Finite Group Databasebeta

← #27293272

Subgroups — #27293272

Order 13568.

434 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 115 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#20277335C211yesmaximal, characteristic
2#20273029C211yesmaximal, characteristic
3#20272662C214yesmaximal
4C53×#1458C211yesmaximal, characteristic
5C2×#13133176C411yescharacteristic
6C22×C4⋊Dic53C411yescharacteristic
7#16810676—21no—
8#16810521—21no—
9#16810521—21no—
10#16810495C412yes—
11#16810495C2212yes—
12#16810492—21no—
13#16810488—21no—
14#16810349—21no—
15#16810347C2212yes—
16C2×#117×C53C2211yescharacteristic
17C53×#431—21no—
18C53×#324—21no—
19C53×#323C2212yes—
20C23×Dic53C2×C412yes—
21#13133176—22no—
22#13133176C814yescomplemented
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—41no—
27C2×C4⋊Dic53—41no—
28C2×#11719919—22no—
29C2×#11719919—22no—
30C2×C22⋊Dic53C2×C412yes—
31C2×C4×Dic53—21no—
32C4⋊#88322—41no—
33C22⋊#88322—22no—
34C4×#88322—41no—
35#13133124—22no—
36#13133124—22no—
37C23×C4×C53C2×C411yescharacteristic
38#117×C53—41no—
39#117×C53—41no—
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C22⋊C4×C53C2×C411yescharacteristic
43C2×C22⋊C4×C53C2311yescharacteristic
44C2×C42×C53D411yescharacteristic
45C2×C42×C53D411yescharacteristic
46C4⋊C8×C53—41no—
47C22⋊C8×C53—22no—
48C4×C8×C53—41no—
49C24×C53C22×C411yescharacteristic
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C2×C812yes—
58C22×C4×C53M1612yes—
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×Dic53—22no—
62C22×Dic53—22no—
63C22×Dic53—42no—
64C22×Dic53—82no—
65C2×C8×C53—42no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—41no—
68C22⋊C4×C53—41no—
69C22⋊C4×C53—41no—
70C42×C53—21no—
71C42×C53—21no—
72C42×C53—21no—
73C42×C53—21no—
74C22⋊Dic53—22no—
75C22⋊Dic53C2×C814yes—
76C4⋊Dic53—24no—
77C4⋊Dic53—42no—
78C4⋊Dic53—41no—
79C4⋊Dic53—41no—
80#11719919—82no—
81#11719919—82no—
82C4×Dic53—24no—
83C2×#88322—42no—
84C23×C53—21no—
85C23×C53—21no—
86C23×C53—41no—
87C23×C53—41no—
88C23×C53C2×C22⋊C411yescharacteristic
89C23×C53C22×C811yescharacteristic
90C23×C53C2×M1611yescharacteristic
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53C22⋊C812yes—
106C2×C4×C53C22⋊C812yes—
107C2×C4×C53#9911yescharacteristic
108C2×C4×C53#9911yescharacteristic
109C2×C4×C53#10112yes—
110C2×Dic53—24no—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C2×Dic53—42no—
114C2×Dic53—42no—
115C2×Dic53—42no—
116C2×Dic53—84no—
117C8×C53—82no—
118#88322—82no—
119#11454—531nomaximal, Sylow
120C22×C53—21no—
121C22×C53—21no—
122C22×C53—21no—
123C22×C53—21no—
124C22×C53—41no—
125C22×C53—41no—
126C22×C53—42no—
127C22×C53—41no—
128C22×C53—41no—
129C22×C53C2×C22⋊C811yescharacteristic
130C22×C53C2×#9911yescharacteristic
131C22×C53C2×#10111yescharacteristic
132C4×C53—22no—
133C4×C53—41no—
134C4×C53—41no—
135C4×C53—41no—
136C4×C53—41no—
137C4×C53#32412yes—
138C4×C53#32412yes—
139Dic53—44no—
140Dic53—82no—
141#2317—531no—
142#1458—531no—
143#1458D5311yescharacteristic
144#1326—534no—
145C2×C53—21no—
146C2×C53—21no—
147C2×C53—41no—
148C2×C53—41no—
149C2×C53C2×#32411yescharacteristic
150C2×C53#212211yescharacteristic
151C2×C53#212311yescharacteristic
152C2×#117—531no—
153C2×#117C2×D5311yescharacteristic
154C22×C4⋊C4—531no—
155#431—21no—
156#431—1061no—
157#389—1061no—
158#389—1061no—
159#385—532no—
160#385—532no—
161#384—1061no—
162#382—1061no—
163#324—21no—
164#324—1061no—
165#323—532no—
166#323C2×D5312yes—
167C53#1145411yescharacteristic, complemented, Sylow
168C23×C4—532no—
169C23×C4C4×D5311yescharacteristic
170#117—41no—
171#117—41no—
172#117—534no—
173#117—1062no—
174C2×C4⋊C4—1061no—
175C2×C4⋊C4—1062no—
176C2×C4⋊C4—1062no—
177C2×C4⋊C4—1061no—
178C2×C4⋊C4—1061no—
179C2×C4⋊C4—2121no—
180C2×C4⋊C4—2121no—
181C2×C4⋊C4D21211yescharacteristic
182C2×C4⋊C4C22⋊D5311yescharacteristic
183C2×C22⋊C4—532no—
184C2×C22⋊C4C4×D5311yescharacteristic
185C2×C22⋊C4C22×D5311yescharacteristic
186C2×C42—1061no—
187C2×C42D21211yescharacteristic
188C2×C42C22⋊D5311yescharacteristic
189C4⋊C8—41no—
190C4⋊C8—2121no—
191C22⋊C8—22no—
192C22⋊C8—1062no—
193C4×C8—41no—
194C4×C8—2121no—
195#95—1062no—
196#95—1062no—
197C24C2×C4×D5311yescharacteristic
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—1062no—
202C22×C4—1062no—
203C22×C4—4242no—
204C22×C4—2122no—
205C22×C4C8×D5312yes—
206C22×C4#1171991212yes—
207C22×C4#1171992111yescharacteristic
208C22×C4#1171992111yescharacteristic
209C22×C4#1171992111yescharacteristic
210C22×C4#1171992111yescharacteristic
211C22×C4C2×D21211yescharacteristic
212C22×C4C2×C22⋊D5311yescharacteristic
213C2×C8—42no—
214C2×C8—2122no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—1064no—
218C4⋊C4—4242no—
219C4⋊C4—4242no—
220C4⋊C4—2122no—
221C4⋊C4—2121no—
222C4⋊C4—2121no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—534no—
226C22⋊C4—1062no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—1064no—
232C23—21no—
233C23—21no—
234C23—41no—
235C23—41no—
236C23C2×#1171992111yescharacteristic
237C23C2×C8×D5311yescharacteristic
238C23C2×#1171991211yescharacteristic
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—1064no—
254C2×C4—4244no—
255C2×C4—2122no—
256C2×C4—2122no—
257C2×C4—2122no—
258C2×C4—2122no—
259C2×C4—2122no—
260C2×C4#1313312912yes—
261C2×C4#1313312912yes—
262C2×C4#1313313111yescharacteristic
263C2×C4#1313313111yescharacteristic
264C2×C4#1313313512yes—
265C8—82no—
266C8—4242no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—42no—
274C22—41no—
275C22—41no—
276C22C2×#1313312911yescharacteristic
277C22C2×#1313313111yescharacteristic
278C22C2×#1313313511yescharacteristic
279C4—22no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—4242no—
285C4—2124no—
286C4#1681034812yes—
287C4#1681034812yes—
288C2—21no—
289C2—21no—
290C2—41no—
291C2—41no—
292C2C2×#1681034811yescharacteristic
293C2#2027636411yescharacteristic
294C2#2027637011yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC23×C53C22×C8
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1458C2
socleC22×C53C2×#99
prime radicalC24×C53C22×C4
cosocleC2×C22⋊C4×C53C23
hypercenter#1458D53
hypocenterC53#11454
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
22. #13133176C8semidirect
167. C53#11454semidirect

Maximal subgroups

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

subgroupindex
#202773352
#202730292
#202726622
C53×#14582
#1145453