Finite Group Databasebeta

← #28308090

Subgroups — #28308090

Order 13568.

440 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 90 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#20289958C212yesmaximal, complemented
2#20289658C212yesmaximal, complemented
3#20289644C212yesmaximal
4#20277429C211yesmaximal, characteristic
5#16811639—22no—
6#16811468—22no—
7#16811468C2214yes—
8C2×#13133206C411yescharacteristic
9C2×Q8×Dic53C2211yescharacteristic
10C2×#13133179C411yescharacteristic
11#16810525C2211yescharacteristic
12#16810512—22no—
13#16810497—22no—
14#16810492C2211yescharacteristic
15#16810469C411yescharacteristic
16C53×#391C411yescharacteristic
17C4⋊#88324—42no—
18C22⋊#88324—24no—
19C4×#88324—42no—
20C22×D212C2×C411yescharacteristic
21C22×C4×D53—21no—
22#13133206—22no—
23#13133206C814yescomplemented
24Q8×Dic53—42no—
25#13133179—42no—
26C2×C4⋊Dic53D412yes—
27C2×C4⋊Dic53C2311yescharacteristic
28C2×#11719919—22no—
29C2×#11719921—22no—
30C2×#11719921—21no—
31C2×#11719921—22no—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53C2×C411yescharacteristic
34C2×C4×Dic53D412yes—
35#13133124—21no—
36#13133124—21no—
37C22×Q8×C53C2×C411yescharacteristic
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C42×C53C2×C411yescharacteristic
40#95×C53—22no—
41C23×D53—21no—
42C2×Q8×C53—41no—
43C2×Q8×C53—41no—
44C2×Q8×C53C2×C812yes—
45C2×Q8×C53M1612yes—
46C22×C4×C53—21no—
47C22×C4×C53—21no—
48C22×C4×C53C22⋊C412yes—
49C22×C4×C53C22×C411yescharacteristic
50C22×Dic53—21no—
51C22×Dic53C22⋊C411yescharacteristic
52C22×Dic53C22⋊C411yescharacteristic
53C22×Dic53C2×D412yes—
54C2×D212—21no—
55C2×D212—41no—
56C2×D212—41no—
57C2×D212C2×C812yes—
58C2×C4×D53—42no—
59C2×C4×D53—82no—
60C2×#88324—44no—
61C4⋊C4×C53—41no—
62C42×C53—41no—
63#11719921—28no—
64#11719921—42no—
65#11719921—82no—
66C4⋊Dic53—42no—
67C4⋊Dic53—41no—
68#11719919—82no—
69C4×Dic53—21no—
70C4×Dic53—22no—
71C4×Dic53—22no—
72C4×Dic53—22no—
73C4×Dic53C2×C812yes—
74C23×C53C2×C22⋊C411yescharacteristic
75C22×D53—22no—
76C22×D53—42no—
77C22×D53—42no—
78C22×D53—41no—
79C22×D53—41no—
80Q8×C53—42no—
81Q8×C53—42no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53—21no—
85C2×C4×C53—22no—
86C2×C4×C53—42no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53C22⋊C814yes—
92C2×C4×C53C22×C811yescharacteristic
93C2×C4×C53C2×M1611yescharacteristic
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—21no—
97C2×Dic53—41no—
98C2×Dic53—41no—
99C2×Dic53—41no—
100C2×Dic53—41no—
101C2×Dic53—42no—
102C2×Dic53#9912yes—
103D212—42no—
104D212—42no—
105C4×D53—84no—
106#88324—84no—
107#11480—531nomaximal, Sylow
108C22×C53—21no—
109C22×C53—21no—
110C22×C53C2×C22⋊C811yescharacteristic
111C22×C53C2×#9911yescharacteristic
112C22×C53#41311yescharacteristic
113C2×D53—44no—
114C2×D53—42no—
115C2×D53—82no—
116C2×D53—81no—
117C2×D53—81no—
118C4×C53—21no—
119C4×C53—21no—
120C4×C53—24no—
121C4×C53—81no—
122C4×C53—81no—
123Dic53—22no—
124Dic53—42no—
125Dic53—42no—
126Dic53—41no—
127#2326—531no—
128#1461—532no—
129#1325—532no—
130#1318—532no—
131C2×C53—21no—
132C2×C53—21no—
133C2×C53#145911yescharacteristic
134C2×C53#212811yescharacteristic
135C2×C53#213311yescharacteristic
136D53—82no—
137C2×#124—531no—
138C2×C4×Q8—531no—
139C2×C4×D4—531no—
140#431—1062no—
141#391#615711yescharacteristic
142#390—531no—
143#388—1062no—
144#385—1062no—
145#384—531no—
146#376—531no—
147#324—534no—
148#324—1062no—
149C53#1148011yescharacteristic, complemented, Sylow
150C22×Q8C2×#615711yescharacteristic
151C22×D4—531no—
152C23×C4—1061no—
153#124—534no—
154#124—1062no—
155C4×Q8—2122no—
156C4×D4—2122no—
157C2×C4⋊C4—532no—
158C2×C4⋊C4—531no—
159C2×C4⋊C4—1062no—
160C2×C4⋊C4C2×#615711yescharacteristic
161C2×C22⋊C4—1062no—
162C2×C22⋊C4—1062no—
163C2×C22⋊C4—1061no—
164C2×C42—531no—
165C2×C42—532no—
166C2×C42—531no—
167C2×C42C2×#615711yescharacteristic
168C4⋊C8—2122no—
169C22⋊C8—1064no—
170C4×C8—2122no—
171#95—22no—
172#95—1061no—
173#95—1061no—
174C24—1061no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8D53⋊C812yes—
178C2×Q8#1171993612yes—
179C2×D4—532no—
180C2×D4—1061no—
181C2×D4—2121no—
182C2×D4—2121no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—531no—
186C22×C4—531no—
187C22×C4—532no—
188C22×C4—1061no—
189C22×C4—4242no—
190C22×C4—2122no—
191C22×C4C22⋊#615712yes—
192C22×C4C22×#615711yescharacteristic
193C2×C8—2124no—
194C4⋊C4—41no—
195C4⋊C4—4242no—
196C4⋊C4—2121no—
197C4⋊C4—2122no—
198C22⋊C4—1068no—
199C22⋊C4—4242no—
200C22⋊C4—2122no—
201C42—41no—
202C42—532no—
203C42—1062no—
204C42—1062no—
205C42—1062no—
206C42—1061no—
207C23—1062no—
208C23—2122no—
209C23—2122no—
210C23—2121no—
211C23—2121no—
212C23C2×C22⋊#615711yescharacteristic
213Q8—42no—
214Q8—42no—
215D4—2122no—
216D4—2122no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—532no—
227C2×C4—1062no—
228C2×C4—1062no—
229C2×C4—1061no—
230C2×C4—4244no—
231C2×C4—2121no—
232C2×C4—2121no—
233C2×C4—2121no—
234C2×C4—2122no—
235C2×C4—2121no—
236C2×C4#1313327414yes—
237C2×C4C2×D53⋊C811yescharacteristic
238C2×C4C2×#1171993611yescharacteristic
239C8—4244no—
240C22—21no—
241C22—21no—
242C22—4242no—
243C22—4241no—
244C22—4241no—
245C22—2124no—
246C22—2122no—
247C22C2×#1313327411yescharacteristic
248C22D53⋊#9911yescharacteristic
249C22#1681161011yescharacteristic
250C4—24no—
251C4—21no—
252C4—21no—
253C4—81no—
254C4—81no—
255C4—1062no—
256C4—2122no—
257C4—2122no—
258C4—2121no—
259C2—21no—
260C2—21no—
261C2—4242no—
262C2#2028995311yescharacteristic
263C2#2029099511yescharacteristic
264C2#2029101311yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#391C4
socleC22×C53C2×#99
prime radicalC22×D212C2×C4
cosocleC2×C4⋊Dic53C23
hypercenter#391#6157
hypocenterC53#11480
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #391

Complements

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

NH ≅ G/Ntype
1. #20289958C2semidirect
2. #20289658C2semidirect
23. #13133206C8semidirect
149. C53#11480semidirect

Maximal subgroups

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

subgroupindex
#202899582
#202896582
#202896442
#202774292
#1148053