Finite Group Databasebeta

← #28310631

Subgroups — #28310631

Order 13568.

532 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C2×#16811531C212yesmaximal
2#20290468C212yesmaximal, complemented
3#20289812C212yesmaximal, complemented
4#20277262C211yesmaximal, characteristic
5C2×#13133278C2214yes—
6C4×#11719940—22no—
7#16811531C418yescomplemented
8#16811503—22no—
9C2×#13133204C2211yescharacteristic
10C2×#13133182C2211yescharacteristic
11C2×D4×Dic53C411yescharacteristic
12#16810529C2211yescharacteristic
13#16810511—22no—
14#16810488C411yescharacteristic
15C22⋊C4×Dic53—22no—
16C42×Dic53C411yescharacteristic
17C53×#390C411yescharacteristic
18C2×#11719940—24no—
19#13133278—24no—
20#13133278C2×C418yescomplemented
21#13133272—42no—
22C4×#88324—42no—
23C22×Dic106C2311yescharacteristic
24C23×Dic53—21no—
25#13133204—22no—
26#13133204C2×C414yes—
27#13133182—41no—
28#13133182—41no—
29D4×Dic53—41no—
30D4×Dic53—41no—
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×#11719919D412yes—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—22no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53C2×C411yescharacteristic
38C2×C4×Dic53C2×C411yescharacteristic
39C2×C4×Dic53D412yes—
40#13133124—22no—
41#13133124—22no—
42C22×D4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C22⋊C4×C53—22no—
45C2×C42×C53C2×C411yescharacteristic
46C24×C53—21no—
47C2×D4×C53—21no—
48C2×D4×C53—41no—
49C2×D4×C53—41no—
50C2×D4×C53C4212yes—
51C22×C4×C53—21no—
52C22×C4×C53C22⋊C412yes—
53C22×C4×C53C22×C411yescharacteristic
54C22×Dic53—21no—
55C22×Dic53—21no—
56C22×Dic53—44no—
57C22×Dic53—41no—
58C22×Dic53—41no—
59C22×Dic53C22⋊C411yescharacteristic
60C22×Dic53C22⋊C411yescharacteristic
61C22×Dic53C2×D412yes—
62C2×Dic106—41no—
63C2×Dic106—41no—
64C2×Dic106C22×C412yes—
65C2×Dic106#4112yes—
66#11719940—28no—
67#11719940—44no—
68C2×#88324—44no—
69C4⋊C4×C53—41no—
70C22⋊C4×C53—44no—
71C42×C53—41no—
72C22⋊Dic53—28no—
73C22⋊Dic53—42no—
74C4⋊Dic53—41no—
75#11719919—42no—
76C4×Dic53—21no—
77C4×Dic53—21no—
78C4×Dic53—21no—
79C4×Dic53—22no—
80C4×Dic53—22no—
81C4×Dic53—42no—
82C4×Dic53—42no—
83C4×Dic53C4212yes—
84C23×C53—22no—
85C23×C53—42no—
86C23×C53—42no—
87C23×C53—41no—
88C23×C53—41no—
89C23×C53C2×C22⋊C411yescharacteristic
90D4×C53—42no—
91D4×C53—42no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—42no—
99C2×C4×C53C2×C4211yescharacteristic
100C2×C4×C53#11711yescharacteristic
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—21no—
106C2×Dic53—41no—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53—42no—
110C2×Dic53—41no—
111C2×Dic53—82no—
112C2×Dic53—82no—
113C2×Dic53#9911yescharacteristic
114C2×Dic53#9911yescharacteristic
115C2×Dic53C4×D414yes—
116Dic106—42no—
117Dic106—42no—
118#88324—48no—
119C4×#355—531nomaximal, Sylow
120C22×C53—21no—
121C22×C53—21no—
122C22×C53—44no—
123C22×C53—42no—
124C22×C53—82no—
125C22×C53—81no—
126C22×C53—81no—
127C22×C53C4×C22⋊C411yescharacteristic
128C22×C53C2×#9911yescharacteristic
129C22×C53#41011yescharacteristic
130C4×C53—21no—
131C4×C53—21no—
132C4×C53—42no—
133C4×C53—42no—
134Dic53—21no—
135Dic53—24no—
136Dic53—42no—
137Dic53—41no—
138Dic53—81no—
139Dic53#35512yescomplemented
140C4×#124—531no—
141C2×#355—532no—
142C4×#101—532no—
143#1394—532no—
144C2×C53—21no—
145C2×C53—21no—
146C2×C53—82no—
147C2×C53C4×#9911yescharacteristic
148C2×C53C2×#35511yescharacteristic
149C2×C53#213211yescharacteristic
150C2×#124—531no—
151C2×C4×Q8—531no—
152C2×C4×D4—531no—
153C2×#101—534no—
154C4×M16—1062no—
155#391—531no—
156#390#615711yescharacteristic
157#388—1062no—
158#382—531no—
159C4×C22⋊C4—1062no—
160C43—531no—
161#355—538no—
162#343—1062no—
163C53C4×#35511yescharacteristic, complemented, Sylow
164C22×Q8—531no—
165C22×D4C2×#615711yescharacteristic
166C23×C4—1061no—
167C2×M16—1064no—
168#124—534no—
169#124—1062no—
170C4×Q8—2121no—
171C4×Q8—2121no—
172C4×D4—2121no—
173C4×D4—2121no—
174C2×C4⋊C4—532no—
175C2×C4⋊C4—531no—
176C2×C4⋊C4C2×#615711yescharacteristic
177C2×C22⋊C4—22no—
178C2×C22⋊C4—1062no—
179C2×C22⋊C4—1061no—
180C2×C42—531no—
181C2×C42—532no—
182C2×C42—531no—
183C2×C42—1061no—
184C2×C42—1061no—
185C2×C42C2×#615711yescharacteristic
186#101—538no—
187#101—1064no—
188#97—2122no—
189C4×C8—2122no—
190#95—1062no—
191#95—1062no—
192C24—21no—
193C2×Q8—532no—
194C2×Q8—532no—
195C2×Q8—2121no—
196C2×Q8—2121no—
197C2×D4—21no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4C4×#615712yes—
201C22×C4—21no—
202C22×C4—531no—
203C22×C4—532no—
204C22×C4—531no—
205C22×C4—1061no—
206C22×C4—1061no—
207C22×C4—2121no—
208C22×C4—2121no—
209C22×C4—2124no—
210C22×C4C22⋊#615712yes—
211C22×C4C22×#615711yescharacteristic
212M16—1068no—
213M16—2124no—
214C2×C8—2124no—
215C4⋊C4—41no—
216C4⋊C4—2122no—
217C4⋊C4—2121no—
218C22⋊C4—44no—
219C22⋊C4—1068no—
220C22⋊C4—2122no—
221C42—41no—
222C42—532no—
223C42—1062no—
224C42—1062no—
225C42—1061no—
226C42—1061no—
227C42—1061no—
228C42—2122no—
229C42—2122no—
230C23—22no—
231C23—42no—
232C23—42no—
233C23—41no—
234C23—41no—
235C23C2×C22⋊#615711yescharacteristic
236Q8—2122no—
237Q8—2122no—
238D4—42no—
239D4—42no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—531no—
248C2×C4—534no—
249C2×C4—531no—
250C2×C4—1062no—
251C2×C4—1062no—
252C2×C4—1062no—
253C2×C4—1062no—
254C2×C4—1061no—
255C2×C4—4242no—
256C2×C4—4242no—
257C2×C4—2122no—
258C2×C4—2121no—
259C2×C4—2121no—
260C2×C4—2121no—
261C2×C4—2121no—
262C2×C4C2×C4×#615711yescharacteristic
263C2×C4#1313329611yescharacteristic
264C8—2128no—
265C22—21no—
266C22—21no—
267C22—44no—
268C22—42no—
269C22—82no—
270C22—81no—
271C22—81no—
272C22C4×C22⋊#615711yescharacteristic
273C22D53⋊#9911yescharacteristic
274C22#1681160311yescharacteristic
275C4—21no—
276C4—21no—
277C4—42no—
278C4—42no—
279C4—532no—
280C4—1064no—
281C4—1061no—
282C4—4241no—
283C4—2122no—
284C4—2121no—
285C2—21no—
286C2—21no—
287C2—82no—
288C2#2029046211yescharacteristic
289C2D53⋊#35511yescharacteristic
290C2#2029100711yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC53×#390C4
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC22×Dic106C23
hypercenter#390#6157
hypocenterC53C4×#355
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #20290468C2semidirect
3. #20289812C2semidirect
7. #16811531C4semidirect
20. #13133278C2×C4semidirect
139. Dic53#355semidirect
163. C53C4×#355semidirect

Maximal subgroups

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

subgroupindex
C2×#168115312
#202904682
#202898122
#202772622
C4×#35553