Finite Group Databasebeta

← #26114005

Subgroups — #26114005

Order 12032.

528 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 99 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#20247862C211yesmaximal, characteristic
2#20245991C211yesmaximal, characteristic
3#20241922C212yesmaximal
4#20241242C212yesmaximal
5C47×#1821C211yesmaximal, characteristic
6C2×#13094003C2211yescharacteristic
7#15689054—22no—
8#15689028C412yescomplemented
9#15689009—24no—
10#15688996—21no—
11#15688996—21no—
12#15688992C412yescomplemented
13#15688986C2212yes—
14#15688983—21no—
15C4×C22⋊Dic47—21no—
16C4×C22⋊Dic47C2211yescharacteristic
17#15688861—24no—
18C22×C4⋊C4×C47C2211yescharacteristic
19C47×#407C2212yes—
20C47×#343—22no—
21C22×Dic94C2×C411yescharacteristic
22#13094003—22no—
23#13094003—22no—
24C2×C4⋊Dic47—22no—
25C2×C4⋊Dic47C2×C411yescharacteristic
26C2×Dic47⋊C4—21no—
27C2×Dic47⋊C4—22no—
28C2×Dic47⋊C4—22no—
29C2×C22⋊Dic47—21no—
30C2×C22⋊Dic47—22no—
31C2×C22⋊Dic47—21no—
32C2×C4×Dic47—22no—
33C2×C4×Dic47C2×C412yes—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C47⋊#95—22no—
37C47⋊#95—22no—
38C47⋊#95—21no—
39C47⋊#95—22no—
40C47⋊#95D414yes—
41C23×C4×C47D411yescharacteristic
42C23×C4×C47D411yescharacteristic
43C23×C4×C47C2311yescharacteristic
44C2×M16×C47—22no—
45C2×C4⋊C4×C47—22no—
46C2×C4⋊C4×C47—22no—
47C2×C4⋊C4×C47C2×C412yes—
48C2×C4⋊C4×C47D412yes—
49C22⋊C8×C47—22no—
50C22⋊C8×C47—22no—
51C24×C47C2×D411yescharacteristic
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—42no—
57C22×C4×C47—81no—
58C22×C4×C47—81no—
59C22×C4×C47C22⋊C412yes—
60C22×C4×C47C22⋊C412yes—
61C22×C4×C47C22×C411yescharacteristic
62C22×C4×C47C2×D411yescharacteristic
63C22×C4×C47C2×D412yes—
64C22×C4×C47#4112yes—
65C22×Dic47—22no—
66C22×Dic47—21no—
67C22×Dic47—21no—
68C22×Dic47—24no—
69C2×Dic94—22no—
70C2×Dic94—42no—
71M16×C47—44no—
72C2×C8×C47—42no—
73C4⋊C4×C47—24no—
74C4⋊C4×C47—24no—
75C22⋊Dic47—24no—
76C22⋊Dic47—44no—
77C22⋊Dic47—41no—
78C22⋊Dic47—41no—
79C4⋊Dic47—22no—
80C4⋊Dic47—82no—
81Dic47⋊C4—42no—
82Dic47⋊C4—82no—
83Dic47⋊C4—82no—
84C4×Dic47—42no—
85C4×Dic47—82no—
86C23×C47—22no—
87C23×C47—22no—
88C23×C47—41no—
89C23×C47C2×C22⋊C411yescharacteristic
90C23×C47#12011yescharacteristic
91C23×C47#12311yescharacteristic
92C2×C4×C47—22no—
93C2×C4×C47—22no—
94C2×C4×C47—22no—
95C2×C4×C47—22no—
96C2×C4×C47—42no—
97C2×C4×C47—42no—
98C2×C4×C47—42no—
99C2×C4×C47—42no—
100C2×C4×C47—42no—
101C2×C4×C47—42no—
102C2×C4×C47—81no—
103C2×C4×C47—81no—
104C2×C4×C47C4×D412yes—
105C2×C4×C47#12112yes—
106C2×Dic47—24no—
107C2×Dic47—42no—
108C2×Dic47—42no—
109C2×Dic47—44no—
110C2×Dic47—42no—
111C2×Dic47—44no—
112C2×Dic47—41no—
113C2×Dic47—41no—
114C2×Dic47—42no—
115C2×Dic47—44no—
116Dic94—82no—
117C8×C47—82no—
118#9461—471nomaximal, Sylow
119C22×C47—21no—
120C22×C47—21no—
121C22×C47—21no—
122C22×C47—21no—
123C22×C47—21no—
124C22×C47—21no—
125C22×C47—21no—
126C22×C47—44no—
127C22×C47#38611yescharacteristic
128C22×C47#41012yes—
129C22×C47#45612yes—
130C4×C47—22no—
131C4×C47—22no—
132C4×C47—42no—
133C4×C47—42no—
134Dic47—44no—
135Dic47—84no—
136Dic47—81no—
137Dic47—81no—
138#2502—471no—
139#2313—471no—
140#1821D4711yescharacteristic
141#1511—472no—
142#1284—472no—
143C2×C47—22no—
144C2×C47—22no—
145C2×C47—41no—
146C2×C47#188611yescharacteristic
147C2×C47#189911yescharacteristic
148C2×C47#191111yescharacteristic
149C2×C22⋊Q8—471no—
150C22×C4⋊C4C2×D4711yescharacteristic
151#407C2×D4712yes—
152#398—942no—
153#391—472no—
154#387—944no—
155#385—941no—
156#385—941no—
157#384—472no—
158#381—472no—
159#380—941no—
160C4×C22⋊C4—471no—
161C4×C22⋊C4—941no—
162#343—22no—
163#328—944no—
164C47#946111yescharacteristic, complemented, Sylow
165C22×Q8—471no—
166C23×C4D18811yescharacteristic
167C23×C4D18811yescharacteristic
168C23×C4C22×D4711yescharacteristic
169C2×M16—22no—
170C22⋊Q8—942no—
171C22⋊Q8—942no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—471no—
175C2×C4⋊C4—942no—
176C2×C4⋊C4—942no—
177C2×C4⋊C4—942no—
178C2×C4⋊C4—941no—
179C2×C4⋊C4C4×D4712yes—
180C2×C4⋊C4C22⋊D4712yes—
181C2×C22⋊C4—941no—
182C2×C22⋊C4—941no—
183C2×C22⋊C4—942no—
184C2×C42—472no—
185C2×C42—942no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188#95—474no—
189#95—942no—
190#95—942no—
191#95—942no—
192#95—942no—
193#95—942no—
194#95—941no—
195C24C2×D18811yescharacteristic
196C2×Q8—942no—
197C2×Q8—1882no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—42no—
203C22×C4—81no—
204C22×C4—81no—
205C22×C4—942no—
206C22×C4—941no—
207C22×C4—941no—
208C22×C4—944no—
209C22×C4(C2×D47)⋊C412yes—
210C22×C4(C2×D47)⋊C412yes—
211C22×C4C2×C4×D4711yescharacteristic
212C22×C4D4×D4712yes—
213C22×C4#1062535212yes—
214C22×C4C2×C22⋊D4711yescharacteristic
215M16—44no—
216C2×C8—42no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—942no—
220C4⋊C4—3762no—
221C4⋊C4—3762no—
222C4⋊C4—3762no—
223C4⋊C4—1882no—
224C22⋊C4—944no—
225C22⋊C4—1884no—
226C22⋊C4—1881no—
227C22⋊C4—1881no—
228C42—3762no—
229C42—1882no—
230C23—22no—
231C23—22no—
232C23—41no—
233C23C2×(C2×D47)⋊C411yescharacteristic
234C23#1309399411yescharacteristic
235C23#1309400811yescharacteristic
236Q8—3762no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—944no—
250C2×C4—1884no—
251C2×C4—1884no—
252C2×C4—1882no—
253C2×C4—1881no—
254C2×C4—1881no—
255C2×C4—1884no—
256C2×C4—1882no—
257C2×C4—1882no—
258C2×C4—1882no—
259C2×C4(C22⋊D47)⋊C412yes—
260C2×C4#1309399612yes—
261C8—82no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—44no—
270C22#1568900011yescharacteristic
271C22#1568909612yes—
272C22#1568930812yes—
273C4—22no—
274C4—22no—
275C4—42no—
276C4—42no—
277C4—3764no—
278C4—3761no—
279C4—3761no—
280C4—1884no—
281C2—22no—
282C2—22no—
283C2—41no—
284C2#2024377911yescharacteristic
285C2#2024384411yescharacteristic
286C2#2024391811yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1821C2
socleC22×C47#386
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#1821D47
hypocenterC47#9461
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
8. #15689028C4semidirect
12. #15688992C4semidirect
164. C47#9461semidirect

Maximal subgroups

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

subgroupindex
#202478622
#202459912
#202419222
#202412422
C47×#18212
#946147