Finite Group Databasebeta

← #28307984

Subgroups — #28307984

Order 13568.

462 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 104 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#20289660C214yesmaximal
3#20277198C211yesmaximal, characteristic
4#16811639—22no—
5#16811468—22no—
6#16811466C2214yes—
7C2×#13133176C411yescharacteristic
8C2×#13133176C2211yescharacteristic
9#16810521—22no—
10#16810496C412yes—
11#16810496C2212yes—
12C4×#11719919—22no—
13C22×C42×C53C411yescharacteristic
14C4⋊#88324—42no—
15C22⋊#88324—24no—
16C4×#88324—42no—
17#13133176—22no—
18#13133176—41no—
19#13133176—41no—
20#13133176C814yescomplemented
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53D412yes—
23C2×#11719919—22no—
24C2×#11719919—22no—
25C2×C22⋊Dic53C2×C412yes—
26C2×C22⋊Dic53C2×C411yescharacteristic
27C2×C22⋊Dic53C2311yescharacteristic
28C2×C4×Dic53—21no—
29C2×C4×Dic53D412yes—
30#13133124—22no—
31#13133124—22no—
32C23×C4×C53C2×C411yescharacteristic
33C23×C4×C53C2×C412yes—
34C2×C42×C53—21no—
35C2×C42×C53—21no—
36C2×C42×C53—41no—
37C2×C42×C53—41no—
38C24×C53C22×C411yescharacteristic
39C22×C4×C53—21no—
40C22×C4×C53—21no—
41C22×C4×C53—22no—
42C22×C4×C53—22no—
43C22×C4×C53—44no—
44C22×C4×C53—42no—
45C22×C4×C53C22⋊C411yescharacteristic
46C22×C4×C53C22⋊C411yescharacteristic
47C22×C4×C53C2×C812yes—
48C22×C4×C53M1612yes—
49C22×Dic53—22no—
50C22×Dic53C22⋊C412yes—
51C22×Dic53C2×D412yes—
52C2×#88324—44no—
53C42×C53—42no—
54C42×C53—41no—
55C42×C53—41no—
56C22⋊Dic53—22no—
57C22⋊Dic53—41no—
58C22⋊Dic53—41no—
59C22⋊Dic53C2×C814yes—
60C4⋊Dic53—24no—
61C4⋊Dic53—42no—
62#11719919—44no—
63#11719919—44no—
64C4×Dic53—24no—
65C4×Dic53—22no—
66C4×Dic53—22no—
67C23×C53—21no—
68C23×C53—21no—
69C23×C53—42no—
70C23×C53C2×C22⋊C411yescharacteristic
71C23×C53C22×C811yescharacteristic
72C23×C53C2×M1611yescharacteristic
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—24no—
80C2×C4×C53—42no—
81C2×C4×C53—44no—
82C2×C4×C53—42no—
83C2×C4×C53—44no—
84C2×C4×C53C22⋊C812yes—
85C2×C4×C53C22⋊C812yes—
86C2×C4×C53#9911yescharacteristic
87C2×C4×C53#9911yescharacteristic
88C2×C4×C53#10112yes—
89C2×Dic53—24no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—42no—
93C2×Dic53—42no—
94C2×Dic53—42no—
95C2×Dic53—42no—
96#88324—84no—
97#11452—531nomaximal, Sylow
98C22×C53—22no—
99C22×C53—21no—
100C22×C53—21no—
101C22×C53—41no—
102C22×C53—41no—
103C22×C53—42no—
104C22×C53—42no—
105C22×C53C2×C22⋊C811yescharacteristic
106C22×C53C2×#9911yescharacteristic
107C22×C53C2×#10111yescharacteristic
108C4×C53—22no—
109C4×C53—44no—
110C4×C53#32412yes—
111C4×C53#32412yes—
112Dic53—44no—
113Dic53—42no—
114Dic53—42no—
115#2296—531no—
116#1458—532no—
117#1326—534no—
118C2×C53—21no—
119C2×C53—21no—
120C2×C53—41no—
121C2×C53—41no—
122C2×C53C2×#32411yescharacteristic
123C2×C53#212012yes—
124C2×#117—531no—
125C2×#117—531no—
126C22×C42#615711yescharacteristic
127#431—1062no—
128#389—1062no—
129#385—532no—
130#385—532no—
131C4×C4⋊C4—1062no—
132#324—1062no—
133#323—534no—
134C53#1145211yescharacteristic, complemented, Sylow
135C23×C4C2×#615712yes—
136C23×C4C2×#615711yescharacteristic
137#117—534no—
138#117—1062no—
139#117—2121no—
140#117—2121no—
141C2×C4⋊C4—532no—
142C2×C4⋊C4—1061no—
143C2×C4⋊C4—1062no—
144C2×C4⋊C4—1062no—
145C2×C22⋊C4—532no—
146C2×C22⋊C4—531no—
147C2×C22⋊C4—531no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—41no—
151C2×C42—41no—
152C2×C42—532no—
153C2×C42—1061no—
154C4⋊C8—2122no—
155C22⋊C8—1064no—
156C4×C8—2122no—
157#95—1062no—
158#95—1062no—
159C24C22×#615711yescharacteristic
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—44no—
165C22×C4—42no—
166C22×C4—532no—
167C22×C4—532no—
168C22×C4—1062no—
169C22×C4D53⋊C812yes—
170C22×C4#1171993612yes—
171C22×C4C22⋊#615711yescharacteristic
172C22×C4C22⋊#615711yescharacteristic
173C2×C8—2124no—
174C4⋊C4—1064no—
175C4⋊C4—2122no—
176C4⋊C4—2124no—
177C4⋊C4—2124no—
178C22⋊C4—534no—
179C22⋊C4—1062no—
180C22⋊C4—2121no—
181C22⋊C4—2121no—
182C42—42no—
183C42—41no—
184C42—41no—
185C42—1064no—
186C42—1062no—
187C42—1062no—
188C23—21no—
189C23—21no—
190C23—42no—
191C23C2×C22⋊#615711yescharacteristic
192C23C2×D53⋊C811yescharacteristic
193C23C2×#1171993611yescharacteristic
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—42no—
202C2×C4—44no—
203C2×C4—42no—
204C2×C4—44no—
205C2×C4—1064no—
206C2×C4—1062no—
207C2×C4—1062no—
208C2×C4—2122no—
209C2×C4—2122no—
210C2×C4—2122no—
211C2×C4—2122no—
212C2×C4#1313327412yes—
213C2×C4#1313327412yes—
214C2×C4#1313327511yescharacteristic
215C2×C4#1313327511yescharacteristic
216C2×C4#1313327812yes—
217C8—4244no—
218C22—22no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—42no—
224C22—42no—
225C22C2×#1313327411yescharacteristic
226C22C2×#1313327511yescharacteristic
227C22C2×#1313327811yescharacteristic
228C4—22no—
229C4—44no—
230C4—2124no—
231C4—2122no—
232C4—2122no—
233C4#1681146912yes—
234C4#1681146912yes—
235C2—21no—
236C2—21no—
237C2—41no—
238C2—41no—
239C2C2×#1681146911yescharacteristic
240C2#2029096612yes—
subgroupG/H
centerC2×C4#13133275
commutatorC23×C53C22×C8
FrattiniC24C22×#6157
FittingC22×C42×C53C4
socleC22×C53C2×#99
prime radicalC24×C53C22×C4
cosocleC2×C22⋊Dic53C23
hypercenterC22×C42#6157
hypocenterC53#11452
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
20. #13133176C8semidirect
134. C53#11452semidirect

Maximal subgroups

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

subgroupindex
#202899462
#202896602
#202771982
#1145253