Finite Group Databasebeta

← #26104057

Subgroups — #26104057

Order 12032.

272 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 63 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×#15688922C211yesmaximal, characteristic
2#20244191C211yesmaximal, characteristic
3C2×C47×#356C211yesmaximal, characteristic
4C2×#13093948—21no—
5C2×#13093948—21no—
6C2×#13093943—21no—
7C2×C4×C47⋊C8C411yescharacteristic
8#15688993C411yescharacteristic
9#15688922—21no—
10#15688922C412yescomplemented
11C2×C4⋊Q8×C47C2211yescharacteristic
12C2×#101×C47—21no—
13C47×#356—21no—
14C47×#356C412yescomplemented
15C2×#10625329—41no—
16C22×C47⋊C8—21no—
17C2×C4⋊Dic47—21no—
18C2×C4⋊Dic47—41no—
19#13093948—42no—
20#13093948—42no—
21#13093943—22no—
22#13093943—41no—
23C4×C47⋊C8—41no—
24C22×Q8×C47D411yescharacteristic
25C22×Q8×C47D411yescharacteristic
26C2×M16×C47—41no—
27C4⋊Q8×C47—41no—
28C4⋊Q8×C47C2×C411yescharacteristic
29C4⋊Q8×C47C2×C411yescharacteristic
30C4⋊Q8×C47D411yescharacteristic
31C4⋊Q8×C47Q811yescharacteristic
32C2×C4⋊C4×C47—41no—
33C2×C42×C47C2×C411yescharacteristic
34#101×C47—22no—
35#101×C47—41no—
36C2×Q8×C47—21no—
37C2×Q8×C47—21no—
38C2×Q8×C47—41no—
39C2×Q8×C47—41no—
40C2×Q8×C47—41no—
41C2×Q8×C47—41no—
42C2×Q8×C47C22⋊C411yescharacteristic
43C2×Q8×C47C22⋊C411yescharacteristic
44C2×Q8×C47C4⋊C411yescharacteristic
45C2×Q8×C47C4⋊C411yescharacteristic
46C22×C4×C47—21no—
47C22×C4×C47—21no—
48C22×C4×C47—21no—
49C22×C4×C47C22⋊C411yescharacteristic
50C22×Dic47—41no—
51M16×C47—42no—
52M16×C47—81no—
53C2×C8×C47—81no—
54C4⋊C4×C47—41no—
55C4⋊C4×C47—41no—
56C4⋊C4×C47—41no—
57C4⋊C4×C47—41no—
58C42×C47—21no—
59C42×C47C4211yescharacteristic
60C42×C47C4⋊C411yescharacteristic
61C4⋊Dic47—41no—
62C4⋊Dic47—41no—
63C4⋊Dic47—82no—
64#10625329—42no—
65#10625329—81no—
66C2×C47⋊C8—22no—
67C2×C47⋊C8—41no—
68C2×C47⋊C8—41no—
69C2×C47⋊C8—81no—
70C23×C47#9911yescharacteristic
71Q8×C47—41no—
72Q8×C47—41no—
73Q8×C47—41no—
74Q8×C47—41no—
75Q8×C47—81no—
76Q8×C47—81no—
77C2×C4×C47—22no—
78C2×C4×C47—22no—
79C2×C4×C47—21no—
80C2×C4×C47—21no—
81C2×C4×C47—21no—
82C2×C4×C47—21no—
83C2×C4×C47—21no—
84C2×C4×C47—21no—
85C2×C4×C47—21no—
86C2×C4×C47—21no—
87C2×C4×C47—41no—
88C2×C4×C47—41no—
89C2×C4×C47—41no—
90C2×C4×C47—41no—
91C2×C4×C47#9511yescharacteristic
92C2×C4×C47#9911yescharacteristic
93C2×Dic47—82no—
94C2×Dic47—81no—
95C8×C47—82no—
96C47⋊C8—42no—
97C47⋊C8—82no—
98#7499—471nomaximal, Sylow
99C22×C47—21no—
100C22×C47—21no—
101C22×C47#34211yescharacteristic
102C22×C47#35311yescharacteristic
103C22×C47#35411yescharacteristic
104C4×C47—22no—
105C4×C47—22no—
106C4×C47—21no—
107C4×C47—21no—
108C4×C47—41no—
109C4×C47—41no—
110C4×C47—41no—
111C4×C47—41no—
112Dic47—161no—
113C2×#356—471no—
114C2×#356D4711yescharacteristic
115#1956—471no—
116C2×C47—21no—
117C2×C47—21no—
118C2×C47#139311yescharacteristic
119C2×C47#141011yescharacteristic
120C2×C47#141111yescharacteristic
121C2×C4⋊Q8C2×D4711yescharacteristic
122C2×#103—941no—
123C2×#103—941no—
124C2×#101—21no—
125C2×#101—941no—
126C2×C4×C8—471no—
127#384—471no—
128#356—21no—
129#356—472no—
130#356—941no—
131#356Dic4712yescomplemented
132C47#749911yescharacteristic, complemented, Sylow
133C22×Q8D18811yescharacteristic
134C22×Q8C22⋊D4711yescharacteristic
135C2×M16—41no—
136C2×M16—1881no—
137C22×C8—941no—
138C4⋊Q8—41no—
139C4⋊Q8Dic9411yescharacteristic
140C4⋊Q8C4×D4711yescharacteristic
141C4⋊Q8C2×Dic4711yescharacteristic
142C4⋊Q8C22⋊D4711yescharacteristic
143C2×C4⋊C4—41no—
144C2×C4⋊C4—941no—
145C2×C4⋊C4—1881no—
146C2×C42C4×D4711yescharacteristic
147#103—1882no—
148#103—1882no—
149#101—22no—
150#101—41no—
151#101—942no—
152#101—1881no—
153C4×C8—1881no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—41no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8Dic47⋊C411yescharacteristic
161C2×Q8C4⋊Dic4711yescharacteristic
162C2×Q8(C2×D47)⋊C411yescharacteristic
163C2×Q8C22⋊Dic4711yescharacteristic
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—1881no—
168C22×C4(C2×D47)⋊C411yescharacteristic
169M16—42no—
170M16—81no—
171M16—3761no—
172M16—1882no—
173C2×C8—81no—
174C2×C8—942no—
175C2×C8—3761no—
176C2×C8—1881no—
177C2×C8—1881no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—3762no—
183C4⋊C4—1881no—
184C4⋊C4—1881no—
185C42—21no—
186C42C4×Dic4711yescharacteristic
187C42Dic47⋊C411yescharacteristic
188C23#1309393811yescharacteristic
189Q8—41no—
190Q8—41no—
191Q8—41no—
192Q8—41no—
193Q8—81no—
194Q8—81no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—3762no—
210C2×C4—3761no—
211C2×C4C47⋊#9511yescharacteristic
212C2×C4#1309393811yescharacteristic
213C8—82no—
214C8—3762no—
215C8—1882no—
216C22—21no—
217C22—21no—
218C22#1568888911yescharacteristic
219C22#1568891511yescharacteristic
220C22#1568891711yescharacteristic
221C4—22no—
222C4—22no—
223C4—21no—
224C4—21no—
225C4—41no—
226C4—41no—
227C4—41no—
228C4—41no—
229C4—7521no—
230C2—21no—
231C2—21no—
232C2#2024151211yescharacteristic
233C2#2024155411yescharacteristic
234C2#2024155611yescharacteristic
subgroupG/H
centerC22#15688915
commutatorC42×C47C42
FrattiniC2×C4⋊Q8C2×D47
FittingC2×C47×#356C2
socleC22×C47#353
prime radicalC23×C47#99
cosocleC2×C4⋊Q8×C47C22
hypercenterC2×#356D47
hypocenterC47#7499
LCSG ▷ C42×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ C2×#356

Complements

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

NH ≅ G/Ntype
10. #15688922C4semidirect
14. C47×#356C4semidirect
131. #356Dic47semidirect
132. C47#7499semidirect

Maximal subgroups

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

subgroupindex
C2×#156889222
#202441912
C2×C47×#3562
#749947