Finite Group Databasebeta

← #28308226

Subgroups — #28308226

Order 13568.

330 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 66 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#20291085C212yesmaximal, complemented
2#20289661C212yesmaximal, complemented
3#20289648C212yesmaximal
4#20283347C211yesmaximal, characteristic
5#16811545—22no—
6#16811519C2214yes—
7#16811516—22no—
8C2×C42×D53C411yescharacteristic
9#16810679—22no—
10#16810675—22no—
11#16810647C2211yescharacteristic
12#16810586C411yescharacteristic
13#16810586C2211yescharacteristic
14M16×Dic53C411yescharacteristic
15M16×Dic53C2211yescharacteristic
16C53×#423C411yescharacteristic
17#13133291—24no—
18C22×C4×D53—21no—
19C22×C4×D53C2×C411yescharacteristic
20C2×#11719917C2×C411yescharacteristic
21C2×#11719917C2311yescharacteristic
22C2×C4×Dic53—21no—
23C2×C4×Dic53C2×C411yescharacteristic
24C42×D53—21no—
25C42×D53—21no—
26C42×D53—21no—
27C42×D53—21no—
28C4⋊#88322—22no—
29#13133129—22no—
30#13133129—22no—
31#13133127—21no—
32#13133127D412yes—
33C8×Dic53—21no—
34C8×Dic53D412yes—
35C2×M16×C53C2×C411yescharacteristic
36C2×M16×C53C2×C411yescharacteristic
37C2×C42×C53C2×C411yescharacteristic
38C4⋊C8×C53—22no—
39C23×D53M1611yescharacteristic
40C22×C4×C53—21no—
41C22×C4×C53C22×C411yescharacteristic
42C22×Dic53—21no—
43C22×Dic53M1611yescharacteristic
44C2×C4×D53—21no—
45C2×C4×D53—21no—
46C2×C4×D53—21no—
47C2×C4×D53—21no—
48C2×C4×D53—21no—
49C2×C4×D53—21no—
50C2×C4×D53—44no—
51C2×C4×D53—41no—
52C2×C4×D53—41no—
53M16×C53—41no—
54M16×C53—41no—
55C2×C8×C53—21no—
56C2×C8×C53C22⋊C412yes—
57C42×C53—21no—
58C42×C53M1611yescharacteristic
59C42×C53M1611yescharacteristic
60C4×Dic53—21no—
61C4×Dic53—21no—
62C4×Dic53—21no—
63C4×Dic53—21no—
64C4×Dic53—21no—
65C4×Dic53C22⋊C411yescharacteristic
66C4×Dic53C22⋊C411yescharacteristic
67#11719917—41no—
68#11719917—41no—
69C2×#88322—21no—
70C2×#88322C2×D412yes—
71#11719910—44no—
72C23×C53C2×M1611yescharacteristic
73C22×D53—21no—
74C22×D53—42no—
75C22×D53—41no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—42no—
81C2×C4×C53C2×C22⋊C411yescharacteristic
82C2×C4×C53C2×M1611yescharacteristic
83C2×Dic53—21no—
84C2×Dic53—21no—
85C2×Dic53—21no—
86C2×Dic53—42no—
87C2×Dic53—41no—
88C4×D53—24no—
89C4×D53—42no—
90C4×D53—42no—
91C4×D53—42no—
92C4×D53—42no—
93C4×D53—41no—
94C4×D53—41no—
95C8×C53—22no—
96C8×C53—41no—
97#88322—22no—
98#88322—41no—
99#11521—531nomaximal, Sylow
100C22×C53—21no—
101C22×C53—41no—
102C22×C53#40711yescharacteristic
103C2×D53—22no—
104C2×D53—41no—
105C2×D53—42no—
106C2×D53—41no—
107C2×D53—42no—
108C4×C53—22no—
109C4×C53—21no—
110C4×C53—41no—
111C4×C53#41011yescharacteristic
112C4×C53#41311yescharacteristic
113Dic53—22no—
114Dic53—41no—
115Dic53—41no—
116#2920—531no—
117#2165—532no—
118#1327—532no—
119#1320—532no—
120C2×C53—21no—
121C2×C53—41no—
122C2×C53#146411yescharacteristic
123D53—42no—
124C22×C42—531no—
125#432—1062no—
126#431—1062no—
127#423—531no—
128#423#615711yescharacteristic
129#407—531no—
130#407—531no—
131C4×M16—531no—
132C4×M16—531no—
133#364—1062no—
134#349—534no—
135#347—1062no—
136C53#1152111yescharacteristic, complemented, Sylow
137C23×C4—531no—
138C23×C4—1061no—
139C2×M16—531no—
140C2×M16—531no—
141C2×M16C2×#615711yescharacteristic
142C2×M16C2×#615711yescharacteristic
143C2×C42—531no—
144C2×C42—1061no—
145C2×C42—1061no—
146C2×C42—1061no—
147C2×C42—1061no—
148C2×C42—1061no—
149C2×C42C2×#615711yescharacteristic
150M32—1064no—
151C4⋊C8—22no—
152C4⋊C8—1062no—
153C22⋊C8—1062no—
154C22⋊C8—1062no—
155#97—532no—
156#97—1061no—
157C4×C8—532no—
158C4×C8—1061no—
159C24—531no—
160C22×C4—21no—
161C22×C4—531no—
162C22×C4—1061no—
163C22×C4—1061no—
164C22×C4—1061no—
165C22×C4—1061no—
166C22×C4—1061no—
167C22×C4—1061no—
168C22×C4—1061no—
169C22×C4—2121no—
170C22×C4—2121no—
171C22×C4—2124no—
172C22×C4C22×#615711yescharacteristic
173M16—41no—
174M16—41no—
175M16—2121no—
176M16—2121no—
177C2×C8—21no—
178C2×C8—532no—
179C2×C8—1061no—
180C2×C8C22⋊#615712yes—
181C42—21no—
182C42—531no—
183C42—531no—
184C42—1061no—
185C42—1061no—
186C42—1061no—
187C42—1061no—
188C42—1061no—
189C42#1171993611yescharacteristic
190C42#1171993611yescharacteristic
191C16—2124no—
192C23—1061no—
193C23—2121no—
194C23—2122no—
195C23D53⋊M1611yescharacteristic
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—42no—
201C2×C4—1064no—
202C2×C4—1061no—
203C2×C4—1061no—
204C2×C4—1061no—
205C2×C4—2122no—
206C2×C4—2122no—
207C2×C4—2122no—
208C2×C4—2122no—
209C2×C4—2122no—
210C2×C4—2121no—
211C2×C4—2121no—
212C2×C4—2121no—
213C2×C4C2×C22⋊#615711yescharacteristic
214C2×C4C2×#1171993611yescharacteristic
215C8—22no—
216C8—41no—
217C8—1062no—
218C8—2121no—
219C22—21no—
220C22—41no—
221C22—1062no—
222C22—2121no—
223C22—2122no—
224C22—2122no—
225C22—2121no—
226C22#1681159411yescharacteristic
227C4—22no—
228C4—21no—
229C4—41no—
230C4—1062no—
231C4—2121no—
232C4—2121no—
233C4#1681160411yescharacteristic
234C4#1681161011yescharacteristic
235C2—21no—
236C2—41no—
237C2—2122no—
238C2#2028996911yescharacteristic
subgroupG/H
centerC4#16811604
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#423C4
socleC2×C53#1464
prime radicalC23×D53M16
cosocleC2×#11719917C23
hypercenter#423#6157
hypocenterC53#11521
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #423

Complements

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

NH ≅ G/Ntype
1. #20291085C2semidirect
2. #20289661C2semidirect
136. C53#11521semidirect

Maximal subgroups

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

subgroupindex
#202910852
#202896612
#202896482
#202833472
#1152153