Finite Group Databasebeta

← #28308216

Subgroups — #28308216

Order 13568.

446 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 76 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#20289966C212yesmaximal
2#20289660C214yesmaximal
3#20277198C211yesmaximal, characteristic
4#16811643—22no—
5#16811468—22no—
6#16811466C2214yes—
7C2×#13133176C411yescharacteristic
8C2×#13133176C2211yescharacteristic
9#16810521—22no—
10#16810496C412yes—
11#16810496C2212yes—
12C4×#11719919—22no—
13C22×C42×C53C411yescharacteristic
14#13133284—42no—
15C22⋊#88324—24no—
16#13133272—42no—
17#13133176—24no—
18#13133176—42no—
19C2×C4⋊Dic53—21no—
20C2×C4⋊Dic53D412yes—
21C2×#11719919—22no—
22C2×#11719919—22no—
23C2×C22⋊Dic53C2×C412yes—
24C2×C22⋊Dic53C2×C411yescharacteristic
25C2×C22⋊Dic53C2311yescharacteristic
26C2×C4×Dic53—21no—
27C2×C4×Dic53D412yes—
28#13133124—22no—
29#13133124—22no—
30C23×C4×C53C2×C411yescharacteristic
31C23×C4×C53C2×C412yes—
32C2×C42×C53—21no—
33C2×C42×C53—21no—
34C2×C42×C53—42no—
35C24×C53C22×C411yescharacteristic
36C22×C4×C53—22no—
37C22×C4×C53—22no—
38C22×C4×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53—44no—
41C22×C4×C53—42no—
42C22×C4×C53C22⋊C411yescharacteristic
43C22×C4×C53C22⋊C411yescharacteristic
44C22×Dic53—22no—
45C22×Dic53C22⋊C412yes—
46C22×Dic53C2×D412yes—
47C2×#88324—44no—
48C42×C53—42no—
49C42×C53—42no—
50C22⋊Dic53—22no—
51C22⋊Dic53—41no—
52C22⋊Dic53—41no—
53C22⋊Dic53M1614yes—
54C4⋊Dic53—24no—
55C4⋊Dic53—42no—
56#11719919—44no—
57#11719919—44no—
58C4×Dic53—24no—
59C4×Dic53—22no—
60C4×Dic53—22no—
61C23×C53—21no—
62C23×C53—21no—
63C23×C53—42no—
64C23×C53C2×C22⋊C411yescharacteristic
65C23×C53C2×M1612yes—
66C2×C4×C53—28no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—24no—
72C2×C4×C53—42no—
73C2×C4×C53—44no—
74C2×C4×C53—42no—
75C2×C4×C53—44no—
76C2×C4×C53#9912yes—
77C2×C4×C53#10112yes—
78C2×Dic53—24no—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—42no—
82C2×Dic53—42no—
83C2×Dic53—42no—
84C2×Dic53—42no—
85#88324—84no—
86#11518—531nomaximal, Sylow
87C22×C53—22no—
88C22×C53—22no—
89C22×C53—42no—
90C22×C53—42no—
91C22×C53—42no—
92C22×C53#40711yescharacteristic
93C22×C53C2×#9911yescharacteristic
94C22×C53C2×#10111yescharacteristic
95C4×C53—22no—
96C4×C53—22no—
97C4×C53—44no—
98Dic53—44no—
99Dic53—42no—
100Dic53—42no—
101#2296—531no—
102#1464—532no—
103#1326—534no—
104C2×C53—21no—
105C2×C53—21no—
106C2×C53—42no—
107C2×C53#146311yescharacteristic
108C2×C53#212012yes—
109C2×#117—531no—
110C2×#117—531no—
111C22×C42#615711yescharacteristic
112#432—1062no—
113#389—1062no—
114#385—532no—
115#385—532no—
116C4×C4⋊C4—1062no—
117#324—1062no—
118#323—534no—
119C53#1151811yescharacteristic, complemented, Sylow
120C23×C4C2×#615712yes—
121C23×C4C2×#615711yescharacteristic
122#117—1064no—
123#117—2122no—
124C2×C4⋊C4—532no—
125C2×C4⋊C4—1061no—
126C2×C4⋊C4—1062no—
127C2×C4⋊C4—1062no—
128C2×C22⋊C4—532no—
129C2×C22⋊C4—531no—
130C2×C22⋊C4—531no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—42no—
134C2×C42—532no—
135C2×C42—1061no—
136C4⋊C8—2122no—
137C22⋊C8—1064no—
138#97—2122no—
139#95—1062no—
140#95—1062no—
141C24C22×#615711yescharacteristic
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—44no—
147C22×C4—42no—
148C22×C4—532no—
149C22×C4—532no—
150C22×C4—1062no—
151C22×C4C22⋊#615711yescharacteristic
152C22×C4C22⋊#615711yescharacteristic
153C2×C8—2124no—
154C4⋊C4—1064no—
155C4⋊C4—2122no—
156C4⋊C4—2124no—
157C4⋊C4—2124no—
158C22⋊C4—534no—
159C22⋊C4—1062no—
160C22⋊C4—2121no—
161C22⋊C4—2121no—
162C42—42no—
163C42—42no—
164C42—1064no—
165C42—1062no—
166C42—1062no—
167C23—21no—
168C23—21no—
169C23—42no—
170C23C2×C22⋊#615711yescharacteristic
171C23D53⋊M1612yes—
172C2×C4—28no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—24no—
178C2×C4—42no—
179C2×C4—44no—
180C2×C4—42no—
181C2×C4—44no—
182C2×C4—1064no—
183C2×C4—1062no—
184C2×C4—1062no—
185C2×C4—2122no—
186C2×C4—2122no—
187C2×C4—2122no—
188C2×C4—2122no—
189C2×C4#1313327512yes—
190C2×C4#1313327812yes—
191C8—4244no—
192C22—22no—
193C22—22no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22#1681159511yescharacteristic
198C22C2×#1313327511yescharacteristic
199C22C2×#1313327811yescharacteristic
200C4—22no—
201C4—22no—
202C4—44no—
203C4—2124no—
204C4—2122no—
205C4—2122no—
206C2—21no—
207C2—21no—
208C2—42no—
209C2#2028996511yescharacteristic
210C2#2029096612yes—
subgroupG/H
centerC22C2×#13133275
commutator, prime radicalC24×C53C22×C4
FrattiniC24C22×#6157
FittingC22×C42×C53C4
socleC22×C53C2×#99
cosocleC2×C22⋊Dic53C23
hypercenterC22×C42#6157
hypocenterC53#11518
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
119. C53#11518semidirect

Maximal subgroups

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

subgroupindex
#202899662
#202896602
#202771982
#1151853