Finite Group Databasebeta

← #27271464

Subgroups — #27271464

Order 13568.

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

normal subgroups only

This group has 302 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#20274787C211yesmaximal, characteristic
2#20272643C211yesmaximal, characteristic
3C53×#1316C211yesmaximal, characteristic
4C22×C22⋊Dic53C411yescharacteristic
5#16810479C411yescharacteristic
6#16810389—21no—
7#16810389—21no—
8#16810347—21no—
9C53×#379C2211yescharacteristic
10C53×#323—21no—
11C23×Dic53—21no—
12C2×C22⋊Dic53—21no—
13C2×C22⋊Dic53—21no—
14C2×C22⋊Dic53—41no—
15C2×C22⋊Dic53—41no—
16C2×C22⋊Dic53—41no—
17C2×C22⋊Dic53C812yescomplemented
18C22⋊#88322—41no—
19C25×C53C2×C411yescharacteristic
20C2×C22⋊C4×C53—41no—
21C2×C22⋊C4×C53D411yescharacteristic
22C2×C22⋊C4×C53D411yescharacteristic
23C22⋊C8×C53—41no—
24C24×C53—21no—
25C24×C53—21no—
26C24×C53—41no—
27C24×C53—41no—
28C24×C53—82no—
29C24×C53C22⋊C411yescharacteristic
30C24×C53C2×C811yescharacteristic
31C24×C53M1611yescharacteristic
32C22×C4×C53—21no—
33C22×C4×C53—21no—
34C22×Dic53—22no—
35C22×Dic53—41no—
36C22×Dic53—41no—
37C22×Dic53—41no—
38C22×Dic53—82no—
39C2×C8×C53—81no—
40C22⋊C4×C53—41no—
41C22⋊C4×C53—41no—
42C22⋊C4×C53—82no—
43C22⋊Dic53—41no—
44C22⋊Dic53—41no—
45C22⋊Dic53—42no—
46C22⋊Dic53—41no—
47C22⋊Dic53—41no—
48C22⋊Dic53—82no—
49C2×#88322—81no—
50C23×C53—21no—
51C23×C53—21no—
52C23×C53—21no—
53C23×C53—21no—
54C23×C53—42no—
55C23×C53—41no—
56C23×C53—42no—
57C23×C53—42no—
58C23×C53—42no—
59C23×C53—41no—
60C23×C53—41no—
61C23×C53—41no—
62C23×C53—81no—
63C23×C53—81no—
64C23×C53—81no—
65C23×C53—81no—
66C23×C53—82no—
67C23×C53—82no—
68C23×C53—82no—
69C23×C53—82no—
70C23×C53C22⋊C811yescharacteristic
71C23×C53#9911yescharacteristic
72C23×C53#10011yescharacteristic
73C2×C4×C53—41no—
74C2×C4×C53—41no—
75C2×C4×C53—81no—
76C2×C4×C53—81no—
77C2×Dic53—42no—
78C2×Dic53—82no—
79C2×Dic53—82no—
80C2×Dic53—82no—
81C2×Dic53—81no—
82C2×Dic53—82no—
83C8×C53—161no—
84#88322—161no—
85#7349—531nomaximal, Sylow
86C22×C53—22no—
87C22×C53—21no—
88C22×C53—21no—
89C22×C53—41no—
90C22×C53—41no—
91C22×C53—41no—
92C22×C53—41no—
93C22×C53—41no—
94C22×C53—41no—
95C22×C53—42no—
96C22×C53—42no—
97C22×C53—41no—
98C22×C53—41no—
99C22×C53—81no—
100C22×C53—81no—
101C22×C53—82no—
102C22×C53—82no—
103C22×C53—84no—
104C22×C53—81no—
105C22×C53—81no—
106C22×C53#32311yescharacteristic
107C22×C53#35211yescharacteristic
108C22×C53#35211yescharacteristic
109C4×C53—81no—
110C4×C53—81no—
111Dic53—82no—
112Dic53—161no—
113#1819—531no—
114#1316—531no—
115#1316D5311yescharacteristic
116C2×C53—21no—
117C2×C53—21no—
118C2×C53—21no—
119C2×C53—41no—
120C2×C53—42no—
121C2×C53—41no—
122C2×C53—81no—
123C2×C53#131711yescharacteristic
124C22×C22⋊C4—531no—
125#379—531no—
126#379C2×D5311yescharacteristic
127#342—1061no—
128#342—1061no—
129#323—21no—
130#323—1061no—
131C53#734911yescharacteristic, complemented, Sylow
132C25C4×D5311yescharacteristic
133C23×C4—1061no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—532no—
136C2×C22⋊C4—1061no—
137C2×C22⋊C4—1061no—
138C2×C22⋊C4—2121no—
139C2×C22⋊C4—2121no—
140C2×C22⋊C4—2121no—
141C2×C22⋊C4D21211yescharacteristic
142C2×C22⋊C4C22⋊D5311yescharacteristic
143C22⋊C8—41no—
144C22⋊C8—2121no—
145C24—21no—
146C24—21no—
147C24—41no—
148C24—41no—
149C24—82no—
150C24C8×D5311yescharacteristic
151C24#1171991211yescharacteristic
152C24#1171992111yescharacteristic
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—1062no—
156C22×C4—4242no—
157C22×C4—2121no—
158C22×C4—2121no—
159C22×C4—2121no—
160C2×C8—81no—
161C2×C8—4241no—
162C22⋊C4—41no—
163C22⋊C4—41no—
164C22⋊C4—82no—
165C22⋊C4—4242no—
166C22⋊C4—2121no—
167C22⋊C4—2121no—
168C22⋊C4—2121no—
169C22⋊C4—2121no—
170C22⋊C4—2122no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—42no—
176C23—41no—
177C23—42no—
178C23—42no—
179C23—42no—
180C23—41no—
181C23—41no—
182C23—41no—
183C23—81no—
184C23—81no—
185C23—81no—
186C23—81no—
187C23—82no—
188C23—82no—
189C23—82no—
190C23—82no—
191C23#1313312911yescharacteristic
192C23#1313313111yescharacteristic
193C23#1313313311yescharacteristic
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—4241no—
199C2×C4—4242no—
200C2×C4—4242no—
201C2×C4—4242no—
202C2×C4—4242no—
203C2×C4—2122no—
204C8—161no—
205C8—8481no—
206C22—22no—
207C22—21no—
208C22—21no—
209C22—41no—
210C22—41no—
211C22—41no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22—42no—
216C22—42no—
217C22—41no—
218C22—41no—
219C22—81no—
220C22—81no—
221C22—82no—
222C22—82no—
223C22—84no—
224C22—81no—
225C22—81no—
226C22#1681034611yescharacteristic
227C22#1681041211yescharacteristic
228C22#1681041411yescharacteristic
229C4—81no—
230C4—81no—
231C4—8481no—
232C4—4242no—
233C2—21no—
234C2—21no—
235C2—21no—
236C2—41no—
237C2—42no—
238C2—41no—
239C2—81no—
240C2#2027264411yescharacteristic
subgroupG/H
centerC2#20272644
commutatorC24×C53C2×C8
Frattini#379C2×D53
FittingC53×#1316C2
socleC2×C53#1317
prime radicalC25×C53C2×C4
cosocleC53×#379C22
hypercenter#1316D53
hypocenterC53#7349
LCSG ▷ C24×C53 ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ #379 ◁ #1316

Complements

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

NH ≅ G/Ntype
17. C2×C22⋊Dic53C8semidirect
131. C53#7349semidirect

Maximal subgroups

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

subgroupindex
#202747872
#202726432
C53×#13162
#734953