Finite Group Databasebeta

← #28299665

Subgroups — #28299665

Order 13568.

590 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 104 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#20290378C212yesmaximal
2#20289533C214yesmaximal, complemented
3#20277199C211yesmaximal, characteristic
4C22×#11719940—22no—
5#16811489—22no—
6#16811489C2214yes—
7C2×#13133176C411yescharacteristic
8C22×C4×Dic53C2211yescharacteristic
9#16810522C412yes—
10#16810522C2212yes—
11#16810498—22no—
12C4×C4⋊Dic53—22no—
13C2×#117×C53C411yescharacteristic
14C2×#11719940—28no—
15C2×#11719940—44no—
16C2×#11719940—44no—
17C22×#88324—24no—
18C23×Dic53D412yes—
19#13133176—42no—
20C2×C4⋊Dic53—22no—
21C2×C4⋊Dic53—22no—
22C2×C4⋊Dic53C2×C412yes—
23C2×C22⋊Dic53—21no—
24C2×C4×Dic53—21no—
25C2×C4×Dic53—42no—
26C2×C4×Dic53C2×C411yescharacteristic
27C2×C4×Dic53D412yes—
28C2×C4×Dic53C2311yescharacteristic
29#13133124—22no—
30#13133124—22no—
31C23×C4×C53C2×C411yescharacteristic
32#117×C53—24no—
33C2×C4⋊C4×C53C2×C412yes—
34C2×C22⋊C4×C53—21no—
35C2×C42×C53—21no—
36C24×C53C22⋊C411yescharacteristic
37C22×C4×C53—22no—
38C22×C4×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53—22no—
41C22×C4×C53C22⋊C411yescharacteristic
42C22×C4×C53C22×C411yescharacteristic
43C22×Dic53—24no—
44C22×Dic53—24no—
45C22×Dic53—22no—
46C22×Dic53—22no—
47C22×Dic53—22no—
48C22×Dic53C22⋊C412yes—
49C22×Dic53C2×D412yes—
50#11719940—416no—
51C2×#88324—28no—
52C2×#88324—44no—
53C2×#88324—44no—
54C4⋊C4×C53—22no—
55C4⋊C4×C53M1614yescomplemented
56C22⋊C4×C53—24no—
57C42×C53—24no—
58C22⋊Dic53—42no—
59C4⋊Dic53—44no—
60C4⋊Dic53—42no—
61C4⋊Dic53—42no—
62C4⋊Dic53—42no—
63C4×Dic53—41no—
64C4×Dic53—42no—
65C4×Dic53—41no—
66C4×Dic53—42no—
67C23×C53—21no—
68C23×C53—21no—
69C23×C53—22no—
70C23×C53—22no—
71C23×C53#9912yes—
72C23×C53C2×C22⋊C411yescharacteristic
73C2×C4×C53—28no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—24no—
79C2×C4×C53—42no—
80C2×C4×C53—42no—
81C2×C4×C53#10112yes—
82C2×C4×C53C2×M1612yes—
83C2×Dic53—28no—
84C2×Dic53—28no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—42no—
90C2×Dic53—42no—
91C2×Dic53—42no—
92C2×Dic53#10414yes—
93C2×Dic53AGL(1,ℤ/8)12yes—
94C2×Dic53#13712yes—
95#88324—48no—
96#7632—531nomaximal, Sylow
97C22×C53—22no—
98C22×C53—24no—
99C22×C53—41no—
100C22×C53—42no—
101C22×C53—41no—
102C22×C53#40711yescharacteristic
103C22×C53C2×#9911yescharacteristic
104C22×C53C2×#10111yescharacteristic
105C22×C53#41712yes—
106C22×C53C2×#10412yes—
107C4×C53—22no—
108C4×C53—22no—
109C4×C53—44no—
110Dic53—22no—
111Dic53—24no—
112Dic53—22no—
113Dic53—82no—
114#2296—531no—
115#1728—532no—
116#1271—534no—
117C2×C53—42no—
118C2×C53#146311yescharacteristic
119C2×C53#148912yes—
120C2×C53#150412yes—
121C2×C53#152611yescharacteristic
122C2×C53#156111yescharacteristic
123C22×M16—1062no—
124C2×#117—531no—
125C2×#117#615711yescharacteristic
126C22×C42—531no—
127#389—532no—
128#389—532no—
129#385—1062no—
130C4×C4⋊C4—1062no—
131#336—534no—
132#336—1062no—
133C53#763211yescharacteristic, complemented, Sylow
134C23×C4—532no—
135C23×C4C2×#615711yescharacteristic
136C2×M16—1068no—
137C2×M16—2124no—
138C2×M16—2124no—
139C22×C8—1064no—
140#117—24no—
141#117—2122no—
142C2×C4⋊C4—532no—
143C2×C4⋊C4—1062no—
144C2×C4⋊C4—1062no—
145C2×C4⋊C4C2×#615712yes—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—1061no—
148C2×C42—21no—
149C2×C42—532no—
150C2×C42—531no—
151C2×C42—531no—
152C2×C42—1061no—
153C2×C42—2122no—
154#95—1062no—
155#95—1062no—
156C24C22⋊#615711yescharacteristic
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—532no—
162C22×C4—532no—
163C22×C4—1064no—
164C22×C4—1062no—
165C22×C4—1062no—
166C22×C4—1064no—
167C22×C4—1062no—
168C22×C4C22⋊#615711yescharacteristic
169C22×C4C22×#615711yescharacteristic
170M16—21216no—
171C2×C8—1068no—
172C2×C8—2124no—
173C2×C8—2124no—
174C4⋊C4—22no—
175C4⋊C4—2122no—
176C4⋊C4—2122no—
177C4⋊C4—2122no—
178C4⋊C4—2124no—
179C4⋊C4#1171994014yescomplemented
180C22⋊C4—24no—
181C22⋊C4—2122no—
182C42—24no—
183C42—2122no—
184C42—2121no—
185C42—2121no—
186C42—2122no—
187C23—21no—
188C23—21no—
189C23—22no—
190C23—22no—
191C23#1313327612yes—
192C23C2×C22⋊#615711yescharacteristic
193C2×C4—28no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—24no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—532no—
202C2×C4—532no—
203C2×C4—534no—
204C2×C4—1068no—
205C2×C4—1068no—
206C2×C4—1062no—
207C2×C4—1062no—
208C2×C4—1062no—
209C2×C4—1062no—
210C2×C4—2122no—
211C2×C4—2122no—
212C2×C4—2122no—
213C2×C4#1313327812yes—
214C2×C4C2×#1171994012yes—
215C8—2128no—
216C22—24no—
217C22—22no—
218C22—42no—
219C22—41no—
220C22—41no—
221C22#1681159311yescharacteristic
222C22C2×#1313327611yescharacteristic
223C22C2×#1313327811yescharacteristic
224C22#1681161412yes—
225C22D53⋊#10412yes—
226C4—22no—
227C4—22no—
228C4—44no—
229C4—1062no—
230C4—1062no—
231C4—1064no—
232C4—4242no—
233C2—42no—
234C2#2028996311yescharacteristic
235C2#2029005012yes—
236C2#2029010312yes—
237C2#2029017211yescharacteristic
238C2#2029025411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×#117#6157
hypocenterC53#7632
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #20289533C2semidirect
55. C4⋊C4×C53M16semidirect
133. C53#7632semidirect
179. C4⋊C4#11719940semidirect

Maximal subgroups

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

subgroupindex
#202903782
#202895332
#202771992
#763253