Finite Group Databasebeta

← #28302985

Subgroups — #28302985

Order 13568.

540 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 108 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#20290500C212yesmaximal, complemented
2C2×#16811479C212yesmaximal
3#20289591C212yesmaximal, complemented
4#20277258C211yesmaximal, characteristic
5C2×#13133284—22no—
6C2×#13133272C2214yes—
7#16811489—22no—
8#16811479—24no—
9C2×#13133206C2211yescharacteristic
10C2×Q8×Dic53C411yescharacteristic
11C2×#13133179C411yescharacteristic
12#16810525C411yescharacteristic
13#16810515—22no—
14#16810488C2211yescharacteristic
15#16810470—22no—
16C42×Dic53C2211yescharacteristic
17C53×#391C411yescharacteristic
18C22×#88324—24no—
19#13133284—28no—
20#13133272—24no—
21#13133272—24no—
22C22×D212C2×C411yescharacteristic
23C22×C4×D53—21no—
24#13133206—22no—
25#13133206D412yes—
26#13133206Q812yes—
27Q8×Dic53—42no—
28#13133179—42no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719921—22no—
32C2×#11719921—21no—
33C2×#11719921—22no—
34C2×C4×Dic53—21no—
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53D412yes—
37C2×C4×Dic53D412yes—
38C2×C4×Dic53C2311yescharacteristic
39#13133124—22no—
40C22×Q8×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C42×C53C2×C411yescharacteristic
43#95×C53—22no—
44C23×D53—21no—
45C2×Q8×C53—21no—
46C2×Q8×C53—41no—
47C2×Q8×C53—41no—
48C2×Q8×C53C4⋊C412yes—
49C22×C4×C53—21no—
50C22×C4×C53—21no—
51C22×C4×C53C22⋊C412yes—
52C22×C4×C53C22×C411yescharacteristic
53C22×Dic53—21no—
54C22×Dic53C22⋊C411yescharacteristic
55C22×Dic53C22⋊C411yescharacteristic
56C22×Dic53C2×D412yes—
57C2×D212—21no—
58C2×D212—41no—
59C2×D212—41no—
60C2×D212C4⋊C412yes—
61C2×C4×D53—42no—
62C2×C4×D53—44no—
63C2×#88324—28no—
64C2×#88324—44no—
65C2×#88324—44no—
66C4⋊C4×C53—41no—
67C42×C53—41no—
68#11719921—24no—
69#11719921—24no—
70#11719921—42no—
71#11719921—44no—
72C4⋊Dic53—42no—
73C4⋊Dic53—41no—
74C4×Dic53—22no—
75C4×Dic53—24no—
76C4×Dic53—42no—
77C4×Dic53—42no—
78C4×Dic53C2×D411yescharacteristic
79C4×Dic53C2×Q811yescharacteristic
80C4×Dic53#4112yes—
81C23×C53C2×C22⋊C411yescharacteristic
82C22×D53—22no—
83C22×D53—41no—
84C22×D53—41no—
85C22×D53—42no—
86C22×D53—42no—
87Q8×C53—42no—
88Q8×C53—42no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—42no—
99C2×C4×C53C2×C4⋊C411yescharacteristic
100C2×C4×C53#11711yescharacteristic
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—21no—
104C2×Dic53—22no—
105C2×Dic53—41no—
106C2×Dic53—42no—
107C2×Dic53#10012yes—
108C2×Dic53#10414yes—
109C2×Dic53#12112yes—
110C2×Dic53C22⋊Q812yes—
111D212—42no—
112D212—42no—
113C4×D53—84no—
114#88324—48no—
115#8579—531nomaximal, Sylow
116C22×C53#38011yescharacteristic
117C22×C53C2×#10011yescharacteristic
118C22×C53#41311yescharacteristic
119C22×C53C2×#10412yes—
120C22×C53#42112yes—
121C2×D53—42no—
122C2×D53—44no—
123C2×D53—81no—
124C2×D53—81no—
125C2×D53—82no—
126C4×C53—21no—
127C4×C53—21no—
128C4×C53—42no—
129C4×C53—42no—
130C4×C53—81no—
131Dic53—22no—
132Dic53—22no—
133Dic53—41no—
134Dic53—81no—
135Dic53#32914yescomplemented
136C4×#124—531no—
137#1767—532no—
138C2×#329—532no—
139#1295—532no—
140C2×C53C2×#32911yescharacteristic
141C2×C53#152912yes—
142C2×C53#153611yescharacteristic
143C2×C53#180011yescharacteristic
144C2×C53#180612yes—
145D53—82no—
146C2×#124—531no—
147C2×C4×Q8—531no—
148C2×C4×D4—531no—
149C2×C4⋊C8—1062no—
150C2×#97—534no—
151#391#615711yescharacteristic
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×Q8C2×#615711yescharacteristic
161C22×D4—531no—
162C23×C4—1061no—
163C22×C8—1064no—
164#124—532no—
165#124—532no—
166#124—1062no—
167C4×Q8—2122no—
168C4×D4—2122no—
169C2×C4⋊C4—531no—
170C2×C4⋊C4—1061no—
171C2×C4⋊C4C2×#615711yescharacteristic
172C2×C22⋊C4—1062no—
173C2×C22⋊C4—1061no—
174C2×C22⋊C4—1062no—
175C2×C42—531no—
176C2×C42—532no—
177C2×C42—531no—
178C2×C42—532no—
179C2×C42—1061no—
180C2×C42C2×#615711yescharacteristic
181C4⋊C8—1068no—
182#97—1064no—
183#97—1064no—
184#95—22no—
185#95—1062no—
186C24—1061no—
187C2×Q8—21no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8C4⋊#615712yes—
191C2×D4—532no—
192C2×D4—1061no—
193C2×D4—2121no—
194C2×D4—2121no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—531no—
198C22×C4—532no—
199C22×C4—531no—
200C22×C4—1061no—
201C22×C4—2122no—
202C22×C4—2124no—
203C22×C4C22⋊#615712yes—
204C22×C4C22×#615711yescharacteristic
205C2×C8—1068no—
206C2×C8—2124no—
207C2×C8—2124no—
208C4⋊C4—41no—
209C4⋊C4—2121no—
210C4⋊C4—2122no—
211C22⋊C4—1064no—
212C22⋊C4—1064no—
213C22⋊C4—2124no—
214C22⋊C4—2122no—
215C42—41no—
216C42—531no—
217C42—531no—
218C42—532no—
219C42—1064no—
220C42—1062no—
221C42—2122no—
222C42—2122no—
223C23—1062no—
224C23—2121no—
225C23—2121no—
226C23—2122no—
227C23—2122no—
228C23C2×C22⋊#615711yescharacteristic
229Q8—42no—
230Q8—42no—
231D4—2122no—
232D4—2122no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—532no—
244C2×C4—534no—
245C2×C4—532no—
246C2×C4—532no—
247C2×C4—1062no—
248C2×C4—1062no—
249C2×C4—1061no—
250C2×C4—1062no—
251C2×C4—4244no—
252C2×C4—2121no—
253C2×C4—2122no—
254C2×C4C2×C4⋊#615711yescharacteristic
255C2×C4#1313329611yescharacteristic
256C8—2128no—
257C22—4241no—
258C22—4242no—
259C22—4241no—
260C22—2122no—
261C22—2124no—
262C22#1681156511yescharacteristic
263C22D53⋊#10011yescharacteristic
264C22#1681161011yescharacteristic
265C22D53⋊#10412yes—
266C22#1681162212yes—
267C4—21no—
268C4—21no—
269C4—42no—
270C4—42no—
271C4—81no—
272C4—534no—
273C4—1062no—
274C4—1062no—
275C4—4241no—
276C4—2121no—
277C2—4242no—
278C2D53⋊#32911yescharacteristic
279C2#2029018312yes—
280C2#2029020511yescharacteristic
281C2#2029058011yescharacteristic
282C2#2029059012yes—
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#391C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C4×Dic53C23
hypercenter#391#6157
hypocenterC53#8579
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
1. #20290500C2semidirect
3. #20289591C2semidirect
135. Dic53#329semidirect
159. C53#8579semidirect

Maximal subgroups

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

subgroupindex
#202905002
C2×#168114792
#202895912
#202772582
#857953