Finite Group Databasebeta

← #27277678

Subgroups — #27277678

Order 13568.

378 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2356C211yesmaximal, characteristic
6#16810676C2212yes—
7#16810647C2212yes—
8C2×#13133136—21no—
9#16810392—21no—
10#16810349—22no—
11C2×C4×Q8×C53C2211yescharacteristic
12C53×#391—21no—
13C53×#387—22no—
14C53×#383C412yes—
15C53×#382C2211yescharacteristic
16C4×C4⋊C4×C53—21no—
17C4×C4⋊C4×C53C2211yescharacteristic
18C53×#376C412yes—
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—
29C2×C4⋊C4×C53—21no—
30C2×C4⋊C4×C53—21no—
31C2×C4⋊C4×C53D412yes—
32C2×C4⋊C4×C53D411yescharacteristic
33C2×C42×C53—22no—
34C2×C42×C53—21no—
35C2×C42×C53C2×C412yes—
36C2×C42×C53C2311yescharacteristic
37#95×C53—22no—
38#95×C53—22no—
39#95×C53—22no—
40#95×C53—21no—
41#95×C53C2×C411yescharacteristic
42#95×C53C2×C412yes—
43#95×C53C2×C411yescharacteristic
44C2×Q8×C53—22no—
45C2×Q8×C53—22no—
46C2×Q8×C53—21no—
47C2×Q8×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—22no—
50C22×C4×C53—21no—
51C22×C4×C53C22⋊C412yes—
52C22×C4×C53C22⋊C412yes—
53C22×C4×C53C22×C411yescharacteristic
54C22×C4×C53C2×D411yescharacteristic
55C22×C4×C53C2×D411yescharacteristic
56C4⋊C4×C53—24no—
57C4⋊C4×C53—22no—
58C4⋊C4×C53—42no—
59C4⋊C4×C53—42no—
60C42×C53—22no—
61C42×C53—22no—
62C42×C53—44no—
63C42×C53—41no—
64C42×C53—41no—
65C42×C53#4112yes—
66C42×C53#4112yes—
67#11719917—42no—
68#11719917—42no—
69C2×#88322—42no—
70C2×#88322—42no—
71C23×C53C2×C22⋊C411yescharacteristic
72Q8×C53—42no—
73Q8×C53—42no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—41no—
80C2×C4×C53—41no—
81C2×C4×C53—41no—
82C2×C4×C53—41no—
83C2×C4×C53—42no—
84C2×C4×C53—41no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53—41no—
88C2×C4×C53—42no—
89C2×C4×C53—41no—
90C2×C4×C53#11711yescharacteristic
91C2×C4×C53#11711yescharacteristic
92C2×C4×C53#12312yes—
93C2×C4×C53#12312yes—
94C2×C4×C53#13512yes—
95C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
96C2×C4×C53#13711yescharacteristic
97#88322—82no—
98#88322—82no—
99#8768—531nomaximal, Sylow
100C22×C53—21no—
101C22×C53—21no—
102C22×C53#38111yescharacteristic
103C22×C53#41611yescharacteristic
104C22×C53#41711yescharacteristic
105C4×C53—22no—
106C4×C53—21no—
107C4×C53—21no—
108C4×C53—42no—
109C4×C53—42no—
110C4×C53—42no—
111C4×C53—81no—
112C4×C53—82no—
113C4×C53#48212yes—
114C4×C53#48512yes—
115#2356D5311yescharacteristic
116#1801—531no—
117#1461—531no—
118#1303—532no—
119#1286—532no—
120C2×C53—21no—
121C2×C53—21no—
122C2×C53#182511yescharacteristic
123C2×C53#182711yescharacteristic
124C2×C53#183011yescharacteristic
125C2×C4×Q8C2×D5311yescharacteristic
126#431—532no—
127#423—532no—
128C2×#101—1061no—
129#391—21no—
130#387—22no—
131#383Dic5312yes—
132#382C2×D5311yescharacteristic
133C4×C4⋊C4—21no—
134C4×C4⋊C4C2×D5311yescharacteristic
135#376Dic5312yes—
136#343—1061no—
137#324—1062no—
138C53#876811yescharacteristic, complemented, Sylow
139C22×Q8C22⋊D5311yescharacteristic
140C2×M16—1062no—
141C4×Q8—22no—
142C4×Q8—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4C22⋊D5312yes—
146C2×C4⋊C4C22⋊D5311yescharacteristic
147C2×C42—21no—
148C2×C42—22no—
149C2×C42C2×Dic5312yes—
150C2×C42C22×D5311yescharacteristic
151C4⋊C8—1062no—
152C4⋊C8—1062no—
153C4⋊C8—1062no—
154#101—2122no—
155C22⋊C8—1062no—
156C4×C8—1062no—
157#95—21no—
158#95—22no—
159#95—22no—
160#95—22no—
161#95C2×Dic5311yescharacteristic
162#95C2×Dic5312yes—
163#95C2×Dic5311yescharacteristic
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—21no—
167C2×Q8—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4C22⋊Dic5312yes—
172C22×C4C22⋊Dic5312yes—
173C22×C4C22×Dic5311yescharacteristic
174C22×C4C2×C22⋊D5311yescharacteristic
175C22×C4C2×C22⋊D5311yescharacteristic
176M16—2122no—
177M16—2122no—
178C2×C8—2122no—
179C2×C8—2122no—
180C4⋊C4—22no—
181C4⋊C4—24no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C42—22no—
185C42—22no—
186C42—44no—
187C42—41no—
188C42—41no—
189C42#1171994512yes—
190C42#1171994512yes—
191C23C2×C22⋊Dic5311yescharacteristic
192Q8—42no—
193Q8—42no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4#1313317611yescharacteristic
211C2×C4#1313317611yescharacteristic
212C2×C4#1313320012yes—
213C2×C4#1313320012yes—
214C2×C4#1313323712yes—
215C2×C4#1313324211yescharacteristic
216C2×C4#1313324711yescharacteristic
217C8—4242no—
218C8—4242no—
219C22—21no—
220C22—21no—
221C22#1681048611yescharacteristic
222C22#1681061911yescharacteristic
223C22#1681062411yescharacteristic
224C4—22no—
225C4—21no—
226C4—21no—
227C4—42no—
228C4—42no—
229C4—42no—
230C4—81no—
231C4—82no—
232C4#1681094712yes—
233C4#1681096412yes—
234C2—21no—
235C2—21no—
236C2#2027481011yescharacteristic
237C2#2027481911yescharacteristic
238C2#2027482911yescharacteristic
subgroupG/H
centerC22#16810486
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2356C2
socleC22×C53#381
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2356D53
hypocenterC53#8768
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2356

Complements

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

NH ≅ G/Ntype
138. C53#8768semidirect

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×#23562
#876853