Finite Group Databasebeta

← #28302275

Subgroups — #28302275

Order 13568.

930 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 144 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#20290019C214yesmaximal, complemented
2#20289946C212yesmaximal
3C2×#16811100C211yesmaximal, characteristic
4#16811639C2214yes—
5#16811473—28no—
6#16811468—22no—
7#16811466—22no—
8C23×Dic106C411yescharacteristic
9#16811100—28no—
10C2×C4×Dic106—22no—
11C2×#13133182C412yes—
12C2×#13133182C2212yes—
13C2×#13133176C2211yescharacteristic
14C2×#13133173—22no—
15C2×#117×C53C411yescharacteristic
16C4⋊#88324—24no—
17C22⋊#88324—24no—
18C4×#88324—24no—
19C22×Dic106—21no—
20C22×Dic106—22no—
21C22×Dic106—24no—
22C22×Dic106—24no—
23C22×Dic106C814yescomplemented
24C22×Dic106C2×C412yes—
25C23×Dic53—21no—
26C4×Dic106—216no—
27#13133182—24no—
28#13133182—24no—
29#13133182—24no—
30#13133182D418yes—
31#13133176—24no—
32#13133173—216no—
33C2×C4⋊Dic53D412yes—
34C2×#11719919—22no—
35C2×#11719919—22no—
36C2×C22⋊Dic53D412yes—
37C2×C4×Dic53—22no—
38C2×C4×Dic53—22no—
39C2×C4×Dic53C2×C411yescharacteristic
40C2×C4×Dic53C2311yescharacteristic
41C23×C4×C53C2×C411yescharacteristic
42#117×C53—24no—
43C2×C4⋊C4×C53C2×C412yes—
44C2×C22⋊C4×C53—21no—
45C2×C42×C53—21no—
46C24×C53C22⋊C411yescharacteristic
47C22×C4×C53—22no—
48C22×C4×C53—22no—
49C22×C4×C53—22no—
50C22×C4×C53C22⋊C411yescharacteristic
51C22×C4×C53C2×C812yes—
52C22×C4×C53M1612yes—
53C22×C4×C53C22×C411yescharacteristic
54C22×Dic53—24no—
55C22×Dic53—24no—
56C22×Dic53—24no—
57C22×Dic53—22no—
58C22×Dic53C22⋊C412yes—
59C22×Dic53C2×D412yes—
60C2×Dic106—24no—
61C2×Dic106—24no—
62C2×Dic106—216no—
63C2×Dic106—24no—
64C2×Dic106—22no—
65C2×Dic106—22no—
66C2×Dic106—22no—
67C2×Dic106—216no—
68C2×Dic106C22⋊C414yes—
69C2×Dic106C2×C814yes—
70C2×Dic106M1614yes—
71C2×#88324—44no—
72C4⋊C4×C53—22no—
73C4⋊C4×C53C22⋊C414yes—
74C22⋊C4×C53—24no—
75C42×C53—24no—
76C22⋊Dic53—24no—
77C4⋊Dic53—24no—
78#11719919—28no—
79#11719919—28no—
80C4×Dic53—28no—
81C4×Dic53—22no—
82C4×Dic53—28no—
83C4×Dic53C2×D414yes—
84C23×C53—21no—
85C23×C53—21no—
86C23×C53—42no—
87C23×C53C22⋊C812yes—
88C23×C53C2×C22⋊C411yescharacteristic
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—24no—
92C2×C4×C53—24no—
93C2×C4×C53—24no—
94C2×C4×C53—44no—
95C2×C4×C53C22⋊C812yes—
96C2×C4×C53C2×C22⋊C412yes—
97C2×C4×C53C22×C811yescharacteristic
98C2×C4×C53C2×M1612yes—
99C2×C4×C53C2×M1611yescharacteristic
100C2×Dic53—216no—
101C2×Dic53—24no—
102C2×Dic53—24no—
103C2×Dic53—24no—
104C2×Dic53—22no—
105C2×Dic53—42no—
106C2×Dic53—42no—
107C2×Dic53#12014yes—
108Dic106—216no—
109Dic106—24no—
110Dic106—24no—
111Dic106—24no—
112Dic106C22⋊C818yes—
113#88324—84no—
114#8323—531nomaximal, Sylow
115C22×C53—22no—
116C22×C53—22no—
117C22×C53—42no—
118C22×C53—42no—
119C22×C53—42no—
120C22×C53#37911yescharacteristic
121C22×C53C2×C22⋊C811yescharacteristic
122C22×C53#40711yescharacteristic
123C4×C53—22no—
124C4×C53—44no—
125C4×C53C2×C22⋊C812yes—
126C4×C53#40712yes—
127Dic53—24no—
128Dic53—28no—
129Dic53—42no—
130C2×#519—531no—
131#1479—534no—
132#1458—532no—
133C2×C53—21no—
134C2×C53—21no—
135C2×C53—42no—
136C2×C53#177911yescharacteristic
137C2×C53#179612yes—
138C23×Q8—531no—
139#519—1068no—
140C2×C4×Q8—532no—
141C2×C4×Q8—532no—
142C2×C4×Q8—1062no—
143C2×#117—531no—
144C2×#117—1062no—
145C2×#117#615711yescharacteristic
146#431—534no—
147Q8⋊C8—1068no—
148#324—1062no—
149#323—1062no—
150C53#832311yescharacteristic, complemented, Sylow
151C22×Q8—532no—
152C22×Q8—534no—
153C22×Q8—1064no—
154C22×Q8—1064no—
155C22×Q8—1062no—
156C22×Q8—1061no—
157C23×C4—1061no—
158C23×C4C2×#615711yescharacteristic
159C4×Q8—538no—
160C4×Q8—10616no—
161C4×Q8—1064no—
162C4×Q8—1064no—
163C4×Q8—1064no—
164#117—24no—
165#117—1064no—
166#117—10616no—
167C2×C4⋊C4—532no—
168C2×C4⋊C4—1062no—
169C2×C4⋊C4—1062no—
170C2×C4⋊C4C2×#615712yes—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—532no—
173C2×C42—21no—
174C2×C42—531no—
175C2×C42—531no—
176C2×C42—1062no—
177C2×C42—1062no—
178C4⋊C8—1064no—
179C22⋊C8—1064no—
180C4×C8—1064no—
181C24C22⋊#615711yescharacteristic
182C2×Q8—534no—
183C2×Q8—534no—
184C2×Q8—534no—
185C2×Q8—10616no—
186C2×Q8—1064no—
187C2×Q8—1064no—
188C2×Q8—10616no—
189C2×Q8—1064no—
190C2×Q8—1062no—
191C2×Q8—1062no—
192C2×Q8—1062no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—532no—
197C22×C4—532no—
198C22×C4—1064no—
199C22×C4—1064no—
200C22×C4—1064no—
201C22×C4—1062no—
202C22×C4D53⋊C812yes—
203C22×C4#1171993612yes—
204C22×C4C22⋊#615711yescharacteristic
205C22×C4C22×#615711yescharacteristic
206C2×C8—2124no—
207C4⋊C4—22no—
208C4⋊C4—1064no—
209C4⋊C4—1068no—
210C4⋊C4—1068no—
211C4⋊C4C22⋊#615714yes—
212C22⋊C4—24no—
213C22⋊C4—1064no—
214C42—24no—
215C42—534no—
216C42—1068no—
217C42—1068no—
218C42—1062no—
219C23—21no—
220C23—21no—
221C23—42no—
222C23#1313327412yes—
223C23C2×C22⋊#615711yescharacteristic
224Q8—538no—
225Q8—10616no—
226Q8—1064no—
227Q8—1064no—
228Q8—1064no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—44no—
235C2×C4—534no—
236C2×C4—10616no—
237C2×C4—1064no—
238C2×C4—1064no—
239C2×C4—1064no—
240C2×C4—1062no—
241C2×C4—2122no—
242C2×C4—2122no—
243C2×C4#1313327412yes—
244C2×C4C2×C22⋊#615712yes—
245C2×C4C2×D53⋊C811yescharacteristic
246C2×C4D53⋊M1612yes—
247C2×C4C2×#1171993611yescharacteristic
248C8—4244no—
249C22—22no—
250C22—22no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22#1681156411yescharacteristic
255C22C2×#1313327411yescharacteristic
256C22#1681159511yescharacteristic
257C4—22no—
258C4—44no—
259C4—1064no—
260C4—1068no—
261C4—2122no—
262C4D53⋊(C22⋊C8)12yes—
263C4#1681159412yes—
264C2—21no—
265C2—21no—
266C2—42no—
267C2#2029053811yescharacteristic
268C2#2029056812yes—
subgroupG/H
centerC22#16811564
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC22×C53#379
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×#117#6157
hypocenterC53#8323
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #20290019C2semidirect
23. C22×Dic106C8semidirect
150. C53#8323semidirect

Maximal subgroups

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

subgroupindex
#202900192
#202899462
C2×#168111002
#832353