Finite Group Databasebeta

← #27277668

Subgroups — #27277668

Order 13568.

422 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 131 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#20274711C211yesmaximal, characteristic
2#20273039C211yesmaximal, characteristic
3#20272618C212yesmaximal
4#20272573C212yesmaximal
5C53×#2351C211yesmaximal, characteristic
6#16810676C2212yes—
7#16810647C2212yes—
8C2×#13133136—21no—
9#16810392—21no—
10#16810349—22no—
11C2×C4×Q8×C53—21no—
12C2×C4×Q8×C53C2211yescharacteristic
13C53×#389—21no—
14C53×#389—22no—
15C53×#382C412yes—
16C53×#382C2211yescharacteristic
17C4×C4⋊C4×C53C412yes—
18C4×C4⋊C4×C53C2211yescharacteristic
19C2×#11719917—22no—
20C4⋊#88322—22no—
21C4⋊#88322—22no—
22C4⋊#88322—22no—
23#13133136—42no—
24C22⋊#88322—22no—
25C4×#88322—22no—
26C22×Q8×C53D411yescharacteristic
27C4×Q8×C53—22no—
28C4×Q8×C53—22no—
29C4×Q8×C53—42no—
30C4×Q8×C53—42no—
31C2×C4⋊C4×C53—22no—
32C2×C4⋊C4×C53—22no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53D412yes—
36C2×C4⋊C4×C53D411yescharacteristic
37C2×C42×C53—22no—
38C2×C42×C53—21no—
39C2×C42×C53C2×C412yes—
40C2×C42×C53C2311yescharacteristic
41#95×C53—22no—
42#95×C53—21no—
43#95×C53C2×C411yescharacteristic
44#95×C53C2×C412yes—
45#95×C53C2×C411yescharacteristic
46C2×Q8×C53—22no—
47C2×Q8×C53—22no—
48C2×Q8×C53D812yes—
49C2×Q8×C53SD1612yes—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53C22⋊C412yes—
54C22×C4×C53C22⋊C412yes—
55C22×C4×C53C22×C411yescharacteristic
56C22×C4×C53C2×D411yescharacteristic
57C22×C4×C53C2×D411yescharacteristic
58C4⋊C4×C53—24no—
59C4⋊C4×C53—22no—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—44no—
62C4⋊C4×C53—44no—
63C4⋊C4×C53—41no—
64C4⋊C4×C53—41no—
65C42×C53—22no—
66C42×C53—22no—
67C42×C53—44no—
68C42×C53—41no—
69C42×C53—41no—
70C42×C53#4112yes—
71C42×C53#4112yes—
72#11719917—42no—
73#11719917—42no—
74C2×#88322—42no—
75C2×#88322—42no—
76C23×C53C2×C22⋊C411yescharacteristic
77Q8×C53—24no—
78Q8×C53—24no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—41no—
84C2×C4×C53—41no—
85C2×C4×C53—41no—
86C2×C4×C53—41no—
87C2×C4×C53—42no—
88C2×C4×C53—41no—
89C2×C4×C53—42no—
90C2×C4×C53—41no—
91C2×C4×C53—42no—
92C2×C4×C53—42no—
93C2×C4×C53—41no—
94C2×C4×C53#10214yes—
95C2×C4×C53#11711yescharacteristic
96C2×C4×C53#11711yescharacteristic
97C2×C4×C53#12312yes—
98C2×C4×C53#12312yes—
99C2×C4×C53C2×D811yescharacteristic
100C2×C4×C53C2×SD1611yescharacteristic
101C2×C4×C53#13712yes—
102#88322—82no—
103#88322—82no—
104#8766—531nomaximal, Sylow
105C22×C53—21no—
106C22×C53—21no—
107C22×C53#38111yescharacteristic
108C22×C53C2×#10211yescharacteristic
109C22×C53#41911yescharacteristic
110C4×C53—24no—
111C4×C53—22no—
112C4×C53—21no—
113C4×C53—21no—
114C4×C53—42no—
115C4×C53—42no—
116C4×C53—82no—
117C4×C53—81no—
118C4×C53#48012yes—
119C4×C53#48312yes—
120#2351D5311yescharacteristic
121#1801—531no—
122#1461—531no—
123#1303—532no—
124#1286—532no—
125C2×C53—21no—
126C2×C53—21no—
127C2×C53#182311yescharacteristic
128C2×C53#182711yescharacteristic
129C2×C53#183011yescharacteristic
130C2×C4×Q8—21no—
131C2×C4×Q8C2×D5311yescharacteristic
132#431—532no—
133#423—532no—
134C2×#101—1061no—
135#389—22no—
136#389—21no—
137#382Dic5312yes—
138#382C2×D5311yescharacteristic
139C4×C4⋊C4Dic5312yes—
140C4×C4⋊C4C2×D5311yescharacteristic
141#343—1061no—
142#324—1062no—
143C53#876611yescharacteristic, complemented, Sylow
144C22×Q8C22⋊D5311yescharacteristic
145C2×M16—1062no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—42no—
149C4×Q8—42no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4C22⋊D5312yes—
155C2×C4⋊C4C22⋊D5311yescharacteristic
156C2×C42—21no—
157C2×C42—22no—
158C2×C42C2×Dic5312yes—
159C2×C42C22×D5311yescharacteristic
160C4⋊C8—1062no—
161C4⋊C8—1062no—
162C4⋊C8—1062no—
163#101—2122no—
164C22⋊C8—1062no—
165C4×C8—1062no—
166#95—21no—
167#95—22no—
168#95C2×Dic5311yescharacteristic
169#95C2×Dic5312yes—
170#95C2×Dic5311yescharacteristic
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8#1171992212yes—
174C2×Q8#1171992312yes—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4C22⋊Dic5312yes—
179C22×C4C22⋊Dic5312yes—
180C22×C4C22×Dic5311yescharacteristic
181C22×C4C2×C22⋊D5311yescharacteristic
182C22×C4C2×C22⋊D5311yescharacteristic
183M16—2122no—
184M16—2122no—
185C2×C8—2122no—
186C2×C8—2122no—
187C4⋊C4—24no—
188C4⋊C4—22no—
189C4⋊C4—42no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C42—22no—
195C42—22no—
196C42—44no—
197C42—41no—
198C42—41no—
199C42#1171994512yes—
200C42#1171994512yes—
201C23C2×C22⋊Dic5311yescharacteristic
202Q8—24no—
203Q8—24no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4#1313313814yes—
220C2×C4#1313317611yescharacteristic
221C2×C4#1313317611yescharacteristic
222C2×C4#1313320012yes—
223C2×C4#1313320012yes—
224C2×C4C2×#1171992211yescharacteristic
225C2×C4C2×#1171992311yescharacteristic
226C2×C4#1313324812yes—
227C8—4242no—
228C8—4242no—
229C22—21no—
230C22—21no—
231C22#1681048611yescharacteristic
232C22C2×#1313313811yescharacteristic
233C22#1681063211yescharacteristic
234C4—24no—
235C4—22no—
236C4—21no—
237C4—21no—
238C4—42no—
239C4—42no—
240C4—82no—
241C4—81no—
242C4#1681093612yes—
243C4#1681095312yes—
244C2—21no—
245C2—21no—
246C2#2027480111yescharacteristic
247C2#2027481911yescharacteristic
248C2#2027482911yescharacteristic
subgroupG/H
centerC22#16810486
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2351C2
socleC22×C53#381
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2351D53
hypocenterC53#8766
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2351

Complements

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

NH ≅ G/Ntype
143. C53#8766semidirect

Maximal subgroups

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

subgroupindex
#202747112
#202730392
#202726182
#202725732
C53×#23512
#876653