Finite Group Databasebeta

← #28302982

Subgroups — #28302982

Order 13568.

540 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 118 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#20290501C212yesmaximal, complemented
2C2×#16811478C212yesmaximal
3#20289591C212yesmaximal, complemented
4C4×#13133204C211yesmaximal, characteristic
5C2×C4⋊#88324—22no—
6C2×#13133272C2214yes—
7#16811489—22no—
8#16811478—24no—
9C2×#13133204C2211yescharacteristic
10C2×C4×Dic106C411yescharacteristic
11#16810528C411yescharacteristic
12#16810526C411yescharacteristic
13#16810517—22no—
14#16810488C2211yescharacteristic
15C4×C22⋊Dic53—22no—
16C42×Dic53C2211yescharacteristic
17C2×C4×D4×C53C411yescharacteristic
18C22×#88324—24no—
19C4⋊#88324—28no—
20#13133272—24no—
21#13133272—24no—
22C22×Dic106C2×C411yescharacteristic
23#13133204—22no—
24#13133204D412yes—
25#13133204Q812yes—
26C4×Dic106—42no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—21no—
29C2×C22⋊Dic53—22no—
30C2×C22⋊Dic53—22no—
31C2×C4×Dic53—21no—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53D412yes—
34C2×C4×Dic53D412yes—
35C2×C4×Dic53C2311yescharacteristic
36#13133124—22no—
37#13133124—22no—
38C22×D4×C53C2×C411yescharacteristic
39C23×C4×C53—21no—
40C4×D4×C53—42no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C22⋊C4×C53—21no—
43C2×C42×C53C2×C411yescharacteristic
44C24×C53—21no—
45C2×D4×C53—21no—
46C2×D4×C53—41no—
47C2×D4×C53—41no—
48C2×D4×C53C4⋊C412yes—
49C22×C4×C53—21no—
50C22×C4×C53—42no—
51C22×C4×C53—44no—
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22×C411yescharacteristic
55C22×Dic53—21no—
56C22×Dic53—21no—
57C22×Dic53C22⋊C412yes—
58C22×Dic53C2×D412yes—
59C2×Dic106—21no—
60C2×Dic106—41no—
61C2×Dic106—41no—
62C2×Dic106C4⋊C412yes—
63C2×#88324—28no—
64C2×#88324—44no—
65C2×#88324—44no—
66C4⋊C4×C53—41no—
67C22⋊C4×C53—42no—
68C42×C53—22no—
69C22⋊Dic53—24no—
70C22⋊Dic53—24no—
71C22⋊Dic53—44no—
72C4⋊Dic53—41no—
73#11719919—42no—
74C4×Dic53—24no—
75C4×Dic53—41no—
76C4×Dic53—42no—
77C4×Dic53—42no—
78C4×Dic53C2×D411yescharacteristic
79C4×Dic53C2×Q811yescharacteristic
80C4×Dic53#4112yes—
81C23×C53—22no—
82C23×C53—41no—
83C23×C53—41no—
84C23×C53—42no—
85C23×C53—42no—
86C23×C53C2×C22⋊C411yescharacteristic
87D4×C53—42no—
88D4×C53—42no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—22no—
93C2×C4×C53—41no—
94C2×C4×C53—42no—
95C2×C4×C53—84no—
96C2×C4×C53#10012yes—
97C2×C4×C53#10414yes—
98C2×C4×C53C2×C4⋊C411yescharacteristic
99C2×C4×C53#11711yescharacteristic
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—41no—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53—42no—
110C2×Dic53#12112yes—
111C2×Dic53C22⋊Q812yes—
112Dic106—42no—
113Dic106—42no—
114#88324—48no—
115#8579—531nomaximal, Sylow
116C22×C53—42no—
117C22×C53—44no—
118C22×C53—81no—
119C22×C53—81no—
120C22×C53—82no—
121C22×C53#38011yescharacteristic
122C22×C53C2×#10011yescharacteristic
123C22×C53#41311yescharacteristic
124C22×C53C2×#10412yes—
125C22×C53#42112yes—
126C4×C53—21no—
127C4×C53—21no—
128C4×C53—41no—
129C4×C53—81no—
130C4×C53#32914yescomplemented
131Dic53—22no—
132Dic53—22no—
133Dic53—42no—
134Dic53—42no—
135Dic53—81no—
136C4×#124—531no—
137#1767—532no—
138C2×#329—532no—
139#1295—532no—
140C2×C53—82no—
141C2×C53C2×#32911yescharacteristic
142C2×C53#152912yes—
143C2×C53#153611yescharacteristic
144C2×C53#180011yescharacteristic
145C2×C53#180612yes—
146C2×#124—531no—
147C2×C4×Q8—531no—
148C2×C4×D4#615711yescharacteristic
149C2×C4⋊C8—1062no—
150C2×#97—534no—
151#391—531no—
152#390—531no—
153#388—1062no—
154#382—531no—
155C4×C22⋊C4—1062no—
156C43—531no—
157#336—1062no—
158#329—1064no—
159C53#857911yescharacteristic, complemented, Sylow
160C22×Q8—531no—
161C22×D4C2×#615711yescharacteristic
162C23×C4—21no—
163C22×C8—1064no—
164#124—532no—
165#124—532no—
166#124—1062no—
167C4×Q8—2122no—
168C4×D4—42no—
169C2×C4⋊C4—531no—
170C2×C4⋊C4—1061no—
171C2×C4⋊C4C2×#615711yescharacteristic
172C2×C22⋊C4—21no—
173C2×C22⋊C4—1062no—
174C2×C22⋊C4—1062no—
175C2×C42—531no—
176C2×C42—532no—
177C2×C42—532no—
178C2×C42—531no—
179C2×C42—1061no—
180C2×C42C2×#615711yescharacteristic
181C4⋊C8—1068no—
182#97—1064no—
183#97—1064no—
184#95—1062no—
185#95—1062no—
186C24—21no—
187C2×Q8—532no—
188C2×Q8—1061no—
189C2×Q8—2121no—
190C2×Q8—2121no—
191C2×D4—21no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4C4⋊#615712yes—
195C22×C4—21no—
196C22×C4—42no—
197C22×C4—44no—
198C22×C4—532no—
199C22×C4—532no—
200C22×C4—1061no—
201C22×C4—1061no—
202C22×C4C22⋊#615711yescharacteristic
203C22×C4C22⋊#615711yescharacteristic
204C22×C4C22×#615711yescharacteristic
205C2×C8—1068no—
206C2×C8—2124no—
207C2×C8—2124no—
208C4⋊C4—41no—
209C4⋊C4—2121no—
210C4⋊C4—2122no—
211C22⋊C4—42no—
212C22⋊C4—1064no—
213C22⋊C4—1064no—
214C22⋊C4—2124no—
215C42—22no—
216C42—531no—
217C42—531no—
218C42—532no—
219C42—1064no—
220C42—2121no—
221C42—2122no—
222C42—2122no—
223C23—22no—
224C23—41no—
225C23—41no—
226C23—42no—
227C23—42no—
228C23C2×C22⋊#615711yescharacteristic
229Q8—2122no—
230Q8—2122no—
231D4—42no—
232D4—42no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—84no—
240C2×C4—532no—
241C2×C4—532no—
242C2×C4—1062no—
243C2×C4—1062no—
244C2×C4—1062no—
245C2×C4—1062no—
246C2×C4—1062no—
247C2×C4—1062no—
248C2×C4—2121no—
249C2×C4—2122no—
250C2×C4—2121no—
251C2×C4—2121no—
252C2×C4#1313327712yes—
253C2×C4#1313328114yes—
254C2×C4C2×C4⋊#615711yescharacteristic
255C2×C4#1313329611yescharacteristic
256C8—2128no—
257C22—42no—
258C22—44no—
259C22—81no—
260C22—82no—
261C22—81no—
262C22#1681156511yescharacteristic
263C22C2×#1313327711yescharacteristic
264C22#1681160911yescharacteristic
265C22C2×#1313328112yes—
266C22#1681162012yes—
267C4—21no—
268C4—21no—
269C4—41no—
270C4—81no—
271C4—1062no—
272C4—1062no—
273C4—4241no—
274C4—2122no—
275C4—2122no—
276C4#1681147814yescomplemented
277C2—82no—
278C2C2×#1681147811yescharacteristic
279C2#2029018012yes—
280C2#2029020311yescharacteristic
281C2#2029057911yescharacteristic
282C2#2029058712yes—
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4×D4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D4×C53C2×C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4×D4#6157
hypocenterC53#8579
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #20290501C2semidirect
3. #20289591C2semidirect
130. C4×C53#329semidirect
159. C53#8579semidirect
276. C4#16811478semidirect

Maximal subgroups

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

subgroupindex
#202905012
C2×#168114782
#202895912
C4×#131332042
#857953