Finite Group Databasebeta

← #26184029

Subgroups — #26184029

Order 12032.

938 conjugacy classes of subgroups; only the 355 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 355 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#20257198C211yesmaximal, characteristic
2#20252469C211yesmaximal, characteristic
3#20241951C216yesmaximal
4#20241886C216yesmaximal
5C47×#2351C211yesmaximal, characteristic
6Q8×Dic94C414yescomplemented
7#15689852C414yescomplemented
8#15689849—24no—
9#15689754—23no—
10#15689680C2216yes—
11#15689601—21no—
12Q8×C47⋊C8—24no—
13#15689176C2216yes—
14#15688859C22112yes—
15#15688859C2214yes—
16C2×C4×Q8×C47—21no—
17C2×C4×Q8×C47C2211yescharacteristic
18C47×#389—23no—
19C47×#382C2213yes—
20C4×C4⋊C4×C47C2213yes—
21#13094024C2×C416yes—
22#13094024C2×C416yes—
23#13094023—212no—
24#13094016—26no—
25#13094015—212no—
26#13094003—26no—
27#13094003C2314yes—
28Q8×Dic47—24no—
29Q8×Dic47—24no—
30C4×Dic94—26no—
31C4×Dic94C2×C4112yes—
32#13093983—26no—
33C4⋊(C47⋊C8)—26no—
34C22⋊(C47⋊C8)C2314yes—
35C4×C47⋊C8—26no—
36C22×Q8×C47D411yescharacteristic
37C4×Q8×C47—44no—
38C4×Q8×C47C2×C414yes—
39C4×Q8×C47D414yes—
40C2×C4⋊C4×C47—23no—
41C2×C4⋊C4×C47—23no—
42C2×C4⋊C4×C47D413yes—
43C2×C42×C47—23no—
44C2×C42×C47C2313yes—
45#95×C47—23no—
46#95×C47C2311yescharacteristic
47#95×C47C2313yes—
48C2×Q8×C47C22⋊C414yes—
49C2×Q8×C47SD1614yes—
50C2×Q8×C47Q1614yes—
51C22×C4×C47—21no—
52C22×C4×C47—23no—
53C22×C4×C47C2×D413yes—
54C22×C4×C47C2×D413yes—
55C22×C4×C47C2411yescharacteristic
56C2×Dic94—26no—
57C2×Dic94C22×C414yes—
58C4⋊C4×C47—46no—
59C4⋊C4×C47—46no—
60C4⋊C4×C47C22⋊C4112yes—
61C42×C47—46no—
62C42×C47C22×C416yes—
63C42×C47C2×D416yes—
64C22⋊Dic47—24no—
65C4⋊Dic47—26no—
66C4⋊Dic47—26no—
67C4⋊Dic47—26no—
68C4⋊Dic47C22×C414yes—
69Dic47⋊C4—212no—
70Dic47⋊C4—24no—
71C4×Dic47—212no—
72C2×C47⋊C8—24no—
73C23×C47C22×D411yescharacteristic
74Q8×C47#103116yes—
75C2×C4×C47—22no—
76C2×C4×C47—43no—
77C2×C4×C47—41no—
78C2×C4×C47—41no—
79C2×C4×C47—41no—
80C2×C4×C47—43no—
81C2×C4×C47—43no—
82C2×C4×C47—43no—
83C2×C4×C47C2×C22⋊C416yes—
84C2×C4×C47C2×C22⋊C416yes—
85C2×C4×C47C2×SD1616yes—
86C2×C4×C47C2×Q1616yes—
87C2×C4×C47C23×C411yescharacteristic
88C2×C4×C47C22×D411yescharacteristic
89C2×Dic47—24no—
90C2×Dic47—24no—
91Dic94—44no—
92Dic94—86no—
93C47⋊C8—44no—
94#17619—471nomaximal, Sylow
95C22×C47—21no—
96C22×C47—21no—
97C22×C47C22×C22⋊C411yescharacteristic
98C22×C47C22×SD1611yescharacteristic
99C22×C47C22×Q1611yescharacteristic
100C4×C47—21no—
101C4×C47—21no—
102C4×C47—26no—
103C4×C47—42no—
104C4×C47—83no—
105C4×C47C2×#103112yes—
106Dic47—44no—
107Dic47—44no—
108#3502—471no—
109#2982—471no—
110#2351D4711yescharacteristic
111#1518—476no—
112#1503—476no—
113C2×C47—21no—
114C2×C47—21no—
115C2×C47#288411yescharacteristic
116C2×C47C22×#10311yescharacteristic
117C2×C47#290111yescharacteristic
118Q82—474no—
119#557—474no—
120#557—944no—
121#544—943no—
122#533—476no—
123#519—941no—
124C2×C4×Q8—21no—
125C2×C4×Q8C2×D4711yescharacteristic
126C8×Q8—944no—
127#431—476no—
128#389—23no—
129#382C2×D4713yes—
130C4×C4⋊C4C2×D4713yes—
131#328—474no—
132#328—4712no—
133C47#1761911yescharacteristic, complemented, Sylow
134C22×Q8C22⋊D4711yescharacteristic
135C4⋊Q8—476no—
136C4⋊Q8—476no—
137C4⋊Q8—9412no—
138#125—9412no—
139#125—946no—
140C22⋊Q8—474no—
141C22⋊Q8—946no—
142C4×Q8—44no—
143C4×Q8—4712no—
144C4×Q8—944no—
145C4×Q8—944no—
146C4×Q8—946no—
147C4×Q8C4×D4714yes—
148C4×Q8D18814yes—
149#117—946no—
150C2×C4⋊C4—23no—
151C2×C4⋊C4—23no—
152C2×C4⋊C4C22⋊D4713yes—
153C2×C42—23no—
154C2×C42C22×D4713yes—
155C4⋊C8—946no—
156C22⋊C8—474no—
157C4×C8—946no—
158#95—23no—
159#95C22×D4711yescharacteristic
160#95C22×D4713yes—
161C2×Q8—474no—
162C2×Q8—946no—
163C2×Q8(C2×D47)⋊C414yes—
164C2×Q8#1062533514yes—
165C2×Q8#1062533714yes—
166C22×C4—21no—
167C22×C4—23no—
168C22×C4C2×C22⋊D4713yes—
169C22×C4C2×C22⋊D4713yes—
170C22×C4C23×D4711yescharacteristic
171C2×C8—944no—
172C4⋊C4—46no—
173C4⋊C4—46no—
174C4⋊C4—474no—
175C4⋊C4—9412no—
176C4⋊C4—946no—
177C4⋊C4—946no—
178C4⋊C4—946no—
179C4⋊C4—944no—
180C4⋊C4(C2×D47)⋊C4112yes—
181C22⋊C4—944no—
182C42—46no—
183C42—9412no—
184C42C2×C4×D4716yes—
185C42C2×D18816yes—
186C23C22×C22⋊D4711yescharacteristic
187Q8—3766no—
188Q8—1884no—
189Q8#13093947116yes—
190C2×C4—22no—
191C2×C4—43no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—43no—
196C2×C4—43no—
197C2×C4—43no—
198C2×C4—944no—
199C2×C4—944no—
200C2×C4C2×(C2×D47)⋊C416yes—
201C2×C4C2×(C2×D47)⋊C416yes—
202C2×C4C2×#1062533516yes—
203C2×C4C2×#1062533716yes—
204C2×C4C22×C4×D4711yescharacteristic
205C2×C4C22×D18811yescharacteristic
206C8—1884no—
207C22—21no—
208C22—21no—
209C22C22×(C2×D47)⋊C411yescharacteristic
210C22C22×#1062533511yescharacteristic
211C22C22×#1062533711yescharacteristic
212C4—26no—
213C4—21no—
214C4—21no—
215C4—42no—
216C4—83no—
217C4—1884no—
218C4—1884no—
219C4C2×#13093947112yes—
220C2—21no—
221C2—21no—
222C2#2025185111yescharacteristic
223C2C22×#1309394711yescharacteristic
224C2#2025193611yescharacteristic
subgroupG/H
centerC22C22×(C2×D47)⋊C4
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2351C2
socleC22×C47C22×C22⋊C4
prime radicalC23×C47C22×D4
cosocleC22×C4×C47C24
hypercenter#2351D47
hypocenterC47#17619
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2351

Complements

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

NH ≅ G/Ntype
6. Q8×Dic94C4semidirect
7. #15689852C4semidirect
133. C47#17619semidirect

Maximal subgroups

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

subgroupindex
#202571982
#202524692
#202419512
#202418862
C47×#23512
#1761947