4×C127C2312yes—
30C23×C4×C127C2312yes—
31C23×C4×C127C2314yes—
32C2×C4⋊C4×C127—28no—
33C2×C4⋊C4×C127—24no—
34C2×C4⋊C4×C127—28no—
35C2×C4⋊C4×C127—28no—
36C2×C4⋊C4×C127—42no—
37C2×C4⋊C4×C127—44no—
38C2×C4⋊C4×C127—42no—
39C2×C4⋊C4×C127—42no—
40C2×C22⋊C4×C127—28no—
41C2×C22⋊C4×C127—28no—
42C2×C22⋊C4×C127—41no—
43C2×C22⋊C4×C127—41no—
44C2×C22⋊C4×C127—41no—
45C2×C22⋊C4×C127—41no—
46C2×C22⋊C4×C127—44no—
47C2×C22⋊C4×C127D414yes—
48C2×C22⋊C4×C127Q814yes—
49#95×C127—24no—
50#95×C127—44no—
51#95×C127—44no—
52#95×C127—44no—
53#95×C127—42no—
54#95×C127—42no—
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
66C22×C4×C127—24no—
67C22×C4×C127—24no—
68C22×C4×C127—24no—
69C22×C4×C127—24no—
70C22×C4×C127—24no—
71C22×C4×C127—24no—
72C22×C4×C127—22no—
73C22×C4×C127—24no—
74C22×C4×C127—24no—
75C22×C4×C127—22no—
76C22×C4×C127—22no—
77C22×C4×C127—24no—
78C22×C4×C127—24no—
79C22×C4×C127—22no—
80C22×C4×C127—22no—
81C22×C4×C127—22no—
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—44no—
92C22×C4×C127—48no—
93C22×C4×C127—48no—
94C22×C4×C127—48no—
95C22×C4×C127C2×D414yes—
96C22×C4×C127C2×Q814yes—
97C22×C4×C127C2×Q814yes—
98C22×C4×C127#4118yes—
99C4⋊C4×C127—416no—
100C4⋊C4×C127—48no—
101C4⋊C4×C127—48no—
102C22⋊C4×C127—44no—
103C22⋊C4×C127—44no—
104C22⋊C4×C127—416no—
105C23×C127—24no—
106C23×C127—22no—
107C23×C127—22no—
108C23×C127—22no—
109C23×C127—22no—
110C23×C127—24no—
111C23×C127—22no—
112C23×C127—22no—
113C23×C127—28no—
114C23×C127—28no—
115C23×C127—44no—
116C23×C127—44no—
117C23×C127—42no—
118C23×C127—42no—
119C23×C127—42no—
120C23×C127—42no—
121C23×C127C4⋊Q814yes—
122C23×C127C22×D411yescharacteristic
123C23×C127C22×Q811yescharacteristic
124C23×C127C22×Q811yescharacteristic
125C23×C127C2×#4114yes—
126C23×C127H(5,𝔽2)11yescharacteristic
127C23×C127H(5,𝔽2)11yescharacteristic
128C23×C127H(5,𝔽2)11yescharacteristic
129C23×C127H(5,𝔽2)14yes—
130C23×C1272−1+411yescharacteristic
131C2×C4×C127—24no—
132C2×C4×C127—28no—
133C2×C4×C127—24no—
134C2×C4×C127—24no—
135C2×C4×C127—24no—
136C2×C4×C127—24no—
137C2×C4×C127—28no—
138C2×C4×C127—28no—
139C2×C4×C127—28no—
140C2×C4×C127—28no—
141C2×C4×C127—416no—
142C2×C4×C127—44no—
143C2×C4×C127—44no—
144C2×C4×C127—44no—
145C2×C4×C127—44no—
146C2×C4×C127—44no—
147C2×C4×C127—42no—
148C2×C4×C127—42no—
149C2×C4×C127—44no—
150C2×C4×C127—42no—
151C2×C4×C127—42no—
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—
161C22×C127—24no—
162C22×C127—28no—
163C22×C127—42no—
164C22×C127—42no—
165C22×C127—44no—
166C22×C127—44no—
167C22×C127—44no—
168C22×C127—44no—
169C22×C127—44no—
170C22×C127C2×C4⋊Q811yescharacteristic
171C22×C127#53312yes—
172C22×C127#53411yescharacteristic
173C22×C127#53711yescharacteristic
174C22×C127#53914yes—
175C22×C127#54014yes—
176C22×C127#54311yescharacteristic
177C22×C127#54312yes—
178C22×C127#54311yescharacteristic
179C22×C127#54411yescharacteristic
180C22×C127#54411yescharacteristic
181C22×C127#54614yes—
182C22×C127#55112yes—
183C22×C127#55414yes—
184C22×C127#55912yes—
185C22×C127#56012yes—
186C22×C127#56012yes—
187C4×C127—44no—
188C4×C127—48no—
189C4×C127—44no—
190C4×C127—44no—
191C4×C127—84no—
192#16824C12711yesmaximal, characteristic, direct product factor, Sylow
193C2×C127—44no—
194C2×C127#254912yes—
195C2×C127#260711yescharacteristic
196C2×C127#265911yescharacteristic
197C2×C127#266712yes—
198C2×C127#269312yes—
199C2×C127#270912yes—
200C2×C127#274812yes—
201C2×C127#280712yes—
202C2×C127#284211yescharacteristic
203C2×#399C2×C12712yes—
204#1441C2×C12714yescomplemented
205#1441C2×C12714yescomplemented
206#1439C2×C12712yes—
207C22⋊#95C2×C12712yes—
208C22⋊#95C2×C12711yescharacteristic
209C127#1682411yescharacteristic, direct product factor, Sylow
210C24×C4C22×C12711yescharacteristic
211C22×C4⋊C4C22×C12714yes—
212C22×C4⋊C4C22×C12712yes—
213C22×C4⋊C4C22×C12712yes—
214C22×C22⋊C4C22×C12714yes—
215C22×C22⋊C4C22×C12711yescharacteristic
216C22×C22⋊C4C22×C12711yescharacteristic
217#399—28no—
218C2×#95C22×C12712yes—
219C2×#95C22×C12712yes—
220C2×#95C22×C12714yes—
221C2×#95C22×C12714yes—
222C2×#95C22×C12712yes—
223C2×#95C22×C12712yes—
224C2×#95C22×C12714yes—
225C25C23×C12711yescharacteristic
226C23×C4—41no—
227C23×C4—41no—
228C23×C4—44no—
229C23×C4Q8×C12714yes—
230C23×C4C23×C12714yes—
231C23×C4C23×C12712yes—
232C23×C4C23×C12712yes—
233C23×C4C23×C12712yes—
234C23×C4C23×C12714yes—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4D4×C12714yes—
251C2×C22⋊C4Q8×C12714yes—
252#95—24no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—42no—
257#95—42no—
258#95—42no—
259C24—24no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—24no—
264C24—21no—
265C24C2×D4×C12712yes—
266C24C2×Q8×C12712yes—
267C24C2×Q8×C12712yes—
268C24C24×C12711yescharacteristic
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
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—44no—
295C22×C4—48no—
296C22×C4—48no—
297C22×C4—48no—
298C22×C4C2×D4×C12714yes—
299C22×C4C2×Q8×C12714yes—
300C22×C4C2×Q8×C12714yes—
301C22×C4#41×C12718yes—
302C4⋊C4—416no—
303C4⋊C4—48no—
304C4⋊C4—48no—
305C22⋊C4—44no—
306C22⋊C4—44no—
307C22⋊C4—416no—
308C23—24no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—24no—
314C23—22no—
315C23—22no—
316C23—28no—
317C23—28no—
318C23—44no—
319C23—44no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23C4⋊Q8×C12714yes—
325C23C22×D4×C12711yescharacteristic
326C23C22×Q8×C12711yescharacteristic
327C23C22×Q8×C12711yescharacteristic
328C23C2×#41×C12714yes—
329C23H(5,𝔽2)×C12711yescharacteristic
330C23H(5,𝔽2)×C12711yescharacteristic
331C23H(5,𝔽2)×C12711yescharacteristic
332C23H(5,𝔽2)×C12714yes—
333C232−1+4×C12711yescharacteristic
334C2×C4—24no—
335C2×C4—28no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—28no—
341C2×C4—28no—
342C2×C4—28no—
343C2×C4—28no—
344C2×C4—416no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C22—24no—
365C22—28no—
366C22—42no—
367C22—42no—
368C22—44no—
369C22—44no—
370C22—44no—
371C22—44no—
372C22—44no—
373C22C2×C4⋊Q8×C12711yescharacteristic
374C22#533×C12712yes—
375C22#534×C12711yescharacteristic
376C22#537×C12711yescharacteristic
377C22#539×C12714yes—
378C22#540×C12714yes—
379C22#543×C12711yescharacteristic
380C22#543×C12712yes—
381C22#543×C12711yescharacteristic
382C22#544×C12711yescharacteristic
383C22#544×C12711yescharacteristic
384C22#546×C12714yes—
385C22#551×C12712yes—
386C22#554×C12714yes—
387C22#559×C12712yes—
388C22#560×C12712yes—
389C22#560×C12712yes—
390C4—44no—
391C4—48no—
392C4—44no—
393C4—44no—
394C4—84no—
395C2—44no—
396C2C127×#254912yes—
397C2C127×#260711yescharacteristic
398C2C127×#265911yescharacteristic
399C2C127×#266712yes—
400C2C127×#269312yes—
401C2C127×#270912yes—
402C2C127×#274812yes—
403C2C127×#280712yes—
404C2C127×#284211yescharacteristic
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×#1441C2semidirect
3. C127×#1441C2semidirect
192. #16824C127direct (N × H)
204. #1441C2×C127semidirect
205. #1441C2×C127semidirect
209. C127#16824direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#399×C1272
C127×#14412
C127×#14392
C127×C22⋊#952
#16824127