Finite Group Databasebeta

← #28307973

Subgroups — #28307973

Order 13568.

434 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 112 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#20289946C212yesmaximal
2#20289655C214yesmaximal, complemented
3#20277338C211yesmaximal, characteristic
4#16811639—22no—
5#16811468C2214yes—
6#16811466—22no—
7C2×#13133176C2211yescharacteristic
8C22×C4⋊Dic53C411yescharacteristic
9#16810522—22no—
10#16810498—22no—
11#16810493C412yes—
12#16810489C2212yes—
13C2×#117×C53C411yescharacteristic
14C4⋊#88324—42no—
15C22⋊#88324—24no—
16C4×#88324—42no—
17C23×Dic53—21no—
18#13133176—42no—
19C2×C4⋊Dic53—21no—
20C2×C4⋊Dic53—24no—
21C2×C4⋊Dic53—42no—
22C2×C4⋊Dic53C2×C412yes—
23C2×C4⋊Dic53C2×C411yescharacteristic
24C2×C4⋊Dic53C2311yescharacteristic
25C2×C22⋊Dic53D412yes—
26C2×C4×Dic53D412yes—
27#13133124—24no—
28C23×C4×C53C2×C411yescharacteristic
29#117×C53—22no—
30#117×C53C814yescomplemented
31C2×C4⋊C4×C53—21no—
32C2×C22⋊C4×C53—21no—
33C2×C42×C53C2×C412yes—
34C24×C53C22⋊C411yescharacteristic
35C22×C4×C53—22no—
36C22×C4×C53—22no—
37C22×C4×C53—22no—
38C22×C4×C53C22⋊C411yescharacteristic
39C22×C4×C53C2×C812yes—
40C22×C4×C53M1612yes—
41C22×C4×C53C22×C411yescharacteristic
42C22×Dic53—22no—
43C22×Dic53—42no—
44C22×Dic53—82no—
45C22×Dic53C22⋊C412yes—
46C22×Dic53C2×D412yes—
47C2×#88324—44no—
48C4⋊C4×C53—24no—
49C22⋊C4×C53—24no—
50C42×C53—22no—
51C42×C53C2×C814yes—
52C22⋊Dic53—42no—
53C4⋊Dic53—41no—
54C4⋊Dic53—41no—
55C4⋊Dic53—42no—
56C4⋊Dic53—42no—
57C4⋊Dic53—84no—
58C4×Dic53—22no—
59C4×Dic53—22no—
60C23×C53—21no—
61C23×C53—21no—
62C23×C53—41no—
63C23×C53—41no—
64C23×C53C22⋊C812yes—
65C23×C53C2×C22⋊C411yescharacteristic
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—24no—
69C2×C4×C53—24no—
70C2×C4×C53—24no—
71C2×C4×C53—42no—
72C2×C4×C53—42no—
73C2×C4×C53C22⋊C812yes—
74C2×C4×C53#9912yes—
75C2×C4×C53#10112yes—
76C2×C4×C53C22×C811yescharacteristic
77C2×C4×C53C2×M1611yescharacteristic
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53—42no—
81C2×Dic53—42no—
82C2×Dic53—44no—
83C2×Dic53—42no—
84C2×Dic53—84no—
85#88324—84no—
86#11449—531nomaximal, Sylow
87C22×C53—21no—
88C22×C53—21no—
89C22×C53—22no—
90C22×C53—42no—
91C22×C53—42no—
92C22×C53—41no—
93C22×C53—41no—
94C22×C53C2×C22⋊C811yescharacteristic
95C22×C53C2×#9911yescharacteristic
96C22×C53C2×#10111yescharacteristic
97C4×C53—21no—
98C4×C53—21no—
99C4×C53—44no—
100C4×C53#32414yes—
101Dic53—42no—
102Dic53—42no—
103Dic53—82no—
104#2317—531no—
105#1458—532no—
106#1324—534no—
107C2×C53—21no—
108C2×C53—21no—
109C2×C53—42no—
110C2×C53C2×#32411yescharacteristic
111C2×C53#212911yescharacteristic
112C2×C53#213411yescharacteristic
113C2×#117—531no—
114C2×#117#615711yescharacteristic
115C22×C4⋊C4—531no—
116#431—1062no—
117#389—1062no—
118#385—1062no—
119#384—532no—
120#382—532no—
121#324—534no—
122#323—1062no—
123C53#1144911yescharacteristic, complemented, Sylow
124C23×C4—1061no—
125C23×C4C2×#615711yescharacteristic
126#117—22no—
127#117—2122no—
128#117#8832414yescomplemented
129C2×C4⋊C4—21no—
130C2×C4⋊C4—531no—
131C2×C4⋊C4—531no—
132C2×C4⋊C4—532no—
133C2×C4⋊C4—1064no—
134C2×C4⋊C4—1061no—
135C2×C4⋊C4—2122no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—532no—
138C2×C42—532no—
139C2×C42C2×#615712yes—
140C4⋊C8—2122no—
141C22⋊C8—1064no—
142C4×C8—2122no—
143#95—1064no—
144C24C22⋊#615711yescharacteristic
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—532no—
149C22×C4—532no—
150C22×C4—1062no—
151C22×C4—4242no—
152C22×C4—2122no—
153C22×C4C2×#8832412yes—
154C22×C4#1171994012yes—
155C22×C4C22⋊#615711yescharacteristic
156C22×C4C22×#615711yescharacteristic
157C2×C8—2124no—
158C4⋊C4—24no—
159C4⋊C4—4244no—
160C4⋊C4—2121no—
161C4⋊C4—2121no—
162C4⋊C4—2122no—
163C4⋊C4—2122no—
164C22⋊C4—24no—
165C22⋊C4—2122no—
166C42—22no—
167C42—1062no—
168C42—1062no—
169C42C2×#8832414yes—
170C23—21no—
171C23—21no—
172C23—41no—
173C23—41no—
174C23C22⋊#8832412yes—
175C23C2×C22⋊#615711yescharacteristic
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—24no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—1062no—
184C2×C4—1062no—
185C2×C4—4244no—
186C2×C4—2124no—
187C2×C4—2122no—
188C2×C4—2122no—
189C2×C4—2122no—
190C2×C4C22⋊#8832412yes—
191C2×C4#1313327612yes—
192C2×C4#1313327812yes—
193C2×C4C22×#8832411yescharacteristic
194C2×C4C2×#1171994011yescharacteristic
195C8—4244no—
196C22—21no—
197C22—21no—
198C22—22no—
199C22—42no—
200C22—42no—
201C22—41no—
202C22—41no—
203C22C2×C22⋊#8832411yescharacteristic
204C22C2×#1313327611yescharacteristic
205C22C2×#1313327811yescharacteristic
206C4—21no—
207C4—21no—
208C4—44no—
209C4—4242no—
210C4—2122no—
211C4—2122no—
212C4#1681146814yes—
213C2—21no—
214C2—21no—
215C2—42no—
216C2C2×#1681146811yescharacteristic
217C2#2029099911yescharacteristic
218C2#2029101611yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenterC2×#117#6157
hypocenterC53#11449
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #20289655C2semidirect
30. #117×C53C8semidirect
123. C53#11449semidirect
128. #117#88324semidirect

Maximal subgroups

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

subgroupindex
#202899462
#202896552
#202773382
#1144953