Finite Group Databasebeta

← #27273291

Subgroups — #27273291

Order 13568.

398 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 87 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#20282708C211yesmaximal, characteristic
2#20276258C211yesmaximal, characteristic
3#20276125C212yesmaximal
4#20272540C212yesmaximal
5C53×#1272C211yesmaximal, characteristic
6C2×#13133148—21no—
7C2×#13133142—22no—
8C2×#13133141—21no—
9#16810544—22no—
10#16810543—21no—
11#16810530—21no—
12#16810529C2212yes—
13#16810527—22no—
14#16810493C2211yescharacteristic
15#16810386—22no—
16#16810378C2211yescharacteristic
17C53×#391C2211yescharacteristic
18C53×#336C2212yes—
19C22×Dic106—21no—
20C22×Dic106D412yes—
21C22×#88322—21no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—22no—
25C2×C4⋊Dic53D412yes—
26C2×#11719919—22no—
27C2×C4×Dic53—21no—
28#13133148—42no—
29#13133142—44no—
30#13133141—42no—
31#13133124—21no—
32#13133124—22no—
33#13133124—22no—
34#13133124—22no—
35C22×Q8×C53D411yescharacteristic
36C22×C8×C53—22no—
37C2×C4⋊C4×C53D411yescharacteristic
38C2×C42×C53C2311yescharacteristic
39#95×C53—22no—
40C2×Q8×C53—21no—
41C2×Q8×C53—41no—
42C2×Q8×C53—41no—
43C2×Q8×C53Q1612yes—
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53C2×D412yes—
47C22×C4×C53C2×D411yescharacteristic
48C22×Dic53—22no—
49C22×Dic53—22no—
50C22×Dic53—41no—
51C2×Dic106—22no—
52C2×Dic106—42no—
53C2×Dic106—42no—
54C2×Dic106—81no—
55C2×Dic106—82no—
56C2×Dic106SD1614yes—
57C2×C8×C53—42no—
58C2×C8×C53—42no—
59C2×C8×C53—42no—
60C4⋊C4×C53—41no—
61C42×C53—41no—
62C4⋊Dic53—42no—
63C4⋊Dic53—41no—
64C4⋊Dic53—41no—
65C4⋊Dic53—42no—
66C4⋊Dic53—42no—
67C4⋊Dic53—81no—
68#11719919—82no—
69C4×Dic53—81no—
70C2×#88322—22no—
71C2×#88322—41no—
72C2×#88322—41no—
73C23×C53#12011yescharacteristic
74Q8×C53—42no—
75Q8×C53—81no—
76C2×C4×C53—21no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—21no—
80C2×C4×C53—22no—
81C2×C4×C53—41no—
82C2×C4×C53—41no—
83C2×C4×C53—41no—
84C2×C4×C53—42no—
85C2×C4×C53C2×SD1612yes—
86C2×C4×C53C2×Q1611yescharacteristic
87C2×C4×C53#13712yes—
88C2×C4×C53#13711yescharacteristic
89C2×Dic53—24no—
90C2×Dic53—42no—
91C2×Dic53—42no—
92C2×Dic53—42no—
93C2×Dic53—42no—
94C2×Dic53—42no—
95C2×Dic53—82no—
96C2×Dic53—81no—
97Dic106—44no—
98Dic106—82no—
99Dic106—84no—
100C8×C53—82no—
101#88322—42no—
102#7812—531nomaximal, Sylow
103C22×C53#44812yes—
104C22×C53#45111yescharacteristic
105C22×C53#45311yescharacteristic
106C22×C53#45512yes—
107C22×C53#45811yescharacteristic
108C4×C53—22no—
109C4×C53—22no—
110C4×C53—21no—
111C4×C53—21no—
112C4×C53—42no—
113C4×C53—81no—
114Dic53—44no—
115Dic53—82no—
116Dic53—161no—
117#2811—531no—
118#2096—531no—
119#2072—532no—
120#1272—532no—
121#1272D5311yescharacteristic
122C2×C53#160812yes—
123C2×C53#161411yescharacteristic
124C2×C53#162311yescharacteristic
125C2×C53#164512yes—
126C2×C53#165911yescharacteristic
127C2×#106—1061no—
128C2×#103—1062no—
129C2×#103—1061no—
130#395—1062no—
131#395—1061no—
132#391—532no—
133#391—1062no—
134#391—1061no—
135#391C2×D5311yescharacteristic
136#384—531no—
137#340—1062no—
138#336—531no—
139#336C2×D5312yes—
140C53#781211yescharacteristic, complemented, Sylow
141C22×Q8—532no—
142C22×Q8—1061no—
143C22×Q8D21211yescharacteristic
144C22×C8—22no—
145C22×C8—1061no—
146C2×C4⋊C4—532no—
147C2×C4⋊C4—1062no—
148C2×C4⋊C4—1061no—
149C2×C4⋊C4—1061no—
150C2×C4⋊C4—1062no—
151C2×C4⋊C4D21211yescharacteristic
152C2×C42—1061no—
153C2×C42C22×D5311yescharacteristic
154#106—2122no—
155#103—2124no—
156#103—2122no—
157#95—22no—
158#95—1062no—
159#95—1061no—
160#95—1062no—
161#95—1062no—
162C2×Q8—21no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—534no—
166C2×Q8—1062no—
167C2×Q8—4241no—
168C2×Q8—4242no—
169C2×Q8—2122no—
170C2×Q8—2122no—
171C2×Q8Dic21212yes—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—1062no—
175C22×C4—1062no—
176C22×C4—2121no—
177C22×C4C2×D21211yescharacteristic
178C22×C4D4×D5312yes—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—1062no—
183C2×C8—2121no—
184C2×C8—2121no—
185C4⋊C4—41no—
186C4⋊C4—4242no—
187C4⋊C4—4241no—
188C4⋊C4—2121no—
189C4⋊C4—2121no—
190C4⋊C4—2122no—
191C4⋊C4—2122no—
192C4⋊C4—2122no—
193C42—41no—
194C42—4241no—
195C23#1313318711yescharacteristic
196Q8—42no—
197Q8—81no—
198Q8—4242no—
199Q8—4244no—
200Q8—2124no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—1064no—
211C2×C4—4242no—
212C2×C4—4241no—
213C2×C4—2122no—
214C2×C4—2122no—
215C2×C4—2122no—
216C2×C4—2122no—
217C2×C4—2122no—
218C2×C4SD16×D5312yes—
219C2×C4C2×Dic21211yescharacteristic
220C2×C4#1313324912yes—
221C2×C4#1313325011yescharacteristic
222C8—82no—
223C8—2122no—
224C22#1681076312yes—
225C22#1681078111yescharacteristic
226C22#1681079311yescharacteristic
227C22#1681080212yes—
228C22#1681081411yescharacteristic
229C4—22no—
230C4—22no—
231C4—21no—
232C4—21no—
233C4—42no—
234C4—81no—
235C4—8481no—
236C4—4242no—
237C4—2124no—
238C2#2027370712yes—
239C2#2027374211yescharacteristic
240C2#2027379411yescharacteristic
241C2#2027393912yes—
242C2#2027402211yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1272C2
socle, prime radicalC23×C53#120
hypercenter#1272D53
hypocenterC53#7812
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
140. C53#7812semidirect

Maximal subgroups

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

subgroupindex
#202827082
#202762582
#202761252
#202725402
C53×#12722
#781253