Finite Group Databasebeta

← #28299654

Subgroups — #28299654

Order 13568.

658 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 116 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#20290441C212yesmaximal
2#20289535C214yesmaximal, complemented
3C2×#16810530C211yesmaximal, characteristic
4C2×C22⋊#88324—22no—
5C2×#13133272—22no—
6#16811489C2214yes—
7C22×C4⋊Dic53C411yescharacteristic
8C22×C4×Dic53C2211yescharacteristic
9#16810530—22no—
10#16810530C412yes—
11#16810530C2212yes—
12C2×#13133124—22no—
13C23×Q8×C53C411yescharacteristic
14C22×#88324—24no—
15C22⋊#88324—28no—
16#13133272—44no—
17C23×Dic53—21no—
18C23×Dic53D412yes—
19C2×C4⋊Dic53—21no—
20C2×C4⋊Dic53—42no—
21C2×C4⋊Dic53C2×C412yes—
22C2×C4×Dic53—42no—
23C2×C4×Dic53C2×C411yescharacteristic
24C2×C4×Dic53D412yes—
25C2×C4×Dic53C2311yescharacteristic
26#13133124—24no—
27#13133124—24no—
28C22×Q8×C53—24no—
29C22×Q8×C53—21no—
30C22×Q8×C53—42no—
31C22×Q8×C53—42no—
32C22×Q8×C53C2×C412yes—
33C23×C4×C53—21no—
34C23×C4×C53C2×C411yescharacteristic
35C24×C53C22⋊C411yescharacteristic
36C2×Q8×C53—24no—
37C2×Q8×C53—22no—
38C2×Q8×C53—416no—
39C2×Q8×C53—42no—
40C2×Q8×C53—42no—
41C2×Q8×C53—42no—
42C2×Q8×C53—42no—
43C2×Q8×C53M1614yes—
44C22×C4×C53—22no—
45C22×C4×C53—22no—
46C22×C4×C53—22no—
47C22×C4×C53—24no—
48C22×C4×C53—22no—
49C22×C4×C53—42no—
50C22×C4×C53—42no—
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22×C411yescharacteristic
53C22×Dic53—24no—
54C22×Dic53—24no—
55C22×Dic53—22no—
56C22×Dic53—42no—
57C22×Dic53—42no—
58C22×Dic53—44no—
59C22×Dic53C22⋊C412yes—
60C22×Dic53C2×D412yes—
61C2×#88324—28no—
62C2×#88324—44no—
63C2×#88324—44no—
64C4⋊Dic53—42no—
65C4⋊Dic53—42no—
66C4×Dic53—24no—
67C4×Dic53—41no—
68C4×Dic53—41no—
69C23×C53—21no—
70C23×C53—21no—
71C23×C53—22no—
72C23×C53#10012yes—
73C23×C53#10414yes—
74C23×C53C2×C22⋊C411yescharacteristic
75Q8×C53—48no—
76Q8×C53—44no—
77Q8×C53—44no—
78C2×C4×C53—28no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—24no—
84C2×C4×C53—48no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53#9912yes—
88C2×C4×C53C2×M1612yes—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—48no—
95C2×Dic53—42no—
96C2×Dic53—44no—
97C2×Dic53—84no—
98C2×Dic53#13714yes—
99#88324—48no—
100#7629—531nomaximal, Sylow
101C22×C53—22no—
102C22×C53—22no—
103C22×C53—24no—
104C22×C53—24no—
105C22×C53#32914yes—
106C22×C53#40711yescharacteristic
107C22×C53C2×#9911yescharacteristic
108C22×C53C2×#10011yescharacteristic
109C22×C53#41912yes—
110C22×C53C2×#10412yes—
111C4×C53—22no—
112C4×C53—22no—
113C4×C53—44no—
114Dic53—24no—
115Dic53—42no—
116Dic53—82no—
117C2×#391—531no—
118#1749—532no—
119#1272—534no—
120C2×C53—24no—
121C2×C53#146211yescharacteristic
122C2×C53#149112yes—
123C2×C53#150812yes—
124C2×C53C2×#32911yescharacteristic
125C2×C53#155811yescharacteristic
126C23×Q8#615711yescharacteristic
127C22×C4⋊C4—531no—
128C22×C42—531no—
129C2×C22⋊C8—1062no—
130C2×#97—1062no—
131#391—532no—
132#391—532no—
133#391—1062no—
134C2×#95—1062no—
135#336—534no—
136C53#762911yescharacteristic, complemented, Sylow
137C22×Q8—24no—
138C22×Q8—21no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8C2×#615712yes—
142C23×C4—21no—
143C23×C4—532no—
144C23×C4—1061no—
145C23×C4C2×#615711yescharacteristic
146C22×C8—1064no—
147C2×C4⋊C4—532no—
148C2×C4⋊C4—1061no—
149C2×C4⋊C4—2122no—
150C2×C42—532no—
151C2×C42—531no—
152C2×C42—531no—
153C2×C42—2122no—
154C22⋊C8—1068no—
155#97—2124no—
156#95—1064no—
157#95—1064no—
158C24C22⋊#615711yescharacteristic
159C2×Q8—24no—
160C2×Q8—22no—
161C2×Q8—416no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8#1171994014yes—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—22no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—532no—
175C22×C4—532no—
176C22×C4—1064no—
177C22×C4—1064no—
178C22×C4—1062no—
179C22×C4—2122no—
180C22×C4—2122no—
181C22×C4—2124no—
182C22×C4C22⋊#615711yescharacteristic
183C22×C4C22×#615711yescharacteristic
184C2×C8—1068no—
185C2×C8—2124no—
186C2×C8—2124no—
187C4⋊C4—2122no—
188C4⋊C4—2122no—
189C42—1064no—
190C42—2121no—
191C42—2121no—
192C23—21no—
193C23—21no—
194C23—22no—
195C23#1313327712yes—
196C23#1313328214yes—
197C23C2×C22⋊#615711yescharacteristic
198Q8—44no—
199Q8—48no—
200Q8—44no—
201C2×C4—28no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—24no—
207C2×C4—48no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—534no—
211C2×C4—1062no—
212C2×C4—1062no—
213C2×C4—1062no—
214C2×C4—1062no—
215C2×C4—1062no—
216C2×C4—4244no—
217C2×C4—2128no—
218C2×C4—2122no—
219C2×C4—2124no—
220C2×C4#1313327512yes—
221C2×C4C2×#1171994012yes—
222C8—2128no—
223C22—24no—
224C22—24no—
225C22—22no—
226C22—22no—
227C22#1681147914yes—
228C22#1681159311yescharacteristic
229C22C2×#1313327511yescharacteristic
230C22C2×#1313327711yescharacteristic
231C22#1681161612yes—
232C22C2×#1313328212yes—
233C4—22no—
234C4—22no—
235C4—44no—
236C4—1064no—
237C4—4242no—
238C4—2122no—
239C2—24no—
240C2#2028996011yescharacteristic
241C2#2029005712yes—
242C2#2029011512yes—
243C2C2×#1681147911yescharacteristic
244C2#2029024811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC23×Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC23×Q8#6157
hypocenterC53#7629
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
2. #20289535C2semidirect
136. C53#7629semidirect

Maximal subgroups

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

subgroupindex
#202904412
#202895352
C2×#168105302
#762953