Finite Group Databasebeta

← #28311326

Subgroups — #28311326

Order 13568.

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

normal subgroups only

This group has 670 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#20289810C214yesmaximal, complemented
3#20277203C211yesmaximal, characteristic
4C2×#13133276C2214yes—
5C2×#13133275—22no—
6#16811566—22no—
7C22×#11719945C2211yescharacteristic
8C22×C4×Dic53C411yescharacteristic, complemented
9#16810528C412yescomplemented
10#16810524C2212yes—
11#16810500—22no—
12C4×#11719921—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×#11719921—24no—
27C2×C4×Dic53—21no—
28C2×C4×Dic53—42no—
29C2×C4×Dic53C2×C412yes—
30#13133124—24no—
31C23×C4×C53C2×C411yescharacteristic
32#117×C53—42no—
33C2×C4⋊C4×C53C2×C412yes—
34C2×C22⋊C4×C53—21no—
35C2×C42×C53—21no—
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—44no—
50C22×Dic53—42no—
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—48no—
68C4×Dic53—42no—
69C4×Dic53—42no—
70C23×C53—21no—
71C23×C53—21no—
72C23×C53—42no—
73C23×C53#9912yes—
74C23×C53C2×C22⋊C411yescharacteristic
75C22×D53—22no—
76C22×D53—22no—
77C22×D53—44no—
78C22×D53#12314yes—
79C2×#6157—84no—
80C2×#6157—88no—
81C2×C4×C53—28no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—44no—
87C2×C4×C53—42no—
88C2×C4×C53#9912yes—
89C2×C4×C53#11712yes—
90C22⋊D53—48no—
91C2×Dic53—24no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—42no—
96C2×Dic53—48no—
97C2×Dic53—42no—
98D212—44no—
99C4×D53—48no—
100Dic106—44no—
101#12857—531nomaximal, Sylow
102C22×C53—22no—
103C22×C53—22no—
104C22×C53—42no—
105C22×C53—42no—
106C22×C53—42no—
107C22×C53#38111yescharacteristic
108C22×C53C2×#9911yescharacteristic
109C22×C53C2×#9911yescharacteristic
110C2×D53—24no—
111C2×D53—44no—
112C2×D53—44no—
113C2×D53—82no—
114#6157—164no—
115C4×C53—22no—
116C4×C53—22no—
117C4×C53—82no—
118Dic53—44no—
119Dic53—42no—
120#2297—531no—
121#1782—532no—
122#1393—534no—
123C2×C53—21no—
124C2×C53—21no—
125C2×C53—42no—
126C2×C53#181911yescharacteristic
127C2×C53#212612yes—
128D53—44no—
129C22×#41—531no—
130C2×#117#615711yescharacteristic, complemented
131C22×C42—531no—
132C2×#99—534no—
133C2×#99—1062no—
134#391—532no—
135#390—532no—
136#386—1062no—
137#381—1062no—
138C4×C22⋊C4—1062no—
139C53#1285711yescharacteristic, complemented, Sylow
140C2×#41—1064no—
141C2×#41—2122no—
142C2×#41—2122no—
143C22×Q8—531no—
144C22×D4—532no—
145C22×D4—531no—
146C23×C4—532no—
147C23×C4—1061no—
148C23×C4C2×#615711yescharacteristic
149#117—42no—
150C2×C4⋊C4C2×#615712yes—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—1064no—
153C2×C22⋊C4—1064no—
154C2×C42—21no—
155C2×C42—532no—
156C2×C42—1061no—
157C2×C42—2122no—
158#99—1068no—
159#99—2124no—
160#95—1064no—
161#95—2124no—
162C24—532no—
163C24C22⋊#615711yescharacteristic
164#41—21216no—
165C2×Q8—1062no—
166C2×Q8—2121no—
167C2×Q8—2121no—
168C2×D4—534no—
169C2×D4—1064no—
170C2×D4—1064no—
171C2×D4—2121no—
172C2×D4—2121no—
173C2×D4—2122no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—532no—
179C22×C4—1064no—
180C22×C4—1062no—
181C22×C4—1064no—
182C22×C4—2122no—
183C22×C4—2124no—
184C22×C4—2124no—
185C22×C4—2122no—
186C22×C4C22⋊#615711yescharacteristic
187C22×C4C22×#615711yescharacteristic
188C4⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—2128no—
191C22⋊C4—2124no—
192C22⋊C4—2124no—
193C42—42no—
194C42—2122no—
195C42—2122no—
196C23—21no—
197C23—21no—
198C23—42no—
199C23—534no—
200C23—1062no—
201C23—1062no—
202C23—2124no—
203C23#1313327612yes—
204C23C2×C22⋊#615711yescharacteristic
205Q8—2124no—
206D4—2124no—
207D4—2128no—
208C2×C4—28no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—44no—
214C2×C4—42no—
215C2×C4—1064no—
216C2×C4—1062no—
217C2×C4—1062no—
218C2×C4—1062no—
219C2×C4—4244no—
220C2×C4—4248no—
221C2×C4—2128no—
222C2×C4—2122no—
223C2×C4—2128no—
224C2×C4—2122no—
225C2×C4#1313327512yes—
226C2×C4#1313329612yes—
227C22—22no—
228C22—22no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—1064no—
233C22—4242no—
234C22—2124no—
235C22—2124no—
236C22#1681156611yescharacteristic
237C22C2×#1313327511yescharacteristic
238C22C2×#1313327611yescharacteristic
239C4—22no—
240C4—22no—
241C4—82no—
242C4—8484no—
243C4—2124no—
244C4—2122no—
245C2—21no—
246C2—21no—
247C2—42no—
248C2—2124no—
249C2#2029061011yescharacteristic
250C2#2029098612yes—
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC22×C53C2×#99
prime radicalC22×#11719945C22
cosocleC22×D212C23
hypercenterC2×#117#6157
hypocenterC53#12857
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. #20289810C2semidirect
8. C22×C4×Dic53C4semidirect
9. #16810528C4semidirect
13. C2×#117×C53C4semidirect
130. C2×#117#6157semidirect
139. C53#12857semidirect

Maximal subgroups

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

subgroupindex
#202905442
#202898102
#202772032
#1285753