Finite Group Databasebeta

← #27293524

Subgroups — #27293524

Order 13568.

330 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 87 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#20276527C212yesmaximal, complemented
2#20272666C212yesmaximal, complemented
3#20272651C212yesmaximal
4C53×#2920C211yesmaximal, characteristic
5#16810444—22no—
6#16810405C2214yes—
7#16810401—22no—
8C22×C42×C53C411yescharacteristic
9C53×#432—22no—
10C53×#431—22no—
11C53×#423C411yescharacteristic
12C53×#423C2211yescharacteristic
13C53×#407C411yescharacteristic
14C53×#407C2211yescharacteristic
15C4×M16×C53C411yescharacteristic
16C4×M16×C53C2211yescharacteristic
17#13133157—24no—
18C23×C4×C53—21no—
19C23×C4×C53C2×C411yescharacteristic
20C2×M16×C53C2×C411yescharacteristic
21C2×M16×C53C2×C411yescharacteristic
22C2×M16×C53C2×C411yescharacteristic
23C2×M16×C53C2311yescharacteristic
24C2×C42×C53—21no—
25C2×C42×C53—21no—
26C2×C42×C53—21no—
27C2×C42×C53—21no—
28C2×C42×C53—21no—
29C2×C42×C53C2×C411yescharacteristic
30C2×C42×C53C2×C411yescharacteristic
31C4⋊C8×C53—22no—
32C4⋊C8×C53—22no—
33C22⋊C8×C53—22no—
34C22⋊C8×C53—22no—
35#97×C53—21no—
36#97×C53D412yes—
37C4×C8×C53—21no—
38C4×C8×C53D412yes—
39C24×C53M1611yescharacteristic
40C22×C4×C53—21no—
41C22×C4×C53—21no—
42C22×C4×C53—21no—
43C22×C4×C53—21no—
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53—21no—
48C22×C4×C53—44no—
49C22×C4×C53—41no—
50C22×C4×C53—41no—
51C22×C4×C53M1611yescharacteristic
52C22×C4×C53C22×C411yescharacteristic
53M16×C53—41no—
54M16×C53—41no—
55M16×C53—41no—
56M16×C53—41no—
57C2×C8×C53—21no—
58C2×C8×C53—21no—
59C2×C8×C53C22⋊C412yes—
60C2×C8×C53C2×D412yes—
61C42×C53—21no—
62C42×C53—21no—
63C42×C53—21no—
64C42×C53—21no—
65C42×C53—21no—
66C42×C53—21no—
67C42×C53C22⋊C411yescharacteristic
68C42×C53C22⋊C411yescharacteristic
69C42×C53M1611yescharacteristic
70C42×C53M1611yescharacteristic
71#11719908—44no—
72C23×C53—21no—
73C23×C53—42no—
74C23×C53—41no—
75C23×C53C2×M1611yescharacteristic
76C2×C4×C53—24no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—42no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53C2×C22⋊C411yescharacteristic
94C2×C4×C53C2×M1611yescharacteristic
95C8×C53—22no—
96C8×C53—22no—
97C8×C53—41no—
98C8×C53—41no—
99#11521—531nomaximal, Sylow
100C22×C53—22no—
101C22×C53—21no—
102C22×C53—41no—
103C22×C53—41no—
104C22×C53—42no—
105C22×C53—42no—
106C22×C53—41no—
107C22×C53#40711yescharacteristic
108C4×C53—22no—
109C4×C53—22no—
110C4×C53—21no—
111C4×C53—41no—
112C4×C53—41no—
113C4×C53—41no—
114C4×C53#41011yescharacteristic
115C4×C53#41311yescharacteristic
116#2920D5311yescharacteristic
117#2165—532no—
118#1327—532no—
119#1320—532no—
120C2×C53—21no—
121C2×C53—42no—
122C2×C53—41no—
123C2×C53#146411yescharacteristic
124C22×C42Dic5311yescharacteristic
125#432—22no—
126#431—22no—
127#423Dic5311yescharacteristic
128#423C2×D5311yescharacteristic
129#407Dic5311yescharacteristic
130#407C2×D5311yescharacteristic
131C4×M16Dic5311yescharacteristic
132C4×M16C2×D5311yescharacteristic
133#364—1062no—
134#349—534no—
135#347—1062no—
136C53#1152111yescharacteristic, complemented, Sylow
137C23×C4—21no—
138C23×C4C2×Dic5311yescharacteristic
139C2×M16C2×Dic5311yescharacteristic
140C2×M16C2×Dic5311yescharacteristic
141C2×M16C2×Dic5311yescharacteristic
142C2×M16C22×D5311yescharacteristic
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42C2×Dic5311yescharacteristic
149C2×C42C2×Dic5311yescharacteristic
150M32—1064no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C22⋊C8—22no—
154C22⋊C8—22no—
155#97—21no—
156#97C22⋊D5312yes—
157C4×C8—21no—
158C4×C8C22⋊D5312yes—
159C24#1171991711yescharacteristic
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—44no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4#1171991711yescharacteristic
172C22×C4C22×Dic5311yescharacteristic
173M16—41no—
174M16—41no—
175M16—41no—
176M16—41no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8C22⋊Dic5312yes—
180C2×C8C2×C22⋊D5312yes—
181C42—21no—
182C42—21no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42#1171991711yescharacteristic
188C42#1171991711yescharacteristic
189C42C22⋊Dic5311yescharacteristic
190C42C22⋊Dic5311yescharacteristic
191C16—2124no—
192C23—21no—
193C23—42no—
194C23—41no—
195C23C2×#1171991711yescharacteristic
196C2×C4—24no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4C2×C22⋊Dic5311yescharacteristic
214C2×C4C2×#1171991711yescharacteristic
215C8—22no—
216C8—22no—
217C8—41no—
218C8—41no—
219C22—22no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—42no—
224C22—42no—
225C22—41no—
226C22#1681058711yescharacteristic
227C4—22no—
228C4—22no—
229C4—21no—
230C4—41no—
231C4—41no—
232C4—41no—
233C4#1681059711yescharacteristic
234C4#1681060711yescharacteristic
235C2—21no—
236C2—42no—
237C2—41no—
238C2#2027304911yescharacteristic
subgroupG/H
centerC4#16810597
commutatorC22×C4×C53C22×C4
FrattiniC2×M16C22×D53
FittingC53×#2920C2
socleC2×C53#1464
prime radicalC24×C53M16
cosocleC2×M16×C53C23
hypercenter#2920D53
hypocenterC53#11521
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C42 ◁ C4×M16 ◁ #2920

Complements

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

NH ≅ G/Ntype
1. #20276527C2semidirect
2. #20272666C2semidirect
136. C53#11521semidirect

Maximal subgroups

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

subgroupindex
#202765272
#202726662
#202726512
C53×#29202
#1152153