Finite Group Databasebeta

← #28298615

Subgroups — #28298615

Order 13568.

322 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 58 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#20289974C211yesmaximal, characteristic
2#20289970C211yesmaximal, characteristic
3#20275005C211yesmaximal, characteristic
4#16811644—21no—
5#16811594—21no—
6#16811469C412yes—
7#16811467C412yes—
8#16811466—21no—
9#16811466—21no—
10C23×#88322C411yescharacteristic
11C2×C4×C22⋊D53C2211yescharacteristic
12C2×#13133129—21no—
13#16810377—21no—
14C2×C22⋊C8×C53C411yescharacteristic
15C2×#11719936—41no—
16C4⋊#88324—41no—
17#13133274—41no—
18C22⋊#88324—22no—
19C22⋊#88324—22no—
20#13133272—41no—
21C22×C22⋊D53Q811yescharacteristic
22C22×C4×D53D411yescharacteristic
23C22×#88322—21no—
24C22×#88322—41no—
25C22×#88322—42no—
26C4×C22⋊D53—41no—
27C4×C22⋊D53—41no—
28C2×#11719919C2×C411yescharacteristic
29C2×C22⋊Dic53D411yescharacteristic
30C2×#11719921C2×C411yescharacteristic
31C2×C4×Dic53D411yescharacteristic
32#13133129—42no—
33C23×C4×C53C2×C411yescharacteristic
34C22×C8×C53—21no—
35C22⋊C8×C53—41no—
36C24×C53C4⋊C411yescharacteristic
37C23×D53C4⋊C411yescharacteristic
38C22×C4×C53—21no—
39C22×C4×C53—21no—
40C22×C4×C53—41no—
41C22×C4×C53—41no—
42C22×C4×C53C4211yescharacteristic
43C22×C4×C53C22⋊C411yescharacteristic
44C2×C22⋊D53—21no—
45C2×C22⋊D53—21no—
46C2×C22⋊D53—41no—
47C2×C22⋊D53—41no—
48C22×Dic53C22⋊C411yescharacteristic
49C22×Dic53C22⋊C411yescharacteristic
50C22×Dic53C4⋊C411yescharacteristic
51C2×C4×D53—21no—
52C2×C4×D53—21no—
53C2×C4×D53—41no—
54C2×C4×D53—41no—
55C2×#88324—42no—
56C2×#88324—42no—
57#11719936—81no—
58#11719936—81no—
59C2×C8×C53—41no—
60C2×C8×C53—41no—
61C2×C8×C53—41no—
62C22⋊Dic53—41no—
63#11719921—41no—
64#11719919—41no—
65C4×Dic53—21no—
66C4×Dic53—21no—
67C2×#88322—22no—
68C2×#88322—41no—
69C2×#88322—42no—
70C2×#88322—41no—
71C2×#88322—42no—
72C23×C53—21no—
73C23×C53—21no—
74C23×C53—41no—
75C23×C53—41no—
76C23×C53#9511yescharacteristic
77C23×C53#10811yescharacteristic
78C23×C53#10811yescharacteristic
79C22×D53—21no—
80C22×D53—21no—
81C22×D53—21no—
82C22×D53—41no—
83C22×D53—41no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—42no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53#9911yescharacteristic
92C2×C4×C53#9911yescharacteristic
93C22⋊D53—42no—
94C22⋊D53—42no—
95C2×Dic53—21no—
96C2×Dic53—21no—
97C2×Dic53—21no—
98C2×Dic53—21no—
99C2×Dic53—21no—
100C2×Dic53—41no—
101C2×Dic53—41no—
102C4×D53—41no—
103C4×D53—41no—
104C4×D53—81no—
105#88324—82no—
106#88324—82no—
107C8×C53—81no—
108#88322—42no—
109#7247—531nomaximal, Sylow
110C22×C53—21no—
111C22×C53—21no—
112C22×C53—21no—
113C22×C53—21no—
114C22×C53—41no—
115C22×C53—41no—
116C22×C53—42no—
117C22×C53—41no—
118C22×C53—41no—
119C22×C53#33811yescharacteristic
120C22×C53#34111yescharacteristic
121C22×C53#34211yescharacteristic
122C2×D53—42no—
123C2×D53—41no—
124C2×D53—41no—
125C2×D53—41no—
126C2×D53—81no—
127C4×C53—21no—
128C4×C53—41no—
129C4×C53AΣL(1,𝔽16)11yescharacteristic
130C4×C53#35211yescharacteristic
131Dic53—41no—
132Dic53—41no—
133Dic53—41no—
134#1869—531no—
135#1466—531no—
136#1465—531no—
137C2×C53—21no—
138C2×C53—21no—
139C2×C53—41no—
140C2×C53—41no—
141C2×C53#129311yescharacteristic
142C2×C53#139011yescharacteristic
143C2×C53#139511yescharacteristic
144D53—81no—
145C23×C8—531no—
146C2×C4×D4—531no—
147#432—1061no—
148#407—1061no—
149C2×C22⋊C8—1061no—
150C2×C22⋊C8#615711yescharacteristic
151#336—1061no—
152#324—532no—
153#323—532no—
154#323—1061no—
155#323—1061no—
156C53#724711yescharacteristic, complemented, Sylow
157C22×D4—531no—
158C23×C4—531no—
159C23×C4C2×#615711yescharacteristic
160C2×M16—2121no—
161C22×C8—21no—
162C22×C8—1061no—
163C22×C8—2122no—
164C22×C8—2121no—
165C4×D4—2121no—
166C4×D4—2121no—
167C2×C4⋊C4—531no—
168C2×C22⋊C4—531no—
169C2×C22⋊C4—531no—
170C2×C42—531no—
171C4⋊C8—2121no—
172C22⋊C8—41no—
173C22⋊C8—1062no—
174C22⋊C8—1062no—
175C22⋊C8—2122no—
176C22⋊C8—2121no—
177#97—2121no—
178C24—531no—
179C24C4⋊#615711yescharacteristic
180C2×D4—1061no—
181C2×D4—1061no—
182C2×D4—2121no—
183C2×D4—2121no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—531no—
189C22×C4—531no—
190C22×C4—531no—
191C22×C4—1061no—
192C22×C4—1061no—
193C22×C4—2121no—
194C22×C4—2121no—
195C22×C4C4×#615711yescharacteristic
196C22×C4C22⋊#615711yescharacteristic
197M16—4241no—
198M16—4241no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—1062no—
203C2×C8—2122no—
204C2×C8—2122no—
205C2×C8—2121no—
206C2×C8—2122no—
207C2×C8—2121no—
208C2×C8—2122no—
209C4⋊C4—2121no—
210C22⋊C4—2121no—
211C22⋊C4—2121no—
212C42—1061no—
213C42—1061no—
214C23—21no—
215C23—21no—
216C23—41no—
217C23—41no—
218C23—1061no—
219C23—1061no—
220C23—1061no—
221C23—2121no—
222C23—2121no—
223C23(C2×#6157)⋊C411yescharacteristic
224C23#1313328711yescharacteristic
225C23#1313328811yescharacteristic
226D4—2122no—
227D4—2122no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—1061no—
236C2×C4—1061no—
237C2×C4—1061no—
238C2×C4—1061no—
239C2×C4—1061no—
240C2×C4—4241no—
241C2×C4—2121no—
242C2×C4—2121no—
243C2×C4—2121no—
244C2×C4—2121no—
245C2×C4#1313327511yescharacteristic
246C2×C4#1313327611yescharacteristic
247C8—81no—
248C8—4242no—
249C8—4242no—
250C8—2122no—
251C22—21no—
252C22—21no—
253C22—21no—
254C22—21no—
255C22—41no—
256C22—41no—
257C22—42no—
258C22—41no—
259C22—41no—
260C22—4241no—
261C22—2122no—
262C22—2121no—
263C22—2121no—
264C22—2121no—
265C22#1681149211yescharacteristic
266C22#1681149911yescharacteristic
267C22#1681150111yescharacteristic
268C4—21no—
269C4—41no—
270C4—2121no—
271C4—2121no—
272C4—2121no—
273C4#1681152411yescharacteristic
274C4#1681152611yescharacteristic
275C2—21no—
276C2—21no—
277C2—41no—
278C2—41no—
279C2—4241no—
280C2#2028958711yescharacteristic
281C2#2028980011yescharacteristic
282C2#2028981611yescharacteristic
subgroupG/H
centerC22#16811501
commutatorC22×C4×C53C42
FrattiniC23×C4C2×#6157
FittingC2×C22⋊C8×C53C4
socleC22×C53#342
prime radicalC22×C22⋊D53Q8
cosocleC2×C4×C22⋊D53C22
hypercenterC2×C22⋊C8#6157
hypocenterC53#7247
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
156. C53#7247semidirect

Maximal subgroups

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

subgroupindex
#202899742
#202899702
#202750052
#724753