Finite Group Databasebeta

← #26103491

Subgroups — #26103491

Order 12032.

402 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 75 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#20244187C211yesmaximal, characteristic
2C4×#13093939C211yesmaximal, characteristic
3C4×#99×C47C211yesmaximal, characteristic
4C2×C4⋊(C47⋊C8)C411yescharacteristic, complemented
5C2×#13093939—21no—
6C2×C22⋊(C47⋊C8)—21no—
7C2×C22⋊(C47⋊C8)—21no—
8C2×#13093935C411yescharacteristic, complemented
9C4×C22⋊Dic47—21no—
10#15688890C412yescomplemented
11C2×C4×D4×C47C2211yescharacteristic
12C2×#99×C47—21no—
13C4×C22⋊C4×C47—21no—
14C47×#342C412yescomplemented
15C22×C47⋊C8—21no—
16C22×C47⋊C8—21no—
17C2×C22⋊Dic47—22no—
18C2×C4×Dic47—41no—
19C4⋊(C47⋊C8)—41no—
20#13093939—24no—
21C22⋊(C47⋊C8)—42no—
22C22⋊(C47⋊C8)—42no—
23#13093935—41no—
24C47⋊#95—41no—
25C22×D4×C47D411yescharacteristic
26C23×C4×C47D411yescharacteristic
27C23×C4×C47D411yescharacteristic
28C4×D4×C47—41no—
29C4×D4×C47—41no—
30C2×C4⋊C4×C47Q811yescharacteristic
31C2×C22⋊C4×C47—22no—
32C2×C22⋊C4×C47C2×C411yescharacteristic
33C2×C22⋊C4×C47C2×C411yescharacteristic
34C2×C42×C47—41no—
35C2×C42×C47C2×C411yescharacteristic
36#99×C47—24no—
37#95×C47—41no—
38C24×C47C22⋊C411yescharacteristic
39C24×C47C22⋊C411yescharacteristic
40C2×D4×C47—22no—
41C2×D4×C47—22no—
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×C47—42no—
52C22×C4×C47C4211yescharacteristic
53C22×C4×C47C22⋊C411yescharacteristic
54C22×C4×C47C4⋊C411yescharacteristic
55C22×C4×C47C4⋊C411yescharacteristic
56C22×C4×C47C4⋊C411yescharacteristic
57C22×Dic47—42no—
58C4⋊C4×C47—41no—
59C22⋊C4×C47—24no—
60C22⋊C4×C47—41no—
61C22⋊C4×C47—41no—
62C22⋊C4×C47—42no—
63C42×C47—21no—
64C42×C47—21no—
65C42×C47—81no—
66C42×C47—81no—
67C22⋊Dic47—24no—
68C22⋊Dic47—42no—
69C4×Dic47—81no—
70C4×Dic47—81no—
71C2×C47⋊C8—22no—
72C2×C47⋊C8—41no—
73C2×C47⋊C8—41no—
74C2×C47⋊C8—41no—
75C2×C47⋊C8—41no—
76C2×C47⋊C8—41no—
77C23×C47—21no—
78C23×C47—21no—
79C23×C47—21no—
80C23×C47—21no—
81C23×C47—41no—
82C23×C47—41no—
83C23×C47—41no—
84C23×C47—41no—
85C23×C47#9511yescharacteristic
86C23×C47#10412yes—
87C23×C47#10412yes—
88D4×C47—44no—
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—44no—
105C2×C4×C47—81no—
106C2×C4×C47—81no—
107C2×C4×C47—82no—
108C2×C4×C47—82no—
109C2×C4×C47#9911yescharacteristic
110C2×C4×C47#9911yescharacteristic
111C2×Dic47—44no—
112C2×Dic47—82no—
113C2×Dic47—82no—
114C47⋊C8—42no—
115C47⋊C8—81no—
116#7283—471nomaximal, Sylow
117C22×C47—21no—
118C22×C47—21no—
119C22×C47—22no—
120C22×C47—22no—
121C22×C47—42no—
122C22×C47—42no—
123C22×C47—41no—
124C22×C47—41no—
125C22×C47—41no—
126C22×C47—41no—
127C22×C47—81no—
128C22×C47—81no—
129C22×C47#33911yescharacteristic
130C22×C47#33911yescharacteristic
131C22×C47#34211yescharacteristic
132C4×C47—21no—
133C4×C47—41no—
134C4×C47—41no—
135C4×C47—41no—
136C4×C47—41no—
137C4×C47—84no—
138C4×C47#35311yescharacteristic
139C4×C47#35411yescharacteristic
140Dic47—84no—
141#1955—471no—
142C4×#99—471no—
143C4×#99D4711yescharacteristic
144C2×C47—21no—
145C2×C47—21no—
146C2×C47—42no—
147C2×C47—42no—
148C2×C47#129711yescharacteristic
149C2×C47#139111yescharacteristic
150C2×C47#139311yescharacteristic
151C2×C4×D4C2×D4711yescharacteristic
152C2×C4⋊C8—471no—
153C2×#99—21no—
154C2×#99—941no—
155C2×C22⋊C8—941no—
156C2×C22⋊C8—941no—
157C2×#97—471no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—941no—
160#342—472no—
161#342Dic4712yescomplemented
162C47#728311yescharacteristic, complemented, Sylow
163C22×D4C22⋊D4711yescharacteristic
164C23×C4D18811yescharacteristic
165C23×C4C22⋊D4711yescharacteristic
166C22×C8—941no—
167C22×C8—941no—
168C4×D4—41no—
169C4×D4—41no—
170C2×C4⋊C4Dic9411yescharacteristic
171C2×C22⋊C4—22no—
172C2×C22⋊C4—942no—
173C2×C22⋊C4C4×D4711yescharacteristic
174C2×C22⋊C4C2×Dic4711yescharacteristic
175C2×C42—41no—
176C2×C42—1881no—
177C2×C42C4×D4711yescharacteristic
178C4⋊C8—1881no—
179#99—24no—
180#99—944no—
181C22⋊C8—1882no—
182C22⋊C8—1882no—
183#97—1881no—
184#95—41no—
185#95—1881no—
186C24(C2×D47)⋊C411yescharacteristic
187C24C22⋊Dic4711yescharacteristic
188C2×D4—22no—
189C2×D4—22no—
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—42no—
200C22×C4—1882no—
201C22×C4C4×Dic4711yescharacteristic
202C22×C4Dic47⋊C411yescharacteristic
203C22×C4Dic47⋊C411yescharacteristic
204C22×C4C4⋊Dic4711yescharacteristic
205C22×C4(C2×D47)⋊C411yescharacteristic
206C2×C8—942no—
207C2×C8—1881no—
208C2×C8—1881no—
209C2×C8—1881no—
210C2×C8—1881no—
211C2×C8—1881no—
212C4⋊C4—41no—
213C22⋊C4—24no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—42no—
217C22⋊C4—944no—
218C22⋊C4—1882no—
219C42—21no—
220C42—21no—
221C42—81no—
222C42—81no—
223C42—3761no—
224C42—3761no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23C47⋊#9511yescharacteristic
234C23#1309395112yes—
235C23#1309395212yes—
236D4—44no—
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—44no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—3762no—
258C2×C4—3762no—
259C2×C4—1884no—
260C2×C4#1309393811yescharacteristic
261C2×C4#1309393811yescharacteristic
262C8—3761no—
263C8—1882no—
264C22—21no—
265C22—21no—
266C22—22no—
267C22—22no—
268C22—42no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—81no—
275C22—81no—
276C22#1568888211yescharacteristic
277C22#1568888311yescharacteristic
278C22#1568888911yescharacteristic
279C4—21no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—84no—
285C4—3764no—
286C4#1568891511yescharacteristic
287C4#1568891711yescharacteristic
288C2—21no—
289C2—21no—
290C2—42no—
291C2—42no—
292C2#2024127411yescharacteristic
293C2#2024150811yescharacteristic
294C2#2024151211yescharacteristic
subgroupG/H
centerC2×C4#13093938
commutatorC22×C4×C47C42
FrattiniC2×C4×D4C2×D47
FittingC4×#99×C47C2
socleC22×C47#342
prime radicalC22×D4×C47D4
cosocleC2×C4×D4×C47C22
hypercenterC4×#99D47
hypocenterC47#7283
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
4. C2×C4⋊(C47⋊C8)C4semidirect
8. C2×#13093935C4semidirect
10. #15688890C4semidirect
14. C47×#342C4semidirect
161. #342Dic47semidirect
162. C47#7283semidirect

Maximal subgroups

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

subgroupindex
#202441872
C4×#130939392
C4×#99×C472
#728347