Finite Group Databasebeta

← #27293528

Subgroups — #27293528

Order 13568.

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

normal subgroups only

This group has 284 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#20276527C212yesmaximal, complemented
2#20272666C212yesmaximal, complemented
3#20272651C212yesmaximal
4C53×#2921C211yesmaximal, characteristic
5#16810444—22no—
6#16810405C2214yes—
7#16810401—22no—
8C2×#117×C53C411yescharacteristic
9C53×#433—22no—
10C53×#433—22no—
11C53×#423C411yescharacteristic
12C53×#423C2211yescharacteristic
13C53×#407C411yescharacteristic
14C53×#407C2211yescharacteristic
15C4×M16×C53C411yescharacteristic
16C4×M16×C53C2211yescharacteristic
17#13133157—24no—
18C23×C4×C53C2×C411yescharacteristic
19C2×M16×C53C2×C411yescharacteristic
20C2×M16×C53C2×C411yescharacteristic
21C2×M16×C53C2×C411yescharacteristic
22C2×M16×C53C2311yescharacteristic
23#117×C53—21no—
24#117×C53—21no—
25#117×C53—22no—
26C2×C4⋊C4×C53—21no—
27C2×C22⋊C4×C53—21no—
28C2×C42×C53C2×C411yescharacteristic
29C2×C42×C53C2×C411yescharacteristic
30C4⋊C8×C53—22no—
31C4⋊C8×C53—22no—
32C22⋊C8×C53—22no—
33C22⋊C8×C53—22no—
34#97×C53—21no—
35#97×C53D412yes—
36C4×C8×C53—21no—
37C4×C8×C53D412yes—
38C24×C53M1611yescharacteristic
39C22×C4×C53—21no—
40C22×C4×C53—21no—
41C22×C4×C53—21no—
42C22×C4×C53—21no—
43C22×C4×C53—41no—
44C22×C4×C53—41no—
45C22×C4×C53M1611yescharacteristic
46C22×C4×C53C22×C411yescharacteristic
47M16×C53—41no—
48M16×C53—41no—
49M16×C53—41no—
50M16×C53—41no—
51C2×C8×C53—21no—
52C2×C8×C53—21no—
53C2×C8×C53C22⋊C412yes—
54C2×C8×C53C2×D412yes—
55C4⋊C4×C53—22no—
56C4⋊C4×C53—22no—
57C22⋊C4×C53—22no—
58C22⋊C4×C53—22no—
59C42×C53—21no—
60C42×C53—21no—
61C42×C53C22⋊C411yescharacteristic
62C42×C53C22⋊C411yescharacteristic
63C42×C53M1612yes—
64#11719908—44no—
65C23×C53—21no—
66C23×C53—41no—
67C23×C53—81no—
68C23×C53C2×M1611yescharacteristic
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—22no—
75C2×C4×C53—42no—
76C2×C4×C53—41no—
77C2×C4×C53—41no—
78C2×C4×C53—41no—
79C2×C4×C53—81no—
80C2×C4×C53—81no—
81C2×C4×C53C2×C22⋊C411yescharacteristic
82C2×C4×C53C2×M1611yescharacteristic
83C8×C53—22no—
84C8×C53—22no—
85C8×C53—41no—
86C8×C53—41no—
87#11522—531nomaximal, Sylow
88C22×C53—21no—
89C22×C53—41no—
90C22×C53—42no—
91C22×C53—41no—
92C22×C53—82no—
93C22×C53#40711yescharacteristic
94C4×C53—21no—
95C4×C53—41no—
96C4×C53—41no—
97C4×C53—42no—
98C4×C53—41no—
99C4×C53#41011yescharacteristic
100C4×C53#41311yescharacteristic
101#2921D5311yescharacteristic
102#2165—532no—
103#1327—532no—
104#1320—532no—
105C2×C53—21no—
106C2×C53—41no—
107C2×C53—81no—
108C2×C53#146411yescharacteristic
109C2×#117Dic5311yescharacteristic
110#433—22no—
111#433—22no—
112#423Dic5311yescharacteristic
113#423C2×D5311yescharacteristic
114#407Dic5311yescharacteristic
115#407C2×D5311yescharacteristic
116C4×M16Dic5311yescharacteristic
117C4×M16C2×D5311yescharacteristic
118#364—1062no—
119#349—534no—
120#347—1062no—
121C53#1152211yescharacteristic, complemented, Sylow
122C23×C4C2×Dic5311yescharacteristic
123C2×M16C2×Dic5311yescharacteristic
124C2×M16C2×Dic5311yescharacteristic
125C2×M16C2×Dic5311yescharacteristic
126C2×M16C22×D5311yescharacteristic
127#117—22no—
128#117—21no—
129#117—21no—
130C2×C4⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C42C2×Dic5311yescharacteristic
133C2×C42C2×Dic5311yescharacteristic
134M32—1064no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C22⋊C8—22no—
138C22⋊C8—22no—
139#97—21no—
140#97C22⋊D5312yes—
141C4×C8—21no—
142C4×C8C22⋊D5312yes—
143C24#1171991711yescharacteristic
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4#1171991711yescharacteristic
151C22×C4C22×Dic5311yescharacteristic
152M16—41no—
153M16—41no—
154M16—41no—
155M16—41no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8C22⋊Dic5312yes—
159C2×C8C2×C22⋊D5312yes—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C42—21no—
165C42—21no—
166C42#1171991712yes—
167C42C22⋊Dic5311yescharacteristic
168C42C22⋊Dic5311yescharacteristic
169C16—2124no—
170C23—21no—
171C23—41no—
172C23—81no—
173C23C2×#1171991711yescharacteristic
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4C2×C22⋊Dic5311yescharacteristic
187C2×C4C2×#1171991711yescharacteristic
188C8—22no—
189C8—22no—
190C8—41no—
191C8—41no—
192C22—21no—
193C22—41no—
194C22—42no—
195C22—41no—
196C22—82no—
197C22#1681058711yescharacteristic
198C4—21no—
199C4—42no—
200C4—41no—
201C4—41no—
202C4—41no—
203C4#1681059711yescharacteristic
204C4#1681060711yescharacteristic
205C2—21no—
206C2—41no—
207C2—81no—
208C2#2027304911yescharacteristic
subgroupG/H
centerC4#16810597
commutatorC22×C4×C53C22×C4
FrattiniC2×M16C22×D53
FittingC53×#2921C2
socleC2×C53#1464
prime radicalC24×C53M16
cosocleC2×M16×C53C23
hypercenter#2921D53
hypocenterC53#11522
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C42 ◁ C4×M16 ◁ #2921

Complements

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

NH ≅ G/Ntype
1. #20276527C2semidirect
2. #20272666C2semidirect
121. C53#11522semidirect

Maximal subgroups

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

subgroupindex
#202765272
#202726662
#202726512
C53×#29212
#1152253