Finite Group Databasebeta

← #27281906

Subgroups — #27281906

Order 13568.

610 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 111 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#20279002C211yesmaximal, characteristic
2#20274815C212yesmaximal
3#20273045C211yesmaximal, characteristic
4#20272568C212yesmaximal
5C53×#2274C211yesmaximal, characteristic
6C2×#13133176C2211yescharacteristic
7#16810587C2212yes—
8#16810562C412yescomplemented
9#16810553C412yescomplemented
10#16810504—22no—
11#16810496—21no—
12#16810486—21no—
13#16810486C2211yescharacteristic
14#16810383—24no—
15#16810349—22no—
16C2×#117×C53—21no—
17C22×C4⋊C4×C53C2211yescharacteristic
18C4×C4⋊C4×C53—24no—
19C4×C22⋊C4×C53—21no—
20C4×C22⋊C4×C53C2212yes—
21C2×#11719917—22no—
22#13133176—22no—
23#13133176—22no—
24C2×C4⋊Dic53C2×C411yescharacteristic
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×#11719919—22no—
27C2×C22⋊Dic53—21no—
28C2×C22⋊Dic53—21no—
29C2×C4×Dic53—21no—
30C22⋊#88322—22no—
31C22⋊#88322—22no—
32#13133124—22no—
33#13133124—22no—
34#13133124—22no—
35#13133124—21no—
36#13133124C2×C412yes—
37C23×C4×C53D411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C23×C4×C53C2311yescharacteristic
40#117×C53—28no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53—22no—
43C2×C4⋊C4×C53—22no—
44C2×C4⋊C4×C53C2×C412yes—
45C2×C4⋊C4×C53D412yes—
46C2×C22⋊C4×C53—22no—
47C2×C42×C53—22no—
48C2×C42×C53—22no—
49C2×C42×C53D414yes—
50#95×C53—22no—
51#95×C53—22no—
52C24×C53C2×D411yescharacteristic
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—24no—
56C22×C4×C53—24no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—42no—
60C22×C4×C53—42no—
61C22×C4×C53C22⋊C412yes—
62C22×C4×C53C22⋊C412yes—
63C22×C4×C53C22×C411yescharacteristic
64C22×C4×C53C2×D412yes—
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53#4112yes—
67C22×Dic53—22no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C4⋊C4×C53—28no—
71C4⋊C4×C53—24no—
72C4⋊C4×C53—24no—
73C22⋊C4×C53—28no—
74C42×C53—28no—
75C42×C53—44no—
76C42×C53—82no—
77C22⋊Dic53—42no—
78C22⋊Dic53—41no—
79C22⋊Dic53—41no—
80C4⋊Dic53—22no—
81C4⋊Dic53—22no—
82#11719919—44no—
83C4×Dic53—42no—
84#11719917—44no—
85C2×#88322—42no—
86C23×C53—22no—
87C23×C53—22no—
88C23×C53—41no—
89C23×C53C2×C22⋊C411yescharacteristic
90C23×C53#12011yescharacteristic
91C23×C53#12311yescharacteristic
92C2×C4×C53—24no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—24no—
98C2×C4×C53—28no—
99C2×C4×C53—22no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—42no—
103C2×C4×C53—44no—
104C2×C4×C53—42no—
105C2×C4×C53—44no—
106C2×C4×C53—42no—
107C2×C4×C53—81no—
108C2×C4×C53#10414yes—
109C2×C4×C53C4×D412yes—
110C2×C4×C53#12112yes—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C2×Dic53—42no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—42no—
117C2×Dic53—42no—
118#88322—82no—
119#9478—531nomaximal, Sylow
120C22×C53—21no—
121C22×C53—21no—
122C22×C53—21no—
123C22×C53—21no—
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—44no—
128C22×C53#38611yescharacteristic
129C22×C53C2×#10412yes—
130C22×C53#45812yes—
131C4×C53—22no—
132C4×C53—22no—
133C4×C53—24no—
134C4×C53—48no—
135C4×C53—42no—
136Dic53—82no—
137Dic53—81no—
138Dic53—81no—
139#2484—531no—
140#2274D5311yescharacteristic
141#1826—532no—
142#1463—531no—
143#1284—532no—
144C2×C53—22no—
145C2×C53—22no—
146C2×C53—41no—
147C2×C53#189011yescharacteristic
148C2×C53#189611yescharacteristic
149C2×C53#191211yescharacteristic
150C2×#117—21no—
151C2×#117—531no—
152C22×C4⋊C4C2×D5311yescharacteristic
153#407—532no—
154#400—532no—
155#398—532no—
156#387—1062no—
157#385—1061no—
158#381—531no—
159#381—1061no—
160C4×C4⋊C4—24no—
161C4×C22⋊C4—21no—
162C4×C22⋊C4C2×D5312yes—
163#339—1064no—
164#324—1062no—
165C53#947811yescharacteristic, complemented, Sylow
166C23×C4C22⋊D5311yescharacteristic
167C23×C4C22⋊D5311yescharacteristic
168C23×C4C22×D5311yescharacteristic
169C2×M16—1062no—
170#117—28no—
171#117—1062no—
172#117—1062no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—531no—
177C2×C4⋊C4—531no—
178C2×C4⋊C4—1062no—
179C2×C4⋊C4C4×D5312yes—
180C2×C4⋊C4D21212yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—1061no—
183C2×C22⋊C4—1061no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—1061no—
187C2×C42C22⋊D5314yes—
188C22⋊C8—1062no—
189C22⋊C8—1062no—
190#95—22no—
191#95—22no—
192#95—532no—
193#95—1061no—
194#95—1062no—
195#95—1062no—
196#95—1062no—
197C24C2×C22⋊D5311yescharacteristic
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—1062no—
207C22×C4—1061no—
208C22×C4—1061no—
209C22×C4#1171992112yes—
210C22×C4#1171992112yes—
211C22×C4C2×C4×D5311yescharacteristic
212C22×C4C2×D21211yescharacteristic
213C22×C4#1171994512yes—
214C22×C4C2×C22⋊D5312yes—
215M16—2124no—
216C2×C8—2122no—
217C4⋊C4—28no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—1062no—
221C4⋊C4—1062no—
222C4⋊C4—2124no—
223C22⋊C4—28no—
224C22⋊C4—2122no—
225C22⋊C4—2121no—
226C22⋊C4—2121no—
227C42—28no—
228C42—44no—
229C42—82no—
230C42—2122no—
231C23—22no—
232C23—22no—
233C23—41no—
234C23C2×#1171992111yescharacteristic
235C23#1313318611yescharacteristic
236C23#1313320011yescharacteristic
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—28no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—81no—
253C2×C4—2122no—
254C2×C4—2122no—
255C2×C4—2122no—
256C2×C4—2121no—
257C2×C4—2121no—
258C2×C4—2122no—
259C2×C4—2122no—
260C2×C4#1313314514yes—
261C2×C4C4×C22⋊D5312yes—
262C2×C4#1313319112yes—
263C8—4242no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22#1681050211yescharacteristic
273C22C2×#1313314512yes—
274C22#1681081312yes—
275C4—22no—
276C4—22no—
277C4—24no—
278C4—48no—
279C4—42no—
280C4—4242no—
281C4—4241no—
282C4—4241no—
283C2—22no—
284C2—22no—
285C2—41no—
286C2#2027512811yescharacteristic
287C2#2027515811yescharacteristic
288C2#2027525211yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2274C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#2274D53
hypocenterC53#9478
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2274

Complements

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

NH ≅ G/Ntype
8. #16810562C4semidirect
9. #16810553C4semidirect
165. C53#9478semidirect

Maximal subgroups

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

subgroupindex
#202790022
#202748152
#202730452
#202725682
C53×#22742
#947853