Finite Group Databasebeta

← #27270840

Subgroups — #27270840

Order 13568.

368 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 87 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#20272558C212yesmaximal
2C53×#1445C211yesmaximal, characteristic
3C2×C22⋊#88322—22no—
4C22×C4⋊C4×C53C411yescharacteristic
5C2×#95×C53—22no—
6C2×#95×C53C411yescharacteristic
7C2×#95×C53C2211yescharacteristic
8C22×#88322—42no—
9C22⋊#88322—24no—
10C22⋊#88322—42no—
11C23×C4×C53—21no—
12C23×C4×C53—21no—
13C23×C4×C53C2×C411yescharacteristic
14C23×C4×C53D412yes—
15C2×C4⋊C4×C53—21no—
16C2×C4⋊C4×C53—41no—
17C2×C4⋊C4×C53—41no—
18C2×C4⋊C4×C53C812yescomplemented
19#95×C53—41no—
20#95×C53—41no—
21#95×C53—42no—
22#95×C53—42no—
23C24×C53C22⋊C411yescharacteristic
24C22×C4×C53—21no—
25C22×C4×C53—22no—
26C22×C4×C53—21no—
27C22×C4×C53—22no—
28C22×C4×C53—22no—
29C22×C4×C53—22no—
30C22×C4×C53—42no—
31C22×C4×C53—42no—
32C22×C4×C53—42no—
33C22×C4×C53—42no—
34C22×C4×C53—41no—
35C22×C4×C53—41no—
36C22×C4×C53—41no—
37C22×C4×C53—41no—
38C22×C4×C53—42no—
39C22×C4×C53—42no—
40C22×C4×C53—42no—
41C22×C4×C53C2×C811yescharacteristic
42C22×C4×C53M1611yescharacteristic
43C22×C4×C53D812yes—
44C22×C4×C53SD1612yes—
45C4⋊C4×C53—42no—
46C4⋊C4×C53—42no—
47C2×#88322—44no—
48C2×#88322—82no—
49C2×#88322—82no—
50C23×C53—22no—
51C23×C53—21no—
52C23×C53—21no—
53C23×C53C22⋊C811yescharacteristic
54C23×C53#9911yescharacteristic
55C23×C53#10011yescharacteristic
56C23×C53#10212yes—
57C23×C53#10412yes—
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—22no—
61C2×C4×C53—22no—
62C2×C4×C53—42no—
63C2×C4×C53—42no—
64C2×C4×C53—42no—
65C2×C4×C53—42no—
66C2×C4×C53—41no—
67C2×C4×C53—41no—
68C2×C4×C53—42no—
69C2×C4×C53—42no—
70C2×C4×C53—42no—
71C2×C4×C53—44no—
72C2×C4×C53—42no—
73C2×C4×C53—44no—
74C2×C4×C53—82no—
75C2×C4×C53—81no—
76C2×C4×C53—82no—
77C2×C4×C53—82no—
78C2×C4×C53—81no—
79C2×C4×C53—81no—
80C2×C4×C53—81no—
81#88322—84no—
82#7104—531nomaximal, Sylow
83C22×C53—21no—
84C22×C53—21no—
85C22×C53—21no—
86C22×C53—21no—
87C22×C53—22no—
88C22×C53—22no—
89C22×C53—22no—
90C22×C53—22no—
91C22×C53#32311yescharacteristic
92C22×C53D4⋊C812yes—
93C22×C53#32712yes—
94C22×C53#32911yescharacteristic
95C22×C53#33111yescharacteristic
96C22×C53#35212yes—
97C22×C53#35411yescharacteristic
98C22×C53#35611yescharacteristic
99C4×C53—41no—
100C4×C53—41no—
101C4×C53—42no—
102C4×C53—42no—
103C4×C53—82no—
104C4×C53—82no—
105#1445D5311yescharacteristic
106#1280—532no—
107C2×C53—21no—
108C2×C53—22no—
109C2×C53—21no—
110C2×C53#127011yescharacteristic
111C2×C53#131711yescharacteristic
112C2×C53#131911yescharacteristic
113C2×C53#134412yes—
114C2×C53#134612yes—
115C22×C4⋊C4Dic5311yescharacteristic
116C2×C22⋊C8—1062no—
117C2×#95—22no—
118C2×#95Dic5311yescharacteristic
119C2×#95C2×D5311yescharacteristic
120C53#710411yescharacteristic, complemented, Sylow
121C23×C4—21no—
122C23×C4—21no—
123C23×C4C2×Dic5311yescharacteristic
124C23×C4C22⋊D5312yes—
125C22×C8—2122no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4#8832212yescomplemented
130C22⋊C8—1064no—
131C22⋊C8—2122no—
132#95—41no—
133#95—41no—
134#95—42no—
135#95—42no—
136C24C22⋊Dic5311yescharacteristic
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4C2×#8832211yescharacteristic
155C22×C4#1171991711yescharacteristic
156C22×C4#1171992212yes—
157C22×C4#1171992312yes—
158C2×C8—4242no—
159C2×C8—4242no—
160C2×C8—2124no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C23—22no—
164C23—21no—
165C23—21no—
166C23C22⋊#8832211yescharacteristic
167C23#1313313211yescharacteristic
168C23#1313313411yescharacteristic
169C23#1313313812yes—
170C23#1313314412yes—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—44no—
185C2×C4—42no—
186C2×C4—44no—
187C2×C4—82no—
188C2×C4—81no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—81no—
194C8—4244no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—22no—
203C22#1681034711yescharacteristic
204C22D4⋊#8832212yes—
205C22#1681035712yes—
206C22#1681036311yescharacteristic
207C22#1681036711yescharacteristic
208C22#1681041312yes—
209C22#1681041811yescharacteristic
210C22#1681042211yescharacteristic
211C4—41no—
212C4—41no—
213C4—42no—
214C4—42no—
215C4—82no—
216C4—82no—
217C2—21no—
218C2—22no—
219C2—21no—
220C2#2027253611yescharacteristic
221C2#2027264511yescharacteristic
222C2#2027264911yescharacteristic
223C2#2027271512yes—
224C2#2027272112yes—
subgroupG/H
centerC23#13133132
commutatorC22×C4×C53C2×C8
FrattiniC2×#95C2×D53
FittingC53×#1445C2
socleC23×C53#99
prime radicalC24×C53C22⋊C4
cosocleC2×#95×C53C22
hypercenter#1445D53
hypocenterC53#7104
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24 ◁ C2×#95 ◁ #1445

Complements

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

NH ≅ G/Ntype
18. C2×C4⋊C4×C53C8semidirect
120. C53#7104semidirect
129. C2×C4⋊C4#88322semidirect

Maximal subgroups

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

subgroupindex
#202725582
C53×#14452
#710453