Finite Group Databasebeta

← #28308701

Subgroups — #28308701

Order 13568.

430 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 92 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#20291073C211yesmaximal, characteristic, complemented
2#20290272C211yesmaximal, characteristic, complemented
3#20289670C214yesmaximal, complemented
4C2×#16810705C211yesmaximal, characteristic
5#16811626—21no—
6#16811544—21no—
7C22⋊#11719910—21no—
8#16811509C2212yescomplemented
9#16811487—21no—
10#16811486C2212yescomplemented
11C22×D424C411yescharacteristic
12C2×#13133179—21no—
13#16810705—21no—
14#16810705—21no—
15#16810705C412yes—
16#16810705C2212yes—
17C2×#13133137—21no—
18C2×C8×Dic53C2211yescharacteristic
19C2×#107×C53C411yescharacteristic
20C2×#11719940—41no—
21C2×#11719910—22no—
22#13133284—41no—
23C4⋊#88324—22no—
24C22×D212—21no—
25C22×C4×D53—41no—
26C2×D424—21no—
27C2×D424—24no—
28C2×D424C2×C412yes—
29C22×#88322D411yescharacteristic
30#13133179—22no—
31#13133179—22no—
32#13133179—41no—
33#13133179—41no—
34C2×#11719921—41no—
35C2×C4×Dic53D411yescharacteristic
36#13133137—22no—
37#13133137—22no—
38C8×Dic53C2×C411yescharacteristic
39C8×Dic53D411yescharacteristic
40C8×Dic53D411yescharacteristic
41C8×Dic53C2311yescharacteristic
42C22×C8×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53—21no—
44#107×C53—21no—
45#107×C53C2×C412yes—
46C23×D53—41no—
47C22×C4×C53—41no—
48C22×C4×C53C22⋊C411yescharacteristic
49C22×Dic53D811yescharacteristic
50C22×Dic53SD1611yescharacteristic
51C2×D212—24no—
52C2×D212—22no—
53C2×D212—42no—
54C2×D212—41no—
55C2×C4×D53—42no—
56C2×C4×D53—42no—
57C2×C4×D53—82no—
58C2×C4×D53—82no—
59#11719940—82no—
60C2×#88324—42no—
61C2×C8×C53—22no—
62C2×C8×C53C22⋊C411yescharacteristic
63C2×C8×C53C22×C411yescharacteristic
64C4⋊C4×C53—22no—
65C4⋊C4×C53—41no—
66#11719921—42no—
67#11719921—42no—
68C4×Dic53C22⋊C411yescharacteristic
69C4×Dic53D811yescharacteristic
70C4×Dic53SD1611yescharacteristic
71C4×Dic53C2×D411yescharacteristic
72C2×#88322—41no—
73C2×#88322C22⋊C411yescharacteristic
74C2×#88322C2×D411yescharacteristic
75D424—24no—
76D424—22no—
77D424M1614yescomplemented
78#11719910—42no—
79C23×C53#10211yescharacteristic
80C22×D53—44no—
81C22×D53—42no—
82C22×D53—82no—
83C22×D53—82no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—42no—
87C2×C4×C53—82no—
88C2×C4×C53#10211yescharacteristic
89C2×C4×C53C2×C22⋊C411yescharacteristic
90C2×Dic53—41no—
91C2×Dic53#10211yescharacteristic
92C2×Dic53#10211yescharacteristic
93C2×Dic53D1612yes—
94C2×Dic53SD3212yes—
95C2×Dic53C2×D811yescharacteristic
96C2×Dic53C2×SD1611yescharacteristic
97D212—24no—
98D212—44no—
99D212—42no—
100C4×D53—84no—
101C4×D53—82no—
102C4×D53—82no—
103#88324—82no—
104C8×C53—21no—
105C8×C53C2×M1612yes—
106#88322—21no—
107#88322—21no—
108#11657—531nomaximal, Sylow
109C22×C53—21no—
110C22×C53—21no—
111C22×C53#35712yes—
112C22×C53C2×#10211yescharacteristic
113C2×D53—44no—
114C2×D53—88no—
115C2×D53—82no—
116C2×D53—82no—
117C4×C53—21no—
118C4×C53—82no—
119C4×C53#40711yescharacteristic
120C4×C53#42111yescharacteristic
121Dic53—21no—
122Dic53—21no—
123Dic53#35712yes—
124Dic53C2×D1611yescharacteristic
125Dic53C2×SD3211yescharacteristic
126C2×C4×D8—531no—
127#2162—531no—
128#1567—531no—
129#1331—534no—
130C2×C53—22no—
131C2×C53#148611yescharacteristic
132C2×C53C2×#35711yescharacteristic
133C2×C53#214311yescharacteristic
134D53—84no—
135C22×D8—531no—
136C2×C4×D4—1061no—
137C4×D8—532no—
138C4×D8—532no—
139C4×D8—1061no—
140C4×D8—1061no—
141C2×#107#615711yescharacteristic
142#423—1061no—
143C2×#102—1061no—
144C2×C4×C8—531no—
145C4⋊C16—1061no—
146C22⋊C16—1061no—
147C4×C16—532no—
148#335—532no—
149#335—1061no—
150C53#1165711yescharacteristic, complemented, Sylow
151C22×D4—1061no—
152C23×C4—2121no—
153C2×D8—532no—
154C2×D8—1064no—
155C2×D8—1061no—
156C2×M16—2121no—
157C22×C8—531no—
158C22×C8C2×#615711yescharacteristic
159C4×D4—1062no—
160C4×D4—1062no—
161C4×D4—2121no—
162C4×D4—2121no—
163C2×C4⋊C4—21no—
164C2×C22⋊C4—2121no—
165C2×C42—531no—
166C2×C16—1062no—
167#107—21no—
168#107C2×#615712yes—
169C4⋊C8—1062no—
170C4⋊C8—2121no—
171#102—1062no—
172#102—1062no—
173C4×C8—531no—
174C4×C8—531no—
175C4×C8—531no—
176C4×C8—531no—
177C24—2121no—
178C2×D4—1064no—
179C2×D4—1062no—
180C2×D4—2122no—
181C2×D4—2121no—
182C22×C4—41no—
183C22×C4—531no—
184C22×C4—531no—
185C22×C4—4242no—
186C22×C4—4242no—
187C22×C4—2122no—
188C22×C4—2122no—
189C22×C4C22⋊#615711yescharacteristic
190D8—534no—
191D8—1064no—
192D8—1062no—
193M16—4242no—
194C2×C8—22no—
195C2×C8—531no—
196C2×C8—531no—
197C2×C8—2121no—
198C2×C8—2122no—
199C2×C8C22⋊#615711yescharacteristic
200C2×C8C22×#615711yescharacteristic
201C4⋊C4—22no—
202C4⋊C4—41no—
203C22⋊C4—2122no—
204C22⋊C4—2122no—
205C42—531no—
206C42—531no—
207C42—531no—
208C42—531no—
209C16—2122no—
210C23—4242no—
211C23—4242no—
212C23—2124no—
213C23—2122no—
214C23#1313327911yescharacteristic
215D4—1064no—
216D4—2124no—
217D4—2122no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—42no—
221C2×C4—82no—
222C2×C4—532no—
223C2×C4—532no—
224C2×C4—531no—
225C2×C4—531no—
226C2×C4—531no—
227C2×C4—531no—
228C2×C4—4242no—
229C2×C4—4244no—
230C2×C4—4242no—
231C2×C4—2121no—
232C2×C4#1313327911yescharacteristic
233C2×C4C2×C22⋊#615711yescharacteristic
234C8—21no—
235C8—1061no—
236C8—1061no—
237C8—4242no—
238C8D53⋊M1612yes—
239C22—21no—
240C22—21no—
241C22—4248no—
242C22—4242no—
243C22—4242no—
244C22—2124no—
245C22#1681153512yes—
246C22C2×#1313327911yescharacteristic
247C4—21no—
248C4—82no—
249C4—531no—
250C4—532no—
251C4—531no—
252C4—1061no—
253C4—1061no—
254C4#1681159511yescharacteristic
255C4#1681162211yescharacteristic
256C2—22no—
257C2—4244no—
258C2#2029004211yescharacteristic
259C2C2×#1681153511yescharacteristic
260C2#2029103111yescharacteristic
subgroupG/H
centerC22C2×#13133279
commutatorC2×C8×C53C22×C4
FrattiniC2×C8C22×#6157
FittingC2×#107×C53C4
socleC22×C53C2×#102
prime radicalC22×D424C4
cosocleC8×Dic53C23
hypercenterC2×#107#6157
hypocenterC53#11657
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
1. #20291073C2semidirect
2. #20290272C2semidirect
3. #20289670C2semidirect
8. #16811509C22semidirect
10. #16811486C22semidirect
77. D424M16semidirect
150. C53#11657semidirect

Maximal subgroups

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

subgroupindex
#202910732
#202902722
#202896702
C2×#168107052
#1165753