Finite Group Databasebeta

← C2×#20241409 #26126549

Subgroups — C2×#20241409 #26126549

Order 12032.

776 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688890C211yesmaximal, characteristic
2C2×#15688847C211yesmaximal, characteristic
3#20241409C214yesmaximal, direct product factor
4C2×C47×#393C211yesmaximal, characteristic
5C22×C22⋊Dic47—21no—
6C2×C22⋊(C47⋊C8)—21no—
7#15688890—21no—
8#15688890C2212yes—
9#15688847—21no—
10#15688847C2212yes—
11C23×Q8×C47C411yescharacteristic, complemented
12C22×C22⋊C4×C47—21no—
13C22×C22⋊C4×C47C2211yescharacteristic
14C47×#393—21no—
15C47×#393—21no—
16C47×#393C412yescomplemented
17C47×#393C2212yes—
18C2×#95×C47—21no—
19C2×#95×C47C411yescharacteristic, complemented
20C23×Dic47—41no—
21C22×C47⋊C8—41no—
22C2×C22⋊Dic47—24no—
23C2×C22⋊Dic47—22no—
24C2×C22⋊Dic47—41no—
25C2×C22⋊Dic47—42no—
26C22⋊(C47⋊C8)—22no—
27C22⋊(C47⋊C8)—41no—
28C25×C47D411yescharacteristic
29C22×Q8×C47—21no—
30C22×Q8×C47—44no—
31C22×Q8×C47—41no—
32C22×Q8×C47—41no—
33C22×Q8×C47C2×C412yescomplemented
34C23×C4×C47—21no—
35C23×C4×C47—21no—
36C23×C4×C47C2×C411yescharacteristic
37C23×C4×C47D411yescharacteristic
38C2×C22⋊C4×C47—22no—
39C2×C22⋊C4×C47—22no—
40C2×C22⋊C4×C47—41no—
41C2×C22⋊C4×C47—41no—
42C2×C22⋊C4×C47—44no—
43C2×C22⋊C4×C47C2×C411yescharacteristic
44C2×C22⋊C4×C47D411yescharacteristic
45C2×C22⋊C4×C47D411yescharacteristic
46C2×C22⋊C4×C47C2311yescharacteristic
47#95×C47—22no—
48#95×C47—21no—
49#95×C47—22no—
50#95×C47C2×C412yescomplemented
51C24×C47—21no—
52C24×C47—21no—
53C24×C47—22no—
54C24×C47—42no—
55C24×C47—42no—
56C24×C47C22⋊C411yescharacteristic
57C24×C47C22⋊C411yescharacteristic
58C24×C47SD1612yes—
59C24×C47Q1612yes—
60C24×C47C2×D411yescharacteristic
61C2×Q8×C47—44no—
62C2×Q8×C47—42no—
63C2×Q8×C47—44no—
64C2×Q8×C47—42no—
65C2×Q8×C47—48no—
66C2×Q8×C47—44no—
67C2×Q8×C47—44no—
68C22×C4×C47—24no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—22no—
73C22×C4×C47—21no—
74C22×C4×C47—22no—
75C22×C4×C47—42no—
76C22×C4×C47—41no—
77C22×C4×C47—41no—
78C22×C4×C47—42no—
79C22×C4×C47—42no—
80C22×C4×C47—42no—
81C22×C4×C47—42no—
82C22×C4×C47—42no—
83C22×C4×C47C22⋊C411yescharacteristic
84C22×C4×C47C22⋊C411yescharacteristic
85C22×C4×C47C22×C411yescharacteristic
86C22×C4×C47C2×D411yescharacteristic
87C22×Dic47—44no—
88C22×Dic47—42no—
89C22×Dic47—82no—
90C22×Dic47—82no—
91C22⋊C4×C47—41no—
92C22⋊C4×C47—41no—
93C22⋊C4×C47—41no—
94C22⋊C4×C47—41no—
95C22⋊C4×C47—44no—
96C22⋊C4×C47—44no—
97C22⋊Dic47—24no—
98C22⋊Dic47—44no—
99C22⋊Dic47—42no—
100C2×C47⋊C8—42no—
101C2×C47⋊C8—82no—
102C23×C47—22no—
103C23×C47—22no—
104C23×C47—22no—
105C23×C47—22no—
106C23×C47—21no—
107C23×C47—21no—
108C23×C47—22no—
109C23×C47—22no—
110C23×C47—21no—
111C23×C47—21no—
112C23×C47—41no—
113C23×C47—41no—
114C23×C47—41no—
115C23×C47—44no—
116C23×C47—44no—
117C23×C47—41no—
118C23×C47—42no—
119C23×C47—42no—
120C23×C47—81no—
121C23×C47—84no—
122C23×C47—81no—
123C23×C47#9911yescharacteristic
124C23×C47#9911yescharacteristic
125C23×C47#10312yes—
126C23×C47#10312yes—
127C23×C47#10412yes—
128C23×C47C2×C22⋊C411yescharacteristic
129C23×C47C2×SD1611yescharacteristic
130C23×C47C2×Q1611yescharacteristic
131Q8×C47—44no—
132Q8×C47—48no—
133Q8×C47—44no—
134C2×C4×C47—24no—
135C2×C4×C47—21no—
136C2×C4×C47—21no—
137C2×C4×C47—21no—
138C2×C4×C47—21no—
139C2×C4×C47—42no—
140C2×C4×C47—44no—
141C2×C4×C47—42no—
142C2×C4×C47—44no—
143C2×C4×C47—42no—
144C2×C4×C47—42no—
145C2×C4×C47—42no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—41no—
150C2×C4×C47—42no—
151C2×C4×C47—42no—
152C2×C4×C47—42no—
153C2×C4×C47—81no—
154C2×C4×C47—84no—
155C2×Dic47—44no—
156C2×Dic47—82no—
157C2×Dic47—88no—
158C2×Dic47—82no—
159C47⋊C8—82no—
160C2×#1351—471nomaximal, Sylow
161C22×C47—21no—
162C22×C47—21no—
163C22×C47—24no—
164C22×C47—24no—
165C22×C47—21no—
166C22×C47—21no—
167C22×C47—21no—
168C22×C47—22no—
169C22×C47—21no—
170C22×C47—42no—
171C22×C47—41no—
172C22×C47—41no—
173C22×C47—41no—
174C22×C47—41no—
175C22×C47—44no—
176C22×C47—44no—
177C22×C47—41no—
178C22×C47—41no—
179C22×C47—81no—
180C22×C47—81no—
181C22×C47—82no—
182C22×C47—82no—
183C22×C47#32812yes—
184C22×C47#32812yes—
185C22×C47AΣL(1,𝔽16)12yes—
186C22×C47#35412yes—
187C22×C47C2×#9911yescharacteristic
188C22×C47C2×#10311yescharacteristic
189C22×C47C2×#10411yescharacteristic
190C4×C47—44no—
191C4×C47—41no—
192C4×C47—41no—
193C4×C47—41no—
194C4×C47—41no—
195C4×C47—82no—
196Dic47—84no—
197C2×#393D4711yescharacteristic
198C2×#342—471no—
199C2×#323—471no—
200#1351—474no—
201C2×C47—21no—
202C2×C47—21no—
203C2×C47—22no—
204C2×C47—41no—
205C2×C47—42no—
206C2×C47—41no—
207C2×C47#135114yescomplemented
208C2×C47C2×#32811yescharacteristic
209C2×C47C2×AΣL(1,𝔽16)11yescharacteristic
210C2×C47C2×#35411yescharacteristic
211C23×Q8Dic4711yescharacteristic, complemented
212C22×C22⋊C4—21no—
213C22×C22⋊C4—941no—
214C22×C22⋊C4C2×D4711yescharacteristic
215C2×C22⋊C8—941no—
216#393—21no—
217#393—21no—
218#393Dic4712yescomplemented
219#393C2×D4712yes—
220C2×#95—21no—
221C2×#95Dic4711yescharacteristic, complemented
222#342—472no—
223#342—941no—
224#323—472no—
225#323—941no—
226C47C2×#135111yescharacteristic, complemented, Sylow
227C25C22⋊D4711yescharacteristic
228C22×Q8—21no—
229C22×Q8—44no—
230C22×Q8—41no—
231C22×Q8—41no—
232C22×Q8C2×Dic4712yescomplemented
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—1881no—
236C23×C4C2×Dic4711yescharacteristic
237C23×C4C22⋊D4711yescharacteristic
238C22×C8—1881no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—944no—
245C2×C22⋊C4—942no—
246C2×C22⋊C4—1881no—
247C2×C22⋊C4—1882no—
248C2×C22⋊C4C2×Dic4711yescharacteristic
249C2×C22⋊C4C22⋊D4711yescharacteristic
250C2×C22⋊C4C22⋊D4711yescharacteristic
251C2×C22⋊C4C22×D4711yescharacteristic
252C22⋊C8—942no—
253C22⋊C8—1881no—
254#95—22no—
255#95—21no—
256#95—22no—
257#95C2×Dic4712yescomplemented
258C24—21no—
259C24—21no—
260C24—22no—
261C24—42no—
262C24—42no—
263C24#1062533612yes—
264C24#1062533712yes—
265C24C22⋊Dic4711yescharacteristic
266C24C22⋊Dic4711yescharacteristic
267C24C2×C22⋊D4711yescharacteristic
268C2×Q8—44no—
269C2×Q8—44no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—48no—
273C2×Q8—44no—
274C2×Q8—44no—
275C22×C4—24no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—3762no—
291C22×C4—3762no—
292C22×C4—1882no—
293C22×C4—1884no—
294C22×C4C22⋊Dic4711yescharacteristic
295C22×C4C22⋊Dic4711yescharacteristic
296C22×C4C22×Dic4711yescharacteristic
297C22×C4C2×C22⋊D4711yescharacteristic
298C2×C8—3762no—
299C2×C8—1882no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—44no—
305C22⋊C4—44no—
306C22⋊C4—944no—
307C22⋊C4—1884no—
308C22⋊C4—1882no—
309C23—22no—
310C23—22no—
311C23—21no—
312C23—21no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—21no—
318C23—21no—
319C23—41no—
320C23—41no—
321C23—44no—
322C23—44no—
323C23—41no—
324C23—41no—
325C23—42no—
326C23—42no—
327C23—84no—
328C23—81no—
329C23—81no—
330C23#1309393911yescharacteristic
331C23#1309393911yescharacteristic
332C23#1309394812yes—
333C23#1309394812yes—
334C23#1309395112yes—
335C23C2×C22⋊Dic4711yescharacteristic
336C23C2×#1062533611yescharacteristic
337C23C2×#1062533711yescharacteristic
338Q8—44no—
339Q8—44no—
340Q8—48no—
341C2×C4—24no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4—84no—
362C2×C4—3768no—
363C2×C4—3762no—
364C2×C4—3762no—
365C2×C4—1884no—
366C8—3762no—
367C22—24no—
368C22—24no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—22no—
375C22—21no—
376C22—42no—
377C22—41no—
378C22—41no—
379C22—44no—
380C22—44no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—81no—
386C22—81no—
387C22—82no—
388C22—82no—
389C22#1568886012yes—
390C22#1568886012yes—
391C22#1568891012yes—
392C22#1568891812yes—
393C22C2×#1309393911yescharacteristic
394C22C2×#1309394811yescharacteristic
395C22C2×#1309395111yescharacteristic
396C4—44no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4—82no—
402C4—3764no—
403C2—21no—
404C2—21no—
405C2—22no—
406C2—41no—
407C2—42no—
408C2—41no—
409C2#2024140914yesdirect product factor
410C2C2×#1568886011yescharacteristic
411C2C2×#1568891011yescharacteristic
412C2C2×#1568891811yescharacteristic
subgroupG/H
centerC23#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC2×C47×#393C2
socleC23×C47#99
prime radicalC25×C47D4
cosocleC2×C22⋊C4×C47C23
hypercenterC2×#393D47
hypocenterC47C2×#1351
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
3. #20241409C2direct (N × H)
11. C23×Q8×C47C4semidirect
16. C47×#393C4semidirect
19. C2×#95×C47C4semidirect
33. C22×Q8×C47C2×C4semidirect
50. #95×C47C2×C4semidirect
207. C2×C47#1351semidirect
211. C23×Q8Dic47semidirect
218. #393Dic47semidirect
221. C2×#95Dic47semidirect
226. C47C2×#1351semidirect
232. C22×Q8C2×Dic47semidirect
257. #95C2×Dic47semidirect
409. C2#20241409direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156888902
C2×#156888472
#202414092
C2×C47×#3932
C2×#135147