Finite Group Databasebeta

← #27271578

Subgroups — #27271578

Order 13568.

284 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 39 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#20272645C211yesmaximal, characteristic, complemented
2#20272623C211yesmaximal, characteristic, complemented
3C53×#2022C211yesmaximal, characteristic
4#16810587—21no—
5#16810347—21no—
6C53×#534C411yescharacteristic
7C2×#123×C53—21no—
8C2×#99×C53—21no—
9C53×#381C2211yescharacteristic
10C53×#342C411yescharacteristic
11C2×#11719917—41no—
12C22⋊#88322—22no—
13C22⋊#88322—41no—
14C22⋊#88322—41no—
15C22×D4×C53—21no—
16C23×C4×C53D411yescharacteristic
17#123×C53—41no—
18#123×C53—42no—
19#123×C53—41no—
20#123×C53—41no—
21#121×C53—41no—
22#120×C53—41no—
23C2×C4⋊C4×C53—41no—
24C2×C22⋊C4×C53—21no—
25C2×C22⋊C4×C53—41no—
26C2×C22⋊C4×C53C2×C411yescharacteristic
27C2×C22⋊C4×C53D411yescharacteristic
28#99×C53—42no—
29#95×C53—41no—
30C24×C53—21no—
31C24×C53C22⋊C411yescharacteristic
32C2×D4×C53—41no—
33C2×D4×C53—41no—
34C2×D4×C53—41no—
35C2×D4×C53—41no—
36C2×D4×C53—41no—
37C2×D4×C53—41no—
38C2×D4×C53—41no—
39C22×C4×C53—21no—
40C22×C4×C53—21no—
41C22×C4×C53—21no—
42C22×C4×C53—21no—
43C22×C4×C53—21no—
44C22×C4×C53—41no—
45C22×C4×C53—41no—
46C22×C4×C53—41no—
47C22×C4×C53D811yescharacteristic
48C22×C4×C53SD1611yescharacteristic
49C4⋊C4×C53—42no—
50C4⋊C4×C53—42no—
51C4⋊C4×C53—41no—
52C22⋊C4×C53—22no—
53C22⋊C4×C53—42no—
54C22⋊C4×C53—42no—
55C22⋊C4×C53—42no—
56C22⋊C4×C53—41no—
57C22⋊C4×C53—41no—
58C22⋊C4×C53—41no—
59#11719917—81no—
60#11719917—81no—
61C2×#88322—42no—
62C2×#88322—81no—
63C23×C53—21no—
64C23×C53—21no—
65C23×C53—41no—
66C23×C53—41no—
67C23×C53—41no—
68C23×C53—81no—
69C23×C53—82no—
70C23×C53#9911yescharacteristic
71C23×C53#10211yescharacteristic
72C23×C53#10411yescharacteristic
73D4×C53—81no—
74D4×C53—81no—
75D4×C53—81no—
76D4×C53—81no—
77D4×C53—161no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—41no—
81C2×C4×C53—41no—
82C2×C4×C53—41no—
83C2×C4×C53—41no—
84C2×C4×C53—41no—
85C2×C4×C53—41no—
86C2×C4×C53—42no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—81no—
91C2×C4×C53—82no—
92C2×C4×C53—81no—
93C2×C4×C53—81no—
94#88322—82no—
95#88322—161no—
96#7395—531nomaximal, Sylow
97C22×C53—21no—
98C22×C53—21no—
99C22×C53—41no—
100C22×C53—41no—
101C22×C53—41no—
102C22×C53—42no—
103C22×C53—81no—
104C22×C53—82no—
105C22×C53—82no—
106C22×C53—81no—
107C22×C53—161no—
108C22×C53#32711yescharacteristic
109C22×C53AΣL(1,𝔽16)11yescharacteristic
110C22×C53#35311yescharacteristic
111C4×C53—41no—
112C4×C53—41no—
113C4×C53—81no—
114C4×C53—81no—
115C4×C53—82no—
116C4×C53—81no—
117#2022D5311yescharacteristic
118#1317—531no—
119#1306—531no—
120C2×C53—21no—
121C2×C53—21no—
122C2×C53—21no—
123C2×C53—41no—
124C2×C53—41no—
125C2×C53—161no—
126C2×C53#134311yescharacteristic
127#534Dic5311yescharacteristic
128C2×#123—21no—
129C2×#99—21no—
130#407—1061no—
131#381C2×D5311yescharacteristic
132#342Dic5311yescharacteristic
133#323—1061no—
134C53#739511yescharacteristic, complemented, Sylow
135C22×D4—21no—
136C23×C4C22⋊D5311yescharacteristic
137C2×M16—2121no—
138#123—42no—
139#123—41no—
140#123—41no—
141#123—41no—
142#121—41no—
143#120—41no—
144C2×C4⋊C4—41no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—41no—
147C2×C22⋊C4C2×Dic5311yescharacteristic
148C2×C22⋊C4C22⋊D5311yescharacteristic
149#99—42no—
150C22⋊C8—1062no—
151C22⋊C8—2121no—
152C22⋊C8—2121no—
153#95—41no—
154C24—21no—
155C24C22⋊Dic5311yescharacteristic
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—41no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4#1171992211yescharacteristic
172C22×C4#1171992311yescharacteristic
173M16—4241no—
174M16—4241no—
175C2×C8—4241no—
176C2×C8—2122no—
177C4⋊C4—41no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C22⋊C4—22no—
181C22⋊C4—42no—
182C22⋊C4—42no—
183C22⋊C4—42no—
184C22⋊C4—41no—
185C22⋊C4—41no—
186C22⋊C4—41no—
187C23—21no—
188C23—21no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23—81no—
193C23—82no—
194C23#1313313211yescharacteristic
195C23#1313313811yescharacteristic
196C23#1313314411yescharacteristic
197D4—81no—
198D4—81no—
199D4—81no—
200D4—81no—
201D4—161no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—81no—
215C2×C4—82no—
216C2×C4—81no—
217C2×C4—81no—
218C8—8481no—
219C8—4242no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—42no—
226C22—81no—
227C22—82no—
228C22—82no—
229C22—81no—
230C22—161no—
231C22#1681035711yescharacteristic
232C22#1681041011yescharacteristic
233C22#1681041611yescharacteristic
234C4—41no—
235C4—41no—
236C4—81no—
237C4—81no—
238C4—82no—
239C4—81no—
240C2—21no—
241C2—21no—
242C2—21no—
243C2—41no—
244C2—41no—
245C2—161no—
246C2#2027271211yescharacteristic
subgroupG/H
centerC2#20272712
commutatorC2×C22⋊C4×C53C2×C4
Frattini#381C2×D53
FittingC53×#2022C2
socleC2×C53#1343
prime radicalC53×#534C4
cosocleC53×#381C22
hypercenter#2022D53
hypocenterC53#7395
LCSG ▷ C2×C22⋊C4×C53 ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ #381 ◁ #2022

Complements

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

NH ≅ G/Ntype
1. #20272645C2semidirect
2. #20272623C2semidirect
134. C53#7395semidirect

Maximal subgroups

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

subgroupindex
#202726452
#202726232
C53×#20222
#739553