Finite Group Databasebeta

← #26114033

Subgroups — #26114033

Order 12032.

726 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#20245974C211yesmaximal, characteristic, complemented
2#20243312C211yesmaximal, characteristic, complemented
3#20241917C212yesmaximal
4#20241241C212yesmaximal, complemented
5C47×#2436C211yesmaximal, characteristic
6C2×#13093983—21no—
7C2×#13093943—22no—
8#15689087C2212yes—
9#15689030—24no—
10#15688986—21no—
11#15688986C2212yes—
12#15688857—24no—
13C23×Q8×C47C2211yescharacteristic
14C2×#124×C47C412yescomplemented
15C2×#121×C47C2211yescharacteristic
16C47×#393—22no—
17C47×#391C412yescomplemented
18C47×#386—21no—
19C47×#386—21no—
20C4×C22⋊C4×C47C2211yescharacteristic
21C2×#10625329—22no—
22#13093983—44no—
23C2×C4⋊Dic47—22no—
24C2×C22⋊Dic47—22no—
25C2×C4×Dic47—22no—
26#13093943—44no—
27C22⋊(C47⋊C8)—22no—
28C22⋊(C47⋊C8)—22no—
29C47⋊#95—24no—
30C47⋊#95—22no—
31C47⋊#95D414yes—
32C22×Q8×C47—22no—
33C22×Q8×C47—22no—
34C22×Q8×C47—44no—
35C22×Q8×C47C2×C412yes—
36C22×Q8×C47D412yes—
37C22×D4×C47—21no—
38C22×D4×C47C2×C411yescharacteristic
39C23×C4×C47D411yescharacteristic
40C23×C4×C47D411yescharacteristic
41C23×C4×C47C2311yescharacteristic
42#124×C47—44no—
43#121×C47—22no—
44#121×C47—22no—
45C2×C4⋊C4×C47C2×C411yescharacteristic
46C2×C22⋊C4×C47—21no—
47C2×C22⋊C4×C47—22no—
48C2×C22⋊C4×C47—22no—
49C2×C22⋊C4×C47—21no—
50C2×C42×C47C2×C412yes—
51#95×C47—22no—
52#95×C47—22no—
53#95×C47—21no—
54C24×C47—21no—
55C24×C47C2×D411yescharacteristic
56C2×Q8×C47—24no—
57C2×Q8×C47—24no—
58C2×Q8×C47—24no—
59C2×Q8×C47—24no—
60C2×Q8×C47—42no—
61C2×Q8×C47—42no—
62C2×Q8×C47—44no—
63C2×Q8×C47—44no—
64C2×Q8×C47—48no—
65C2×D4×C47—22no—
66C2×D4×C47—42no—
67C2×D4×C47—44no—
68C2×D4×C47—41no—
69C2×D4×C47—41no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—24no—
73C22×C4×C47—21no—
74C22×C4×C47—21no—
75C22×C4×C47—22no—
76C22×C4×C47—21no—
77C22×C4×C47—42no—
78C22×C4×C47—42no—
79C22×C4×C47C22⋊C412yes—
80C22×C4×C47C22⋊C412yes—
81C22×C4×C47C22×C411yescharacteristic
82C22×C4×C47C2×D411yescharacteristic
83C22×C4×C47C2×D412yes—
84C22×C4×C47#4112yes—
85C22×Dic47—24no—
86C4⋊C4×C47—22no—
87C22⋊C4×C47—44no—
88C22⋊C4×C47—42no—
89C22⋊C4×C47—44no—
90C22⋊C4×C47—42no—
91C42×C47—42no—
92C22⋊Dic47—44no—
93C4⋊Dic47—44no—
94C4×Dic47—44no—
95#10625329—44no—
96C2×C47⋊C8—42no—
97C23×C47—22no—
98C23×C47—22no—
99C23×C47—41no—
100C23×C47—42no—
101C23×C47—42no—
102C23×C47—41no—
103C23×C47—42no—
104C23×C47C2×C22⋊C411yescharacteristic
105C23×C47#12011yescharacteristic
106C23×C47#12311yescharacteristic
107Q8×C47—48no—
108Q8×C47—48no—
109D4×C47—82no—
110D4×C47—82no—
111D4×C47—82no—
112C2×C4×C47—22no—
113C2×C4×C47—22no—
114C2×C4×C47—24no—
115C2×C4×C47—22no—
116C2×C4×C47—22no—
117C2×C4×C47—22no—
118C2×C4×C47—22no—
119C2×C4×C47—22no—
120C2×C4×C47—22no—
121C2×C4×C47—22no—
122C2×C4×C47—42no—
123C2×C4×C47—42no—
124C2×C4×C47—42no—
125C2×C4×C47—42no—
126C2×C4×C47—42no—
127C2×C4×C47—42no—
128C2×C4×C47—42no—
129C2×C4×C47—41no—
130C2×C4×C47—42no—
131C2×C4×C47#9914yes—
132C2×C4×C47C4×D412yes—
133C2×C4×C47#12112yes—
134C2×Dic47—44no—
135C2×Dic47—44no—
136C2×Dic47—44no—
137C47⋊C8—82no—
138#9466—471nomaximal, Sylow
139C22×C47—21no—
140C22×C47—21no—
141C22×C47—21no—
142C22×C47—21no—
143C22×C47—21no—
144C22×C47—21no—
145C22×C47—21no—
146C22×C47—44no—
147C22×C47—84no—
148C22×C47—81no—
149C22×C47—81no—
150C22×C47—81no—
151C22×C47#38611yescharacteristic
152C22×C47C2×#9912yes—
153C22×C47#45412yes—
154C4×C47—22no—
155C4×C47—24no—
156C4×C47—22no—
157C4×C47—42no—
158C4×C47—82no—
159C4×C47—81no—
160Dic47—84no—
161#2436D4711yescharacteristic
162#2310—471no—
163#1784—471no—
164#1510—472no—
165#1284—472no—
166C2×C47—22no—
167C2×C47—22no—
168C2×C47—41no—
169C2×C47—161no—
170C2×C47#188811yescharacteristic
171C2×C47#189811yescharacteristic
172C2×C47#191211yescharacteristic
173C23×Q8C2×D4711yescharacteristic
174C2×#124Dic4712yescomplemented
175C2×#121C2×D4711yescharacteristic
176C2×#117—941no—
177C2×#101—942no—
178#407—472no—
179#393—22no—
180#391—944no—
181#391Dic4712yescomplemented
182#386—21no—
183#386—21no—
184#381—472no—
185#381—941no—
186C4×C22⋊C4C2×D4711yescharacteristic
187#327—944no—
188C47#946611yescharacteristic, complemented, Sylow
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—44no—
192C22×Q8C2×Dic4712yes—
193C22×Q8C22⋊D4712yes—
194C22×D4—21no—
195C22×D4C2×Dic4711yescharacteristic
196C23×C4C22⋊D4711yescharacteristic
197C23×C4C22⋊D4711yescharacteristic
198C23×C4C22×D4711yescharacteristic
199C2×M16—942no—
200#124—44no—
201#121—22no—
202#121—22no—
203#117—1884no—
204C2×C4⋊C4—942no—
205C2×C4⋊C4C2×Dic4711yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—942no—
211C2×C42—942no—
212C2×C42C2×Dic4712yes—
213#101—1884no—
214C22⋊C8—942no—
215C22⋊C8—942no—
216#95—22no—
217#95—22no—
218#95—21no—
219#95—474no—
220#95—944no—
221#95—942no—
222C24—21no—
223C24C2×C22⋊D4711yescharacteristic
224C2×Q8—24no—
225C2×Q8—24no—
226C2×Q8—24no—
227C2×Q8—24no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—44no—
231C2×Q8—44no—
232C2×Q8—48no—
233C2×D4—22no—
234C2×D4—42no—
235C2×D4—44no—
236C2×D4—41no—
237C2×D4—41no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—24no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—944no—
248C22×C4C22⋊Dic4712yes—
249C22×C4C22⋊Dic4712yes—
250C22×C4D4×D4712yes—
251C22×C4#1062535212yes—
252C22×C4C22×Dic4711yescharacteristic
253C22×C4C2×C22⋊D4711yescharacteristic
254M16—1884no—
255C2×C8—1882no—
256C4⋊C4—22no—
257C4⋊C4—1884no—
258C22⋊C4—44no—
259C22⋊C4—42no—
260C22⋊C4—44no—
261C22⋊C4—42no—
262C22⋊C4—1884no—
263C42—42no—
264C42—1884no—
265C23—22no—
266C23—22no—
267C23—41no—
268C23—42no—
269C23—42no—
270C23—41no—
271C23—42no—
272C23C2×C22⋊Dic4711yescharacteristic
273C23#1309399211yescharacteristic
274C23#1309400511yescharacteristic
275Q8—48no—
276Q8—48no—
277D4—82no—
278D4—82no—
279D4—82no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—1884no—
300C2×C4—1884no—
301C2×C4—1884no—
302C2×C4#1309393914yes—
303C2×C4D4×Dic4712yes—
304C2×C4#1309399612yes—
305C8—3762no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—44no—
314C22—84no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22#1568900311yescharacteristic
319C22C2×#1309393912yes—
320C22#1568929512yes—
321C4—22no—
322C4—24no—
323C4—22no—
324C4—42no—
325C4—82no—
326C4—81no—
327C4—3764no—
328C2—22no—
329C2—22no—
330C2—41no—
331C2—161no—
332C2#2024379211yescharacteristic
333C2#2024384211yescharacteristic
334C2#2024392611yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2436C2
socleC22×C47#386
prime radicalC2×#121×C47C22
cosocleC23×C4×C47C23
hypercenter#2436D47
hypocenterC47#9466
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2436

Complements

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

NH ≅ G/Ntype
1. #20245974C2semidirect
2. #20243312C2semidirect
4. #20241241C2semidirect
14. C2×#124×C47C4semidirect
17. C47×#391C4semidirect
174. C2×#124Dic47semidirect
181. #391Dic47semidirect
188. C47#9466semidirect

Maximal subgroups

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

subgroupindex
#202459742
#202433122
#202419172
#202412412
C47×#24362
#946647