Finite Group Databasebeta

← #26108042

Subgroups — #26108042

Order 12032.

782 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 143 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×#15689830C211yesmaximal, characteristic
2#20242074C212yesmaximal, complemented
3#20241816C212yesmaximal, complemented
4#20241709C211yesmaximal, characteristic
5C47×#1800C211yesmaximal, characteristic
6#15689830—24no—
7#15689830—24no—
8C2×#13094024C2211yescharacteristic
9C2×#13094015—21no—
10C2×#13094003C412yes—
11C2×#13094001—22no—
12C2×C4×Dic94C2211yescharacteristic
13C2×D4×Dic47—21no—
14C22×C4⋊Dic47C412yes—
15#15689176C2212yes—
16#15688868—24no—
17#15688853—24no—
18#15688847—22no—
19C2×C4×D4×C47C2211yescharacteristic
20C47×#423C2212yes—
21C2×#100×C47—21no—
22C47×#343—21no—
23C22×Dic94—21no—
24C22×Dic94C2×C411yescharacteristic
25C23×Dic47—22no—
26#13094024—22no—
27#13094024D414yes—
28#13094015—28no—
29#13094003—24no—
30#13094003—24no—
31#13094001—216no—
32C4×Dic94—22no—
33C4×Dic94D414yes—
34D4×Dic47—28no—
35C2×C4⋊Dic47—22no—
36C2×C4⋊Dic47—21no—
37C2×C4⋊Dic47—24no—
38C2×C4⋊Dic47—24no—
39C2×C4⋊Dic47C2×C411yescharacteristic
40C2×C4⋊Dic47C2×C412yes—
41C2×Dic47⋊C4—22no—
42C2×Dic47⋊C4—21no—
43C2×C22⋊Dic47—22no—
44C2×C4×Dic47—21no—
45C4⋊(C47⋊C8)—22no—
46C22⋊(C47⋊C8)—22no—
47C4×C47⋊C8—22no—
48C22×D4×C47D411yescharacteristic
49C23×C4×C47C2×C412yes—
50C2×M16×C47—22no—
51C4×D4×C47—22no—
52C4×D4×C47—22no—
53C2×C4⋊C4×C47D411yescharacteristic
54C2×C22⋊C4×C47D412yes—
55C2×C42×C47C2311yescharacteristic
56C4⋊C8×C47—22no—
57C4⋊C8×C47—22no—
58#100×C47—42no—
59C24×C47C22⋊C412yes—
60C2×D4×C47—22no—
61C2×D4×C47—22no—
62C2×D4×C47SD1612yes—
63C2×D4×C47Q1612yes—
64C22×C4×C47—22no—
65C22×C4×C47—22no—
66C22×C4×C47—44no—
67C22×C4×C47C22⋊C412yes—
68C22×C4×C47C22×C411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×Dic47—21no—
72C22×Dic47—21no—
73C22×Dic47—22no—
74C22×Dic47—28no—
75C22×Dic47—84no—
76C2×Dic94—21no—
77C2×Dic94—24no—
78C2×Dic94—42no—
79C2×Dic94—82no—
80C2×Dic94C22⋊C412yes—
81M16×C47—42no—
82M16×C47—42no—
83C2×C8×C47—42no—
84C4⋊C4×C47—22no—
85C22⋊C4×C47—24no—
86C42×C47C2×D412yes—
87C42×C47C2×D412yes—
88C22⋊Dic47—28no—
89C4⋊Dic47—28no—
90C4⋊Dic47—21no—
91C4⋊Dic47—22no—
92C4⋊Dic47—24no—
93C4⋊Dic47C22⋊C412yes—
94C4⋊Dic47C22⋊C414yes—
95Dic47⋊C4—28no—
96Dic47⋊C4—24no—
97C4×Dic47—24no—
98C2×C47⋊C8—42no—
99C23×C47—22no—
100C23×C47—22no—
101C23×C47—42no—
102C23×C47—42no—
103C23×C47#10314yes—
104C23×C47C2×C22⋊C411yescharacteristic
105D4×C47—24no—
106D4×C47—24no—
107C2×C4×C47—22no—
108C2×C4×C47—22no—
109C2×C4×C47—41no—
110C2×C4×C47—44no—
111C2×C4×C47—44no—
112C2×C4×C47—42no—
113C2×C4×C47C2×C22⋊C411yescharacteristic
114C2×C4×C47C2×C22⋊C411yescharacteristic
115C2×C4×C47#12012yes—
116C2×C4×C47#12012yes—
117C2×C4×C47C2×SD1611yescharacteristic
118C2×C4×C47C2×Q1611yescharacteristic
119C2×C4×C47#13712yes—
120C2×Dic47—22no—
121C2×Dic47—22no—
122C2×Dic47—24no—
123C2×Dic47—42no—
124C2×Dic47—44no—
125C2×Dic47—42no—
126C2×Dic47—88no—
127Dic94—24no—
128Dic94—42no—
129Dic94—84no—
130C8×C47—82no—
131C47⋊C8—82no—
132#8395—471nomaximal, Sylow
133C22×C47—21no—
134C22×C47—21no—
135C22×C47—24no—
136C22×C47—44no—
137C22×C47—44no—
138C22×C47—42no—
139C22×C47—42no—
140C22×C47#37911yescharacteristic
141C22×C47C2×#10311yescharacteristic
142C22×C47#41911yescharacteristic
143C4×C47—22no—
144C4×C47—21no—
145C4×C47—42no—
146C4×C47#41312yes—
147C4×C47#44812yes—
148C4×C47#45112yes—
149Dic47—44no—
150Dic47—42no—
151Dic47—42no—
152C2×#554—471no—
153#1800D4711yescharacteristic
154#1549—472no—
155#1487—472no—
156#1460—471no—
157C2×C47—21no—
158C2×C47—21no—
159C2×C47—44no—
160C2×C47#178511yescharacteristic
161C2×C47#178711yescharacteristic
162C2×C47#179511yescharacteristic
163#554—944no—
164#554—944no—
165C2×C4⋊Q8—471no—
166C2×#125—941no—
167C2×C22⋊Q8—472no—
168C2×C22⋊Q8—942no—
169C2×C4×Q8—471no—
170C2×C4×D4—941no—
171C2×C4×D4C2×D4711yescharacteristic
172C22×C4⋊C4—472no—
173#431—472no—
174#423C2×D4712yes—
175C2×#100—21no—
176#343—21no—
177#332—944no—
178Q8⋊C8—944no—
179#323—942no—
180C47#839511yescharacteristic, complemented, Sylow
181C22×Q8—471no—
182C22×Q8—941no—
183C22×D4C22⋊D4711yescharacteristic
184C23×C4—942no—
185C23×C4C4×D4712yes—
186C2×M16—22no—
187C4⋊Q8—474no—
188C4⋊Q8—942no—
189#125—948no—
190C22⋊Q8—9416no—
191C22⋊Q8—944no—
192C22⋊Q8—944no—
193C4×Q8—474no—
194C4×Q8—942no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—948no—
198C2×C4⋊C4—471no—
199C2×C4⋊C4—472no—
200C2×C4⋊C4—944no—
201C2×C4⋊C4—944no—
202C2×C4⋊C4—942no—
203C2×C4⋊C4—942no—
204C2×C4⋊C4—941no—
205C2×C4⋊C4—941no—
206C2×C4⋊C4C22⋊D4711yescharacteristic
207C2×C22⋊C4—942no—
208C2×C22⋊C4D18812yes—
209C2×C42—941no—
210C2×C42C22×D4711yescharacteristic
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—942no—
214#100—42no—
215C22⋊C8—942no—
216C4×C8—942no—
217C24(C2×D47)⋊C412yes—
218C2×Q8—472no—
219C2×Q8—941no—
220C2×Q8—944no—
221C2×Q8—3762no—
222C2×Q8—1882no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4#1062533512yes—
226C2×D4#1062533712yes—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—44no—
230C22×C4—948no—
231C22×C4—941no—
232C22×C4—941no—
233C22×C4—942no—
234C22×C4—3764no—
235C22×C4(C2×D47)⋊C412yes—
236C22×C4C2×C4×D4711yescharacteristic
237C22×C4C2×D18811yescharacteristic
238C22×C4C2×C22⋊D4711yescharacteristic
239M16—42no—
240M16—42no—
241C2×C8—42no—
242C2×C8—1882no—
243C4⋊C4—22no—
244C4⋊C4—472no—
245C4⋊C4—474no—
246C4⋊C4—948no—
247C4⋊C4—948no—
248C4⋊C4—941no—
249C4⋊C4—942no—
250C4⋊C4—944no—
251C4⋊C4—944no—
252C22⋊C4—24no—
253C22⋊C4—948no—
254C42—944no—
255C42D4×D4712yes—
256C42D4×D4712yes—
257C23—22no—
258C23—22no—
259C23—42no—
260C23—42no—
261C23#1309394714yes—
262C23C2×(C2×D47)⋊C411yescharacteristic
263Q8—944no—
264Q8—3764no—
265Q8—1882no—
266D4—24no—
267D4—24no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—42no—
274C2×C4—942no—
275C2×C4—942no—
276C2×C4—944no—
277C2×C4—3768no—
278C2×C4—1882no—
279C2×C4—1884no—
280C2×C4—1882no—
281C2×C4C22⋊C4×D4711yescharacteristic
282C2×C4C22⋊C4×D4711yescharacteristic
283C2×C4#1309399212yes—
284C2×C4#1309399412yes—
285C2×C4C2×#1062533511yescharacteristic
286C2×C4C2×#1062533711yescharacteristic
287C2×C4#1309405412yes—
288C8—82no—
289C8—3762no—
290C22—21no—
291C22—21no—
292C22—24no—
293C22—44no—
294C22—44no—
295C22—42no—
296C22—42no—
297C22#1568897811yescharacteristic
298C22C2×#1309394711yescharacteristic
299C22#1568913111yescharacteristic
300C4—21no—
301C4—22no—
302C4—42no—
303C4—1882no—
304C4—1882no—
305C4—1884no—
306C4#1568910412yes—
307C4#1568926112yes—
308C4#1568927812yes—
309C2—21no—
310C2—21no—
311C2—44no—
312C2#2024331311yescharacteristic
313C2#2024332211yescharacteristic
314C2#2024335911yescharacteristic
subgroupG/H
centerC22#15688978
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1800C2
socleC22×C47#379
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenter#1800D47
hypocenterC47#8395
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #20242074C2semidirect
3. #20241816C2semidirect
180. C47#8395semidirect

Maximal subgroups

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

subgroupindex
C2×#156898302
#202420742
#202418162
#202417092
C47×#18002
#839547