Finite Group Databasebeta

← #28302273

Subgroups — #28302273

Order 13568.

930 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 164 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#20290017C214yesmaximal, complemented
2#20289946C212yesmaximal
3C2×#16811101C211yesmaximal, characteristic
4#16811639C2214yes—
5Q8⋊#88324—28no—
6#16811468—22no—
7#16811466—22no—
8#16811101—28no—
9C2×Q8×Dic53—22no—
10C2×Q8×Dic53C412yes—
11C2×Q8×Dic53C2212yes—
12C2×#13133176—22no—
13C2×#13133176C411yescharacteristic
14C2×#13133176C2211yescharacteristic
15C23×Q8×C53C411yescharacteristic
16C4⋊#88324—24no—
17C22⋊#88324—24no—
18C4×#88324—24no—
19Q8×Dic53—216no—
20Q8×Dic53—24no—
21Q8×Dic53—24no—
22Q8×Dic53—24no—
23Q8×Dic53D418yes—
24#13133176—216no—
25#13133176—24no—
26#13133176—24no—
27C2×C4⋊Dic53—22no—
28C2×C4⋊Dic53—22no—
29C2×C4⋊Dic53C2×C412yes—
30C2×C4⋊Dic53D412yes—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53D412yes—
33C2×C4×Dic53—21no—
34C2×C4×Dic53—22no—
35C2×C4×Dic53—22no—
36C2×C4×Dic53C2×C411yescharacteristic
37C2×C4×Dic53C2311yescharacteristic
38C22×Q8×C53—24no—
39C22×Q8×C53—24no—
40C22×Q8×C53—21no—
41C22×Q8×C53—22no—
42C22×Q8×C53C814yescomplemented
43C22×Q8×C53C2×C412yes—
44C23×C4×C53—21no—
45C23×C4×C53C2×C411yescharacteristic
46C24×C53C22⋊C411yescharacteristic
47C2×Q8×C53—216no—
48C2×Q8×C53—24no—
49C2×Q8×C53—24no—
50C2×Q8×C53—216no—
51C2×Q8×C53—24no—
52C2×Q8×C53—22no—
53C2×Q8×C53—22no—
54C2×Q8×C53—22no—
55C2×Q8×C53C22⋊C414yes—
56C2×Q8×C53C2×C814yes—
57C2×Q8×C53M1614yes—
58C22×C4×C53—24no—
59C22×C4×C53—24no—
60C22×C4×C53—24no—
61C22×C4×C53—22no—
62C22×C4×C53—22no—
63C22×C4×C53—22no—
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C2×C812yes—
66C22×C4×C53M1612yes—
67C22×C4×C53C22×C411yescharacteristic
68C22×Dic53—22no—
69C22×Dic53C22⋊C412yes—
70C22×Dic53C2×D412yes—
71C2×#88324—44no—
72C22⋊Dic53—24no—
73C22⋊Dic53—24no—
74C4⋊Dic53—28no—
75C4⋊Dic53—22no—
76C4⋊Dic53—28no—
77C4⋊Dic53—24no—
78C4⋊Dic53C22⋊C414yes—
79C4×Dic53—24no—
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
89Q8×C53—216no—
90Q8×C53—24no—
91Q8×C53—24no—
92Q8×C53—24no—
93Q8×C53C22⋊C818yes—
94C2×C4×C53—216no—
95C2×C4×C53—24no—
96C2×C4×C53—21no—
97C2×C4×C53—24no—
98C2×C4×C53—24no—
99C2×C4×C53—21no—
100C2×C4×C53—24no—
101C2×C4×C53—24no—
102C2×C4×C53C22⋊C812yes—
103C2×C4×C53C2×C22⋊C412yes—
104C2×C4×C53C22×C811yescharacteristic
105C2×C4×C53C2×M1612yes—
106C2×C4×C53C2×M1611yescharacteristic
107C2×Dic53—24no—
108C2×Dic53—22no—
109C2×Dic53—42no—
110C2×Dic53—44no—
111C2×Dic53—42no—
112C2×Dic53#12014yes—
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—28no—
124C4×C53—22no—
125C4×C53C2×C22⋊C812yes—
126C4×C53#40712yes—
127Dic53—24no—
128Dic53—42no—
129Dic53—44no—
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#615711yescharacteristic
139#519—1068no—
140C2×C4×Q8—532no—
141C2×C4×Q8—532no—
142C2×C4×Q8—1062no—
143C2×#117—531no—
144C2×#117—531no—
145C2×#117—1062no—
146#431—534no—
147Q8⋊C8—1068no—
148#324—1062no—
149#323—1062no—
150C53#832311yescharacteristic, complemented, Sylow
151C22×Q8—24no—
152C22×Q8—24no—
153C22×Q8—22no—
154C22×Q8—21no—
155C22×Q8#8832414yescomplemented
156C22×Q8C2×#615712yes—
157C23×C4—21no—
158C23×C4C2×#615711yescharacteristic
159C4×Q8—538no—
160C4×Q8—1064no—
161C4×Q8—1064no—
162C4×Q8—10616no—
163C4×Q8—1064no—
164#117—1064no—
165#117—1064no—
166#117—10616no—
167C2×C4⋊C4—532no—
168C2×C4⋊C4—532no—
169C2×C4⋊C4—1062no—
170C2×C4⋊C4—1062no—
171C2×C22⋊C4—532no—
172C2×C22⋊C4—1061no—
173C2×C42—531no—
174C2×C42—531no—
175C2×C42—1062no—
176C2×C42—1062no—
177C2×C42—1061no—
178C4⋊C8—1064no—
179C22⋊C8—1064no—
180C4×C8—1064no—
181C24C22⋊#615711yescharacteristic
182C2×Q8—24no—
183C2×Q8—216no—
184C2×Q8—24no—
185C2×Q8—216no—
186C2×Q8—24no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8C2×#8832414yes—
191C2×Q8#1171994014yes—
192C2×Q8C22⋊#615714yes—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—532no—
200C22×C4—532no—
201C22×C4—1062no—
202C22×C4C2×#8832412yes—
203C22×C4#1171994012yes—
204C22×C4C22⋊#615711yescharacteristic
205C22×C4C22×#615711yescharacteristic
206C2×C8—2124no—
207C4⋊C4—534no—
208C4⋊C4—1064no—
209C4⋊C4—1068no—
210C4⋊C4—1068no—
211C4⋊C4—1062no—
212C22⋊C4—1064no—
213C22⋊C4—1064no—
214C42—534no—
215C42—1068no—
216C42—1068no—
217C42—1064no—
218C42—1062no—
219C23—21no—
220C23—21no—
221C23—42no—
222C23C22⋊#8832412yes—
223C23C2×C22⋊#615711yescharacteristic
224Q8—216no—
225Q8—24no—
226Q8—24no—
227Q8—24no—
228Q8C22⋊#8832418yes—
229C2×C4—216no—
230C2×C4—24no—
231C2×C4—21no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—21no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—534no—
238C2×C4—1064no—
239C2×C4—1062no—
240C2×C4—2122no—
241C2×C4—2124no—
242C2×C4—2122no—
243C2×C4C22⋊#8832412yes—
244C2×C4C2×C22⋊#615712yes—
245C2×C4C22×#8832411yescharacteristic
246C2×C4C2×#1171994012yes—
247C2×C4C2×#1171994011yescharacteristic
248C8—4244no—
249C22—22no—
250C22—22no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22#1681156411yescharacteristic
255C22C2×C22⋊#8832411yescharacteristic
256C22#1681159311yescharacteristic
257C4—28no—
258C4—22no—
259C4—1064no—
260C4—2124no—
261C4—2122no—
262C4C2×C22⋊#8832412yes—
263C4#1681159312yes—
264C2—21no—
265C2—21no—
266C2—42no—
267C2#2029053711yescharacteristic
268C2#2029056712yes—
subgroupG/H
centerC22#16811564
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC23×Q8×C53C4
socleC22×C53#379
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC23×Q8#6157
hypocenterC53#8323
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. #20290017C2semidirect
42. C22×Q8×C53C8semidirect
150. C53#8323semidirect
155. C22×Q8#88324semidirect

Maximal subgroups

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

subgroupindex
#202900172
#202899462
C2×#168111012
#832353