Finite Group Databasebeta

← #26125784

Subgroups — #26125784

Order 12032.

270 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 79 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#20252320C211yesmaximal, characteristic
2#20245204C211yesmaximal, characteristic
3#20241340C212yesmaximal
4#20241338C212yesmaximal
5C47×#2166C211yesmaximal, characteristic
6#15689676C411yescharacteristic
7#15689251—21no—
8#15689251—21no—
9C8×Dic94—21no—
10C8×Dic94—21no—
11#15689151C2211yescharacteristic
12#15689150C411yescharacteristic
13#15689146C2211yescharacteristic
14#15689078C411yescharacteristic
15C2×C8×Dic47C411yescharacteristic
16#15688944—21no—
17#15688905C2212yes—
18#15688901—21no—
19C47×#405C2211yescharacteristic
20C47×#364—21no—
21C47×#349C2212yes—
22C47×#347—21no—
23C2×#10625329C2×C411yescharacteristic
24C22×C47⋊C8C2×C411yescharacteristic
25#13094023—21no—
26#13094015—21no—
27#13094001—21no—
28#13094001—21no—
29C4×Dic94—21no—
30C4×Dic94—21no—
31#13093983C2×C411yescharacteristic
32C2×C4×Dic47C2×C411yescharacteristic
33#13093964—22no—
34C4⋊(C47⋊C8)—21no—
35C4⋊(C47⋊C8)—21no—
36Dic47⋊C8—21no—
37Dic47⋊C8—21no—
38Dic47⋊C8—21no—
39Dic47⋊C8—21no—
40#13093934—21no—
41C8×Dic47—21no—
42C8×Dic47—21no—
43C8×Dic47—21no—
44C2×M16×C47C2311yescharacteristic
45C22×C8×C47C2×C411yescharacteristic
46#117×C47C2×C411yescharacteristic
47M32×C47—22no—
48#97×C47D411yescharacteristic
49#97×C47D411yescharacteristic
50C4×C8×C47D411yescharacteristic
51C4×C8×C47D411yescharacteristic
52C22×C4×C47C22×C411yescharacteristic
53C22×Dic47—21no—
54C2×Dic94—21no—
55C2×Dic94—21no—
56M16×C47—41no—
57C2×C8×C47—22no—
58C2×C8×C47C22⋊C411yescharacteristic
59C2×C8×C47C22⋊C411yescharacteristic
60C2×C8×C47C2×D411yescharacteristic
61C2×C8×C47C2×D411yescharacteristic
62C4⋊C4×C47—21no—
63C22⋊C4×C47—21no—
64C42×C47C22⋊C411yescharacteristic
65C42×C47C22⋊C411yescharacteristic
66C22⋊Dic47M1611yescharacteristic
67C22⋊Dic47M1611yescharacteristic
68C4⋊Dic47—21no—
69C4⋊Dic47—21no—
70C4⋊Dic47—21no—
71Dic47⋊C4—21no—
72Dic47⋊C4—21no—
73Dic47⋊C4—21no—
74Dic47⋊C4—21no—
75C4×Dic47—21no—
76C4×Dic47—21no—
77C4×Dic47M1611yescharacteristic
78C4×Dic47M1611yescharacteristic
79#10625329—41no—
80C2×C47⋊C8—21no—
81C2×C47⋊C8—21no—
82C2×C47⋊C8—41no—
83C16×C47—42no—
84C47⋊C16—42no—
85C23×C47C2×M1611yescharacteristic
86C2×C4×C47—21no—
87C2×C4×C47—21no—
88C2×C4×C47—21no—
89C2×C4×C47—21no—
90C2×C4×C47C2×C22⋊C411yescharacteristic
91C2×C4×C47C2×M1611yescharacteristic
92C2×Dic47—21no—
93C2×Dic47—21no—
94C2×Dic47—21no—
95C2×Dic47—21no—
96C2×Dic47—42no—
97Dic94—42no—
98Dic94—42no—
99C8×C47—21no—
100C8×C47—21no—
101C8×C47—21no—
102C8×C47#10112yes—
103C47⋊C8—41no—
104C47⋊C8—41no—
105#11539—471nomaximal, Sylow
106C22×C47—21no—
107C22×C47—41no—
108C22×C47#40711yescharacteristic
109C4×C47—21no—
110C4×C47—41no—
111C4×C47—41no—
112C4×C47#41011yescharacteristic
113C4×C47C2×#10111yescharacteristic
114Dic47—22no—
115Dic47—41no—
116Dic47—41no—
117Dic47—41no—
118#2960—471no—
119#2166—471no—
120#2166D4711yescharacteristic
121#1328—472no—
122#1327—472no—
123C2×C47—21no—
124C2×C47—41no—
125C2×C47#146711yescharacteristic
126#533—471no—
127#446—941no—
128#446—941no—
129C8×Q8—941no—
130C8×Q8—941no—
131#424—471no—
132#424—471no—
133#423—471no—
134#405—471no—
135#405C2×D4711yescharacteristic
136C2×C4×C8—471no—
137#364—21no—
138#364—941no—
139#349—472no—
140#349C2×D4712yes—
141#347—21no—
142#347—941no—
143C47#1153911yescharacteristic, complemented, Sylow
144C2×M16—471no—
145C2×M16C22×D4711yescharacteristic
146C22×C8—471no—
147C22×C8C4×D4711yescharacteristic
148C4⋊Q8—941no—
149#125—941no—
150C22⋊Q8—941no—
151C22⋊Q8—941no—
152C4×Q8—941no—
153C4×Q8—941no—
154#117—471no—
155#117C4×D4711yescharacteristic
156C2×C42—471no—
157M32—22no—
158M32—942no—
159C4⋊C8—941no—
160C4⋊C8—941no—
161C4⋊C8—941no—
162C4⋊C8—941no—
163C4⋊C8—941no—
164C4⋊C8—941no—
165#97—941no—
166#97D18811yescharacteristic
167#97C22⋊D4711yescharacteristic
168C4×C8—941no—
169C4×C8—941no—
170C4×C8—941no—
171C4×C8D18811yescharacteristic
172C4×C8C22⋊D4711yescharacteristic
173C2×Q8—941no—
174C2×Q8—941no—
175C22×C4—941no—
176C22×C4C2×C4×D4711yescharacteristic
177M16—41no—
178M16—1881no—
179C2×C8—22no—
180C2×C8—941no—
181C2×C8—941no—
182C2×C8—1881no—
183C2×C8(C2×D47)⋊C411yescharacteristic
184C2×C8(C2×D47)⋊C411yescharacteristic
185C2×C8C2×D18811yescharacteristic
186C2×C8C2×C22⋊D4711yescharacteristic
187C4⋊C4—21no—
188C4⋊C4—941no—
189C4⋊C4—941no—
190C4⋊C4—941no—
191C4⋊C4—941no—
192C4⋊C4—941no—
193C4⋊C4—941no—
194C4⋊C4—941no—
195C22⋊C4—21no—
196C22⋊C4—471no—
197C22⋊C4—471no—
198C42—471no—
199C42—471no—
200C42—941no—
201C42—941no—
202C42(C2×D47)⋊C411yescharacteristic
203C42(C2×D47)⋊C411yescharacteristic
204C16—42no—
205C16—1882no—
206C23M16×D4711yescharacteristic
207Q8—1882no—
208Q8—1882no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—941no—
214C2×C4—941no—
215C2×C4—941no—
216C2×C4—941no—
217C2×C4—1882no—
218C2×C4C2×(C2×D47)⋊C411yescharacteristic
219C2×C4M16×D4711yescharacteristic
220C8—21no—
221C8—21no—
222C8—21no—
223C8—1881no—
224C8—1881no—
225C8#1309394212yes—
226C22—21no—
227C22—41no—
228C22#1568908611yescharacteristic
229C4—21no—
230C4—41no—
231C4—41no—
232C4—942no—
233C4—1881no—
234C4—1881no—
235C4—1881no—
236C4#1568909611yescharacteristic
237C4C2×#1309394211yescharacteristic
238C2—21no—
239C2—41no—
240C2#2024173111yescharacteristic
subgroupG/H
centerC4#15689096
commutatorC22×C4×C47C22×C4
FrattiniC2×M16C22×D47
FittingC47×#2166C2
socleC2×C47#1467
prime radicalC23×C47C2×M16
cosocleC2×M16×C47C23
hypercenter#2166D47
hypocenterC47#11539
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×C8 ◁ #405 ◁ #2166

Complements

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

NH ≅ G/Ntype
143. C47#11539semidirect

Maximal subgroups

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

subgroupindex
#202523202
#202452042
#202413402
#202413382
C47×#21662
#1153947