Finite Group Databasebeta

← #28298735

Subgroups — #28298735

Order 13568.

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

normal subgroups only

This group has 324 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#20289954C211yesmaximal, characteristic
2#20289950C211yesmaximal, characteristic
3#20275012C211yesmaximal, characteristic
4#16811640—21no—
5D53⋊(C22⋊C8)—21no—
6#16811469C412yes—
7#16811467C412yes—
8#16811466—21no—
9#16811466—21no—
10C2×C4×C22⋊D53C2211yescharacteristic
11C2×C22⋊#88322C411yescharacteristic
12C2×#13133129—21no—
13#16810377—21no—
14C22×M16×C53C411yescharacteristic
15C2×D53⋊C8—41no—
16#13133284—41no—
17#13133274—41no—
18C22⋊#88324—22no—
19C22⋊#88324—22no—
20C4×#88324—41no—
21C22×C22⋊D53Q811yescharacteristic
22C22×C4×D53D411yescharacteristic
23C22×#88322—21no—
24C4×C22⋊D53—41no—
25C4×C22⋊D53—41no—
26C2×#11719919C2×C411yescharacteristic
27C2×C22⋊Dic53D411yescharacteristic
28C2×#11719921C2×C411yescharacteristic
29C2×C4×Dic53D411yescharacteristic
30C22⋊#88322—41no—
31#13133129—42no—
32C23×C4×C53C2×C411yescharacteristic
33C2×M16×C53—41no—
34C2×M16×C53—42no—
35C22×C8×C53—21no—
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—22no—
45C2×C22⋊D53—22no—
46C2×C22⋊D53—41no—
47C22×Dic53C22⋊C411yescharacteristic
48C22×Dic53C22⋊C411yescharacteristic
49C22×Dic53C4⋊C411yescharacteristic
50C2×C4×D53—21no—
51C2×C4×D53—21no—
52C2×C4×D53—41no—
53C2×C4×D53—41no—
54C2×#88324—42no—
55C2×#88324—42no—
56D53⋊C8—81no—
57D53⋊C8—81no—
58M16×C53—42no—
59M16×C53—42no—
60C2×C8×C53—22no—
61C2×C8×C53—41no—
62C2×C8×C53—41no—
63C22⋊Dic53—41no—
64#11719921—41no—
65#11719919—41no—
66C4×Dic53—21no—
67C4×Dic53—21no—
68C2×#88322—41no—
69C2×#88322—41no—
70C2×#88322—41no—
71C23×C53—21no—
72C23×C53—21no—
73C23×C53—21no—
74C23×C53—41no—
75C23×C53—41no—
76C23×C53#9511yescharacteristic
77C22×D53—21no—
78C22×D53—21no—
79C22×D53—41no—
80C22×D53—41no—
81C22×D53#10812yes—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—42no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53#9911yescharacteristic
90C2×C4×C53#9911yescharacteristic
91C22⋊D53—44no—
92C2×Dic53—21no—
93C2×Dic53—21no—
94C2×Dic53—21no—
95C2×Dic53—21no—
96C2×Dic53—41no—
97C2×Dic53—41no—
98C2×Dic53#10812yes—
99C4×D53—41no—
100C4×D53—41no—
101C4×D53—81no—
102#88324—82no—
103#88324—82no—
104C8×C53—42no—
105#88322—81no—
106#7285—531nomaximal, Sylow
107C22×C53—21no—
108C22×C53—21no—
109C22×C53—41no—
110C22×C53—41no—
111C22×C53—41no—
112C22×C53—81no—
113C22×C53—81no—
114C22×C53#34111yescharacteristic
115C22×C53#34111yescharacteristic
116C22×C53#34211yescharacteristic
117C2×D53—22no—
118C2×D53—41no—
119C2×D53—41no—
120C2×D53—42no—
121C2×D53—81no—
122C4×C53—21no—
123C4×C53—41no—
124C4×C53AΣL(1,𝔽16)11yescharacteristic
125C4×C53#35211yescharacteristic
126Dic53—22no—
127Dic53—41no—
128Dic53—41no—
129#1870—531no—
130#1460—531no—
131#1459—531no—
132C2×C53—21no—
133C2×C53—21no—
134C2×C53—81no—
135C2×C53#129811yescharacteristic
136C2×C53#139011yescharacteristic
137C2×C53#139511yescharacteristic
138D53—42no—
139C22×M16#615711yescharacteristic
140C2×C4×D4—531no—
141#431—1061no—
142C2×C22⋊C8—531no—
143C2×C22⋊C8—1061no—
144C2×C22⋊C8—1061no—
145#336—1061no—
146#324—532no—
147#323—532no—
148#323—1061no—
149#323—1061no—
150C53#728511yescharacteristic, complemented, Sylow
151C22×D4—531no—
152C23×C4—531no—
153C23×C4C2×#615711yescharacteristic
154C2×M16—42no—
155C2×M16—41no—
156C22×C8—21no—
157C22×C8—1061no—
158C22×C8—2121no—
159C4×D4—2121no—
160C4×D4—2121no—
161C2×C4⋊C4—531no—
162C2×C22⋊C4—531no—
163C2×C22⋊C4—531no—
164C2×C42—531no—
165C4⋊C8—2121no—
166C22⋊C8—1062no—
167C22⋊C8—1062no—
168C22⋊C8—2122no—
169C22⋊C8—2121no—
170C22⋊C8—2121no—
171C4×C8—2121no—
172C24—531no—
173C24C4⋊#615711yescharacteristic
174C2×D4—1062no—
175C2×D4—1062no—
176C2×D4—2121no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—531no—
182C22×C4—531no—
183C22×C4—531no—
184C22×C4—1061no—
185C22×C4—1061no—
186C22×C4—2121no—
187C22×C4—2121no—
188C22×C4C4×#615711yescharacteristic
189C22×C4C22⋊#615711yescharacteristic
190M16—42no—
191M16—42no—
192C2×C8—22no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—4241no—
196C2×C8—4241no—
197C2×C8—2121no—
198C2×C8—2121no—
199C2×C8—2122no—
200C2×C8—2122no—
201C2×C8—2121no—
202C4⋊C4—2121no—
203C22⋊C4—2121no—
204C22⋊C4—2121no—
205C42—1061no—
206C42—1061no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—41no—
211C23—41no—
212C23—532no—
213C23—1061no—
214C23—1061no—
215C23—2121no—
216C23—2121no—
217C23(C2×#6157)⋊C411yescharacteristic
218D4—2124no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—532no—
227C2×C4—1061no—
228C2×C4—1061no—
229C2×C4—1061no—
230C2×C4—1061no—
231C2×C4—4241no—
232C2×C4—2121no—
233C2×C4—2121no—
234C2×C4—2121no—
235C2×C4—2121no—
236C2×C4#1313327511yescharacteristic
237C2×C4#1313327611yescharacteristic
238C8—42no—
239C8—4241no—
240C8—4242no—
241C8—4242no—
242C22—21no—
243C22—21no—
244C22—41no—
245C22—41no—
246C22—41no—
247C22—81no—
248C22—81no—
249C22—1062no—
250C22—4241no—
251C22—2122no—
252C22—2121no—
253C22—2121no—
254C22#1681149811yescharacteristic
255C22#1681149811yescharacteristic
256C22#1681150111yescharacteristic
257C4—21no—
258C4—41no—
259C4—1062no—
260C4—2121no—
261C4—2121no—
262C4#1681152311yescharacteristic
263C4#1681152511yescharacteristic
264C2—21no—
265C2—21no—
266C2—81no—
267C2—2122no—
268C2#2028959711yescharacteristic
269C2#2028979911yescharacteristic
270C2#2028981611yescharacteristic
subgroupG/H
centerC2×C4#13133275
commutatorC22×C4×C53C42
FrattiniC23×C4C2×#6157
FittingC22×M16×C53C4
socleC22×C53#342
prime radicalC22×C22⋊D53Q8
cosocleC2×C4×C22⋊D53C22
hypercenterC22×M16#6157
hypocenterC53#7285
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
150. C53#7285semidirect

Maximal subgroups

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

subgroupindex
#202899542
#202899502
#202750122
#728553