Finite Group Databasebeta

← #28303390

Subgroups — #28303390

Order 13568.

514 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 120 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
1C2×#16811468C212yesmaximal
2#20289558C214yesmaximal
3#20277071C211yesmaximal, characteristic
4C2×C22⋊#88324C2214yes—
5#16811468—22no—
6#16811468—22no—
7C2×#13133176C411yescharacteristic
8C22×C4⋊Dic53C2211yescharacteristic
9C4⋊C4×Dic53—22no—
10C4×C4⋊Dic53—22no—
11C22⋊C4×Dic53C412yes—
12C22⋊C4×Dic53C2212yes—
13C2×#117×C53C411yescharacteristic
14C22×#88324—24no—
15C22⋊#88324—24no—
16C22⋊#88324—24no—
17C23×Dic53C2×C411yescharacteristic
18C23×Dic53C2311yescharacteristic
19#13133176—22no—
20#13133176C814yescomplemented
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53—24no—
23C2×C4⋊Dic53D412yes—
24C2×C4⋊Dic53D412yes—
25C2×C22⋊Dic53C2×C412yes—
26C2×C4×Dic53—21no—
27C2×C4×Dic53—22no—
28C2×C4×Dic53—22no—
29#13133124—22no—
30#13133124—22no—
31C23×C4×C53C2×C411yescharacteristic
32#117×C53—24no—
33C2×C4⋊C4×C53—21no—
34C2×C22⋊C4×C53C2×C412yes—
35C2×C42×C53—21no—
36C24×C53C22×C411yescharacteristic
37C22×C4×C53—22no—
38C22×C4×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53C22⋊C411yescharacteristic
41C22×C4×C53C22⋊C411yescharacteristic
42C22×C4×C53C2×C812yes—
43C22×C4×C53M1612yes—
44C22×Dic53—22no—
45C22×Dic53—22no—
46C22×Dic53—81no—
47C22×Dic53—81no—
48C22×Dic53C22⋊C412yes—
49C22×Dic53C2×D412yes—
50C22×Dic53#4114yes—
51C2×#88324—44no—
52C2×#88324—84no—
53C4⋊C4×C53—24no—
54C22⋊C4×C53—22no—
55C22⋊C4×C53M1614yes—
56C42×C53—24no—
57C22⋊Dic53—22no—
58C22⋊Dic53C2×C814yes—
59C4⋊Dic53—24no—
60C4⋊Dic53—24no—
61C4⋊Dic53—24no—
62C4×Dic53—24no—
63C4×Dic53—82no—
64C4×Dic53—82no—
65C23×C53—41no—
66C23×C53—41no—
67C23×C53C2×C22⋊C411yescharacteristic
68C23×C53#11712yes—
69C23×C53C22×C811yescharacteristic
70C23×C53C2×M1612yes—
71C23×C53C2×M1611yescharacteristic
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—24no—
77C2×C4×C53—24no—
78C2×C4×C53—24no—
79C2×C4×C53—42no—
80C2×C4×C53—42no—
81C2×C4×C53C22⋊C812yes—
82C2×C4×C53C22⋊C812yes—
83C2×Dic53—22no—
84C2×Dic53—24no—
85C2×Dic53—44no—
86C2×Dic53—42no—
87C2×Dic53—42no—
88C2×Dic53—42no—
89C2×Dic53—42no—
90C2×Dic53—82no—
91C2×Dic53#12314yes—
92#88324—84no—
93#8735—531nomaximal, Sylow
94C22×C53—21no—
95C22×C53—22no—
96C22×C53—21no—
97C22×C53—22no—
98C22×C53—41no—
99C22×C53—42no—
100C22×C53—41no—
101C22×C53#38111yescharacteristic
102C22×C53C2×C22⋊C811yescharacteristic
103C22×C53#40711yescharacteristic
104C22×C53#43112yes—
105C22×C53#43212yes—
106C4×C53—22no—
107C4×C53—22no—
108C4×C53—44no—
109Dic53—24no—
110Dic53—44no—
111Dic53—42no—
112#2274—531no—
113C2×#324—532no—
114#1282—534no—
115C2×C53—21no—
116C2×C53—21no—
117C2×C53—22no—
118C2×C53—22no—
119C2×C53#181711yescharacteristic
120C2×C53#182811yescharacteristic
121C2×C53#183011yescharacteristic
122C2×#117—531no—
123C2×#117#615711yescharacteristic
124C22×C4⋊C4—531no—
125C2×C22⋊C8—534no—
126C4×C4⋊C4—1062no—
127C4×C4⋊C4—1062no—
128C4×C22⋊C4—532no—
129C4×C22⋊C4—532no—
130#324—1062no—
131#324—1062no—
132C53#873511yescharacteristic, complemented, Sylow
133C23×C4—531no—
134C23×C4—531no—
135C23×C4C2×#615711yescharacteristic
136C22×C8—1064no—
137#117—24no—
138#117—534no—
139#117—1062no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—532no—
142C2×C4⋊C4—532no—
143C2×C4⋊C4—1064no—
144C2×C4⋊C4—1061no—
145C2×C22⋊C4—532no—
146C2×C22⋊C4C2×#615712yes—
147C2×C42—21no—
148C2×C42—1061no—
149C2×C42—1062no—
150C2×C42—1062no—
151C22⋊C8—1064no—
152C22⋊C8—1064no—
153#95—1062no—
154#95—1062no—
155C24C22×#615711yescharacteristic
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—532no—
160C22×C4—534no—
161C22×C4—532no—
162C22×C4—1062no—
163C22×C4—1062no—
164C22×C4—4241no—
165C22×C4—4241no—
166C22×C4D53⋊C812yes—
167C22×C4#1171993612yes—
168C22×C4C22⋊#615711yescharacteristic
169C22×C4C22⋊#615711yescharacteristic
170C2×C8—4244no—
171C2×C8—2124no—
172C4⋊C4—24no—
173C4⋊C4—1064no—
174C4⋊C4—1064no—
175C4⋊C4—1064no—
176C22⋊C4—22no—
177C22⋊C4—534no—
178C22⋊C4—1062no—
179C22⋊C4#1171994014yes—
180C42—24no—
181C42—1064no—
182C42—4242no—
183C42—4242no—
184C23—41no—
185C23—41no—
186C23C2×C22⋊#615711yescharacteristic
187C23#1313329612yes—
188C23C2×D53⋊C811yescharacteristic
189C23C2×#1171994012yes—
190C23C2×#1171993611yescharacteristic
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—534no—
201C2×C4—1062no—
202C2×C4—1064no—
203C2×C4—4242no—
204C2×C4—2122no—
205C2×C4—2124no—
206C2×C4—2122no—
207C2×C4—2122no—
208C2×C4—2122no—
209C2×C4#1313327412yes—
210C2×C4#1313327412yes—
211C8—4244no—
212C22—21no—
213C22—22no—
214C22—21no—
215C22—22no—
216C22—41no—
217C22—42no—
218C22—41no—
219C22#1681156611yescharacteristic
220C22C2×#1313327411yescharacteristic
221C22#1681159311yescharacteristic
222C22#1681164012yes—
223C22#1681164412yes—
224C4—22no—
225C4—22no—
226C4—44no—
227C4—1064no—
228C4—2124no—
229C4—2122no—
230C2—21no—
231C2—21no—
232C2—22no—
233C2—22no—
234C2#2029060611yescharacteristic
235C2#2029062711yescharacteristic
236C2#2029062911yescharacteristic
subgroupG/H
centerC22#16811566
commutatorC23×C53C22×C8
FrattiniC24C22×#6157
FittingC2×#117×C53C4
socleC22×C53#381
prime radicalC24×C53C22×C4
cosocleC23×Dic53C23
hypercenterC2×#117#6157
hypocenterC53#8735
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
20. #13133176C8semidirect
132. C53#8735semidirect

Maximal subgroups

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

subgroupindex
C2×#168114682
#202895582
#202770712
#873553