Finite Group Databasebeta

← #27365955

Subgroups — #27365955

Order 13568.

840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 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#20285929C212yesmaximal, complemented
2#20285060C211yesmaximal, characteristic
3#20274049C218yesmaximal, complemented
4#20273350C211yesmaximal, characteristic
5#20273330C212yesmaximal, complemented
6C53×#3550C211yesmaximal, characteristic
7#16811180C2211yescharacteristic
8#16811033C2214yes—
9#16810993C2214yes—
10#16810991—21no—
11#16810987—21no—
12#16810982C2214yes—
13#16810964—24no—
14#16810915—22no—
15#16810905—22no—
16#16810680C2212yes—
17#16810619—22no—
18C2×#13133128C2211yescharacteristic
19#16810365C2218yes—
20#16810363C2214yes—
21C53×#563C2214yes—
22C53×#559—21no—
23Q82×C53—21no—
24C53×#556—22no—
25D4×Q8×C53—22no—
26C53×#542—24no—
27C53×#536—22no—
28C53×#533C2212yes—
29C53×#532C2211yescharacteristic
30C22×#88322—21no—
31#13133217C2312yes—
32#13133209—21no—
33#13133196—22no—
34C4×Dic106—22no—
35#13133176—21no—
36#13133150—24no—
37#13133148—24no—
38C4⋊#88322—22no—
39#13133141—24no—
40#13133140—24no—
41#13133140—24no—
42#13133138—24no—
43C22⋊#88322—22no—
44#13133128—21no—
45#13133128C2314yes—
46#13133128C2312yes—
47C2×#41×C53D412yes—
48C22×Q8×C53—22no—
49C4⋊Q8×C53—24no—
50C4⋊Q8×C53—24no—
51C4⋊Q8×C53D412yes—
52C4⋊Q8×C53D412yes—
53C4⋊Q8×C53D411yescharacteristic
54#127×C53D411yescharacteristic
55#126×C53—28no—
56#125×C53C2314yes—
57#124×C53—24no—
58#124×C53—24no—
59#124×C53C2312yes—
60#123×C53—24no—
61#123×C53—24no—
62C22⋊Q8×C53—24no—
63C22⋊Q8×C53—28no—
64C22⋊Q8×C53—22no—
65C22⋊Q8×C53—22no—
66#121×C53—22no—
67C4×Q8×C53—24no—
68C4×Q8×C53—22no—
69C4×Q8×C53—22no—
70C4×D4×C53—24no—
71C4×D4×C53—22no—
72#117×C53—22no—
73#117×C53D414yes—
74C2×C42×C53C2311yescharacteristic
75#41×C53—44no—
76C2×Q8×C53—24no—
77C2×Q8×C53—22no—
78C2×Q8×C53—22no—
79C2×Q8×C53—84no—
80C2×Q8×C53C2×D412yes—
81C2×D4×C53—22no—
82C2×D4×C53—22no—
83C2×D4×C53C2×D412yes—
84C22×C4×C53—24no—
85C22×C4×C53—22no—
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D412yes—
88C2×Dic106—22no—
89C4⋊C4×C53—28no—
90C4⋊C4×C53—22no—
91C4⋊C4×C53—22no—
92C4⋊C4×C53—22no—
93C4⋊C4×C53—24no—
94C4⋊C4×C53—24no—
95C4⋊C4×C53C2×D414yes—
96C4⋊C4×C53C2×D414yes—
97C22⋊C4×C53—24no—
98C22⋊C4×C53—28no—
99C22⋊C4×C53—24no—
100C22⋊C4×C53—24no—
101C42×C53—24no—
102C42×C53—24no—
103C42×C53C2×D412yes—
104C42×C53C2×D411yescharacteristic
105C42×C53C2411yescharacteristic
106C22⋊Dic53—41no—
107C4⋊Dic53—22no—
108C4⋊Dic53—22no—
109#11719919—42no—
110C4×Dic53—41no—
111C2×#88322—22no—
112C2×#88322—24no—
113C2×#88322—42no—
114C23×C53—22no—
115C23×C53#12011yescharacteristic
116Q8×C53—44no—
117Q8×C53—42no—
118Q8×C53—44no—
119Q8×C53—84no—
120D4×C53—42no—
121D4×C53—44no—
122D4×C53—82no—
123C2×C4×C53—22no—
124C2×C4×C53—24no—
125C2×C4×C53—22no—
126C2×C4×C53—24no—
127C2×C4×C53—42no—
128C2×C4×C53—44no—
129C2×C4×C53—84no—
130C2×C4×C53#12012yes—
131C2×C4×C53#12011yescharacteristic
132C2×C4×C53C22×D412yes—
133C2×C4×C53C22×D411yescharacteristic
134C2×Dic53—42no—
135Dic106—44no—
136#88322—44no—
137#88322—42no—
138#19058—531nomaximal, Sylow
139C22×C53—21no—
140C22×C53—22no—
141C22×C53—42no—
142C22×C53—44no—
143C22×C53C2×#12011yescharacteristic
144C4×C53—21no—
145C4×C53—22no—
146C4×C53—22no—
147C4×C53—42no—
148C4×C53—44no—
149C4×C53—44no—
150C4×C53#45512yes—
151Dic53—82no—
152#3550D5311yescharacteristic
153#3237—532no—
154#3135—531no—
155#1664—538no—
156#1536—531no—
157#1531—532no—
158C2×C53—41no—
159C2×C53—82no—
160C2×C53C2×#45511yescharacteristic
161C2×C53#302012yes—
162#563C2×D5314yes—
163#559—21no—
164Q82—21no—
165#556—22no—
166D4×Q8—22no—
167#542—24no—
168#536—22no—
169#533—531no—
170#533C2×D5312yes—
171#532C2×D5311yescharacteristic
172#501—534no—
173#491—534no—
174#491—1061no—
175#490—1061no—
176#489—534no—
177#485—1064no—
178#477—1062no—
179#475—1062no—
180#432—532no—
181#416—1062no—
182C2×#97—531no—
183#330—538no—
184#329—534no—
185C53#1905811yescharacteristic, complemented, Sylow
186C2×#41C22⋊D5312yes—
187C22×Q8—22no—
188C22×C8—1061no—
189C4⋊Q8—24no—
190C4⋊Q8—24no—
191C4⋊Q8—532no—
192C4⋊Q8C22⋊D5311yescharacteristic
193C4⋊Q8C22⋊D5312yes—
194C4⋊Q8C22⋊D5312yes—
195#127C22⋊D5311yescharacteristic
196#126—28no—
197#125—1061no—
198#125C22×D5314yes—
199#124—24no—
200#124—24no—
201#124C22×D5312yes—
202#123—24no—
203#123—24no—
204C22⋊Q8—24no—
205C22⋊Q8—22no—
206C22⋊Q8—22no—
207C22⋊Q8—28no—
208C22⋊Q8—1062no—
209#121—22no—
210C4×Q8—24no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—1062no—
214C4×D4—24no—
215C4×D4—22no—
216#117—22no—
217#117—1061no—
218#117C22⋊D5314yes—
219C2×C42C22×D5311yescharacteristic
220#107—1064no—
221#106—1064no—
222C4⋊C8—1062no—
223#103—1064no—
224#103—1064no—
225#103—1064no—
226#102—1064no—
227C22⋊C8—1062no—
228#97—534no—
229#97—532no—
230#97—1061no—
231#41—44no—
232C2×Q8—24no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—84no—
236C2×Q8—1062no—
237C2×Q8C2×C22⋊D5312yes—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4C2×C22⋊D5312yes—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4C2×C22⋊D5311yescharacteristic
244C22×C4C2×C22⋊D5312yes—
245C2×C8—1062no—
246C2×C8—1064no—
247C2×C8—2122no—
248C4⋊C4—28no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—1062no—
255C4⋊C4—1062no—
256C4⋊C4—2122no—
257C4⋊C4C2×C22⋊D5314yes—
258C4⋊C4C2×C22⋊D5314yes—
259C22⋊C4—24no—
260C22⋊C4—28no—
261C22⋊C4—24no—
262C22⋊C4—24no—
263C22⋊C4—2121no—
264C42—24no—
265C42—24no—
266C42—2121no—
267C42C2×C22⋊D5311yescharacteristic
268C42C2×C22⋊D5312yes—
269C42C23×D5311yescharacteristic
270C23—22no—
271C23#1313318611yescharacteristic
272Q8—42no—
273Q8—44no—
274Q8—44no—
275Q8—84no—
276Q8—2124no—
277D4—42no—
278D4—44no—
279D4—82no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—84no—
287C2×C4—2122no—
288C2×C4#1313318612yes—
289C2×C4#1313318611yescharacteristic
290C2×C4C22×C22⋊D5312yes—
291C2×C4C22×C22⋊D5311yescharacteristic
292C8—2124no—
293C8—2122no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—44no—
298C22C2×#1313318611yescharacteristic
299C4—21no—
300C4—22no—
301C4—22no—
302C4—42no—
303C4—44no—
304C4—44no—
305C4—4242no—
306C4#1681080112yes—
307C2—41no—
308C2—82no—
309C2C2×#1681080111yescharacteristic
310C2#2028407912yes—
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3550C2
socleC22×C53C2×#120
prime radicalC53×#532C22
hypercenter#3550D53
hypocenterC53#19058
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3550

Complements

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

NH ≅ G/Ntype
1. #20285929C2semidirect
3. #20274049C2semidirect
5. #20273330C2semidirect
185. C53#19058semidirect

Maximal subgroups

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

subgroupindex
#202859292
#202850602
#202740492
#202733502
#202733302
C53×#35502
#1905853