Finite Group Databasebeta

← #28298460

Subgroups — #28298460

Order 13568.

286 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 68 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#20290943C211yesmaximal, characteristic
2#20290451C211yesmaximal, characteristic
3#20276325C211yesmaximal, characteristic
4C2×#13133292—21no—
5D53⋊(C22⋊C8)C411yescharacteristic
6C2×#13133271—21no—
7C4×C22⋊#6157C411yescharacteristic
8#16811518—21no—
9C22⋊#11719910C412yescomplemented
10(C2×#6157)⋊C8—21no—
11C22×C8×D53C2211yescharacteristic
12C2×#13133157C411yescharacteristic, complemented
13#16810402—21no—
14#16810402—21no—
15C2×M32×C53C411yescharacteristic, complemented
16C2×D53⋊C8—21no—
17C2×C22⋊#6157C812yes—
18C2×C4×#6157—21no—
19#13133292—41no—
20#13133292—41no—
21C2×#11719910—22no—
22#13133274—21no—
23C22⋊#88324C812yes—
24#13133271—41no—
25#13133271—41no—
26(C2×#6157)⋊C4—21no—
27C22×C4×D53C2×C411yescharacteristic
28C22×#88322C2×C411yescharacteristic
29C2×C8×D53—41no—
30C2×C8×D53—41no—
31C2×C8×D53D411yescharacteristic
32C2×C8×D53D411yescharacteristic
33C2×C8×D53D411yescharacteristic
34C2×C8×D53Q811yescharacteristic
35#13133157—41no—
36C2×#11719908—21no—
37C22×C8×C53C2×C411yescharacteristic
38M32×C53—41no—
39C2×C16×C53—21no—
40C23×D53C2×C811yescharacteristic
41C22×#6157—22no—
42C22×C4×C53C4211yescharacteristic
43C22×Dic53C2×C811yescharacteristic
44C2×C4×D53—41no—
45C2×C4×D53—41no—
46C2×C4×D53C22⋊C411yescharacteristic
47C2×C4×D53C4⋊C411yescharacteristic
48C2×C4×D53M1611yescharacteristic
49C2×C4×D53M1611yescharacteristic
50C22⋊#6157—42no—
51C2×#88324—22no—
52C4×#6157—41no—
53C4×#6157—41no—
54D53⋊C8—41no—
55D53⋊C8—41no—
56C2×C8×C53—21no—
57C2×C8×C53—21no—
58C2×C8×C53C22⋊C411yescharacteristic
59C2×C8×C53C4⋊C411yescharacteristic
60C2×#88322—41no—
61C2×#88322C22⋊C411yescharacteristic
62C2×#88322C4⋊C411yescharacteristic
63C8×D53—21no—
64C8×D53—21no—
65C8×D53—21no—
66C8×D53—21no—
67C8×D53—41no—
68C8×D53—41no—
69#11719910—42no—
70C16×C53—41no—
71#11719908—41no—
72C23×C53C4×C811yescharacteristic
73C22×D53—22no—
74C22×D53—41no—
75C22×D53—41no—
76C22×D53C22⋊C811yescharacteristic
77C22×D53C4⋊C811yescharacteristic
78C2×#6157—42no—
79C2×#6157—42no—
80C2×#6157—42no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53#9511yescharacteristic
84C2×C4×C53#9711yescharacteristic
85C2×Dic53—41no—
86C2×Dic53C22⋊C811yescharacteristic
87C2×Dic53C4⋊C811yescharacteristic
88C4×D53—21no—
89C4×D53—21no—
90C4×D53—21no—
91C4×D53—21no—
92C4×D53—41no—
93C4×D53—41no—
94#88324—42no—
95C8×C53—21no—
96C8×C53#9911yescharacteristic
97C8×C53#9911yescharacteristic
98#88322—21no—
99#88322—21no—
100#7201—531nomaximal, Sylow
101C22×C53—21no—
102C22×C53—21no—
103C22×C53—21no—
104C22×C53#33611yescharacteristic
105C2×D53—21no—
106C2×D53—21no—
107C2×D53—21no—
108C2×D53—21no—
109C2×D53—21no—
110C2×D53—44no—
111C2×D53#34912yes—
112#6157—82no—
113C4×C53—21no—
114C4×C53#34211yescharacteristic
115C4×C53#34411yescharacteristic
116Dic53—21no—
117Dic53—21no—
118#2112—531no—
119#2112—531no—
120#1752—531no—
121C2×C53—41no—
122C2×C53#128211yescharacteristic
123C2×C53#137711yescharacteristic
124C2×C53#137811yescharacteristic
125D53—22no—
126D53—41no—
127C23×C8—531no—
128C2×M32—531no—
129C2×M32—1061no—
130C2×M32#615711yescharacteristic, complemented
131C2×C22⋊C8—531no—
132C2×#97—1061no—
133C4×C22⋊C4—531no—
134C22⋊C16—532no—
135C22⋊C16—1061no—
136C22⋊C16—1061no—
137C22⋊C16—1061no—
138#336—1061no—
139C53#720111yescharacteristic, complemented, Sylow
140C23×C4—531no—
141C22×C8—531no—
142C22×C8—531no—
143C22×C8—531no—
144C22×C8—531no—
145C22×C8—531no—
146C22×C8—1061no—
147C22×C8—2121no—
148C22×C8—2121no—
149C22×C8C2×#615711yescharacteristic
150C2×C22⋊C4—532no—
151C2×C42—1061no—
152M32—41no—
153M32—2121no—
154M32—2121no—
155M32—2121no—
156C2×C16—21no—
157C2×C16—1062no—
158C2×C16—1061no—
159C22⋊C8—532no—
160C22⋊C8—1061no—
161#97—2121no—
162#97—2121no—
163#95—1061no—
164C24—531no—
165C22×C4—531no—
166C22×C4—531no—
167C22×C4—531no—
168C22×C4—531no—
169C22×C4—531no—
170C22×C4—1062no—
171C22×C4—2121no—
172C22×C4—2121no—
173C22×C4C4×#615711yescharacteristic
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—531no—
177C2×C8—531no—
178C2×C8—1061no—
179C2×C8—1061no—
180C2×C8—1061no—
181C2×C8—1061no—
182C2×C8—1062no—
183C2×C8—2121no—
184C2×C8—2121no—
185C2×C8—2121no—
186C2×C8—2121no—
187C2×C8—2121no—
188C2×C8C4⋊#615711yescharacteristic
189C2×C8C22⋊#615711yescharacteristic
190C22⋊C4—2122no—
191C42—2121no—
192C42—2121no—
193C16—41no—
194C16—2122no—
195C16—2121no—
196C23—531no—
197C23—531no—
198C23—1062no—
199C23—2121no—
200C23—2121no—
201C23C8×#615711yescharacteristic
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—531no—
205C2×C4—531no—
206C2×C4—1061no—
207C2×C4—1061no—
208C2×C4—1061no—
209C2×C4—1061no—
210C2×C4—2121no—
211C2×C4—2121no—
212C2×C4—2121no—
213C2×C4—2122no—
214C2×C4—2122no—
215C2×C4—2122no—
216C2×C4(C2×#6157)⋊C411yescharacteristic
217C2×C4#1313327111yescharacteristic
218C8—21no—
219C8—1061no—
220C8—1061no—
221C8—2122no—
222C8#1313327511yescharacteristic
223C8#1313327611yescharacteristic
224C22—21no—
225C22—21no—
226C22—21no—
227C22—532no—
228C22—1061no—
229C22—1061no—
230C22—1061no—
231C22—1061no—
232C22—1061no—
233C22—2124no—
234C22(C2×#6157)⋊C811yescharacteristic
235C4—21no—
236C4—1061no—
237C4—1061no—
238C4—4242no—
239C4#1681150111yescharacteristic
240C4#1681150511yescharacteristic
241C2—41no—
242C2—1062no—
243C2—2121no—
244C2#2028955711yescharacteristic
245C2#2028976111yescharacteristic
246C2#2028976711yescharacteristic
subgroupG/H
centerC2×C4(C2×#6157)⋊C4
commutatorC23×C53C4×C8
FrattiniC22×C8C2×#6157
FittingC2×M32×C53C4
socleC22×C53#336
prime radicalC23×D53C2×C8
cosocleC22×C8×D53C22
hypercenterC2×M32#6157
hypocenterC53#7201
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×M32

Complements

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

NH ≅ G/Ntype
9. C22⋊#11719910C4semidirect
12. C2×#13133157C4semidirect
15. C2×M32×C53C4semidirect
130. C2×M32#6157semidirect
139. C53#7201semidirect

Maximal subgroups

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

subgroupindex
#202909432
#202904512
#202763252
#720153