Finite Group Databasebeta

← #27293260

Subgroups — #27293260

Order 13568.

434 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 131 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#20273029C212yesmaximal
2#20272661C214yesmaximal, complemented
3C53×#2317C211yesmaximal, characteristic
4#16810676—22no—
5#16810349C2214yes—
6#16810347—22no—
7C2×#117×C53C411yescharacteristic
8C2×#117×C53C2211yescharacteristic
9C22×C4⋊C4×C53C411yescharacteristic
10C53×#389—22no—
11C53×#385—22no—
12C53×#384C2212yes—
13C53×#382C412yes—
14C4⋊#88322—42no—
15C22⋊#88322—24no—
16C4×#88322—42no—
17C23×C4×C53—21no—
18C23×C4×C53C2×C411yescharacteristic
19#117×C53—22no—
20#117×C53—42no—
21#117×C53C814yescomplemented
22C2×C4⋊C4×C53—21no—
23C2×C4⋊C4×C53—21no—
24C2×C4⋊C4×C53—24no—
25C2×C4⋊C4×C53—42no—
26C2×C4⋊C4×C53C2×C412yes—
27C2×C4⋊C4×C53C2×C411yescharacteristic
28C2×C4⋊C4×C53C2311yescharacteristic
29C2×C22⋊C4×C53—21no—
30C2×C22⋊C4×C53D412yes—
31C2×C42×C53C2×C412yes—
32C2×C42×C53D412yes—
33#95×C53—24no—
34C24×C53C22⋊C411yescharacteristic
35C22×C4×C53—22no—
36C22×C4×C53—22no—
37C22×C4×C53—22no—
38C22×C4×C53—22no—
39C22×C4×C53—42no—
40C22×C4×C53—82no—
41C22×C4×C53C22⋊C412yes—
42C22×C4×C53C22⋊C411yescharacteristic
43C22×C4×C53C2×C812yes—
44C22×C4×C53M1612yes—
45C22×C4×C53C22×C411yescharacteristic
46C22×C4×C53C2×D412yes—
47C4⋊C4×C53—24no—
48C4⋊C4×C53—41no—
49C4⋊C4×C53—41no—
50C4⋊C4×C53—42no—
51C4⋊C4×C53—42no—
52C4⋊C4×C53—84no—
53C22⋊C4×C53—24no—
54C22⋊C4×C53—42no—
55C42×C53—22no—
56C42×C53—22no—
57C42×C53—22no—
58C42×C53C2×C814yes—
59C2×#88322—44no—
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—22no—
72C2×C4×C53—22no—
73C2×C4×C53—42no—
74C2×C4×C53—42no—
75C2×C4×C53—42no—
76C2×C4×C53—42no—
77C2×C4×C53—44no—
78C2×C4×C53—42no—
79C2×C4×C53—84no—
80C2×C4×C53C22⋊C812yes—
81C2×C4×C53#9912yes—
82C2×C4×C53#10112yes—
83C2×C4×C53C22×C811yescharacteristic
84C2×C4×C53C2×M1611yescharacteristic
85#88322—84no—
86#11450—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—42no—
101C4×C53—42no—
102C4×C53—82no—
103C4×C53#32414yes—
104#2317D5311yescharacteristic
105#1458—532no—
106#1325—534no—
107C2×C53—21no—
108C2×C53—21no—
109C2×C53—42no—
110C2×C53C2×#32411yescharacteristic
111C2×C53#212811yescharacteristic
112C2×C53#213511yescharacteristic
113C2×#117Dic5311yescharacteristic
114C2×#117C2×D5311yescharacteristic
115C22×C4⋊C4Dic5311yescharacteristic
116#431—1062no—
117#389—22no—
118#385—22no—
119#384C2×D5312yes—
120#382Dic5312yes—
121#324—534no—
122#323—1062no—
123C53#1145011yescharacteristic, complemented, Sylow
124C23×C4—21no—
125C23×C4C2×Dic5311yescharacteristic
126#117—22no—
127#117—42no—
128#117#8832214yescomplemented
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4C2×Dic5312yes—
134C2×C4⋊C4C2×Dic5311yescharacteristic
135C2×C4⋊C4C22×D5311yescharacteristic
136C2×C22⋊C4—21no—
137C2×C22⋊C4C22⋊D5312yes—
138C2×C42C2×Dic5312yes—
139C2×C42C22⋊D5312yes—
140C4⋊C8—2122no—
141C22⋊C8—1064no—
142C4×C8—2122no—
143#95—24no—
144C24C22⋊Dic5311yescharacteristic
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—42no—
150C22×C4—82no—
151C22×C4C2×#8832212yes—
152C22×C4#1171991712yes—
153C22×C4C22⋊Dic5312yes—
154C22×C4C22⋊Dic5311yescharacteristic
155C22×C4C22×Dic5311yescharacteristic
156C22×C4C2×C22⋊D5312yes—
157C2×C8—2124no—
158C4⋊C4—24no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—84no—
164C22⋊C4—24no—
165C22⋊C4—42no—
166C42—22no—
167C42—22no—
168C42—22no—
169C42C2×#8832214yes—
170C23—21no—
171C23—21no—
172C23—41no—
173C23—41no—
174C23C22⋊#8832212yes—
175C23C2×C22⋊Dic5311yescharacteristic
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—24no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—44no—
188C2×C4—42no—
189C2×C4—84no—
190C2×C4C22⋊#8832212yes—
191C2×C4#1313313212yes—
192C2×C4#1313313612yes—
193C2×C4C22×#8832211yescharacteristic
194C2×C4C2×#1171991711yescharacteristic
195C8—4244no—
196C22—21no—
197C22—21no—
198C22—22no—
199C22—42no—
200C22—42no—
201C22—41no—
202C22—41no—
203C22C2×C22⋊#8832211yescharacteristic
204C22C2×#1313313211yescharacteristic
205C22C2×#1313313611yescharacteristic
206C4—21no—
207C4—21no—
208C4—44no—
209C4—42no—
210C4—42no—
211C4—82no—
212C4#1681034914yes—
213C2—21no—
214C2—21no—
215C2—42no—
216C2C2×#1681034911yescharacteristic
217C2#2027638711yescharacteristic
218C2#2027641011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC2×C4×C53C22×C8
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2317C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2317D53
hypocenterC53#11450
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. #20272661C2semidirect
21. #117×C53C8semidirect
123. C53#11450semidirect
128. #117#88322semidirect

Maximal subgroups

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

subgroupindex
#202730292
#202726612
C53×#23172
#1145053