Finite Group Databasebeta

← #27271671

Subgroups — #27271671

Order 13568.

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

normal subgroups only

This group has 332 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#20272643C211yesmaximal, characteristic
2#20272619C211yesmaximal, characteristic
3C53×#2029C211yesmaximal, characteristic
4#16810479—21no—
5#16810347—21no—
6C53×#543C411yescharacteristic, complemented
7C53×#379—21no—
8C53×#379C2211yescharacteristic
9C53×#342—21no—
10C53×#342C411yescharacteristic, complemented
11C2×C22⋊Dic53—22no—
12C2×C22⋊Dic53—41no—
13C2×C22⋊Dic53—41no—
14C22⋊#88322—41no—
15C25×C53D411yescharacteristic
16C22⋊Q8×C53—42no—
17C22⋊Q8×C53—41no—
18C2×C22⋊C4×C53—21no—
19C2×C22⋊C4×C53—21no—
20C2×C22⋊C4×C53—41no—
21C2×C22⋊C4×C53—42no—
22C2×C22⋊C4×C53C2×C411yescharacteristic
23C2×C22⋊C4×C53D411yescharacteristic
24C24×C53—21no—
25C24×C53—21no—
26C24×C53—41no—
27C24×C53—41no—
28C24×C53—82no—
29C24×C53C22⋊C411yescharacteristic
30C24×C53SD1611yescharacteristic
31C24×C53Q1611yescharacteristic
32C2×Q8×C53—41no—
33C22×C4×C53—22no—
34C22×C4×C53—21no—
35C22×C4×C53—21no—
36C22×C4×C53—41no—
37C22×Dic53—42no—
38C4⋊C4×C53—41no—
39C4⋊C4×C53—42no—
40C22⋊C4×C53—41no—
41C22⋊C4×C53—41no—
42C22⋊C4×C53—41no—
43C22⋊C4×C53—41no—
44C22⋊C4×C53—41no—
45C22⋊C4×C53—41no—
46C22⋊C4×C53—82no—
47C22⋊C4×C53—84no—
48C22⋊Dic53—82no—
49C22⋊Dic53—81no—
50C22⋊Dic53—81no—
51C22⋊Dic53—82no—
52C2×#88322—81no—
53C23×C53—21no—
54C23×C53—21no—
55C23×C53—21no—
56C23×C53—21no—
57C23×C53—41no—
58C23×C53—41no—
59C23×C53—42no—
60C23×C53—42no—
61C23×C53—42no—
62C23×C53—44no—
63C23×C53—41no—
64C23×C53—41no—
65C23×C53—81no—
66C23×C53—81no—
67C23×C53—81no—
68C23×C53—81no—
69C23×C53—81no—
70C23×C53—82no—
71C23×C53—82no—
72C23×C53—82no—
73C23×C53#9911yescharacteristic
74C23×C53#10311yescharacteristic
75C23×C53#10411yescharacteristic
76Q8×C53—161no—
77C2×C4×C53—42no—
78C2×C4×C53—41no—
79C2×C4×C53—41no—
80C2×C4×C53—81no—
81C2×C4×C53—81no—
82C2×C4×C53—82no—
83C2×C4×C53—81no—
84C2×C4×C53—82no—
85C2×Dic53—82no—
86C2×Dic53—82no—
87C2×Dic53—82no—
88#88322—161no—
89#7426—531nomaximal, Sylow
90C22×C53—22no—
91C22×C53—21no—
92C22×C53—21no—
93C22×C53—41no—
94C22×C53—41no—
95C22×C53—41no—
96C22×C53—41no—
97C22×C53—42no—
98C22×C53—41no—
99C22×C53—41no—
100C22×C53—42no—
101C22×C53—41no—
102C22×C53—42no—
103C22×C53—41no—
104C22×C53—81no—
105C22×C53—82no—
106C22×C53—82no—
107C22×C53—81no—
108C22×C53—84no—
109C22×C53—81no—
110C22×C53#32811yescharacteristic
111C22×C53AΣL(1,𝔽16)11yescharacteristic
112C22×C53#35411yescharacteristic
113C4×C53—81no—
114C4×C53—82no—
115C4×C53—81no—
116C4×C53—161no—
117Dic53—162no—
118#2029D5311yescharacteristic
119#1316—531no—
120#1304—531no—
121C2×C53—21no—
122C2×C53—21no—
123C2×C53—21no—
124C2×C53—42no—
125C2×C53—41no—
126C2×C53—41no—
127C2×C53—81no—
128C2×C53#135111yescharacteristic
129#543Dic5311yescharacteristic, complemented
130#379—21no—
131#379—1061no—
132#379C2×D5311yescharacteristic
133#342—21no—
134#342Dic5311yescharacteristic, complemented
135#323—1061no—
136C53#742611yescharacteristic, complemented, Sylow
137C25C22⋊D5311yescharacteristic
138C22⋊Q8—42no—
139C22⋊Q8—41no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—1062no—
145C2×C22⋊C4—2121no—
146C2×C22⋊C4—2121no—
147C2×C22⋊C4C2×Dic5311yescharacteristic
148C2×C22⋊C4C22⋊D5311yescharacteristic
149C22⋊C8—2121no—
150C24—21no—
151C24—21no—
152C24—41no—
153C24—41no—
154C24—82no—
155C24#1171992411yescharacteristic
156C24#1171992511yescharacteristic
157C24C22⋊Dic5311yescharacteristic
158C2×Q8—41no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—41no—
163C22×C4—2122no—
164C2×C8—4241no—
165C4⋊C4—42no—
166C4⋊C4—41no—
167C22⋊C4—41no—
168C22⋊C4—41no—
169C22⋊C4—41no—
170C22⋊C4—41no—
171C22⋊C4—41no—
172C22⋊C4—41no—
173C22⋊C4—82no—
174C22⋊C4—84no—
175C22⋊C4—4242no—
176C22⋊C4—4241no—
177C22⋊C4—4241no—
178C22⋊C4—4242no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—42no—
184C23—41no—
185C23—41no—
186C23—42no—
187C23—42no—
188C23—44no—
189C23—41no—
190C23—41no—
191C23—81no—
192C23—81no—
193C23—81no—
194C23—81no—
195C23—81no—
196C23—82no—
197C23—82no—
198C23—82no—
199C23#1313313211yescharacteristic
200C23#1313314111yescharacteristic
201C23#1313314411yescharacteristic
202Q8—161no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—82no—
209C2×C4—81no—
210C2×C4—82no—
211C2×C4—4242no—
212C2×C4—4242no—
213C2×C4—4242no—
214C8—8481no—
215C22—22no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—42no—
226C22—41no—
227C22—42no—
228C22—41no—
229C22—81no—
230C22—82no—
231C22—82no—
232C22—81no—
233C22—84no—
234C22—81no—
235C22#1681036011yescharacteristic
236C22#1681041011yescharacteristic
237C22#1681041811yescharacteristic
238C4—81no—
239C4—82no—
240C4—81no—
241C4—161no—
242C4—8482no—
243C2—21no—
244C2—21no—
245C2—21no—
246C2—41no—
247C2—42no—
248C2—41no—
249C2—81no—
250C2#2027273611yescharacteristic
subgroupG/H
centerC2#20272736
commutatorC2×C22⋊C4×C53C2×C4
Frattini#379C2×D53
FittingC53×#2029C2
socleC2×C53#1351
prime radicalC25×C53D4
cosocleC53×#379C22
hypercenter#2029D53
hypocenterC53#7426
LCSG ▷ C2×C22⋊C4×C53 ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ #379 ◁ #2029

Complements

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

NH ≅ G/Ntype
6. C53×#543C4semidirect
10. C53×#342C4semidirect
129. #543Dic53semidirect
134. #342Dic53semidirect
136. C53#7426semidirect

Maximal subgroups

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

subgroupindex
#202726432
#202726192
C53×#20292
#742653