Finite Group Databasebeta

← C127×#8410 #44572083

Subgroups — C127×#8410 #44572083

Order 32512.

692 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3458C211yesmaximal, characteristic
2C127×#1798C211yesmaximal, characteristic
3C127×#1542C212yesmaximal, complemented
4C127×#1481C212yesmaximal, complemented
5C127×#1458C211yesmaximal, characteristic
6#544×C127—22no—
7#544×C127—22no—
8#544×C127—21no—
9#544×C127C412yes—
10#543×C127—21no—
11#543×C127C412yes—
12#533×C127—24no—
13#533×C127—21no—
14#533×C127C2212yes—
15C2×#117×C127—21no—
16C2×#117×C127C2211yescharacteristic
17#431×C127C2212yes—
18C2×C4⋊C8×C127C2211yescharacteristic
19C2×C4⋊C8×C127C2211yescharacteristic
20C2×C22⋊C8×C127—21no—
21#336×C127—21no—
22#333×C127—22no—
23#332×C127—22no—
24Q8⋊C8×C127—24no—
25#324×C127—21no—
26#323×C127—21no—
27C23×C4×C127D411yescharacteristic
28C22×C8×C127—21no—
29C22×C8×C127—21no—
30C4⋊Q8×C127—24no—
31C4⋊Q8×C127—21no—
32C4⋊Q8×C127—21no—
33C4⋊Q8×C127—24no—
34C4⋊Q8×C127D412yes—
35C4⋊Q8×C127D412yes—
36#125×C127—24no—
37#125×C127—22no—
38#125×C127—24no—
39#125×C127—22no—
40C22⋊Q8×C127—24no—
41C22⋊Q8×C127—24no—
42C22⋊Q8×C127—24no—
43C22⋊Q8×C127—22no—
44C22⋊Q8×C127—22no—
45C22⋊Q8×C127—22no—
46C22⋊Q8×C127—24no—
47C22⋊Q8×C127C2×C414yes—
48C4×Q8×C127—22no—
49C4×Q8×C127—22no—
50C4×Q8×C127—22no—
51C4×Q8×C127D414yes—
52#117×C127—22no—
53#117×C127—22no—
54#117×C127—22no—
55#117×C127—22no—
56#117×C127—22no—
57#117×C127—22no—
58C2×C4⋊C4×C127—21no—
59C2×C4⋊C4×C127—21no—
60C2×C4⋊C4×C127C2×C411yescharacteristic
61C2×C4⋊C4×C127D411yescharacteristic
62C2×C22⋊C4×C127—22no—
63C2×C22⋊C4×C127C2×C411yescharacteristic
64C2×C22⋊C4×C127D411yescharacteristic
65C2×C42×C127—22no—
66C2×C42×C127D411yescharacteristic
67C2×C42×C127C2311yescharacteristic
68C4⋊C8×C127—22no—
69C4⋊C8×C127—22no—
70C4⋊C8×C127—22no—
71C22⋊C8×C127—22no—
72C22⋊C8×C127—42no—
73C4×C8×C127—22no—
74C24×C127C22⋊C411yescharacteristic
75C2×Q8×C127—22no—
76C2×Q8×C127—22no—
77C2×Q8×C127—22no—
78C2×Q8×C127C22⋊C414yes—
79C22×C4×C127—21no—
80C22×C4×C127—21no—
81C22×C4×C127—21no—
82C22×C4×C127—21no—
83C22×C4×C127—22no—
84C22×C4×C127—22no—
85C22×C4×C127C22⋊C411yescharacteristic
86C22×C4×C127C22⋊C411yescharacteristic
87C22×C4×C127C22⋊C411yescharacteristic
88C22×C4×C127SD1612yes—
89C22×C4×C127Q1612yes—
90C22×C4×C127C22×C411yescharacteristic
91C22×C4×C127C2×D411yescharacteristic
92C22×C4×C127C2×D411yescharacteristic
93C2×C8×C127—42no—
94C2×C8×C127—41no—
95C2×C8×C127—41no—
96C2×C8×C127—41no—
97C2×C8×C127—41no—
98C2×C8×C127—41no—
99C2×C8×C127—41no—
100C4⋊C4×C127—22no—
101C4⋊C4×C127—22no—
102C4⋊C4×C127—22no—
103C4⋊C4×C127—24no—
104C4⋊C4×C127—24no—
105C4⋊C4×C127—22no—
106C4⋊C4×C127—22no—
107C4⋊C4×C127—24no—
108C4⋊C4×C127—22no—
109C4⋊C4×C127—22no—
110C4⋊C4×C127—22no—
111C4⋊C4×C127—22no—
112C4⋊C4×C127—24no—
113C4⋊C4×C127C22⋊C414yes—
114C22⋊C4×C127—24no—
115C22⋊C4×C127—24no—
116C22⋊C4×C127—22no—
117C22⋊C4×C127—22no—
118C42×C127—24no—
119C42×C127—21no—
120C42×C127—24no—
121C42×C127—21no—
122C42×C127C2×D412yes—
123C42×C127C2×D412yes—
124C23×C127—21no—
125C23×C127—21no—
126C23×C127—41no—
127C23×C127—41no—
128C23×C127#10312yes—
129C23×C127C2×C22⋊C411yescharacteristic
130Q8×C127—44no—
131Q8×C127—82no—
132Q8×C127—82no—
133Q8×C127—82no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—22no—
137C2×C4×C127—22no—
138C2×C4×C127—22no—
139C2×C4×C127—22no—
140C2×C4×C127—24no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—24no—
144C2×C4×C127—41no—
145C2×C4×C127—41no—
146C2×C4×C127—41no—
147C2×C4×C127—41no—
148C2×C4×C127—82no—
149C2×C4×C127—82no—
150C2×C4×C127#10312yes—
151C2×C4×C127C2×C22⋊C412yes—
152C2×C4×C127#12012yes—
153C2×C4×C127#12011yescharacteristic
154C2×C4×C127#12011yescharacteristic
155C2×C4×C127C2×SD1611yescharacteristic
156C2×C4×C127C2×Q1611yescharacteristic
157C2×C4×C127#13711yescharacteristic
158C2×C4×C127#13711yescharacteristic
159C8×C127—82no—
160C8×C127—81no—
161C8×C127—81no—
162C22×C127—21no—
163C22×C127—21no—
164C22×C127—22no—
165C22×C127—42no—
166C22×C127—41no—
167C22×C127—41no—
168C22×C127—81no—
169C22×C127#37911yescharacteristic
170C22×C127C2×#10311yescharacteristic
171C22×C127#41911yescharacteristic
172C4×C127—22no—
173C4×C127—22no—
174C4×C127—22no—
175C4×C127—41no—
176C4×C127—44no—
177C4×C127—41no—
178C4×C127—44no—
179C4×C127#44811yescharacteristic
180C4×C127#44811yescharacteristic
181C4×C127#45111yescharacteristic
182C4×C127#45111yescharacteristic
183#8410C12711yesmaximal, characteristic, direct product factor, Sylow
184C2×C127—21no—
185C2×C127—21no—
186C2×C127—42no—
187C2×C127#178711yescharacteristic
188C2×C127#179111yescharacteristic
189C2×C127#179411yescharacteristic
190#3458C2×C12711yescharacteristic
191#1798C2×C12711yescharacteristic
192#1542C2×C12712yescomplemented
193#1481C2×C12712yescomplemented
194#1458C2×C12711yescharacteristic
195C127#841011yescharacteristic, direct product factor, Sylow
196#544—22no—
197#544—22no—
198#544—21no—
199#544C4×C12712yes—
200#543—21no—
201#543C4×C12712yes—
202#533—24no—
203#533—21no—
204#533C22×C12712yes—
205C2×#117—21no—
206C2×#117C22×C12711yescharacteristic
207#431C22×C12712yes—
208C2×C4⋊C8C22×C12711yescharacteristic
209C2×C4⋊C8C22×C12711yescharacteristic
210C2×C22⋊C8—21no—
211#336—21no—
212#333—22no—
213#332—22no—
214Q8⋊C8—24no—
215#324—21no—
216#323—21no—
217C23×C4D4×C12711yescharacteristic
218C22×C8—21no—
219C22×C8—21no—
220C4⋊Q8—24no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—24no—
224C4⋊Q8D4×C12712yes—
225C4⋊Q8D4×C12712yes—
226#125—24no—
227#125—22no—
228#125—24no—
229#125—22no—
230C22⋊Q8—24no—
231C22⋊Q8—24no—
232C22⋊Q8—24no—
233C22⋊Q8—22no—
234C22⋊Q8—22no—
235C22⋊Q8—22no—
236C22⋊Q8—24no—
237C22⋊Q8C2×C4×C12714yes—
238C4×Q8—22no—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×Q8D4×C12714yes—
242#117—22no—
243#117—22no—
244#117—22no—
245#117—22no—
246#117—22no—
247#117—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4C2×C4×C12711yescharacteristic
251C2×C4⋊C4D4×C12711yescharacteristic
252C2×C22⋊C4—22no—
253C2×C22⋊C4C2×C4×C12711yescharacteristic
254C2×C22⋊C4D4×C12711yescharacteristic
255C2×C42—22no—
256C2×C42D4×C12711yescharacteristic
257C2×C42C23×C12711yescharacteristic
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—42no—
263C4×C8—22no—
264C24C22⋊C4×C12711yescharacteristic
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8C22⋊C4×C12714yes—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4C22⋊C4×C12711yescharacteristic
276C22×C4C22⋊C4×C12711yescharacteristic
277C22×C4C22⋊C4×C12711yescharacteristic
278C22×C4SD16×C12712yes—
279C22×C4Q16×C12712yes—
280C22×C4C22×C4×C12711yescharacteristic
281C22×C4C2×D4×C12711yescharacteristic
282C22×C4C2×D4×C12711yescharacteristic
283C2×C8—42no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—24no—
303C4⋊C4C22⋊C4×C12714yes—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—22no—
307C22⋊C4—22no—
308C42—24no—
309C42—24no—
310C42—21no—
311C42—21no—
312C42C2×D4×C12712yes—
313C42C2×D4×C12712yes—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23#103×C12712yes—
319C23C2×C22⋊C4×C12711yescharacteristic
320Q8—44no—
321Q8—82no—
322Q8—82no—
323Q8—82no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—24no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4#103×C12712yes—
341C2×C4C2×C22⋊C4×C12712yes—
342C2×C4#120×C12712yes—
343C2×C4#120×C12711yescharacteristic
344C2×C4#120×C12711yescharacteristic
345C2×C4C2×SD16×C12711yescharacteristic
346C2×C4C2×Q16×C12711yescharacteristic
347C2×C4#137×C12711yescharacteristic
348C2×C4#137×C12711yescharacteristic
349C8—82no—
350C8—81no—
351C8—81no—
352C22—21no—
353C22—21no—
354C22—22no—
355C22—42no—
356C22—41no—
357C22—41no—
358C22—81no—
359C22#379×C12711yescharacteristic
360C22C2×#103×C12711yescharacteristic
361C22#419×C12711yescharacteristic
362C4—22no—
363C4—22no—
364C4—22no—
365C4—41no—
366C4—44no—
367C4—41no—
368C4—44no—
369C4#448×C12711yescharacteristic
370C4#448×C12711yescharacteristic
371C4#451×C12711yescharacteristic
372C4#451×C12711yescharacteristic
373C2—21no—
374C2—21no—
375C2—42no—
376C2C127×#178711yescharacteristic
377C2C127×#179111yescharacteristic
378C2C127×#179411yescharacteristic
subgroupG/H
centerC2×C4×C127#120
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
socleC22×C127#379
prime radicalC24×C127C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#1542C2semidirect
4. C127×#1481C2semidirect
183. #8410C127direct (N × H)
192. #1542C2×C127semidirect
193. #1481C2×C127semidirect
195. C127#8410direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#34582
C127×#17982
C127×#15422
C127×#14812
C127×#14582
#8410127