Finite Group Databasebeta

← C4×#15688860 #26107157

Subgroups — C4×#15688860 #26107157

Order 12032.

732 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C22⋊(C47⋊C8)C211yesmaximal, characteristic
2C2×#15688860C212yesmaximal
3C4×C47⋊#95C211yesmaximal, characteristic
4#20241266C212yesmaximal
5C4×C22⋊Q8×C47C211yesmaximal, characteristic
6C22×C4×Dic47—21no—
7C2×C22⋊(C47⋊C8)C2212yes—
8C2×C4×C47⋊C8—21no—
9C2×C47⋊#95—21no—
10C2×C47⋊#95C2212yes—
11#15688878—21no—
12#15688860C418yesdirect product factor
13C2×C22⋊Q8×C47C2211yescharacteristic
14C2×C4×Q8×C47C411yescharacteristic, complemented
15C22×C42×C47C2211yescharacteristic
16C47×#391C411yescharacteristic, complemented
17C47×#389C411yescharacteristic, complemented
18C47×#387—21no—
19C47×#385—21no—
20C47×#380C2211yescharacteristic
21C4×C4⋊C4×C47—21no—
22C4×C4⋊C4×C47C411yescharacteristic, complemented
23C4×C22⋊C4×C47—21no—
24C23×Dic47—22no—
25C22×C47⋊C8—22no—
26C2×C4×Dic47—24no—
27C2×C4×Dic47—42no—
28C2×C4×Dic47—42no—
29C22⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)C2×C414yes—
31C4×C47⋊C8—42no—
32C47⋊#95—24no—
33C47⋊#95—22no—
34C47⋊#95C2×C414yes—
35C22×Q8×C47C2×C411yescharacteristic
36C23×C4×C47D411yescharacteristic
37C23×C4×C47D411yescharacteristic
38C23×C4×C47C2311yescharacteristic
39C22⋊Q8×C47—22no—
40C22⋊Q8×C47C2×C414yes—
41C4×Q8×C47—42no—
42C2×C4⋊C4×C47—21no—
43C2×C4⋊C4×C47—21no—
44C2×C4⋊C4×C47—21no—
45C2×C4⋊C4×C47C2×C411yescharacteristic
46C2×C4⋊C4×C47C2×C411yescharacteristic
47C2×C4⋊C4×C47C2×C411yescharacteristic
48C2×C22⋊C4×C47—21no—
49C2×C22⋊C4×C47—21no—
50C2×C42×C47—21no—
51C2×C42×C47—21no—
52C2×C42×C47—42no—
53C2×C42×C47C2×C411yescharacteristic
54C2×C42×C47C2×C411yescharacteristic
55C2×C42×C47D411yescharacteristic
56C2×C42×C47D411yescharacteristic
57#95×C47—21no—
58#95×C47—21no—
59#95×C47—21no—
60#95×C47—21no—
61#95×C47—21no—
62C24×C47C2×D411yescharacteristic
63C2×Q8×C47—21no—
64C2×Q8×C47—41no—
65C2×Q8×C47—41no—
66C2×Q8×C47C4212yescomplemented
67C22×C4×C47—22no—
68C22×C4×C47—22no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—22no—
72C22×C4×C47—22no—
73C22×C4×C47—21no—
74C22×C4×C47—21no—
75C22×C4×C47—41no—
76C22×C4×C47—41no—
77C22×C4×C47—41no—
78C22×C4×C47C22⋊C411yescharacteristic
79C22×C4×C47C22⋊C411yescharacteristic
80C22×C4×C47C22⋊C411yescharacteristic
81C22×C4×C47C22⋊C411yescharacteristic
82C22×C4×C47SD1612yes—
83C22×C4×C47Q1612yes—
84C22×C4×C47C22×C411yescharacteristic
85C22×C4×C47C22×C412yes—
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47#4112yes—
88C22×Dic47—24no—
89C22×Dic47—24no—
90C22×Dic47—24no—
91C22×Dic47—42no—
92C22×Dic47—42no—
93C22×Dic47—42no—
94C22×Dic47—42no—
95C4⋊C4×C47—21no—
96C4⋊C4×C47—24no—
97C4⋊C4×C47—42no—
98C4⋊C4×C47—41no—
99C4⋊C4×C47—41no—
100C4⋊C4×C47—42no—
101C4⋊C4×C47C4212yescomplemented
102C22⋊C4×C47—24no—
103C22⋊C4×C47—42no—
104C42×C47—22no—
105C42×C47—22no—
106C42×C47—41no—
107C42×C47—41no—
108C42×C47—42no—
109C42×C47—42no—
110C4×Dic47—48no—
111C2×C47⋊C8—24no—
112C2×C47⋊C8—42no—
113C2×C47⋊C8—42no—
114C23×C47—21no—
115C23×C47—21no—
116C23×C47—21no—
117C23×C47—21no—
118C23×C47C2×C22⋊C411yescharacteristic
119C23×C47C4×D412yes—
120C23×C47C2×SD1611yescharacteristic
121C23×C47C2×Q1611yescharacteristic
122C23×C47#13512yes—
123Q8×C47—42no—
124Q8×C47—42no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—24no—
134C2×C4×C47—24no—
135C2×C4×C47—22no—
136C2×C4×C47—22no—
137C2×C4×C47—21no—
138C2×C4×C47—21no—
139C2×C4×C47—21no—
140C2×C4×C47—42no—
141C2×C4×C47—42no—
142C2×C4×C47—44no—
143C2×C4×C47—41no—
144C2×C4×C47—41no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—41no—
150C2×C4×C47—41no—
151C2×C4×C47—41no—
152C2×C4×C47—41no—
153C2×C4×C47#9912yes—
154C2×C4×C47#10312yes—
155C2×C4×C47#10312yes—
156C2×C4×C47#10412yes—
157C2×C4×C47#10412yes—
158C2×C4×C47C2×C4211yescharacteristic
159C2×C4×C47#11711yescharacteristic
160C2×C4×C47C4×D412yes—
161C2×Dic47—28no—
162C2×Dic47—44no—
163C2×Dic47—44no—
164C2×Dic47—44no—
165C2×Dic47—44no—
166C2×Dic47—44no—
167C2×Dic47—44no—
168C47⋊C8—44no—
169C4×#328—471nomaximal, Sylow
170C22×C47—22no—
171C22×C47—21no—
172C22×C47—21no—
173C22×C47—22no—
174C22×C47—22no—
175C22×C47—22no—
176C22×C47—21no—
177C22×C47—21no—
178C22×C47C4×C22⋊C411yescharacteristic
179C22×C47C2×#9911yescharacteristic
180C22×C47#41011yescharacteristic
181C22×C47C2×#10311yescharacteristic
182C22×C47#41611yescharacteristic
183C22×C47C2×#10411yescharacteristic
184C22×C47C2×#10411yescharacteristic
185C22×C47C4×SD1612yes—
186C22×C47C4×Q1612yes—
187C4×C47—22no—
188C4×C47—21no—
189C4×C47—21no—
190C4×C47—22no—
191C4×C47—22no—
192C4×C47—42no—
193C4×C47—41no—
194C4×C47—42no—
195C4×C47—81no—
196C4×C47—81no—
197C4×C47#32814yescomplemented
198Dic47—48no—
199C4×C22⋊Q8D4711yescharacteristic
200C4×C22⋊C8—471no—
201C2×#328—472no—
202C4×#95—471no—
203#1294—472no—
204C2×C47—22no—
205C2×C47—22no—
206C2×C47C2×#32811yescharacteristic
207C2×C47#150011yescharacteristic
208C2×C47#150111yescharacteristic
209C2×C47#150311yescharacteristic
210C2×C47C4×#9911yescharacteristic
211C2×C47C4×#10411yescharacteristic
212C2×C47C4×#10311yescharacteristic
213C2×C22⋊Q8C2×D4711yescharacteristic
214C2×C4×Q8Dic4711yescharacteristic, complemented
215C22×C42—941no—
216C22×C42C2×D4711yescharacteristic
217C2×C22⋊C8—472no—
218C2×C4×C8—941no—
219#391Dic4711yescharacteristic, complemented
220#389Dic4711yescharacteristic, complemented
221#387—21no—
222#385—21no—
223#380C2×D4711yescharacteristic
224C4×C4⋊C4—21no—
225C4×C4⋊C4Dic4711yescharacteristic, complemented
226C4×C22⋊C4—21no—
227C2×#95—472no—
228C2×#95—941no—
229#336—941no—
230#328—478no—
231C47C4×#32811yescharacteristic, complemented, Sylow
232C22×Q8C2×Dic4711yescharacteristic
233C23×C4—942no—
234C23×C4C22⋊D4711yescharacteristic
235C23×C4C22⋊D4711yescharacteristic
236C23×C4C22×D4711yescharacteristic
237C22×C8—942no—
238C22⋊Q8—22no—
239C22⋊Q8C4×D4714yes—
240C4×Q8—42no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4C2×Dic4711yescharacteristic
245C2×C4⋊C4C2×Dic4711yescharacteristic
246C2×C4⋊C4C2×Dic4711yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—42no—
252C2×C42—944no—
253C2×C42—1882no—
254C2×C42—1882no—
255C2×C42C2×Dic4711yescharacteristic
256C2×C42C2×Dic4711yescharacteristic
257C2×C42C22⋊D4711yescharacteristic
258C2×C42C22⋊D4711yescharacteristic
259C22⋊C8—474no—
260C22⋊C8—942no—
261C4×C8—1882no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—474no—
268#95—944no—
269#95—942no—
270C24C2×C22⋊D4711yescharacteristic
271C2×Q8—21no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8C4×Dic4712yescomplemented
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—944no—
287C22×C4—944no—
288C22×C4—944no—
289C22×C4—1882no—
290C22×C4—1882no—
291C22×C4—1882no—
292C22×C4—1882no—
293C22×C4#1062533612yes—
294C22×C4#1062533712yes—
295C22×C4C22⋊Dic4711yescharacteristic
296C22×C4C22⋊Dic4711yescharacteristic
297C22×C4C22⋊Dic4711yescharacteristic
298C22×C4C22⋊Dic4711yescharacteristic
299C22×C4C2×C4×D4712yes—
300C22×C4#1062535012yes—
301C22×C4C22×Dic4711yescharacteristic
302C22×C4C2×C22⋊D4711yescharacteristic
303C2×C8—944no—
304C2×C8—1882no—
305C2×C8—1882no—
306C4⋊C4—21no—
307C4⋊C4—24no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4C4×Dic4712yescomplemented
313C22⋊C4—24no—
314C22⋊C4—42no—
315C42—22no—
316C42—22no—
317C42—41no—
318C42—41no—
319C42—42no—
320C42—42no—
321C42—1888no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23C2×C22⋊Dic4711yescharacteristic
327C23C4×C22⋊D4712yes—
328C23C2×#1062533611yescharacteristic
329C23C2×#1062533711yescharacteristic
330C23#1309404412yes—
331Q8—42no—
332Q8—42no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—948no—
362C2×C4—1884no—
363C2×C4—1884no—
364C2×C4—1884no—
365C2×C4—1884no—
366C2×C4—1884no—
367C2×C4—1884no—
368C2×C4#1309393912yes—
369C2×C4#1309394812yes—
370C2×C4#1309394812yes—
371C2×C4#1309395112yes—
372C2×C4#1309395112yes—
373C2×C4C2×C4×Dic4711yescharacteristic
374C2×C4#1309398311yescharacteristic
375C2×C4C4×C22⋊D4712yes—
376C8—1884no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—22no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—22no—
385C22C4×C22⋊Dic4711yescharacteristic
386C22C2×#1309393911yescharacteristic
387C22#1568909711yescharacteristic
388C22C2×#1309394811yescharacteristic
389C22#1568911911yescharacteristic
390C22C2×#1309395111yescharacteristic
391C22C2×#1309395111yescharacteristic
392C22C4×#1062533612yes—
393C22C4×#1062533712yes—
394C4—22no—
395C4—22no—
396C4—22no—
397C4—21no—
398C4—21no—
399C4—41no—
400C4—42no—
401C4—42no—
402C4—81no—
403C4—81no—
404C4—1888no—
405C4#1568886014yesdirect product factor
406C2—22no—
407C2—22no—
408C2C2×#1568886011yescharacteristic
409C2#2024187411yescharacteristic
410C2#2024187911yescharacteristic
411C2#2024188711yescharacteristic
412C2C4×#1309393911yescharacteristic
413C2C4×#1309395111yescharacteristic
414C2C4×#1309394811yescharacteristic
subgroupG/H
centerC22×C4C22⋊Dic47
commutatorC2×C4×C47C2×C42
FrattiniC23×C4C22×D47
FittingC4×C22⋊Q8×C47C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenterC4×C22⋊Q8D47
hypocenterC47C4×#328
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
12. #15688860C4direct (N × H)
14. C2×C4×Q8×C47C4semidirect
16. C47×#391C4semidirect
17. C47×#389C4semidirect
22. C4×C4⋊C4×C47C4semidirect
66. C2×Q8×C47C42semidirect
101. C4⋊C4×C47C42semidirect
197. C4×C47#328semidirect
214. C2×C4×Q8Dic47semidirect
219. #391Dic47semidirect
220. #389Dic47semidirect
225. C4×C4⋊C4Dic47semidirect
231. C47C4×#328semidirect
274. C2×Q8C4×Dic47semidirect
312. C4⋊C4C4×Dic47semidirect
405. C4#15688860direct (N × H)

Maximal subgroups

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

subgroupindex
C4×C22⋊(C47⋊C8)2
C2×#156888602
C4×C47⋊#952
#202412662
C4×C22⋊Q8×C472
C4×#32847