Finite Group Databasebeta

← #27277616

Subgroups — #27277616

Order 13568.

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

normal subgroups only

This group has 566 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#20277300C211yesmaximal, characteristic
2#20274638C212yesmaximal
3#20272624C212yesmaximal
4#20272623C211yesmaximal, characteristic
5C53×#1306C211yesmaximal, characteristic
6C23×Dic106C2211yescharacteristic
7C2×#13133173C412yes—
8C2×#13133135—24no—
9#16810587C2212yes—
10#16810527—24no—
11#16810486C2211yescharacteristic
12#16810484C412yes—
13C53×#407C2212yes—
14C53×#381C2211yescharacteristic
15C22×Dic106—22no—
16C22×Dic106—21no—
17C22×Dic106—21no—
18C22×Dic106—44no—
19C22×Dic106D414yes—
20C23×Dic53C2×C411yescharacteristic
21C23×Dic53C2×C411yescharacteristic
22C2×#11719917—22no—
23#13133173—44no—
24C2×#11719919—22no—
25C2×C22⋊Dic53C2×C412yes—
26C2×C4×Dic53—22no—
27#13133135—48no—
28C22⋊#88322—22no—
29C22⋊#88322—22no—
30#13133124—22no—
31#13133124—24no—
32C23×C4×C53C2311yescharacteristic
33C2×M16×C53—22no—
34C2×C22⋊C4×C53C2×C412yes—
35C22⋊C8×C53—22no—
36C22⋊C8×C53—22no—
37#95×C53—22no—
38C24×C53C22×C411yescharacteristic
39C22×C4×C53—22no—
40C22×C4×C53—42no—
41C22×C4×C53C2×D412yes—
42C22×C4×C53#4112yes—
43C22×C4×C53#4111yescharacteristic
44C22×C4×C53#4111yescharacteristic
45C22×Dic53—21no—
46C22×Dic53—21no—
47C22×Dic53—24no—
48C22×Dic53—21no—
49C22×Dic53—21no—
50C22×Dic53—22no—
51C22×Dic53—42no—
52C22×Dic53C22⋊C412yes—
53C22×Dic53C22⋊C412yes—
54C2×Dic106—24no—
55C2×Dic106—24no—
56C2×Dic106—28no—
57C2×Dic106—44no—
58C2×Dic106—44no—
59C2×Dic106—48no—
60C2×Dic106—44no—
61M16×C53—44no—
62C2×C8×C53—42no—
63C22⋊C4×C53—42no—
64C22⋊Dic53—42no—
65#11719919—44no—
66C4×Dic53—44no—
67#11719917—44no—
68C2×#88322—42no—
69C23×C53—22no—
70C23×C53—42no—
71C23×C53C2×C22⋊C411yescharacteristic
72C23×C53#11711yescharacteristic
73C23×C53#11711yescharacteristic
74C2×C4×C53—24no—
75C2×C4×C53—42no—
76C2×C4×C53—41no—
77C2×C4×C53—41no—
78C2×C4×C53—44no—
79C2×C4×C53—42no—
80C2×C4×C53#12312yes—
81C2×C4×C53#12312yes—
82C2×Dic53—24no—
83C2×Dic53—24no—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—24no—
88C2×Dic53—42no—
89C2×Dic53—42no—
90C2×Dic53—44no—
91C2×Dic53—42no—
92C2×Dic53#9914yes—
93Dic106—48no—
94Dic106—48no—
95C8×C53—82no—
96#88322—82no—
97#8754—531nomaximal, Sylow
98C22×C53—22no—
99C22×C53—21no—
100C22×C53—21no—
101C22×C53—22no—
102C22×C53—22no—
103C22×C53—44no—
104C22×C53#38111yescharacteristic
105C22×C53C2×#9912yes—
106C4×C53—22no—
107C4×C53—22no—
108C4×C53—82no—
109Dic53—24no—
110Dic53—42no—
111Dic53—82no—
112#2310—531no—
113#1784—532no—
114#1306—532no—
115#1306—531no—
116#1306D5311yescharacteristic
117C2×C53—22no—
118C2×C53—41no—
119C2×C53—41no—
120C2×C53#181911yescharacteristic
121C2×C53#183012yes—
122C23×Q8—531no—
123C2×#117—532no—
124C2×#101—1064no—
125#407—532no—
126#407C2×D5312yes—
127#391—1064no—
128#381—531no—
129#381—532no—
130#381C2×D5311yescharacteristic
131C53#875411yescharacteristic, complemented, Sylow
132C22×Q8—534no—
133C22×Q8—1062no—
134C22×Q8—1061no—
135C22×Q8—1061no—
136C22×Q8—2124no—
137C23×C4—531no—
138C23×C4—531no—
139C23×C4C22×D5311yescharacteristic
140C2×M16—22no—
141C2×M16—1062no—
142#117—2124no—
143C2×C4⋊C4—1062no—
144C2×C22⋊C4—532no—
145C2×C22⋊C4C4×D5312yes—
146C2×C42—1062no—
147#101—2128no—
148C22⋊C8—22no—
149C22⋊C8—22no—
150C22⋊C8—1062no—
151C22⋊C8—1062no—
152#95—22no—
153#95—1064no—
154#95—1062no—
155C24C2×C4×D5311yescharacteristic
156C2×Q8—1068no—
157C2×Q8—1064no—
158C2×Q8—1064no—
159C2×Q8—2124no—
160C2×Q8—2128no—
161C2×Q8—2124no—
162C2×Q8—2124no—
163C22×C4—22no—
164C22×C4—42no—
165C22×C4—532no—
166C22×C4—532no—
167C22×C4—1064no—
168C22×C4—1061no—
169C22×C4—1061no—
170C22×C4—1061no—
171C22×C4—1061no—
172C22×C4—1062no—
173C22×C4—2122no—
174C22×C4#1171994512yes—
175C22×C4D4×D5312yes—
176C22×C4#1171994711yescharacteristic
177C22×C4#1171994911yescharacteristic
178M16—44no—
179M16—2124no—
180C2×C8—42no—
181C2×C8—2122no—
182C4⋊C4—2124no—
183C22⋊C4—42no—
184C22⋊C4—2122no—
185C42—2124no—
186C23—22no—
187C23—42no—
188C23C22⋊C4×D5311yescharacteristic
189C23#1313317411yescharacteristic
190C23#1313317511yescharacteristic
191Q8—2128no—
192Q8—2128no—
193C2×C4—24no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—44no—
198C2×C4—42no—
199C2×C4—534no—
200C2×C4—1064no—
201C2×C4—1064no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4—1064no—
205C2×C4—1062no—
206C2×C4—2124no—
207C2×C4—2122no—
208C2×C4—2122no—
209C2×C4—2122no—
210C2×C4#1313319912yes—
211C2×C4#1313320212yes—
212C8—82no—
213C8—4242no—
214C22—22no—
215C22—21no—
216C22—21no—
217C22—22no—
218C22—22no—
219C22—44no—
220C22#1681048511yescharacteristic
221C22#99×D5312yes—
222C4—22no—
223C4—22no—
224C4—82no—
225C4—1064no—
226C4—4242no—
227C4—2122no—
228C2—22no—
229C2—41no—
230C2—41no—
231C2#2027478811yescharacteristic
232C2#2027482812yes—
subgroupG/H
centerC22#16810485
commutator, prime radicalC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1306C2
socleC22×C53#381
cosocleC23×C4×C53C23
hypercenter#1306D53
hypocenterC53#8754
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1306

Complements

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

NH ≅ G/Ntype
131. C53#8754semidirect

Maximal subgroups

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

subgroupindex
#202773002
#202746382
#202726242
#202726232
C53×#13062
#875453