Finite Group Databasebeta

← #27271405

Subgroups — #27271405

Order 13568.

338 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 71 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#20275514C211yesmaximal, characteristic
2#20274703C211yesmaximal, characteristic
3C53×#1465C211yesmaximal, characteristic
4C2×C4⋊#88322C411yescharacteristic, complemented
5C2×#13133132C412yes—
6C2×C22⋊#88322—21no—
7C2×C22⋊#88322—21no—
8C2×#13133128C411yescharacteristic, complemented
9#16810483—21no—
10#16810390—21no—
11C2×C4×D4×C53C2211yescharacteristic
12C53×#407—21no—
13C53×#324C412yescomplemented
14C53×#323—21no—
15C22×#88322—21no—
16C22×#88322—21no—
17C2×C4⋊Dic53—41no—
18C2×C22⋊Dic53—22no—
19C4⋊#88322—41no—
20#13133132—42no—
21C22⋊#88322—42no—
22C22⋊#88322—42no—
23#13133128—41no—
24#13133124—41no—
25C22×D4×C53C2×C411yescharacteristic
26C23×C4×C53D411yescharacteristic
27C23×C4×C53D411yescharacteristic
28C2×M16×C53—41no—
29C4×D4×C53—41no—
30C4×D4×C53—41no—
31C2×C4⋊C4×C53C2×C411yescharacteristic
32C2×C22⋊C4×C53D411yescharacteristic
33C2×C22⋊C4×C53Q811yescharacteristic
34C2×C42×C53C2×C411yescharacteristic
35C22⋊C8×C53—22no—
36C22⋊C8×C53—41no—
37C24×C53C22⋊C411yescharacteristic
38C24×C53C4⋊C411yescharacteristic
39C2×D4×C53—21no—
40C2×D4×C53—21no—
41C2×D4×C53—41no—
42C2×D4×C53—41no—
43C22×C4×C53—21no—
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53—41no—
48C22×C4×C53—41no—
49C22×C4×C53—41no—
50C22×C4×C53—41no—
51C22×C4×C53C4211yescharacteristic
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C4⋊C411yescharacteristic
55C22×C4×C53C4⋊C411yescharacteristic
56C22×Dic53—42no—
57M16×C53—81no—
58M16×C53—81no—
59C2×C8×C53—42no—
60C4⋊C4×C53—41no—
61C22⋊C4×C53—41no—
62C22⋊C4×C53—41no—
63C42×C53—21no—
64C42×C53—21no—
65C22⋊Dic53—42no—
66C22⋊Dic53—42no—
67C4⋊Dic53—81no—
68C4⋊Dic53—81no—
69C2×#88322—22no—
70C2×#88322—41no—
71C2×#88322—41no—
72C2×#88322—41no—
73C2×#88322—41no—
74C2×#88322—41no—
75C23×C53—21no—
76C23×C53—21no—
77C23×C53—21no—
78C23×C53—21no—
79C23×C53—21no—
80C23×C53—41no—
81C23×C53—41no—
82C23×C53—41no—
83C23×C53—41no—
84C23×C53#9511yescharacteristic
85C23×C53#10011yescharacteristic
86C23×C53#10111yescharacteristic
87D4×C53—42no—
88D4×C53—42no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—81no—
105C2×C4×C53—81no—
106C2×C4×C53#9911yescharacteristic
107C2×C4×C53#9911yescharacteristic
108C2×Dic53—82no—
109C2×Dic53—82no—
110C2×Dic53—82no—
111C8×C53—82no—
112#88322—42no—
113#88322—81no—
114#7322—531nomaximal, Sylow
115C22×C53—21no—
116C22×C53—21no—
117C22×C53—21no—
118C22×C53—21no—
119C22×C53—41no—
120C22×C53—41no—
121C22×C53—42no—
122C22×C53—41no—
123C22×C53—41no—
124C22×C53—41no—
125C22×C53—41no—
126C22×C53—41no—
127C22×C53—81no—
128C22×C53—81no—
129C22×C53#34211yescharacteristic
130C22×C53#34311yescharacteristic
131C22×C53#34411yescharacteristic
132C4×C53—21no—
133C4×C53—41no—
134C4×C53—41no—
135C4×C53—41no—
136C4×C53—41no—
137C4×C53#35311yescharacteristic
138C4×C53#35411yescharacteristic
139Dic53—162no—
140#1955—531no—
141#1799—531no—
142#1465D5311yescharacteristic
143C2×C53—21no—
144C2×C53—21no—
145C2×C53—41no—
146C2×C53—41no—
147C2×C53—81no—
148C2×C53#130511yescharacteristic
149C2×C53#139311yescharacteristic
150C2×C53#139511yescharacteristic
151C2×C4×D4C2×D5311yescharacteristic
152C2×C4⋊C8—531no—
153C2×#99—532no—
154#407—21no—
155C2×C22⋊C8—1061no—
156C2×C22⋊C8—1061no—
157C2×#97—531no—
158#380—1061no—
159#342—1061no—
160#324Dic5312yescomplemented
161#323—21no—
162C53#732211yescharacteristic, complemented, Sylow
163C22×D4C4×D5311yescharacteristic
164C23×C4D21211yescharacteristic
165C23×C4C22⋊D5311yescharacteristic
166C2×M16—41no—
167C22×C8—1061no—
168C22×C8—1061no—
169C4×D4—41no—
170C4×D4—41no—
171C2×C4⋊C4—2121no—
172C2×C4⋊C4C2×Dic5311yescharacteristic
173C2×C22⋊C4—1062no—
174C2×C22⋊C4Dic10611yescharacteristic
175C2×C22⋊C4C22⋊D5311yescharacteristic
176C2×C42C4×D5311yescharacteristic
177C4⋊C8—2121no—
178#99—2122no—
179C22⋊C8—22no—
180C22⋊C8—41no—
181C22⋊C8—2122no—
182C22⋊C8—2122no—
183#97—2121no—
184#95—2121no—
185C24#1171991911yescharacteristic
186C24#1171992111yescharacteristic
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—41no—
190C2×D4—41no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—2122no—
200C22×C4C4×Dic5311yescharacteristic
201C22×C4#1171991911yescharacteristic
202C22×C4C4⋊Dic5311yescharacteristic
203C22×C4#1171992111yescharacteristic
204C22×C4C22⋊Dic5311yescharacteristic
205M16—81no—
206M16—81no—
207C2×C8—42no—
208C2×C8—1062no—
209C2×C8—2121no—
210C2×C8—2121no—
211C2×C8—2121no—
212C2×C8—2121no—
213C2×C8—2121no—
214C4⋊C4—41no—
215C4⋊C4—4241no—
216C4⋊C4—4241no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—2122no—
220C22⋊C4—2122no—
221C42—21no—
222C42—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23#1313312411yescharacteristic
233C23#1313313311yescharacteristic
234C23#1313313511yescharacteristic
235D4—42no—
236D4—42no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—4242no—
255C2×C4—4242no—
256C2×C4—4242no—
257C2×C4#1313313111yescharacteristic
258C2×C4#1313313111yescharacteristic
259C8—82no—
260C8—4241no—
261C8—2122no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—81no—
275C22—81no—
276C22#1681038911yescharacteristic
277C22#1681039111yescharacteristic
278C22#1681039411yescharacteristic
279C4—21no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—8482no—
285C4#1681041511yescharacteristic
286C4#1681041711yescharacteristic
287C2—21no—
288C2—21no—
289C2—41no—
290C2—41no—
291C2—81no—
292C2#2027262211yescharacteristic
293C2#2027283911yescharacteristic
294C2#2027284511yescharacteristic
subgroupG/H
centerC22#16810389
commutatorC22×C4×C53C42
FrattiniC2×C4×D4C2×D53
FittingC53×#1465C2
socleC22×C53#342
prime radicalC22×D4×C53C2×C4
cosocleC2×C4×D4×C53C22
hypercenter#1465D53
hypocenterC53#7322
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
4. C2×C4⋊#88322C4semidirect
8. C2×#13133128C4semidirect
13. C53×#324C4semidirect
160. #324Dic53semidirect
162. C53#7322semidirect

Maximal subgroups

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

subgroupindex
#202755142
#202747032
C53×#14652
#732253