Finite Group Databasebeta

← #26121308

Subgroups — #26121308

Order 12032.

296 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 89 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#20244368C211yesmaximal, characteristic
2#20243507C211yesmaximal, characteristic
3#20242160C211yesmaximal, characteristic
4#20241792C211yesmaximal, characteristic
5#20241790C211yesmaximal, characteristic
6#20241216C211yesmaximal, characteristic
7C47×#1571C211yesmaximal, characteristic
8#15689680C2211yescharacteristic
9#15689176C2211yescharacteristic
10C2×C4⋊(C47⋊C8)C2211yescharacteristic
11#15688969C2211yescharacteristic
12#15688884—22no—
13#15688877—22no—
14#15688873—22no—
15#15688855—22no—
16#15688853—22no—
17C47×#431C2211yescharacteristic
18C47×#423C2211yescharacteristic
19C2×C4⋊C8×C47C2211yescharacteristic
20C47×#335—22no—
21C22×C47⋊C8—21no—
22#13094024Q811yescharacteristic
23#13094024Q811yescharacteristic
24#13094016—21no—
25#13094003—21no—
26#13094003—21no—
27C4×Dic94—21no—
28C4×Dic94—21no—
29#13093983—21no—
30C2×C4×Dic47D412yes—
31C4⋊(C47⋊C8)—21no—
32C4⋊(C47⋊C8)—21no—
33C4⋊(C47⋊C8)—21no—
34C22⋊(C47⋊C8)—21no—
35C4×C47⋊C8D412yes—
36C47⋊#95—21no—
37C47⋊#95—21no—
38C2×M16×C47—21no—
39C22×C8×C47—21no—
40C2×C42×C47C2311yescharacteristic
41C4⋊C8×C47—21no—
42C4⋊C8×C47—21no—
43C4⋊C8×C47—21no—
44C4⋊C8×C47—21no—
45C4⋊C8×C47—21no—
46C22⋊C8×C47—21no—
47C4×C8×C47D412yes—
48C22×C4×C47C2×D411yescharacteristic
49C22×C4×C47#4111yescharacteristic
50C22×C4×C47#4111yescharacteristic
51C22×Dic47—22no—
52C2×Dic94—21no—
53C2×Dic94—21no—
54M16×C47—42no—
55C2×C8×C47—22no—
56C2×C8×C47—41no—
57C2×C8×C47—41no—
58C2×C8×C47—42no—
59C42×C47C2×D411yescharacteristic
60C42×C47C2×D411yescharacteristic
61C42×C47C2×Q811yescharacteristic
62C42×C47#4111yescharacteristic
63C22⋊Dic47—41no—
64C4⋊Dic47—21no—
65C4⋊Dic47—21no—
66C4⋊Dic47—21no—
67C4⋊Dic47—21no—
68Dic47⋊C4—41no—
69Dic47⋊C4—41no—
70C4×Dic47—22no—
71C4×Dic47—22no—
72C4×Dic47—41no—
73C2×C47⋊C8—22no—
74C2×C47⋊C8—41no—
75C2×C47⋊C8—42no—
76C23×C47#12311yescharacteristic
77C2×C4×C47—41no—
78C2×C4×C47—41no—
79C2×C4×C47C22⋊Q811yescharacteristic
80C2×C4×C47C22⋊Q811yescharacteristic
81C2×C4×C47C22⋊Q811yescharacteristic
82C2×C4×C47#12311yescharacteristic
83C2×C4×C47#12311yescharacteristic
84C2×C4×C47#12711yescharacteristic
85C2×C4×C47#13512yes—
86C2×C4×C47#13712yes—
87C2×Dic47—42no—
88C2×Dic47—42no—
89C2×Dic47—41no—
90C2×Dic47—41no—
91C2×Dic47—42no—
92Dic94—42no—
93Dic94—42no—
94C8×C47—42no—
95C8×C47—81no—
96C8×C47—81no—
97C47⋊C8—42no—
98C47⋊C8—81no—
99#10669—471nomaximal, Sylow
100C22×C47—21no—
101C22×C47—21no—
102C22×C47#39911yescharacteristic
103C22×C47#48512yes—
104C4×C47—21no—
105C4×C47—21no—
106C4×C47—21no—
107C4×C47—21no—
108C4×C47#47912yes—
109C4×C47#49511yescharacteristic
110C4×C47#49711yescharacteristic
111Dic47—82no—
112Dic47—81no—
113Dic47—81no—
114#1995—471no—
115#1832—471no—
116#1571D4711yescharacteristic
117#1570—471no—
118#1481—471no—
119#1481—471no—
120#1274—471no—
121C2×C47—22no—
122C2×C47#209711yescharacteristic
123C2×C47#209811yescharacteristic
124C2×C47#209811yescharacteristic
125#533—471no—
126#431—471no—
127#431C2×D4711yescharacteristic
128#423C2×D4711yescharacteristic
129C2×C4⋊C8—471no—
130C2×C4⋊C8C2×D4711yescharacteristic
131#376—471no—
132#339—942no—
133#336—942no—
134#335—22no—
135#334—942no—
136Q8⋊C8—942no—
137Q8⋊C8—942no—
138C47#1066911yescharacteristic, complemented, Sylow
139C2×M16—21no—
140C22×C8—21no—
141C22×C8—941no—
142C4⋊Q8—471no—
143C4⋊Q8—471no—
144#125—941no—
145C22⋊Q8—941no—
146C22⋊Q8—941no—
147C4×Q8—941no—
148C4×Q8—941no—
149#117—941no—
150C2×C42—472no—
151C2×C42C22×D4711yescharacteristic
152C4⋊C8—21no—
153C4⋊C8—21no—
154C4⋊C8—21no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—941no—
158C4⋊C8—941no—
159C4⋊C8—941no—
160C22⋊C8—21no—
161C22⋊C8—941no—
162C4×C8—472no—
163C4×C8C22⋊D4712yes—
164#95—941no—
165#95—941no—
166C2×Q8—941no—
167C2×Q8—941no—
168C22×C4—942no—
169C22×C4#1062535011yescharacteristic
170C22×C4D4×D4711yescharacteristic
171C22×C4#1062535411yescharacteristic
172M16—42no—
173C2×C8—22no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—42no—
177C2×C8—942no—
178C2×C8—1881no—
179C2×C8—1882no—
180C4⋊C4—941no—
181C4⋊C4—941no—
182C4⋊C4—941no—
183C4⋊C4—941no—
184C4⋊C4—1881no—
185C4⋊C4—1881no—
186C22⋊C4—1881no—
187C42—942no—
188C42—942no—
189C42—1881no—
190C42#1062535011yescharacteristic
191C42D4×D4711yescharacteristic
192C42Q8×D4711yescharacteristic
193C42C2×C22⋊D4711yescharacteristic
194C23#1309400911yescharacteristic
195Q8—1882no—
196Q8—1882no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—1882no—
200C2×C4—1882no—
201C2×C4—1881no—
202C2×C4—1881no—
203C2×C4—1882no—
204C2×C4#1309400011yescharacteristic
205C2×C4#1309400211yescharacteristic
206C2×C4#1309400211yescharacteristic
207C2×C4#1309400711yescharacteristic
208C2×C4#1309400911yescharacteristic
209C2×C4#1309402011yescharacteristic
210C2×C4#1309404312yes—
211C2×C4#1309405612yes—
212C8—42no—
213C8—81no—
214C8—81no—
215C8—3761no—
216C8—1882no—
217C22—21no—
218C22—21no—
219C22#1568905811yescharacteristic
220C22#1568946612yes—
221C4—21no—
222C4—21no—
223C4—21no—
224C4—21no—
225C4—3762no—
226C4—3761no—
227C4—3761no—
228C4#1568942912yes—
229C4#1568950611yescharacteristic
230C4#1568951611yescharacteristic
231C2—22no—
232C2#2024493311yescharacteristic
233C2#2024494011yescharacteristic
234C2#2024494411yescharacteristic
subgroupG/H
centerC2×C4#13094020
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1571C2
socleC22×C47#399
prime radicalC23×C47#123
hypercenter#1571D47
hypocenterC47#10669
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1571

Complements

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

NH ≅ G/Ntype
138. C47#10669semidirect

Maximal subgroups

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

subgroupindex
#202443682
#202435072
#202421602
#202417922
#202417902
#202412162
C47×#15712
#1066947