Finite Group Databasebeta

← #26126830

Subgroups — #26126830

Order 12032.

786 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×#13093939C211yesmaximal, characteristic, complemented
2#20241729C211yesmaximal, characteristic, complemented
3#20241385C214yesmaximal, complemented
4C47×#2408C211yesmaximal, characteristic
5#15689180—21no—
6C2×#13093939—21no—
7C4×C22⋊Dic47—21no—
8#15688890C2212yescomplemented
9#15688847—21no—
10#15688847C2212yescomplemented
11C23×D4×C47C411yescharacteristic, complemented
12C2×#121×C47—21no—
13C2×#120×C47—21no—
14C2×C4×D4×C47C2211yescharacteristic
15C47×#394C412yescomplemented
16C47×#392C2212yes—
17C47×#390—21no—
18C47×#380C411yescharacteristic, complemented
19C47×#379—21no—
20C2×C22⋊Dic47—22no—
21C2×C4×Dic47—41no—
22C4⋊(C47⋊C8)—41no—
23#13093939—24no—
24C22⋊(C47⋊C8)—22no—
25#13093935—41no—
26C47⋊#95—41no—
27C25×C47—21no—
28C22×D4×C47—22no—
29C22×D4×C47—21no—
30C22×D4×C47—41no—
31C22×D4×C47—41no—
32C22×D4×C47—44no—
33C22×D4×C47C2×C412yes—
34C22×D4×C47D411yescharacteristic
35C23×C4×C47C2×C411yescharacteristic
36C23×C4×C47D411yescharacteristic
37#121×C47—44no—
38#120×C47—44no—
39C4×D4×C47—41no—
40C4×D4×C47—41no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47C2×C411yescharacteristic
43C2×C22⋊C4×C47—21no—
44C2×C22⋊C4×C47—22no—
45C2×C22⋊C4×C47—22no—
46C2×C22⋊C4×C47—22no—
47C2×C22⋊C4×C47D411yescharacteristic
48C2×C22⋊C4×C47C2311yescharacteristic
49C2×C42×C47D411yescharacteristic
50#95×C47C2×C412yes—
51C24×C47—22no—
52C24×C47—42no—
53C24×C47—44no—
54C24×C47—42no—
55C24×C47—42no—
56C24×C47—82no—
57C24×C47C22⋊C411yescharacteristic
58C24×C47C2×D411yescharacteristic
59C2×D4×C47—24no—
60C2×D4×C47—22no—
61C2×D4×C47—22no—
62C2×D4×C47—41no—
63C2×D4×C47—42no—
64C2×D4×C47—42no—
65C2×D4×C47—48no—
66C2×D4×C47—44no—
67C2×D4×C47—44no—
68C2×D4×C47—44no—
69C2×D4×C47—44no—
70C2×D4×C47—44no—
71C2×D4×C47—41no—
72C2×D4×C47—41no—
73C2×D4×C47—82no—
74C22×C4×C47—21no—
75C22×C4×C47—21no—
76C22×C4×C47—21no—
77C22×C4×C47—21no—
78C22×C4×C47—22no—
79C22×C4×C47—42no—
80C22×C4×C47—42no—
81C22×C4×C47C22⋊C411yescharacteristic
82C22×C4×C47C22⋊C411yescharacteristic
83C22×C4×C47C22⋊C411yescharacteristic
84C22×C4×C47C22×C411yescharacteristic
85C22×C4×C47C2×D411yescharacteristic
86C22×Dic47—42no—
87C4⋊C4×C47—41no—
88C4⋊C4×C47—42no—
89C22⋊C4×C47—41no—
90C22⋊C4×C47—41no—
91C22⋊C4×C47—44no—
92C22⋊C4×C47—42no—
93C22⋊C4×C47—44no—
94C22⋊C4×C47—82no—
95C42×C47—21no—
96C42×C47—21no—
97C22⋊Dic47—24no—
98C22⋊Dic47—42no—
99C4×Dic47—82no—
100C2×C47⋊C8—42no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47—21no—
104C23×C47—21no—
105C23×C47—24no—
106C23×C47—41no—
107C23×C47—41no—
108C23×C47—41no—
109C23×C47—41no—
110C23×C47—42no—
111C23×C47—42no—
112C23×C47—42no—
113C23×C47—48no—
114C23×C47—42no—
115C23×C47—42no—
116C23×C47—42no—
117C23×C47—84no—
118C23×C47—88no—
119C23×C47—88no—
120C23×C47—84no—
121C23×C47#10412yes—
122C23×C47C2×C22⋊C411yescharacteristic
123C23×C47AGL(1,ℤ/8)12yes—
124D4×C47—44no—
125D4×C47—44no—
126D4×C47—44no—
127D4×C47—44no—
128D4×C47—44no—
129D4×C47—88no—
130C2×C4×C47—22no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—21no—
135C2×C4×C47—21no—
136C2×C4×C47—21no—
137C2×C4×C47—42no—
138C2×C4×C47—42no—
139C2×C4×C47—42no—
140C2×C4×C47—41no—
141C2×C4×C47—41no—
142C2×C4×C47—41no—
143C2×C4×C47—42no—
144C2×C4×C47—42no—
145C2×C4×C47—42no—
146C2×C4×C47—81no—
147C2×C4×C47#9911yescharacteristic
148C2×C4×C47#9911yescharacteristic
149C2×Dic47—44no—
150C2×Dic47—82no—
151C2×Dic47—82no—
152C47⋊C8—82no—
153#11772—471nomaximal, Sylow
154C22×C47—21no—
155C22×C47—21no—
156C22×C47—22no—
157C22×C47—22no—
158C22×C47—42no—
159C22×C47—42no—
160C22×C47—42no—
161C22×C47—44no—
162C22×C47—44no—
163C22×C47—41no—
164C22×C47—41no—
165C22×C47—41no—
166C22×C47—41no—
167C22×C47—44no—
168C22×C47—81no—
169C22×C47—82no—
170C22×C47—82no—
171C22×C47—816no—
172C22×C47—82no—
173C22×C47—82no—
174C22×C47C2×#9911yescharacteristic
175C22×C47#41811yescharacteristic
176C22×C47C2×#10411yescharacteristic
177C4×C47—21no—
178C4×C47—41no—
179C4×C47—41no—
180C4×C47—41no—
181C4×C47—41no—
182C4×C47—82no—
183C4×C47AΣL(1,𝔽16)12yes—
184Dic47—84no—
185#2408D4711yescharacteristic
186C4×#99—471no—
187#1466—471no—
188#1343—474no—
189C2×C47—21no—
190C2×C47—21no—
191C2×C47—42no—
192C2×C47—42no—
193C2×C47—84no—
194C2×C47#150711yescharacteristic
195C2×C47C2×AΣL(1,𝔽16)11yescharacteristic
196C2×C47#212811yescharacteristic
197C23×D4Dic4711yescharacteristic, complemented
198C2×#121—21no—
199C2×#120—21no—
200C2×C4×D4C2×D4711yescharacteristic
201#432—941no—
202C2×#99—941no—
203#394Dic4712yescomplemented
204#392C2×D4712yes—
205#390—21no—
206#380Dic4711yescharacteristic, complemented
207#379—21no—
208C4×C22⋊C4—941no—
209#342—472no—
210#323—472no—
211#323—941no—
212C47#1177211yescharacteristic, complemented, Sylow
213C25—21no—
214C22×D4—22no—
215C22×D4—21no—
216C22×D4—41no—
217C22×D4—41no—
218C22×D4—44no—
219C22×D4C2×Dic4712yes—
220C22×D4C22⋊D4711yescharacteristic
221C23×C4C2×Dic4711yescharacteristic
222C23×C4C22⋊D4711yescharacteristic
223#121—44no—
224#120—44no—
225C4×D4—41no—
226C4×D4—41no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4C2×Dic4711yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—942no—
234C2×C22⋊C4C22⋊D4711yescharacteristic
235C2×C22⋊C4C22×D4711yescharacteristic
236C2×C42—1881no—
237C2×C42C22⋊D4711yescharacteristic
238C4⋊C8—1881no—
239#99—944no—
240C22⋊C8—942no—
241#97—1881no—
242#95—1881no—
243#95C2×Dic4712yes—
244C24—22no—
245C24—42no—
246C24—44no—
247C24—42no—
248C24—42no—
249C24—82no—
250C24C22⋊Dic4711yescharacteristic
251C24C2×C22⋊D4711yescharacteristic
252C2×D4—24no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—48no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—44no—
262C2×D4—44no—
263C2×D4—44no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—82no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—1882no—
275C22×C4C22⋊Dic4711yescharacteristic
276C22×C4C22⋊Dic4711yescharacteristic
277C22×C4C22⋊Dic4711yescharacteristic
278C22×C4C22×Dic4711yescharacteristic
279C22×C4C2×C22⋊D4711yescharacteristic
280C2×C8—1882no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—44no—
286C22⋊C4—42no—
287C22⋊C4—44no—
288C22⋊C4—82no—
289C22⋊C4—944no—
290C22⋊C4—1882no—
291C42—21no—
292C42—21no—
293C42—3762no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—24no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—48no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—84no—
311C23—88no—
312C23—88no—
313C23—84no—
314C23#1309395112yes—
315C23C2×C22⋊Dic4711yescharacteristic
316C23#1309405012yes—
317D4—44no—
318D4—44no—
319D4—44no—
320D4—44no—
321D4—44no—
322D4—88no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—81no—
340C2×C4—3762no—
341C2×C4—3762no—
342C2×C4—1884no—
343C2×C4#1309393911yescharacteristic
344C2×C4#1309393911yescharacteristic
345C8—3762no—
346C22—21no—
347C22—21no—
348C22—22no—
349C22—22no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—44no—
354C22—44no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—44no—
360C22—81no—
361C22—82no—
362C22—82no—
363C22—816no—
364C22—82no—
365C22—82no—
366C22C2×#1309393911yescharacteristic
367C22#1568912811yescharacteristic
368C22C2×#1309395111yescharacteristic
369C4—21no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—82no—
375C4—3764no—
376C4#1568891012yes—
377C2—21no—
378C2—21no—
379C2—42no—
380C2—42no—
381C2—84no—
382C2#2024190411yescharacteristic
383C2C2×#1568891011yescharacteristic
384C2#2024506011yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
Fitting, prime radicalC47×#2408C2
socleC22×C47C2×#99
cosocleC2×C22⋊C4×C47C23
hypercenter#2408D47
hypocenterC47#11772
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. C4×#13093939C2semidirect
2. #20241729C2semidirect
3. #20241385C2semidirect
8. #15688890C22semidirect
10. #15688847C22semidirect
11. C23×D4×C47C4semidirect
15. C47×#394C4semidirect
18. C47×#380C4semidirect
197. C23×D4Dic47semidirect
203. #394Dic47semidirect
206. #380Dic47semidirect
212. C47#11772semidirect

Maximal subgroups

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

subgroupindex
C4×#130939392
#202417292
#202413852
C47×#24082
#1177247