Finite Group Databasebeta

← #28298516

Subgroups — #28298516

Order 13568.

290 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 50 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#20290613C211yesmaximal, characteristic
2#20290444C211yesmaximal, characteristic
3#20273048C211yesmaximal, characteristic
4C2×#13133274—21no—
5C2×#13133272C411yescharacteristic, complemented
6C2×C4×#88324C411yescharacteristic, complemented
7#16811566—21no—
8#16811501—21no—
9#16811501C412yes—
10#16811489—21no—
11C2×#13133175C2211yescharacteristic
12#16810677—21no—
13#16810349C411yescharacteristic, complemented
14#16810346—21no—
15C53×#324C411yescharacteristic, complemented
16C22×#88324—21no—
17C22×#88324—21no—
18C2×C22⋊#6157—22no—
19#13133274—24no—
20#13133272—41no—
21C4×#88324—41no—
22(C2×#6157)⋊C4—41no—
23(C2×#6157)⋊C4—41no—
24C22×C4×D53D411yescharacteristic
25#13133175—41no—
26#13133175—41no—
27C2×C4⋊Dic53Q811yescharacteristic
28C2×#11719921C2×C411yescharacteristic
29C2×#11719921D411yescharacteristic
30C2×C4×Dic53C2×C411yescharacteristic
31C2×C4×Dic53D411yescharacteristic
32#13133146—41no—
33C22⋊#88322—21no—
34#13133127—41no—
35C2×C4⋊C4×C53C2×C411yescharacteristic
36C22⋊C8×C53—21no—
37C23×D53C22⋊C411yescharacteristic
38C22×#6157—42no—
39C22×C4×C53C4211yescharacteristic
40C22×C4×C53C22⋊C411yescharacteristic
41C22×C4×C53C4⋊C411yescharacteristic
42C22×Dic53C22⋊C411yescharacteristic
43C22×Dic53C4⋊C411yescharacteristic
44C22×Dic53C4⋊C411yescharacteristic
45C2×C4×D53—22no—
46C2×C4×D53—21no—
47C2×C4×D53—21no—
48C2×C4×D53—41no—
49C22⋊#6157—42no—
50C22⋊#6157—42no—
51C2×#88324—22no—
52C2×#88324—22no—
53C2×#88324—41no—
54C2×#88324—41no—
55C2×#88324—41no—
56C2×#88324—41no—
57C2×C8×C53—41no—
58C4⋊C4×C53—41no—
59#11719921—41no—
60#11719921—41no—
61C4⋊Dic53—41no—
62C4×Dic53—21no—
63C4×Dic53—21no—
64C4×Dic53—21no—
65C4×Dic53—21no—
66C2×#88322—41no—
67C23×C53#9511yescharacteristic
68C22×D53—21no—
69C22×D53—21no—
70C22×D53—41no—
71C22×D53—41no—
72C22×D53#10412yes—
73C2×#6157—82no—
74C2×#6157—82no—
75C2×#6157—82no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—41no—
82C2×C4×C53—41no—
83C2×Dic53—21no—
84C2×Dic53—21no—
85C2×Dic53—21no—
86C2×Dic53—21no—
87C2×Dic53—41no—
88C2×Dic53—41no—
89C2×Dic53#9911yescharacteristic
90C2×Dic53#9911yescharacteristic
91C4×D53—42no—
92C4×D53—42no—
93#88324—42no—
94#88324—42no—
95C8×C53—81no—
96#88322—81no—
97#7214—531nomaximal, Sylow
98C22×C53—21no—
99C22×C53—21no—
100C22×C53H(3,ℤ/4)11yescharacteristic
101C22×C53#33911yescharacteristic
102C22×C53#34211yescharacteristic
103C2×D53—22no—
104C2×D53—41no—
105C2×D53—41no—
106C2×D53—42no—
107C2×D53—81no—
108#6157—162no—
109C4×C53—41no—
110C4×C53—41no—
111C4×C53—41no—
112Dic53—21no—
113Dic53—21no—
114Dic53—41no—
115Dic53—41no—
116#1820—531no—
117#1750—531no—
118#1464—531no—
119C2×C53—21no—
120C2×C53—21no—
121C2×C53#128411yescharacteristic
122C2×C53#139111yescharacteristic
123C2×C53#139511yescharacteristic
124D53—42no—
125C2×#117—531no—
126#432—1061no—
127C2×C22⋊C8—1061no—
128C2×#97—531no—
129C2×C4×C8—531no—
130#381—1061no—
131#342—532no—
132#342—1061no—
133#336—1061no—
134#324—531no—
135#324#615711yescharacteristic, complemented
136#323—1061no—
137C53#721411yescharacteristic, complemented, Sylow
138C23×C4—531no—
139C22×C8—1061no—
140C22×C8—1061no—
141#117—2121no—
142#117—2121no—
143C2×C4⋊C4—531no—
144C2×C4⋊C4C2×#615711yescharacteristic
145C2×C22⋊C4—531no—
146C2×C22⋊C4—531no—
147C2×C22⋊C4—1062no—
148C2×C42—531no—
149C2×C42—531no—
150C4⋊C8—2121no—
151C22⋊C8—21no—
152C22⋊C8—1064no—
153C22⋊C8—1061no—
154#97—2121no—
155#97—2121no—
156C4×C8—2121no—
157#95—2121no—
158#95—2121no—
159C24—531no—
160C22×C4—531no—
161C22×C4—531no—
162C22×C4—531no—
163C22×C4—1061no—
164C22×C4—1061no—
165C22×C4—1062no—
166C22×C4—2121no—
167C22×C4—2122no—
168C22×C4C4×#615711yescharacteristic
169C22×C4C4⋊#615711yescharacteristic
170C22×C4C22⋊#615711yescharacteristic
171C2×C8—41no—
172C2×C8—1062no—
173C2×C8—1062no—
174C2×C8—2121no—
175C2×C8—2121no—
176C2×C8—2121no—
177C2×C8—2121no—
178C2×C8—2121no—
179C4⋊C4—41no—
180C4⋊C4—2121no—
181C22⋊C4—2121no—
182C22⋊C4—2121no—
183C22⋊C4—2122no—
184C22⋊C4—2122no—
185C42—1061no—
186C42—1061no—
187C42—1061no—
188C42—1061no—
189C23—532no—
190C23—1061no—
191C23—1061no—
192C23—2121no—
193C23—2121no—
194C23(C2×#6157)⋊C411yescharacteristic
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—531no—
203C2×C4—531no—
204C2×C4—1061no—
205C2×C4—1061no—
206C2×C4—1061no—
207C2×C4—1061no—
208C2×C4—4242no—
209C2×C4—4242no—
210C2×C4—4242no—
211C2×C4—2122no—
212C2×C4—2122no—
213C2×C4—2121no—
214C2×C4—2121no—
215C8—81no—
216C8—4241no—
217C8—2122no—
218C8—2122no—
219C22—21no—
220C22—21no—
221C22—1062no—
222C22—4241no—
223C22—2122no—
224C22—2121no—
225C22—2121no—
226C22#1681149111yescharacteristic
227C22#1681149511yescharacteristic
228C22#1681150011yescharacteristic
229C4—41no—
230C4—41no—
231C4—41no—
232C4—8482no—
233C4—1061no—
234C4—1061no—
235C4—2121no—
236C4—2121no—
237C2—21no—
238C2—21no—
239C2—2122no—
240C2#2028956411yescharacteristic
241C2#2028980211yescharacteristic
242C2#2028981711yescharacteristic
subgroupG/H
centerC22#16811500
commutatorC22×C4×C53C42
FrattiniC2×C4⋊C4C2×#6157
FittingC53×#324C4
socleC22×C53#342
prime radicalC23×D53C22⋊C4
cosocleC2×#13133175C22
hypercenter#324#6157
hypocenterC53#7214
LCSG ▷ C22×C4×C53 ▷ C22×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
5. C2×#13133272C4semidirect
6. C2×C4×#88324C4semidirect
13. #16810349C4semidirect
15. C53×#324C4semidirect
135. #324#6157semidirect
137. C53#7214semidirect

Maximal subgroups

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

subgroupindex
#202906132
#202904442
#202730482
#721453