Finite Group Databasebeta

← #27281727

Subgroups — #27281727

Order 13568.

456 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 103 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#20277821C211yesmaximal, characteristic, complemented
2#20273342C211yesmaximal, characteristic, complemented
3#20273223C212yesmaximal
4#20272609C212yesmaximal, complemented
5C53×#3142C211yesmaximal, characteristic
6#16810680C2212yes—
7#16810509—22no—
8#16810509—22no—
9C4⋊C4×Dic53—22no—
10#16810469C2212yes—
11#16810365—22no—
12#16810357—22no—
13#16810357—22no—
14C53×#532C2211yescharacteristic
15C2×#125×C53C2211yescharacteristic
16C53×#491C412yescomplemented
17C53×#490C412yescomplemented
18C53×#488—22no—
19C53×#417—22no—
20C4×M16×C53C2211yescharacteristic
21C2×C4⋊Dic53—22no—
22C2×C4×Dic53—22no—
23C2×C4×Dic53—22no—
24C4⋊#88322—22no—
25C22⋊#88322—22no—
26#13133128—22no—
27#13133124—22no—
28#13133124—22no—
29#13133124—22no—
30#13133124D412yes—
31#13133124D412yes—
32C2×#41×C53—21no—
33C2×M16×C53—21no—
34C4⋊Q8×C53C2×C411yescharacteristic
35#127×C53C2×C411yescharacteristic
36#125×C53—22no—
37#125×C53C2×C412yes—
38#125×C53D412yes—
39#124×C53—21no—
40#121×C53—22no—
41C4×D4×C53—22no—
42C2×C4⋊C4×C53—22no—
43C2×C4⋊C4×C53D412yes—
44C2×C42×C53C2311yescharacteristic
45#103×C53—24no—
46#102×C53—24no—
47#97×C53C2×C412yes—
48C4×C8×C53—21no—
49#41×C53—44no—
50C2×Q8×C53—21no—
51C2×D4×C53—21no—
52C2×D4×C53—22no—
53C2×D4×C53—41no—
54C22×C4×C53—22no—
55C22×C4×C53—41no—
56C22×C4×C53C2×D411yescharacteristic
57C22×C4×C53C2×D412yes—
58C22×Dic53—22no—
59C22×Dic53—22no—
60M16×C53—42no—
61C2×C8×C53—22no—
62C4⋊C4×C53—24no—
63C4⋊C4×C53—24no—
64C4⋊C4×C53—22no—
65C4⋊C4×C53—22no—
66C4⋊C4×C53C22⋊C414yes—
67C22⋊C4×C53—42no—
68C42×C53C22×C411yescharacteristic
69C42×C53C2×D411yescharacteristic
70C42×C53#4112yes—
71C4⋊Dic53—44no—
72C4×Dic53—44no—
73C4×Dic53—44no—
74C2×#88322—42no—
75C23×C53—41no—
76C23×C53#12011yescharacteristic
77Q8×C53—42no—
78D4×C53—42no—
79D4×C53—82no—
80D4×C53—82no—
81C2×C4×C53—22no—
82C2×C4×C53—24no—
83C2×C4×C53—42no—
84C2×C4×C53—42no—
85C2×C4×C53—42no—
86C2×C4×C53—41no—
87C2×C4×C53—82no—
88C2×C4×C53C2×C22⋊C411yescharacteristic
89C2×C4×C53C4×D412yes—
90C2×C4×C53#12112yes—
91C2×C4×C53#12311yescharacteristic
92C2×Dic53—24no—
93C2×Dic53—42no—
94C2×Dic53—42no—
95C2×Dic53—42no—
96C2×Dic53—42no—
97C2×Dic53—42no—
98C8×C53—42no—
99#88322—82no—
100#9442—531nomaximal, Sylow
101C22×C53—22no—
102C22×C53—81no—
103C22×C53—82no—
104C22×C53#38611yescharacteristic
105C22×C53#45412yes—
106C4×C53—21no—
107C4×C53—21no—
108C4×C53—22no—
109C4×C53—22no—
110C4×C53—44no—
111C4×C53—81no—
112Dic53—44no—
113Dic53—82no—
114#3142D5311yescharacteristic
115#2371—531no—
116#1534—531no—
117#1505—532no—
118#1300—532no—
119C2×C53—22no—
120C2×C53—161no—
121C2×C53#188811yescharacteristic
122C2×C53#190612yes—
123#532C2×D5311yescharacteristic
124C2×#125C2×D5311yescharacteristic
125#491Dic5312yescomplemented
126#490Dic5312yescomplemented
127#488—22no—
128#432—532no—
129#417—22no—
130C4×M16C2×D5311yescharacteristic
131#387—1062no—
132#387—1062no—
133C4×C4⋊C4—1062no—
134#376—532no—
135#330—1062no—
136#327—1062no—
137#327—1062no—
138C53#944211yescharacteristic, complemented, Sylow
139C2×#41—21no—
140C2×M16—21no—
141C4⋊Q8C2×Dic5311yescharacteristic
142#127C2×Dic5311yescharacteristic
143#125—22no—
144#125C2×Dic5312yes—
145#125C22⋊D5312yes—
146#124—21no—
147#121—22no—
148C4×D4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—1062no—
151C2×C4⋊C4C22⋊D5312yes—
152C2×C42—1062no—
153C2×C42—1062no—
154C2×C42C22×D5311yescharacteristic
155C4⋊C8—1062no—
156#103—24no—
157#102—24no—
158C22⋊C8—1062no—
159#97—1062no—
160#97C2×Dic5312yes—
161C4×C8—21no—
162#95—532no—
163#95—532no—
164#95—1062no—
165#95—1062no—
166#95—1062no—
167#41—44no—
168C2×Q8—21no—
169C2×D4—21no—
170C2×D4—22no—
171C2×D4—41no—
172C22×C4—22no—
173C22×C4—41no—
174C22×C4—1062no—
175C22×C4—1062no—
176C22×C4D4×D5312yes—
177C22×C4C2×C22⋊D5311yescharacteristic
178M16—42no—
179C2×C8—22no—
180C2×C8—2122no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—2124no—
186C4⋊C4C22⋊Dic5314yes—
187C22⋊C4—42no—
188C42—2124no—
189C42—2124no—
190C42#1171994712yes—
191C42C22×Dic5311yescharacteristic
192C42C2×C22⋊D5311yescharacteristic
193C23—41no—
194C23#1313318511yescharacteristic
195Q8—42no—
196D4—42no—
197D4—82no—
198D4—82no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—82no—
206C2×C4—1064no—
207C2×C4—2122no—
208C2×C4—2122no—
209C2×C4—2122no—
210C2×C4—2122no—
211C2×C4—2122no—
212C2×C4C2×C22⋊Dic5311yescharacteristic
213C2×C4D4×Dic5312yes—
214C2×C4#1313318912yes—
215C2×C4#1313319811yescharacteristic
216C8—42no—
217C8—4242no—
218C22—22no—
219C22—82no—
220C22—81no—
221C22#1681050311yescharacteristic
222C22#1681079512yes—
223C4—22no—
224C4—22no—
225C4—21no—
226C4—21no—
227C4—44no—
228C4—81no—
229C4—4242no—
230C4—2124no—
231C2—22no—
232C2—161no—
233C2#2027511911yescharacteristic
234C2#2027521812yes—
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3142C2
socleC22×C53#386
prime radicalC53×#532C22
cosocleC2×C42×C53C23
hypercenter#3142D53
hypocenterC53#9442
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3142

Complements

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

NH ≅ G/Ntype
1. #20277821C2semidirect
2. #20273342C2semidirect
4. #20272609C2semidirect
16. C53×#491C4semidirect
17. C53×#490C4semidirect
125. #491Dic53semidirect
126. #490Dic53semidirect
138. C53#9442semidirect

Maximal subgroups

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

subgroupindex
#202778212
#202733422
#202732232
#202726092
C53×#31422
#944253