Finite Group Databasebeta

← #28299715

Subgroups — #28299715

Order 13568.

490 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 96 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#20290606C212yesmaximal
2#20289534C214yesmaximal, complemented
3#20277197C211yesmaximal, characteristic
4C2×C22⋊#88324—22no—
5#16811489—22no—
6#16811489C2214yes—
7C2×#13133176C411yescharacteristic
8C22×C4×Dic53C2211yescharacteristic
9#16810519C412yes—
10#16810519C2212yes—
11#16810495—22no—
12C4×#11719919—22no—
13C2×#117×C53C411yescharacteristic
14C22×#88324—24no—
15C22⋊#88324—44no—
16C23×Dic53D412yes—
17#13133176—24no—
18C2×C4⋊Dic53C2×C412yes—
19C2×#11719919—22no—
20C2×#11719919—22no—
21C2×C22⋊Dic53—21no—
22C2×C4×Dic53—21no—
23C2×C4×Dic53—42no—
24C2×C4×Dic53C2×C411yescharacteristic
25C2×C4×Dic53D412yes—
26C2×C4×Dic53C2311yescharacteristic
27#13133124—22no—
28#13133124—22no—
29C23×C4×C53C2×C411yescharacteristic
30#117×C53—42no—
31C2×C4⋊C4×C53C2×C412yes—
32C2×C22⋊C4×C53—21no—
33C2×C42×C53—21no—
34C24×C53C22⋊C411yescharacteristic
35C22×C4×C53—22no—
36C22×C4×C53—22no—
37C22×C4×C53—22no—
38C22×C4×C53—22no—
39C22×C4×C53C22⋊C411yescharacteristic
40C22×C4×C53C22×C411yescharacteristic
41C22×Dic53—24no—
42C22×Dic53—22no—
43C22×Dic53—24no—
44C22×Dic53—42no—
45C22×Dic53C22⋊C412yes—
46C22×Dic53C2×D412yes—
47C2×#88324—44no—
48C2×#88324—44no—
49C2×#88324—44no—
50C4⋊C4×C53—42no—
51C22⋊C4×C53—42no—
52C42×C53—42no—
53C22⋊Dic53—24no—
54C4⋊Dic53—22no—
55C4⋊Dic53M1614yes—
56#11719919—44no—
57#11719919—44no—
58C4×Dic53—24no—
59C4×Dic53—41no—
60C4×Dic53—42no—
61C4×Dic53—41no—
62C23×C53—21no—
63C23×C53—21no—
64C23×C53—22no—
65C23×C53—22no—
66C23×C53#9912yes—
67C23×C53C2×C22⋊C411yescharacteristic
68C2×C4×C53—28no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—42no—
74C2×C4×C53—42no—
75C2×C4×C53—42no—
76C2×C4×C53#10112yes—
77C2×C4×C53C2×M1612yes—
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53—24no—
81C2×Dic53—22no—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—48no—
85C2×Dic53—42no—
86C2×Dic53—42no—
87C2×Dic53#13514yes—
88#88324—84no—
89#7646—531nomaximal, Sylow
90C22×C53—22no—
91C22×C53—24no—
92C22×C53—41no—
93C22×C53—41no—
94C22×C53—42no—
95C22×C53#40711yescharacteristic
96C22×C53C2×#9911yescharacteristic
97C22×C53C2×#10111yescharacteristic
98C22×C53#41612yes—
99C22×C53#42112yes—
100C4×C53—22no—
101C4×C53—22no—
102C4×C53—82no—
103Dic53—24no—
104Dic53—44no—
105Dic53—42no—
106#2296—531no—
107#1817—532no—
108#1271—534no—
109C2×C53—42no—
110C2×C53#146311yescharacteristic
111C2×C53#148812yes—
112C2×C53#151612yes—
113C2×C53#153711yescharacteristic
114C2×C53#154411yescharacteristic
115C2×#117—531no—
116C2×#117#615711yescharacteristic
117C22×C42—531no—
118C2×C22⋊C8—1062no—
119#389—532no—
120#389—532no—
121#385—1062no—
122C4×C4⋊C4—1062no—
123#336—534no—
124#336—1062no—
125C53#764611yescharacteristic, complemented, Sylow
126C23×C4—532no—
127C23×C4C2×#615711yescharacteristic
128C22×C8—1064no—
129#117—42no—
130#117—1064no—
131C2×C4⋊C4—532no—
132C2×C4⋊C4—1062no—
133C2×C4⋊C4—1062no—
134C2×C4⋊C4C2×#615712yes—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—1061no—
137C2×C42—21no—
138C2×C42—532no—
139C2×C42—531no—
140C2×C42—531no—
141C2×C42—1061no—
142C2×C42—2122no—
143C22⋊C8—2124no—
144#95—1062no—
145#95—1062no—
146C24C22⋊#615711yescharacteristic
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—532no—
152C22×C4—532no—
153C22×C4—1064no—
154C22×C4—1064no—
155C22×C4—1062no—
156C22×C4—2122no—
157C22×C4C22⋊#615711yescharacteristic
158C22×C4C22×#615711yescharacteristic
159C2×C8—2124no—
160C2×C8—2124no—
161C2×C8—2124no—
162C4⋊C4—42no—
163C4⋊C4—534no—
164C4⋊C4—1062no—
165C4⋊C4—2124no—
166C4⋊C4—2124no—
167C22⋊C4—42no—
168C22⋊C4—1064no—
169C42—42no—
170C42—1064no—
171C42—2122no—
172C42—2121no—
173C42—2121no—
174C23—21no—
175C23—21no—
176C23—22no—
177C23—22no—
178C23#1313327512yes—
179C23C2×C22⋊#615711yescharacteristic
180C2×C4—28no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—534no—
189C2×C4—1062no—
190C2×C4—1062no—
191C2×C4—1064no—
192C2×C4—1062no—
193C2×C4—1062no—
194C2×C4—1062no—
195C2×C4—2128no—
196C2×C4—2122no—
197C2×C4—2122no—
198C2×C4#1313327812yes—
199C2×C4D53⋊M1612yes—
200C8—4244no—
201C22—24no—
202C22—22no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22#1681159511yescharacteristic
207C22C2×#1313327511yescharacteristic
208C22C2×#1313327811yescharacteristic
209C22#1681161312yes—
210C22#1681162012yes—
211C4—22no—
212C4—22no—
213C4—82no—
214C4—1064no—
215C4—2124no—
216C4—2122no—
217C2—42no—
218C2#2028996511yescharacteristic
219C2#2029004812yes—
220C2#2029014112yes—
221C2#2029020611yescharacteristic
222C2#2029022011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×#117#6157
hypocenterC53#7646
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #20289534C2semidirect
125. C53#7646semidirect

Maximal subgroups

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

subgroupindex
#202906062
#202895342
#202771972
#764653