Finite Group Databasebeta

← #28298634

Subgroups — #28298634

Order 13568.

344 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 66 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#20290574C211yesmaximal, characteristic
2#20290415C211yesmaximal, characteristic
3#20273028C211yesmaximal, characteristic
4C2×#13133296—21no—
5C2×#13133284C411yescharacteristic, complemented
6C2×C4⋊#88324C411yescharacteristic, complemented
7C2×#13133274—21no—
8#16811501—21no—
9#16811501C412yes—
10#16811489—21no—
11C2×#13133175C2211yescharacteristic
12#16810673—21no—
13#16810349C411yescharacteristic, complemented
14#16810346—21no—
15C53×#324C411yescharacteristic, complemented
16C22×#88324—21no—
17C22×#88324—21no—
18#13133296—22no—
19#13133296—22no—
20#13133296—41no—
21#13133296—41no—
22C2×C4⋊#6157—41no—
23C2×C22⋊#6157—22no—
24C2×C4×#6157—41no—
25#13133284—41no—
26C4⋊#88324—41no—
27#13133274—41no—
28#13133274—41no—
29C22×C4×D53D411yescharacteristic
30#13133175—41no—
31#13133175—41no—
32C2×C4⋊Dic53Q811yescharacteristic
33C2×#11719921C2×C411yescharacteristic
34C2×#11719921D411yescharacteristic
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53D411yescharacteristic
37#13133146—41no—
38C22⋊#88322—21no—
39C8×Dic53—41no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C22⋊C8×C53—21no—
42C23×D53C22⋊C411yescharacteristic
43C22×#6157—42no—
44C22×C4×C53C4211yescharacteristic
45C22×C4×C53C22⋊C411yescharacteristic
46C22×C4×C53C4⋊C411yescharacteristic
47C22×Dic53C22⋊C411yescharacteristic
48C22×Dic53C4⋊C411yescharacteristic
49C22×Dic53C4⋊C411yescharacteristic
50C2×C4×D53—21no—
51C2×C4×D53—21no—
52C2×C4×D53—21no—
53C2×C4×D53—21no—
54C2×C4×D53D811yescharacteristic
55C2×C4×D53SD1611yescharacteristic
56C2×C4×D53SD1611yescharacteristic
57C2×C4×D53Q1611yescharacteristic
58C22⋊#6157—24no—
59C22⋊#6157—42no—
60C2×#88324—41no—
61C2×#88324—41no—
62C2×#88324—41no—
63C2×#88324—41no—
64C2×#88324—41no—
65C2×#88324—41no—
66C4⋊#6157—42no—
67C4⋊#6157—42no—
68C4×#6157—42no—
69C4×#6157—42no—
70C2×C8×C53—41no—
71C4⋊C4×C53—41no—
72#11719921—41no—
73#11719921—41no—
74C4⋊Dic53—41no—
75C4×Dic53—21no—
76C4×Dic53—21no—
77C4×Dic53—21no—
78C4×Dic53—21no—
79C2×#88322—41no—
80C23×C53#9511yescharacteristic
81C22×D53—21no—
82C22×D53—21no—
83C22×D53—41no—
84C22×D53—41no—
85C22×D53#10211yescharacteristic
86C22×D53#10311yescharacteristic
87C2×#6157—44no—
88C2×#6157—82no—
89C2×#6157—82no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53#10611yescharacteristic
97C2×C4×C53#10711yescharacteristic
98C2×Dic53—21no—
99C2×Dic53—21no—
100C2×Dic53—41no—
101C2×Dic53—41no—
102C2×Dic53#9911yescharacteristic
103C2×Dic53#9911yescharacteristic
104C2×Dic53#10211yescharacteristic
105C2×Dic53#10311yescharacteristic
106C2×Dic53#10411yescharacteristic
107C2×Dic53#10411yescharacteristic
108C4×D53—21no—
109C4×D53—21no—
110C4×D53—21no—
111C4×D53—21no—
112C4×D53—21no—
113C4×D53—21no—
114C4×D53—21no—
115C4×D53—21no—
116#88324—81no—
117#88324—81no—
118C8×C53—81no—
119#88322—81no—
120#7252—531nomaximal, Sylow
121C22×C53—21no—
122C22×C53—21no—
123C22×C53#33911yescharacteristic
124C22×C53#34011yescharacteristic
125C22×C53#34211yescharacteristic
126C2×D53—21no—
127C2×D53—21no—
128C2×D53—41no—
129C2×D53—41no—
130C2×D53—41no—
131C2×D53—41no—
132C2×D53—81no—
133#6157—84no—
134C4×C53—21no—
135C4×C53—21no—
136C4×C53—41no—
137C4×C53—41no—
138Dic53—41no—
139Dic53—41no—
140Dic53#32711yescharacteristic
141Dic53#32711yescharacteristic
142Dic53#32811yescharacteristic
143Dic53#32811yescharacteristic
144#1798—531no—
145#1740—531no—
146#1458—531no—
147C2×C53—21no—
148C2×C53—21no—
149C2×C53#129411yescharacteristic
150C2×C53#139111yescharacteristic
151C2×C53#139511yescharacteristic
152D53—41no—
153D53—41no—
154C2×#117—531no—
155C2×#117—1061no—
156#431—1061no—
157C2×C4⋊C8—531no—
158C2×C4⋊C8—531no—
159C2×C22⋊C8—1061no—
160#342—532no—
161#342—1061no—
162#336—1061no—
163#324—531no—
164#324#615711yescharacteristic, complemented
165#323—1061no—
166C53#725211yescharacteristic, complemented, Sylow
167C23×C4—531no—
168C22×C8—1061no—
169C22×C8—1061no—
170#117—1062no—
171#117—1062no—
172#117—2121no—
173#117—2121no—
174#117—2121no—
175#117—2121no—
176C2×C4⋊C4—531no—
177C2×C4⋊C4—2121no—
178C2×C4⋊C4C2×#615711yescharacteristic
179C2×C22⋊C4—531no—
180C2×C22⋊C4—531no—
181C2×C22⋊C4—1062no—
182C2×C42—531no—
183C2×C42—531no—
184C2×C42—2121no—
185C4⋊C8—2121no—
186C4⋊C8—2121no—
187C4⋊C8—2121no—
188C22⋊C8—21no—
189C22⋊C8—1061no—
190C22⋊C8—2121no—
191C22⋊C8—2121no—
192C4×C8—2121no—
193C24—531no—
194C22×C4—531no—
195C22×C4—531no—
196C22×C4—531no—
197C22×C4—531no—
198C22×C4—531no—
199C22×C4—531no—
200C22×C4—531no—
201C22×C4—1061no—
202C22×C4—1061no—
203C22×C4—1061no—
204C22×C4—1061no—
205C22×C4—2122no—
206C22×C4C4×#615711yescharacteristic
207C22×C4C4⋊#615711yescharacteristic
208C22×C4C22⋊#615711yescharacteristic
209C2×C8—41no—
210C2×C8—2121no—
211C2×C8—2121no—
212C2×C8—2121no—
213C2×C8—2121no—
214C2×C8—2121no—
215C2×C8—2121no—
216C2×C8—2121no—
217C4⋊C4—41no—
218C4⋊C4—2121no—
219C4⋊C4—2122no—
220C4⋊C4—2122no—
221C22⋊C4—1064no—
222C22⋊C4—2121no—
223C22⋊C4—2121no—
224C22⋊C4—2122no—
225C42—1061no—
226C42—1061no—
227C42—1061no—
228C42—1061no—
229C42—2122no—
230C42—2122no—
231C23—531no—
232C23—531no—
233C23—1061no—
234C23—1061no—
235C23—2121no—
236C23—2121no—
237C23(C2×#6157)⋊C411yescharacteristic
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—531no—
245C2×C4—531no—
246C2×C4—531no—
247C2×C4—531no—
248C2×C4—531no—
249C2×C4—531no—
250C2×C4—1061no—
251C2×C4—1061no—
252C2×C4—1061no—
253C2×C4—1061no—
254C2×C4—1061no—
255C2×C4—1061no—
256C2×C4—1061no—
257C2×C4—1061no—
258C2×C4—1061no—
259C2×C4—1061no—
260C2×C4—4242no—
261C2×C4—4242no—
262C2×C4—2124no—
263C2×C4—2121no—
264C2×C4—2121no—
265C2×C4#1313328511yescharacteristic
266C2×C4#1313328611yescharacteristic
267C8—81no—
268C8—4241no—
269C8—4241no—
270C8—4241no—
271C22—21no—
272C22—21no—
273C22—1061no—
274C22—1061no—
275C22—4241no—
276C22—2121no—
277C22—2121no—
278C22—2121no—
279C22—2121no—
280C22#1681149511yescharacteristic
281C22#1681149711yescharacteristic
282C22#1681150011yescharacteristic
283C4—21no—
284C4—21no—
285C4—41no—
286C4—41no—
287C4—531no—
288C4—531no—
289C4—531no—
290C4—531no—
291C4—4244no—
292C4—2121no—
293C4—2121no—
294C2—21no—
295C2—21no—
296C2—2121no—
297C2—2121no—
298C2#2028959011yescharacteristic
299C2#2028980211yescharacteristic
300C2#2028981711yescharacteristic
subgroupG/H
centerC22#16811500
commutatorC22×C4×C53C42
FrattiniC2×C4⋊C4C2×#6157
FittingC53×#324C4
socleC22×C53#342
prime radicalC23×D53C22⋊C4
cosocleC2×#13133175C22
hypercenter#324#6157
hypocenterC53#7252
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #324

Complements

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

NH ≅ G/Ntype
5. C2×#13133284C4semidirect
6. C2×C4⋊#88324C4semidirect
13. #16810349C4semidirect
15. C53×#324C4semidirect
164. #324#6157semidirect
166. C53#7252semidirect

Maximal subgroups

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

subgroupindex
#202905742
#202904152
#202730282
#725253