Finite Group Databasebeta

← #26103589

Subgroups — #26103589

Order 12032.

338 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 71 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#20243376C211yesmaximal, characteristic
2#20241726C211yesmaximal, characteristic
3C47×#1955C211yesmaximal, characteristic
4C2×#13093939C412yes—
5#15689087—21no—
6#15688983—21no—
7#15688890—21no—
8#15688849C412yescomplemented
9#15688847—21no—
10C2×C4×D4×C47C2211yescharacteristic
11C2×C4⋊C8×C47C411yescharacteristic, complemented
12C2×C22⋊C8×C47—21no—
13C2×C22⋊C8×C47—21no—
14C2×#97×C47C411yescharacteristic, complemented
15C2×#10625329—41no—
16C2×C4⋊Dic47—41no—
17C2×C22⋊Dic47—22no—
18#13093939—42no—
19C22⋊(C47⋊C8)—22no—
20C22⋊(C47⋊C8)—41no—
21C47⋊#95—41no—
22C22×D4×C47C2×C411yescharacteristic
23C23×C4×C47D411yescharacteristic
24C23×C4×C47D411yescharacteristic
25C22×C8×C47—21no—
26C22×C8×C47—21no—
27C4×D4×C47—41no—
28C4×D4×C47—41no—
29C2×C4⋊C4×C47C2×C411yescharacteristic
30C2×C22⋊C4×C47D411yescharacteristic
31C2×C22⋊C4×C47Q811yescharacteristic
32C2×C42×C47C2×C411yescharacteristic
33C4⋊C8×C47—41no—
34C22⋊C8×C47—42no—
35C22⋊C8×C47—42no—
36#97×C47—41no—
37C24×C47C22⋊C411yescharacteristic
38C24×C47C4⋊C411yescharacteristic
39C2×D4×C47—21no—
40C2×D4×C47—21no—
41C2×D4×C47—41no—
42C2×D4×C47—41no—
43C22×C4×C47—21no—
44C22×C4×C47—21no—
45C22×C4×C47—21no—
46C22×C4×C47—21no—
47C22×C4×C47—41no—
48C22×C4×C47—41no—
49C22×C4×C47—41no—
50C22×C4×C47—41no—
51C22×C4×C47C4211yescharacteristic
52C22×C4×C47C22⋊C411yescharacteristic
53C22×C4×C47C22⋊C411yescharacteristic
54C22×C4×C47C4⋊C411yescharacteristic
55C22×C4×C47C4⋊C411yescharacteristic
56C22×Dic47—42no—
57C2×C8×C47—22no—
58C2×C8×C47—41no—
59C2×C8×C47—41no—
60C2×C8×C47—41no—
61C2×C8×C47—41no—
62C2×C8×C47—41no—
63C4⋊C4×C47—41no—
64C22⋊C4×C47—41no—
65C22⋊C4×C47—41no—
66C42×C47—21no—
67C42×C47—21no—
68C22⋊Dic47—42no—
69C22⋊Dic47—42no—
70C4⋊Dic47—81no—
71C4⋊Dic47—81no—
72#10625329—81no—
73#10625329—81no—
74C2×C47⋊C8—42no—
75C23×C47—21no—
76C23×C47—21no—
77C23×C47—21no—
78C23×C47—21no—
79C23×C47—21no—
80C23×C47—41no—
81C23×C47—41no—
82C23×C47—41no—
83C23×C47—41no—
84C23×C47#9511yescharacteristic
85C23×C47#10011yescharacteristic
86C23×C47#10111yescharacteristic
87D4×C47—42no—
88D4×C47—42no—
89C2×C4×C47—21no—
90C2×C4×C47—21no—
91C2×C4×C47—21no—
92C2×C4×C47—21no—
93C2×C4×C47—21no—
94C2×C4×C47—21no—
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—41no—
98C2×C4×C47—41no—
99C2×C4×C47—41no—
100C2×C4×C47—41no—
101C2×C4×C47—41no—
102C2×C4×C47—41no—
103C2×C4×C47—41no—
104C2×C4×C47—81no—
105C2×C4×C47—81no—
106C2×C4×C47#9911yescharacteristic
107C2×C4×C47#9911yescharacteristic
108C2×Dic47—82no—
109C2×Dic47—82no—
110C2×Dic47—82no—
111C8×C47—42no—
112C8×C47—81no—
113C47⋊C8—82no—
114#7322—471nomaximal, Sylow
115C22×C47—21no—
116C22×C47—21no—
117C22×C47—21no—
118C22×C47—21no—
119C22×C47—41no—
120C22×C47—41no—
121C22×C47—42no—
122C22×C47—41no—
123C22×C47—41no—
124C22×C47—41no—
125C22×C47—41no—
126C22×C47—41no—
127C22×C47—81no—
128C22×C47—81no—
129C22×C47#34211yescharacteristic
130C22×C47#34311yescharacteristic
131C22×C47#34411yescharacteristic
132C4×C47—21no—
133C4×C47—41no—
134C4×C47—41no—
135C4×C47—41no—
136C4×C47—41no—
137C4×C47#35311yescharacteristic
138C4×C47#35411yescharacteristic
139Dic47—162no—
140#1955D4711yescharacteristic
141#1799—471no—
142#1465—471no—
143C2×C47—21no—
144C2×C47—21no—
145C2×C47—41no—
146C2×C47—41no—
147C2×C47—81no—
148C2×C47#130511yescharacteristic
149C2×C47#139311yescharacteristic
150C2×C47#139511yescharacteristic
151C2×C4×D4C2×D4711yescharacteristic
152C2×C4⋊C8Dic4711yescharacteristic, complemented
153C2×#99—472no—
154#407—941no—
155C2×C22⋊C8—21no—
156C2×C22⋊C8—21no—
157C2×#97Dic4711yescharacteristic, complemented
158#380—941no—
159#342—941no—
160#324—472no—
161#323—941no—
162C47#732211yescharacteristic, complemented, Sylow
163C22×D4C4×D4711yescharacteristic
164C23×C4C22⋊D4711yescharacteristic
165C23×C4C22⋊D4711yescharacteristic
166C2×M16—1881no—
167C22×C8—21no—
168C22×C8—21no—
169C4×D4—41no—
170C4×D4—41no—
171C2×C4⋊C4—1881no—
172C2×C4⋊C4C4×D4711yescharacteristic
173C2×C22⋊C4—942no—
174C2×C22⋊C4Dic9411yescharacteristic
175C2×C22⋊C4D18811yescharacteristic
176C2×C42C2×Dic4711yescharacteristic
177C4⋊C8—41no—
178#99—1882no—
179C22⋊C8—42no—
180C22⋊C8—42no—
181C22⋊C8—942no—
182C22⋊C8—1881no—
183#97—41no—
184#95—1881no—
185C24Dic47⋊C411yescharacteristic
186C24(C2×D47)⋊C411yescharacteristic
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—41no—
190C2×D4—41no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—1882no—
200C22×C4C4×Dic4711yescharacteristic
201C22×C4Dic47⋊C411yescharacteristic
202C22×C4C4⋊Dic4711yescharacteristic
203C22×C4(C2×D47)⋊C411yescharacteristic
204C22×C4C22⋊Dic4711yescharacteristic
205M16—3761no—
206M16—3761no—
207C2×C8—22no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—1882no—
214C4⋊C4—41no—
215C4⋊C4—3761no—
216C4⋊C4—3761no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—1882no—
220C22⋊C4—1882no—
221C42—21no—
222C42—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23C47⋊#9511yescharacteristic
233C23#1309394011yescharacteristic
234C23#1309394211yescharacteristic
235D4—42no—
236D4—42no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—3762no—
255C2×C4—3762no—
256C2×C4—3762no—
257C2×C4#1309393911yescharacteristic
258C2×C4#1309393911yescharacteristic
259C8—42no—
260C8—81no—
261C8—3762no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—81no—
275C22—81no—
276C22#1568889011yescharacteristic
277C22#1568889111yescharacteristic
278C22#1568889511yescharacteristic
279C4—21no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—7522no—
285C4#1568891611yescharacteristic
286C4#1568891811yescharacteristic
287C2—21no—
288C2—21no—
289C2—41no—
290C2—41no—
291C2—81no—
292C2#2024129311yescharacteristic
293C2#2024151311yescharacteristic
294C2#2024151711yescharacteristic
subgroupG/H
centerC22#15688890
commutatorC22×C4×C47C42
FrattiniC2×C4×D4C2×D47
FittingC47×#1955C2
socleC22×C47#342
prime radicalC22×D4×C47C2×C4
cosocleC2×C4×D4×C47C22
hypercenter#1955D47
hypocenterC47#7322
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1955

Complements

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

NH ≅ G/Ntype
8. #15688849C4semidirect
11. C2×C4⋊C8×C47C4semidirect
14. C2×#97×C47C4semidirect
152. C2×C4⋊C8Dic47semidirect
157. C2×#97Dic47semidirect
162. C47#7322semidirect

Maximal subgroups

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

subgroupindex
#202433762
#202417262
C47×#19552
#732247