Finite Group Databasebeta

← #28298852

Subgroups — #28298852

Order 13568.

244 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 48 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#20290609C211yesmaximal, characteristic
2#20289969C211yesmaximal, characteristic
3#20273048C211yesmaximal, characteristic
4#16811644—21no—
5C2×#13133274C411yescharacteristic
6C2×#13133274C411yescharacteristic
7#16811489—21no—
8#16811489—21no—
9#16811468C412yes—
10#16811467—21no—
11C2×#13133175C2211yescharacteristic
12#16810677—21no—
13#16810349C411yescharacteristic
14#16810346—21no—
15C53×#324C411yescharacteristic
16C22×#88324—21no—
17C22×#88324—21no—
18C4⋊#88324—41no—
19#13133274—41no—
20#13133274—41no—
21C22⋊#88324—22no—
22#13133272—41no—
23C22×C4×D53C2×C411yescharacteristic
24#13133175—41no—
25#13133175—41no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719921D411yescharacteristic
28C2×#11719921Q811yescharacteristic
29C2×C4×Dic53D411yescharacteristic
30C2×C4×Dic53D411yescharacteristic
31#13133146—41no—
32C22⋊#88322—21no—
33#13133127—41no—
34C2×C4⋊C4×C53C2×C411yescharacteristic
35C22⋊C8×C53—21no—
36C23×D53C4⋊C411yescharacteristic
37C22×C4×C53C4211yescharacteristic
38C22×C4×C53C22⋊C411yescharacteristic
39C22×C4×C53C4⋊C411yescharacteristic
40C22×Dic53C22⋊C411yescharacteristic
41C22×Dic53C22⋊C411yescharacteristic
42C22×Dic53C4⋊C411yescharacteristic
43C2×C4×D53—21no—
44C2×C4×D53—21no—
45C2×C4×D53—41no—
46C2×C4×D53—41no—
47C2×#88324—42no—
48C2×#88324—41no—
49C2×#88324—41no—
50C2×#88324—41no—
51C2×#88324—41no—
52C2×#88324—41no—
53C2×#88324—41no—
54C2×C8×C53—41no—
55C4⋊C4×C53—41no—
56#11719921—41no—
57#11719921—41no—
58C4⋊Dic53—41no—
59C4×Dic53—21no—
60C4×Dic53—21no—
61C4×Dic53—21no—
62C4×Dic53—21no—
63C2×#88322—41no—
64C23×C53#9511yescharacteristic
65C22×D53—21no—
66C22×D53—21no—
67C22×D53—21no—
68C22×D53—41no—
69C22×D53—41no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—41no—
76C2×C4×C53—41no—
77C2×Dic53—21no—
78C2×Dic53—21no—
79C2×Dic53—21no—
80C2×Dic53—21no—
81C2×Dic53—41no—
82C2×Dic53—41no—
83C2×Dic53#9911yescharacteristic
84C2×Dic53#9911yescharacteristic
85C4×D53—42no—
86C4×D53—42no—
87#88324—81no—
88#88324—81no—
89#88324—82no—
90C8×C53—81no—
91#88322—81no—
92#7327—531nomaximal, Sylow
93C22×C53—21no—
94C22×C53—21no—
95C22×C53#34211yescharacteristic
96C22×C53#34411yescharacteristic
97C22×C53#34411yescharacteristic
98C2×D53—41no—
99C2×D53—41no—
100C2×D53—41no—
101C2×D53—81no—
102C2×D53—81no—
103C4×C53—41no—
104C4×C53—41no—
105C4×C53—41no—
106Dic53—21no—
107Dic53—21no—
108Dic53—41no—
109Dic53—41no—
110#1818—531no—
111#1464—531no—
112#1464—531no—
113C2×C53—21no—
114C2×C53—21no—
115C2×C53#130611yescharacteristic
116C2×C53#139511yescharacteristic
117C2×C53#139511yescharacteristic
118D53—81no—
119C2×#117—531no—
120#432—1061no—
121#432—1061no—
122C2×C22⋊C8—531no—
123C2×C22⋊C8—531no—
124#336—1061no—
125#336—1061no—
126#324—532no—
127#324—531no—
128#324#615711yescharacteristic
129#323—1061no—
130#323—1061no—
131C53#732711yescharacteristic, complemented, Sylow
132C23×C4—531no—
133C22×C8—1061no—
134C22×C8—1061no—
135#117—2121no—
136#117—2121no—
137C2×C4⋊C4—531no—
138C2×C4⋊C4C2×#615711yescharacteristic
139C2×C22⋊C4—531no—
140C2×C22⋊C4—531no—
141C2×C42—531no—
142C2×C42—531no—
143C4⋊C8—2121no—
144C4⋊C8—2121no—
145C22⋊C8—21no—
146C22⋊C8—1062no—
147C22⋊C8—1061no—
148C22⋊C8—2121no—
149C22⋊C8—2121no—
150#97—2121no—
151#97—2121no—
152C24—531no—
153C22×C4—531no—
154C22×C4—531no—
155C22×C4—531no—
156C22×C4—1061no—
157C22×C4—1061no—
158C22×C4—2121no—
159C22×C4—2121no—
160C22×C4C4×#615711yescharacteristic
161C22×C4C4⋊#615711yescharacteristic
162C22×C4C22⋊#615711yescharacteristic
163C2×C8—41no—
164C2×C8—2121no—
165C2×C8—2121no—
166C2×C8—2121no—
167C2×C8—2122no—
168C2×C8—2121no—
169C2×C8—2121no—
170C2×C8—2121no—
171C2×C8—2121no—
172C4⋊C4—41no—
173C4⋊C4—2121no—
174C22⋊C4—2121no—
175C22⋊C4—2121no—
176C42—1061no—
177C42—1061no—
178C42—1061no—
179C42—1061no—
180C23—1061no—
181C23—1061no—
182C23—1061no—
183C23—2121no—
184C23—2121no—
185C23(C2×#6157)⋊C411yescharacteristic
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—531no—
194C2×C4—531no—
195C2×C4—1061no—
196C2×C4—1061no—
197C2×C4—1061no—
198C2×C4—1061no—
199C2×C4—2122no—
200C2×C4—2122no—
201C2×C4—2121no—
202C2×C4—2121no—
203C8—81no—
204C8—4241no—
205C8—4241no—
206C8—4241no—
207C8—4242no—
208C22—21no—
209C22—21no—
210C22—4241no—
211C22—4241no—
212C22—2121no—
213C22—2121no—
214C22—2121no—
215C22#1681150011yescharacteristic
216C22#1681150511yescharacteristic
217C22#1681150611yescharacteristic
218C4—41no—
219C4—41no—
220C4—41no—
221C4—1061no—
222C4—1061no—
223C4—2121no—
224C4—2121no—
225C2—21no—
226C2—21no—
227C2—4241no—
228C2#2028961911yescharacteristic
229C2#2028981411yescharacteristic
230C2#2028981711yescharacteristic
subgroupG/H
centerC22#16811500
commutatorC22×C4×C53C42
FrattiniC2×C4⋊C4C2×#6157
FittingC53×#324C4
socleC22×C53#342
prime radicalC23×D53C4⋊C4
cosocleC2×#13133175C22
hypercenter#324#6157
hypocenterC53#7327
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #324

Complements

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

NH ≅ G/Ntype
131. C53#7327semidirect

Maximal subgroups

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

subgroupindex
#202906092
#202899692
#202730482
#732753