Finite Group Databasebeta

← #28311333

Subgroups — #28311333

Order 13568.

642 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 80 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#20290544C212yesmaximal
2#20289809C214yesmaximal, complemented
3#20277303C211yesmaximal, characteristic
4C2×#13133276C2214yes—
5C2×#13133275—22no—
6#16811566—22no—
7C22×#11719945C2211yescharacteristic
8C22×C4⋊Dic53C411yescharacteristic, complemented
9#16810529C412yescomplemented
10#16810525C2212yes—
11#16810500—22no—
12#16810480—22no—
13C2×#117×C53C411yescharacteristic, complemented
14C2×C22⋊#6157—24no—
15#13133276—28no—
16#13133275—44no—
17(C2×#6157)⋊C4—44no—
18C2×#11719945—24no—
19C2×#11719945—42no—
20C2×#11719945—42no—
21C22×Dic106C2×C411yescharacteristic
22C22×D212C2311yescharacteristic
23C22×C22⋊D53D412yes—
24C23×Dic53—21no—
25C22×C4×D53D412yes—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—42no—
28C2×C4⋊Dic53C2×C412yes—
29C2×#11719921—24no—
30#13133124—24no—
31C23×C4×C53C2×C411yescharacteristic
32#117×C53—42no—
33C2×C4⋊C4×C53—21no—
34C2×C22⋊C4×C53—21no—
35C2×C42×C53C2×C412yes—
36C24×C53C22⋊C411yescharacteristic
37C23×D53C2×D412yes—
38C22×#6157—44no—
39C22×C4×C53—22no—
40C22×C4×C53—22no—
41C22×C4×C53—22no—
42C22×C4×C53—22no—
43C22×C4×C53C22⋊C411yescharacteristic
44C22×C4×C53C22×C411yescharacteristic
45C2×C22⋊D53—24no—
46C2×C22⋊D53—24no—
47C2×C22⋊D53—42no—
48C22×Dic53—22no—
49C22×Dic53—42no—
50C22×Dic53—82no—
51C22×Dic53C22⋊C412yes—
52#11719945—416no—
53C2×D212—41no—
54C2×D212—41no—
55C2×D212#4114yes—
56C2×C4×D53—24no—
57C2×C4×D53—24no—
58C2×C4×D53—42no—
59C2×Dic106—22no—
60C2×Dic106—41no—
61C2×Dic106—41no—
62C22⋊#6157—44no—
63C22⋊#6157—44no—
64C4⋊C4×C53—42no—
65C22⋊C4×C53—42no—
66C42×C53—42no—
67#11719921—84no—
68C4⋊Dic53—42no—
69C4⋊Dic53—42no—
70C23×C53—21no—
71C23×C53—21no—
72C23×C53—41no—
73C23×C53—41no—
74C23×C53#9912yes—
75C23×C53C2×C22⋊C411yescharacteristic
76C22×D53—22no—
77C22×D53—22no—
78C22×D53—42no—
79C22×D53—42no—
80C22×D53#12314yes—
81C2×#6157—84no—
82C2×#6157—84no—
83C2×#6157—84no—
84C2×C4×C53—28no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53#9912yes—
93C2×C4×C53#11712yes—
94C22⋊D53—48no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—44no—
99C2×Dic53—42no—
100C2×Dic53—84no—
101D212—44no—
102C4×D53—48no—
103Dic106—44no—
104#12859—531nomaximal, Sylow
105C22×C53—21no—
106C22×C53—21no—
107C22×C53—22no—
108C22×C53—42no—
109C22×C53—42no—
110C22×C53—41no—
111C22×C53—41no—
112C22×C53#38111yescharacteristic
113C22×C53C2×#9911yescharacteristic
114C22×C53C2×#9911yescharacteristic
115C2×D53—24no—
116C2×D53—42no—
117C2×D53—42no—
118C2×D53—44no—
119C2×D53—82no—
120#6157—164no—
121C4×C53—22no—
122C4×C53—22no—
123C4×C53—82no—
124Dic53—42no—
125Dic53—82no—
126#2311—531no—
127#1782—532no—
128#1393—534no—
129C2×C53—21no—
130C2×C53—21no—
131C2×C53—42no—
132C2×C53#181911yescharacteristic
133C2×C53#212811yescharacteristic
134C2×C53#212911yescharacteristic
135D53—44no—
136C22×#41—531no—
137C2×#117#615711yescharacteristic, complemented
138C22×C4⋊C4—531no—
139C2×#99—534no—
140C2×#99—1062no—
141#391—532no—
142#390—532no—
143#386—1062no—
144#381—1062no—
145#380—1062no—
146C53#1285911yescharacteristic, complemented, Sylow
147C2×#41—1064no—
148C2×#41—2122no—
149C2×#41—2122no—
150C22×Q8—531no—
151C22×D4—532no—
152C22×D4—531no—
153C23×C4—532no—
154C23×C4—1061no—
155C23×C4C2×#615711yescharacteristic
156#117—42no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—532no—
159C2×C4⋊C4—1061no—
160C2×C4⋊C4—2122no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—1064no—
163C2×C22⋊C4—1064no—
164C2×C42C2×#615712yes—
165#99—1068no—
166#99—2124no—
167#95—1064no—
168#95—2124no—
169C24—532no—
170C24C22⋊#615711yescharacteristic
171#41—21216no—
172C2×Q8—1062no—
173C2×Q8—2121no—
174C2×Q8—2121no—
175C2×D4—534no—
176C2×D4—1064no—
177C2×D4—1064no—
178C2×D4—2121no—
179C2×D4—2121no—
180C2×D4—2122no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—532no—
186C22×C4—1064no—
187C22×C4—1064no—
188C22×C4—1062no—
189C22×C4—4242no—
190C22×C4—2122no—
191C22×C4—2124no—
192C22×C4—2122no—
193C22×C4C22⋊#615711yescharacteristic
194C22×C4C22×#615711yescharacteristic
195C4⋊C4—42no—
196C4⋊C4—2122no—
197C4⋊C4—2122no—
198C22⋊C4—42no—
199C22⋊C4—4244no—
200C22⋊C4—2124no—
201C22⋊C4—2124no—
202C42—42no—
203C23—21no—
204C23—21no—
205C23—41no—
206C23—41no—
207C23—534no—
208C23—1062no—
209C23—1062no—
210C23—2122no—
211C23—2122no—
212C23#1313327512yes—
213C23C2×C22⋊#615711yescharacteristic
214Q8—2124no—
215D4—2124no—
216D4—2128no—
217C2×C4—28no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—1062no—
226C2×C4—1062no—
227C2×C4—1062no—
228C2×C4—4244no—
229C2×C4—4244no—
230C2×C4—4244no—
231C2×C4—4244no—
232C2×C4—2128no—
233C2×C4—2124no—
234C2×C4—2122no—
235C2×C4#1313327612yes—
236C2×C4#1313329612yes—
237C22—21no—
238C22—21no—
239C22—22no—
240C22—42no—
241C22—42no—
242C22—41no—
243C22—41no—
244C22—1064no—
245C22—4242no—
246C22—2124no—
247C22—2122no—
248C22—2122no—
249C22#1681156611yescharacteristic
250C22C2×#1313327511yescharacteristic
251C22C2×#1313327611yescharacteristic
252C4—22no—
253C4—22no—
254C4—82no—
255C4—8484no—
256C4—4242no—
257C4—2122no—
258C2—21no—
259C2—21no—
260C2—42no—
261C2—2124no—
262C2#2029061011yescharacteristic
263C2#2029099611yescharacteristic
264C2#2029099911yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC22×C53C2×#99
prime radicalC22×#11719945C22
cosocleC22×D212C23
hypercenterC2×#117#6157
hypocenterC53#12859
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #20289809C2semidirect
8. C22×C4⋊Dic53C4semidirect
9. #16810529C4semidirect
13. C2×#117×C53C4semidirect
137. C2×#117#6157semidirect
146. C53#12859semidirect

Maximal subgroups

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

subgroupindex
#202905442
#202898092
#202773032
#1285953