Finite Group Databasebeta

← #27293525

Subgroups — #27293525

Order 13568.

284 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 69 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#20283350C211yesmaximal, characteristic
2#20276524C212yesmaximal, complemented
3#20272665C212yesmaximal, complemented
4#20272651C211yesmaximal, characteristic
5C53×#1320C211yesmaximal, characteristic
6C2×#13133176C411yescharacteristic
7#16810681—22no—
8#16810681—22no—
9#16810646C411yescharacteristic
10#16810646C2211yescharacteristic
11#16810587C411yescharacteristic
12M16×Dic53C411yescharacteristic
13M16×Dic53C2211yescharacteristic
14#16810443—22no—
15#16810405C2212yes—
16#16810400—22no—
17C53×#407C2211yescharacteristic
18C53×#349C2212yes—
19C2×#11719917C2×C411yescharacteristic
20C2×#11719917C2×C411yescharacteristic
21#13133176—21no—
22#13133176—21no—
23#13133176—22no—
24C2×C4⋊Dic53—21no—
25C2×C22⋊Dic53—21no—
26C2×C4×Dic53C2×C411yescharacteristic
27C2×C4×Dic53C2×C411yescharacteristic
28#13133157—22no—
29#13133146—22no—
30#13133146—22no—
31C22⋊#88322—22no—
32#13133127—21no—
33#13133127D412yes—
34C8×Dic53—21no—
35C8×Dic53D412yes—
36C23×C4×C53C2×C411yescharacteristic
37C2×M16×C53C2×C411yescharacteristic
38C2×M16×C53C2311yescharacteristic
39M32×C53—22no—
40C22⋊C8×C53—22no—
41C24×C53M1611yescharacteristic
42C22×C4×C53—21no—
43C22×C4×C53—21no—
44C22×C4×C53—41no—
45C22×C4×C53—41no—
46C22×C4×C53M1611yescharacteristic
47C22×C4×C53C22×C411yescharacteristic
48C22×Dic53—21no—
49C22×Dic53—21no—
50M16×C53—41no—
51M16×C53—41no—
52C2×C8×C53—21no—
53C2×C8×C53C2×D412yes—
54C22⋊Dic53—22no—
55C22⋊Dic53—22no—
56C4⋊Dic53—22no—
57C4⋊Dic53—22no—
58C4×Dic53—21no—
59C4×Dic53—21no—
60C4×Dic53C22⋊C411yescharacteristic
61C4×Dic53C22⋊C411yescharacteristic
62C4×Dic53M1612yes—
63#11719917—41no—
64#11719917—41no—
65C2×#88322—21no—
66C2×#88322C22⋊C412yes—
67C16×C53—42no—
68#11719908—42no—
69C23×C53—21no—
70C23×C53—41no—
71C23×C53—81no—
72C23×C53C2×M1611yescharacteristic
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—42no—
77C2×C4×C53—41no—
78C2×C4×C53—41no—
79C2×C4×C53—41no—
80C2×C4×C53C2×C22⋊C411yescharacteristic
81C2×C4×C53C2×M1611yescharacteristic
82C2×Dic53—21no—
83C2×Dic53—21no—
84C2×Dic53—22no—
85C2×Dic53—81no—
86C2×Dic53—81no—
87C8×C53—22no—
88C8×C53—41no—
89#88322—22no—
90#88322—41no—
91#11522—531nomaximal, Sylow
92C22×C53—21no—
93C22×C53—41no—
94C22×C53—42no—
95C22×C53—41no—
96C22×C53—82no—
97C22×C53#40711yescharacteristic
98C4×C53—21no—
99C4×C53—41no—
100C4×C53#41011yescharacteristic
101C4×C53#41311yescharacteristic
102Dic53—41no—
103Dic53—41no—
104Dic53—42no—
105#2921—531no—
106#2165—532no—
107#1327—532no—
108#1320—531no—
109#1320D5311yescharacteristic
110C2×C53—21no—
111C2×C53—41no—
112C2×C53—81no—
113C2×C53#146411yescharacteristic
114C2×#117—531no—
115#433—1062no—
116#433—1062no—
117#423—531no—
118#423—531no—
119#407—531no—
120#407C2×D5311yescharacteristic
121C4×M16—531no—
122C4×M16—531no—
123#364—1062no—
124#349—532no—
125#349C2×D5312yes—
126#347—1062no—
127C53#1152211yescharacteristic, complemented, Sylow
128C23×C4C4×D5311yescharacteristic
129C2×M16—531no—
130C2×M16—531no—
131C2×M16C4×D5311yescharacteristic
132C2×M16C22×D5311yescharacteristic
133#117—1062no—
134#117—1061no—
135#117—1061no—
136C2×C4⋊C4—1061no—
137C2×C22⋊C4—1061no—
138C2×C42—531no—
139C2×C42—531no—
140M32—22no—
141M32—1062no—
142C4⋊C8—1062no—
143C4⋊C8—1062no—
144C22⋊C8—22no—
145C22⋊C8—1062no—
146#97—532no—
147#97—1061no—
148C4×C8—532no—
149C4×C8—1061no—
150C24#1171991211yescharacteristic
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—1061no—
156C22×C4—1061no—
157C22×C4#1171991211yescharacteristic
158C22×C4C2×C4×D5311yescharacteristic
159M16—41no—
160M16—41no—
161M16—2121no—
162M16—2121no—
163C2×C8—21no—
164C2×C8—532no—
165C2×C8—1061no—
166C2×C8D4×D5312yes—
167C4⋊C4—1062no—
168C4⋊C4—1062no—
169C22⋊C4—1062no—
170C22⋊C4—1062no—
171C42—531no—
172C42—531no—
173C42—532no—
174C42—1061no—
175C42—1061no—
176C16—42no—
177C16—2122no—
178C23—21no—
179C23—41no—
180C23—81no—
181C23C2×#1171991211yescharacteristic
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—1061no—
190C2×C4—1061no—
191C2×C4—1062no—
192C2×C4—4241no—
193C2×C4—4241no—
194C2×C4C22⋊C4×D5311yescharacteristic
195C2×C4M16×D5311yescharacteristic
196C8—22no—
197C8—41no—
198C8—1062no—
199C8—2121no—
200C22—21no—
201C22—41no—
202C22—42no—
203C22—41no—
204C22—82no—
205C22#1681058511yescharacteristic
206C4—21no—
207C4—41no—
208C4—2121no—
209C4—2121no—
210C4—2122no—
211C4#1681059511yescharacteristic
212C4#1681060411yescharacteristic
213C2—21no—
214C2—41no—
215C2—81no—
216C2#2027304611yescharacteristic
subgroupG/H
centerC4#16810595
commutatorC22×C4×C53C22×C4
FrattiniC2×M16C22×D53
FittingC53×#1320C2
socleC2×C53#1464
prime radicalC24×C53M16
cosocleC2×M16×C53C23
hypercenter#1320D53
hypocenterC53#11522
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1320

Complements

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

NH ≅ G/Ntype
2. #20276524C2semidirect
3. #20272665C2semidirect
127. C53#11522semidirect

Maximal subgroups

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

subgroupindex
#202833502
#202765242
#202726652
#202726512
C53×#13202
#1152253