Finite Group Databasebeta

← #26216023

Subgroups — #26216023

Order 12032.

1546 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 1546 conjugacy classes of subgroups — too many to draw. Only its 275 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×#15689596C211yesmaximal, characteristic
2#20245969C212yesmaximal
3#20243462C211yesmaximal, characteristic, complemented
4#20243461C214yesmaximal, complemented
5#20243460C212yesmaximal, complemented
6#20243119C214yesmaximal, complemented
7C47×#1819C211yesmaximal, characteristic, complemented
8C23×C22⋊D47C2211yescharacteristic
9#15689596—28no—
10C2×C4×C22⋊D47C414yescomplemented
11C2×(C22⋊D47)⋊C4C414yescomplemented
12C2×#13093980C2214yes—
13C2×#13093980C2212yes—
14C22×C22⋊Dic47C2211yescharacteristic
15C2×C22⋊C4×D47C2212yes—
16#15689023—28no—
17#15688979C2212yes—
18#15688978C2214yes—
19#15688890C2214yes—
20#15688889C2218yes—
21C22×C22⋊C4×C47C2211yescharacteristic
22C47×#379C2212yes—
23C47×#342C2214yes—
24C24×D47C2311yescharacteristic
25C22×C22⋊D47—28no—
26C22×C22⋊D47C2×C412yes—
27C22×C22⋊D47C2×C412yes—
28C22×C22⋊D47D418yes—
29C23×Dic47C2×C412yes—
30C23×Dic47C2311yescharacteristic
31C22×C4×D47C2×C414yes—
32C4×C22⋊D47—28no—
33C4×C22⋊D47—28no—
34(C22⋊D47)⋊C4—216no—
35#13093980—28no—
36#13093980—216no—
37C2×Dic47⋊C4—24no—
38C2×Dic47⋊C4C2×C414yes—
39C2×C22⋊Dic47—24no—
40C2×C22⋊Dic47—24no—
41C2×C22⋊Dic47C2×C412yes—
42C2×C22⋊Dic47C2314yes—
43C2×C22⋊Dic47C2312yes—
44C2×(C2×D47)⋊C4—28no—
45C2×(C2×D47)⋊C4C2×C414yes—
46C22⋊C4×D47—28no—
47C2×C4×Dic47—24no—
48C2×C4×Dic47C2×C414yes—
49C25×C47C2311yescharacteristic
50C23×C4×C47C2×C412yes—
51C2×C22⋊C4×C47—24no—
52C2×C22⋊C4×C47—24no—
53C2×C22⋊C4×C47C2×C412yes—
54C2×C22⋊C4×C47C2314yes—
55C2×C22⋊C4×C47C2312yes—
56C24×C47—24no—
57C24×C47—82no—
58C24×C47C22×C411yescharacteristic
59C24×C47C22×C411yescharacteristic
60C24×C47C2×D414yes—
61C24×C47C2411yescharacteristic
62C23×D47—24no—
63C23×D47—82no—
64C23×D47C22×C412yes—
65C23×D47C2×D414yes—
66C22×C4×C47—22no—
67C22×C4×C47—22no—
68C22×C4×C47—24no—
69C22×C4×C47—82no—
70C22×C4×C47C22×C414yes—
71C2×C22⋊D47—28no—
72C2×C22⋊D47—28no—
73C2×C22⋊D47—216no—
74C2×C22⋊D47—216no—
75C2×C22⋊D47C22⋊C418yes—
76C2×C22⋊D47C22⋊C418yes—
77C22×Dic47—24no—
78C22×Dic47—24no—
79C22×Dic47—82no—
80C22×Dic47C22×C414yes—
81C22×Dic47C22×C412yes—
82C22×Dic47C2×D414yes—
83C22×Dic47#4118yes—
84C2×C4×D47—28no—
85C2×C4×D47—84no—
86C22⋊C4×C47—24no—
87C22⋊C4×C47—24no—
88C22⋊C4×C47—28no—
89C22⋊C4×C47—84no—
90C22⋊Dic47—24no—
91C22⋊Dic47—24no—
92C22⋊Dic47—28no—
93C22⋊Dic47—84no—
94(C2×D47)⋊C4—28no—
95(C2×D47)⋊C4—88no—
96Dic47⋊C4—28no—
97Dic47⋊C4—48no—
98C4×Dic47—28no—
99C4×Dic47—48no—
100C23×C47—22no—
101C23×C47—22no—
102C23×C47—28no—
103C23×C47—42no—
104C23×C47—42no—
105C23×C47—42no—
106C23×C47—44no—
107C23×C47—44no—
108C23×C47—88no—
109C23×C47C2×C22⋊C412yes—
110C23×C47C2×C22⋊C412yes—
111C23×C47C23×C411yescharacteristic
112C23×C47C22×D412yes—
113C23×C47C2×#4114yes—
114C22×D47—24no—
115C22×D47—28no—
116C22×D47—44no—
117C22×D47—48no—
118C22×D47—88no—
119C22×D47C2×C22⋊C414yes—
120C2×C4×C47—24no—
121C2×C4×C47—44no—
122C2×C4×C47—48no—
123C2×C4×C47—44no—
124C2×C4×C47—82no—
125C2×C4×C47—82no—
126C22⋊D47—216no—
127C22⋊D47—216no—
128C2×Dic47—24no—
129C2×Dic47—44no—
130C2×Dic47—48no—
131C2×Dic47—44no—
132C2×Dic47—84no—
133C2×Dic47C2×C22⋊C414yes—
134C2×Dic47#11718yes—
135C2×Dic47#12318yes—
136C2×Dic47#12318yes—
137C4×D47—88no—
138#20690—471nomaximal, Sylow
139C22×C47—22no—
140C22×C47—24no—
141C22×C47—24no—
142C22×C47—22no—
143C22×C47—22no—
144C22×C47—22no—
145C22×C47—22no—
146C22×C47—44no—
147C22×C47—44no—
148C22×C47—44no—
149C22×C47—88no—
150C22×C47C22×C22⋊C411yescharacteristic
151C22×C47C2×#11712yes—
152C22×C47C2×#12314yes—
153C2×D47—28no—
154C2×D47—24no—
155C2×D47—48no—
156C2×D47—88no—
157C4×C47—44no—
158C4×C47—84no—
159Dic47—44no—
160Dic47—84no—
161Dic47#38118yes—
162C2×#518—471no—
163#2309—472no—
164#1819—472no—
165#1819—474no—
166#1819—471no—
167#1819D4711yescharacteristic, complemented
168#1740—474no—
169C2×C47—24no—
170C2×C47—22no—
171C2×C47—42no—
172C2×C47—42no—
173C2×C47C2×#38111yescharacteristic
174C2×C47#288312yes—
175D47—44no—
176C23×D4—471no—
177#518—948no—
178C2×C4×D4—474no—
179C2×C4×D4—474no—
180C2×#117—472no—
181C2×#117—474no—
182C22×C22⋊C4—471no—
183C22×C22⋊C4—472no—
184C22×C22⋊C4C2×D4711yescharacteristic
185#390—948no—
186#379—472no—
187#379—474no—
188#379C2×D4712yes—
189#342—478no—
190#342—474no—
191#342C2×D4714yes—
192C47#2069011yescharacteristic, complemented, Sylow
193C25—471no—
194C25C22×D4711yescharacteristic
195C22×D4—472no—
196C22×D4—472no—
197C22×D4—478no—
198C22×D4—948no—
199C23×C4—471no—
200C23×C4—472no—
201C23×C4—474no—
202C23×C4C4×D4712yes—
203C4×D4—9416no—
204C4×D4—948no—
205C4×D4—948no—
206#117—948no—
207#117—9416no—
208C2×C4⋊C4—474no—
209C2×C4⋊C4—944no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—472no—
213C2×C22⋊C4—472no—
214C2×C22⋊C4—474no—
215C2×C22⋊C4—474no—
216C2×C22⋊C4—948no—
217C2×C22⋊C4—944no—
218C2×C22⋊C4—948no—
219C2×C22⋊C4—944no—
220C2×C22⋊C4C4×D4712yes—
221C2×C22⋊C4C22×D4714yes—
222C2×C22⋊C4C22×D4712yes—
223C2×C42—474no—
224C2×C42—944no—
225C24—24no—
226C24—82no—
227C24—472no—
228C24—474no—
229C24—944no—
230C24—3762no—
231C24C2×C4×D4711yescharacteristic
232C24C2×C4×D4711yescharacteristic
233C24D4×D4714yes—
234C24C23×D4711yescharacteristic
235C2×D4—478no—
236C2×D4—478no—
237C2×D4—948no—
238C2×D4—948no—
239C2×D4—9416no—
240C2×D4—9416no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—82no—
245C22×C4—472no—
246C22×C4—474no—
247C22×C4—478no—
248C22×C4—474no—
249C22×C4—948no—
250C22×C4—944no—
251C22×C4—944no—
252C22×C4—3764no—
253C22×C4—3762no—
254C22×C4C2×C4×D4714yes—
255C4⋊C4—948no—
256C4⋊C4—1888no—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C22⋊C4—28no—
260C22⋊C4—84no—
261C22⋊C4—948no—
262C22⋊C4—944no—
263C22⋊C4—944no—
264C22⋊C4—948no—
265C22⋊C4—3768no—
266C22⋊C4—3764no—
267C42—948no—
268C42—1888no—
269C23—22no—
270C23—22no—
271C23—28no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—44no—
276C23—44no—
277C23—88no—
278C23—474no—
279C23—948no—
280C23—944no—
281C23—3768no—
282C23—1888no—
283C23—1884no—
284C23C22⋊C4×D4712yes—
285C23C22⋊C4×D4712yes—
286C23C22×C4×D4711yescharacteristic
287C23C2×D4×D4712yes—
288C23#41×D4714yes—
289D4—9416no—
290D4—9416no—
291C2×C4—24no—
292C2×C4—44no—
293C2×C4—48no—
294C2×C4—44no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—474no—
298C2×C4—478no—
299C2×C4—478no—
300C2×C4—478no—
301C2×C4—944no—
302C2×C4—3768no—
303C2×C4—3764no—
304C2×C4—1888no—
305C2×C4—1884no—
306C2×C4—1884no—
307C22—22no—
308C22—24no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—24no—
314C22—44no—
315C22—44no—
316C22—44no—
317C22—88no—
318C22—944no—
319C22—948no—
320C22—3768no—
321C22—1888no—
322C22C2×C22⋊C4×D4711yescharacteristic
323C22#117×D4712yes—
324C22#123×D4714yes—
325C4—44no—
326C4—84no—
327C4—478no—
328C4—3764no—
329C4—1884no—
330C2—24no—
331C2—22no—
332C2—42no—
333C2—42no—
334C2—1884no—
335C2#381×D4711yescharacteristic
336C2#2025184612yes—
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC47×#1819C2
socleC22×C47C22×C22⋊C4
prime radicalC23×C22⋊D47C22
cosocleC24×C47C24
hypercenter#1819D47
hypocenterC47#20690
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
3. #20243462C2semidirect
4. #20243461C2semidirect
5. #20243460C2semidirect
6. #20243119C2semidirect
7. C47×#1819C2semidirect
10. C2×C4×C22⋊D47C4semidirect
11. C2×(C22⋊D47)⋊C4C4semidirect
167. #1819D47semidirect
192. C47#20690semidirect

Maximal subgroups

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

subgroupindex
C2×#156895962
#202459692
#202434622
#202434612
#202434602
#202431192
C47×#18192
#2069047