Finite Group Databasebeta

← #26109700

Subgroups — #26109700

Order 12032.

630 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 155 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#384×D47C211yesmaximal, characteristic
2#20243365C212yesmaximal
3#20241272C212yesmaximal, complemented
4#20241271C211yesmaximal, characteristic, complemented
5C47×#1296C211yesmaximal, characteristic, complemented
6C2×C4⋊C4×D47—22no—
7C2×C4⋊C4×D47C412yes—
8C2×C42×D47C2211yescharacteristic
9C2×C4⋊(C47⋊C8)C2212yes—
10C2×(C2×D47)⋊C8—22no—
11#15688993C2211yescharacteristic
12#15688992—22no—
13#15688992C412yes—
14#15688877—22no—
15C2×C4⋊C8×C47C2212yes—
16C47×#384C2211yescharacteristic
17C22×C4×D47—22no—
18C22×C4×D47C2×C411yescharacteristic
19C22×C4×D47D412yes—
20C22×C47⋊C8—22no—
21C2×C4⋊Dic47—22no—
22C2×C4⋊Dic47C2×C412yes—
23C2×Dic47⋊C4—22no—
24C2×Dic47⋊C4—24no—
25C4⋊C4×D47—44no—
26C4⋊C4×D47—44no—
27C4⋊C4×D47—44no—
28C2×C4×Dic47C2×C411yescharacteristic
29C2×C4×Dic47D412yes—
30C42×D47—22no—
31C42×D47—21no—
32C42×D47—21no—
33C4⋊(C47⋊C8)—22no—
34C4⋊(C47⋊C8)—22no—
35(C2×D47)⋊C8—44no—
36C22×C8×C47—22no—
37C2×C4⋊C4×C47—22no—
38C2×C4⋊C4×C47C2×C412yes—
39C2×C42×C47C2311yescharacteristic
40C4⋊C8×C47—22no—
41C4⋊C8×C47—22no—
42C23×D47C22⋊C411yescharacteristic
43C22×C4×C47—22no—
44C22×C4×C47C22×C411yescharacteristic
45C22×C4×C47C2×D412yes—
46C22×Dic47—22no—
47C22×Dic47C22⋊C411yescharacteristic
48C22×Dic47C22⋊C412yes—
49C2×C4×D47—22no—
50C2×C4×D47—21no—
51C2×C4×D47—21no—
52C2×C4×D47—22no—
53C2×C4×D47—22no—
54C2×C4×D47—22no—
55C2×C4×D47—22no—
56C2×C4×D47—22no—
57C2×C4×D47—48no—
58C2×C4×D47—44no—
59C2×C4×D47SD1614yes—
60C2×C4×D47Q1614yes—
61C2×C8×C47—42no—
62C2×C8×C47—42no—
63C2×C8×C47—42no—
64C4⋊C4×C47—42no—
65C4⋊C4×C47—42no—
66C4⋊C4×C47—42no—
67C42×C47#4112yes—
68C42×C47#4111yescharacteristic
69C42×C47#4111yescharacteristic
70C4⋊Dic47—42no—
71C4⋊Dic47—42no—
72C4⋊Dic47—42no—
73Dic47⋊C4—44no—
74Dic47⋊C4—48no—
75C4×Dic47—21no—
76C4×Dic47—21no—
77C4×Dic47—22no—
78C4×Dic47—22no—
79C2×C47⋊C8—42no—
80C2×C47⋊C8—42no—
81C2×C47⋊C8—42no—
82C23×C47C2×C22⋊C411yescharacteristic
83C22×D47—21no—
84C22×D47—21no—
85C22×D47—22no—
86C22×D47#10012yes—
87C22×D47#10314yes—
88C2×C4×C47—21no—
89C2×C4×C47—21no—
90C2×C4×C47—42no—
91C2×C4×C47—44no—
92C2×C4×C47#11711yescharacteristic
93C2×C4×C47#11711yescharacteristic
94C2×C4×C47#12312yes—
95C2×C4×C47#12312yes—
96C2×C4×C47C2×SD1612yes—
97C2×C4×C47C2×Q1612yes—
98C2×C4×C47AGL(1,ℤ/8)12yes—
99C2×C4×C47AGL(1,ℤ/8)12yes—
100C2×Dic47—21no—
101C2×Dic47—22no—
102C2×Dic47—21no—
103C2×Dic47—22no—
104C2×Dic47—42no—
105C2×Dic47—44no—
106C2×Dic47#10112yes—
107C2×Dic47#10314yes—
108C4×D47—24no—
109C4×D47—24no—
110C4×D47—24no—
111C4×D47—24no—
112C4×D47—24no—
113C4×D47—24no—
114C4×D47—88no—
115C8×C47—82no—
116C47⋊C8—82no—
117#8725—471nomaximal, Sylow
118C22×C47#38111yescharacteristic
119C22×C47C2×#10011yescharacteristic
120C22×C47C2×#10111yescharacteristic
121C22×C47C2×#10312yes—
122C22×C47#41812yes—
123C2×D47—24no—
124C2×D47—22no—
125C2×D47—22no—
126C2×D47—24no—
127C2×D47#33314yes—
128C4×C47—21no—
129C4×C47—21no—
130C4×C47—82no—
131C4×C47#48112yes—
132C4×C47#48112yes—
133C4×C47#48412yes—
134C4×C47#48412yes—
135Dic47—22no—
136Dic47—24no—
137Dic47—82no—
138C2×#384—471no—
139#1797—472no—
140#1296—472no—
141#1296—471no—
142#1296D4711yescharacteristic, complemented
143C2×C47C2×#33311yescharacteristic
144C2×C47#154912yes—
145C2×C47#155511yescharacteristic
146C2×C47#182111yescharacteristic
147C2×C47#182412yes—
148D47—24no—
149C22×C4⋊C4—472no—
150C22×C4⋊C4—942no—
151C22×C42—471no—
152C2×C4⋊C8—472no—
153C2×C4⋊C8C2×D4712yes—
154C2×C22⋊C8—942no—
155#384—472no—
156#384—471no—
157#384—942no—
158#384C2×D4711yescharacteristic
159#336—942no—
160C47#872511yescharacteristic, complemented, Sylow
161C23×C4—472no—
162C23×C4—471no—
163C23×C4—942no—
164C22×C8—22no—
165C22×C8—942no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—472no—
168C2×C4⋊C4—942no—
169C2×C4⋊C4—944no—
170C2×C4⋊C4—942no—
171C2×C4⋊C4—1884no—
172C2×C4⋊C4—1884no—
173C2×C4⋊C4—1884no—
174C2×C4⋊C4C4×D4712yes—
175C2×C42—471no—
176C2×C42—472no—
177C2×C42—942no—
178C2×C42—941no—
179C2×C42—941no—
180C2×C42C22×D4711yescharacteristic
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—942no—
184C4⋊C8—942no—
185C22⋊C8—1884no—
186C24—471no—
187C22×C4—22no—
188C22×C4—471no—
189C22×C4—472no—
190C22×C4—474no—
191C22×C4—474no—
192C22×C4—942no—
193C22×C4—942no—
194C22×C4—941no—
195C22×C4—941no—
196C22×C4—942no—
197C22×C4—942no—
198C22×C4—942no—
199C22×C4—942no—
200C22×C4—942no—
201C22×C4—1884no—
202C22×C4—1888no—
203C22×C4C2×C4×D4711yescharacteristic
204C22×C4D4×D4712yes—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—1882no—
209C2×C8—1882no—
210C2×C8—1882no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—1882no—
215C4⋊C4—1884no—
216C4⋊C4—1888no—
217C4⋊C4—1882no—
218C4⋊C4—1882no—
219C42—941no—
220C42—941no—
221C42—942no—
222C42—942no—
223C42#1062535012yes—
224C42#1062535211yescharacteristic
225C42#1062535411yescharacteristic
226C23—472no—
227C23—474no—
228C23—941no—
229C23—941no—
230C23—942no—
231C23C22⋊C4×D4711yescharacteristic
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—44no—
236C2×C4—472no—
237C2×C4—474no—
238C2×C4—944no—
239C2×C4—944no—
240C2×C4—941no—
241C2×C4—942no—
242C2×C4—941no—
243C2×C4—942no—
244C2×C4—944no—
245C2×C4—944no—
246C2×C4—944no—
247C2×C4—944no—
248C2×C4—3768no—
249C2×C4—1882no—
250C2×C4—1884no—
251C2×C4#1309398111yescharacteristic
252C2×C4#1309398211yescharacteristic
253C2×C4#1309400612yes—
254C2×C4#1309400912yes—
255C2×C4SD16×D4712yes—
256C2×C4Q16×D4712yes—
257C2×C4#1309404812yes—
258C2×C4#1309405112yes—
259C8—82no—
260C8—3762no—
261C22—474no—
262C22—942no—
263C22—944no—
264C22—944no—
265C22—942no—
266C22#1568898511yescharacteristic
267C22#100×D4711yescharacteristic
268C22#101×D4711yescharacteristic
269C22#103×D4712yes—
270C22#1568912612yes—
271C4—21no—
272C4—21no—
273C4—82no—
274C4—942no—
275C4—944no—
276C4—3762no—
277C4#1568944012yes—
278C4#1568944312yes—
279C4#1568945712yes—
280C4#1568946012yes—
281C2—944no—
282C2#333×D4711yescharacteristic
283C2#2024207112yes—
284C2#2024210011yescharacteristic
285C2#2024346711yescharacteristic
286C2#2024347712yes—
subgroupG/H
centerC23C22⋊C4×D47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1296C2
socleC23×C47C2×C22⋊C4
prime radicalC23×D47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1296D47
hypocenterC47#8725
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
3. #20241272C2semidirect
4. #20241271C2semidirect
5. C47×#1296C2semidirect
142. #1296D47semidirect
160. C47#8725semidirect

Maximal subgroups

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

subgroupindex
#384×D472
#202433652
#202412722
#202412712
C47×#12962
#872547