Finite Group Databasebeta

← #27287171

Subgroups — #27287171

Order 13568.

476 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 89 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#20282990C211yesmaximal, characteristic, complemented
2#20275539C212yesmaximal
3#20275538C211yesmaximal, characteristic
4#20272597C212yesmaximal, complemented
5C53×#1295C211yesmaximal, characteristic, complemented
6C2×#13133214C2211yescharacteristic
7C2×#13133139—22no—
8C2×#13133139—22no—
9C2×#13133137—22no—
10C2×#13133128C2211yescharacteristic
11#16810557—22no—
12#16810556—22no—
13#16810555—22no—
14#16810489C2212yes—
15C2×#97×C53C2212yes—
16C53×#382C2211yescharacteristic
17C22×D212—21no—
18C22×D212—22no—
19C22×#88322—21no—
20#13133214—22no—
21#13133214—21no—
22#13133214Q812yes—
23C2×C4⋊Dic53—22no—
24C2×C4⋊Dic53D412yes—
25C2×C4⋊Dic53D412yes—
26C2×#11719919—22no—
27C2×#11719919—22no—
28#13133139—44no—
29#13133139—44no—
30#13133137—44no—
31#13133128—22no—
32#13133124—22no—
33#13133124—22no—
34#13133124—22no—
35C22×C8×C53—22no—
36C2×C4⋊C4×C53D412yes—
37C2×C42×C53C2311yescharacteristic
38#97×C53—22no—
39#97×C53—22no—
40#95×C53—22no—
41C23×D53—41no—
42C22×C4×C53—22no—
43C22×C4×C53C2×D411yescharacteristic
44C22×C4×C53C2×D412yes—
45C22×Dic53—22no—
46C22×Dic53—22no—
47C2×D212—24no—
48C2×D212—22no—
49C2×D212—42no—
50C2×D212—42no—
51C2×D212—42no—
52C2×D212—42no—
53C2×D212—42no—
54C2×D212—42no—
55C2×D212—42no—
56C2×D212—41no—
57C2×C8×C53—24no—
58C2×C8×C53—42no—
59C2×C8×C53—42no—
60C4⋊C4×C53—42no—
61C42×C53C2×Q811yescharacteristic
62C42×C53#4112yes—
63C42×C53#4111yescharacteristic
64C4⋊Dic53—42no—
65C4⋊Dic53—42no—
66C4⋊Dic53—82no—
67#11719919—82no—
68#11719919—82no—
69C2×#88322—22no—
70C2×#88322—41no—
71C2×#88322—41no—
72C23×C53#12011yescharacteristic
73C22×D53—42no—
74C22×D53—82no—
75C22×D53—81no—
76C22×D53—82no—
77C22×D53—81no—
78C2×C4×C53—21no—
79C2×C4×C53—22no—
80C2×C4×C53—21no—
81C2×C4×C53—22no—
82C2×C4×C53—42no—
83C2×C4×C53—42no—
84C2×C4×C53—42no—
85C2×C4×C53C22⋊Q812yes—
86C2×C4×C53C22⋊Q811yescharacteristic
87C2×C4×C53#12412yes—
88C2×C4×C53#12411yescharacteristic
89C2×Dic53—42no—
90C2×Dic53—42no—
91C2×Dic53—42no—
92C2×Dic53—42no—
93C2×Dic53—42no—
94C2×Dic53—42no—
95D212—44no—
96D212—44no—
97D212—44no—
98D212—82no—
99D212—82no—
100D212—82no—
101C8×C53—44no—
102#88322—42no—
103#10360—531nomaximal, Sylow
104C22×C53#39311yescharacteristic
105C22×C53#45312yes—
106C22×C53#45311yescharacteristic
107C22×C53#45412yes—
108C22×C53#45411yescharacteristic
109C2×D53—84no—
110C2×D53—82no—
111C2×D53—162no—
112C2×D53—161no—
113C2×D53—161no—
114C4×C53—21no—
115C4×C53—22no—
116C4×C53—22no—
117C4×C53—21no—
118C4×C53—82no—
119Dic53—82no—
120Dic53—82no—
121#2845—531no—
122#1961—532no—
123#1961—531no—
124#1295—532no—
125#1295D5311yescharacteristic, complemented
126C2×C53#165612yes—
127C2×C53#165611yescharacteristic
128C2×C53#165811yescharacteristic
129C2×C53#202712yes—
130C2×C53#202711yescharacteristic
131D53—162no—
132C2×#127—531no—
133C2×#102—1062no—
134C2×#102—1062no—
135C2×#102—1062no—
136C2×#97—531no—
137C2×#97C2×D5312yes—
138#398—1062no—
139#398—1062no—
140#398—1062no—
141#382—532no—
142#382C2×D5311yescharacteristic
143C53#1036011yescharacteristic, complemented, Sylow
144C22×D4—1061no—
145C22×D4—1062no—
146C22×C8—22no—
147C22×C8—1061no—
148#127—532no—
149#127—1062no—
150#127—1061no—
151C2×C4⋊C4—532no—
152C2×C4⋊C4—532no—
153C2×C4⋊C4—1062no—
154C2×C4⋊C4—1062no—
155C2×C4⋊C4—1062no—
156C2×C4⋊C4D21212yes—
157C2×C42C22×D5311yescharacteristic
158#102—2124no—
159#102—2124no—
160#102—2124no—
161#97—22no—
162#97—22no—
163#97—1062no—
164#95—22no—
165#95—1062no—
166#95—1062no—
167#95—1062no—
168C24—2121no—
169C2×D4—1064no—
170C2×D4—1062no—
171C2×D4—2122no—
172C2×D4—2122no—
173C2×D4—2122no—
174C2×D4—2122no—
175C2×D4—2122no—
176C2×D4—2122no—
177C2×D4—2122no—
178C2×D4—2121no—
179C22×C4—22no—
180C22×C4—1062no—
181C22×C4—1062no—
182C22×C4C2×D21211yescharacteristic
183C22×C4D4×D5312yes—
184C2×C8—24no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—1062no—
188C2×C8—2121no—
189C2×C8—2121no—
190C4⋊C4—42no—
191C4⋊C4—4242no—
192C4⋊C4—4242no—
193C4⋊C4—4242no—
194C4⋊C4—2122no—
195C4⋊C4—2122no—
196C42#1171994512yes—
197C42#1171994711yescharacteristic
198C42Q8×D5311yescharacteristic
199C23—4242no—
200C23—4241no—
201C23—4242no—
202C23—4241no—
203C23—2122no—
204C23#1313318711yescharacteristic
205D4—4242no—
206D4—4242no—
207D4—4242no—
208D4—2124no—
209D4—2124no—
210D4—2124no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—2122no—
219C2×C4—2122no—
220C2×C4—2122no—
221C2×C4—2122no—
222C2×C4—2122no—
223C2×C4—2122no—
224C2×C4#1313319512yes—
225C2×C4#1313319711yescharacteristic
226C2×C4#1313320312yes—
227C2×C4#1313320511yescharacteristic
228C8—44no—
229C8—2122no—
230C22—8482no—
231C22—8481no—
232C22—8481no—
233C22—4244no—
234C22—4242no—
235C22#1681053611yescharacteristic
236C22#1681079212yes—
237C22#1681079311yescharacteristic
238C22#1681079712yes—
239C22#1681079811yescharacteristic
240C4—22no—
241C4—22no—
242C4—21no—
243C4—21no—
244C4—82no—
245C4—4242no—
246C4—4242no—
247C2—8482no—
248C2#2027400912yes—
249C2#2027401011yescharacteristic
250C2#2027401811yescharacteristic
251C2#2027587012yes—
252C2#2027587211yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1295C2
socleC23×C53#120
prime radicalC2×#13133214C22
hypercenter#1295D53
hypocenterC53#10360
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
1. #20282990C2semidirect
4. #20272597C2semidirect
5. C53×#1295C2semidirect
125. #1295D53semidirect
143. C53#10360semidirect

Maximal subgroups

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

subgroupindex
#202829902
#202755392
#202755382
#202725972
C53×#12952
#1036053