Finite Group Databasebeta

← #27277528

Subgroups — #27277528

Order 13568.

514 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 135 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#20277120C211yesmaximal, characteristic
2#20274692C212yesmaximal
3#20272599C212yesmaximal, complemented
4#20272598C211yesmaximal, characteristic, complemented
5C53×#1296C211yesmaximal, characteristic, complemented
6C2×#13133175C412yes—
7C2×C42×D53C2211yescharacteristic
8C2×C4⋊#88322C2212yes—
9C2×#13133129—22no—
10#16810493C2211yescharacteristic
11#16810488C412yes—
12#16810480—22no—
13C4⋊C4×Dic53—22no—
14#16810377—22no—
15C2×C4⋊C8×C53C2212yes—
16C53×#384C2211yescharacteristic
17C22×C4×D53C2×C411yescharacteristic
18C22×C4×D53D412yes—
19C22×#88322—22no—
20#13133175—44no—
21C2×C4⋊Dic53—22no—
22C2×C4⋊Dic53C2×C412yes—
23C2×#11719921—22no—
24C2×C4×Dic53—22no—
25C2×C4×Dic53C2×C411yescharacteristic
26C2×C4×Dic53D412yes—
27C42×D53—22no—
28C42×D53—21no—
29C42×D53—21no—
30C4⋊#88322—22no—
31C4⋊#88322—22no—
32#13133129—44no—
33#13133124—24no—
34C22×C8×C53—22no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53C2×C412yes—
37C2×C42×C53C2311yescharacteristic
38C4⋊C8×C53—22no—
39C4⋊C8×C53—22no—
40C23×D53C22⋊C411yescharacteristic
41C22×C4×C53—22no—
42C22×C4×C53C22×C411yescharacteristic
43C22×C4×C53C2×D412yes—
44C22×Dic53—22no—
45C22×Dic53C22⋊C412yes—
46C22×Dic53C22⋊C411yescharacteristic
47C2×C4×D53—22no—
48C2×C4×D53—21no—
49C2×C4×D53—21no—
50C2×C4×D53—22no—
51C2×C4×D53—22no—
52C2×C4×D53—22no—
53C2×C4×D53—22no—
54C2×C4×D53—22no—
55C2×C4×D53—22no—
56C2×C4×D53—22no—
57C2×C8×C53—42no—
58C2×C8×C53—42no—
59C2×C8×C53—42no—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—42no—
62C4⋊C4×C53—42no—
63C42×C53#4112yes—
64C42×C53#4111yescharacteristic
65C42×C53#4111yescharacteristic
66#11719921—44no—
67C4⋊Dic53—42no—
68C4⋊Dic53—42no—
69C4⋊Dic53—42no—
70C4×Dic53—21no—
71C4×Dic53—21no—
72C4×Dic53—22no—
73C4×Dic53—22no—
74C4×Dic53—44no—
75C2×#88322—42no—
76C2×#88322—42no—
77C2×#88322—42no—
78C23×C53C2×C22⋊C411yescharacteristic
79C22×D53—21no—
80C22×D53—21no—
81C22×D53—22no—
82C22×D53—22no—
83C22×D53#10112yes—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—42no—
87C2×C4×C53—44no—
88C2×C4×C53#11711yescharacteristic
89C2×C4×C53#11711yescharacteristic
90C2×C4×C53#12312yes—
91C2×C4×C53#12312yes—
92C2×C4×C53#13512yes—
93C2×C4×C53#13512yes—
94C2×C4×C53AGL(1,ℤ/8)12yes—
95C2×C4×C53#13712yes—
96C2×Dic53—21no—
97C2×Dic53—21no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—42no—
102C2×Dic53—44no—
103C2×Dic53#10012yes—
104C4×D53—24no—
105C4×D53—24no—
106C4×D53—42no—
107C4×D53—42no—
108C4×D53—42no—
109C4×D53—42no—
110C8×C53—82no—
111#88322—82no—
112#8728—531nomaximal, Sylow
113C22×C53#38111yescharacteristic
114C22×C53C2×#10011yescharacteristic
115C22×C53C2×#10111yescharacteristic
116C22×C53#41612yes—
117C22×C53#41712yes—
118C2×D53—24no—
119C2×D53—22no—
120C2×D53—41no—
121C2×D53—41no—
122C2×D53—42no—
123C4×C53—21no—
124C4×C53—21no—
125C4×C53—82no—
126C4×C53#48212yes—
127C4×C53#48212yes—
128C4×C53#48512yes—
129C4×C53#48512yes—
130Dic53—22no—
131Dic53—42no—
132Dic53—82no—
133#2285—531no—
134#1797—532no—
135#1296—532no—
136#1296—531no—
137#1296D5311yescharacteristic, complemented
138C2×C53#154311yescharacteristic
139C2×C53#155212yes—
140C2×C53#156011yescharacteristic
141C2×C53#182111yescharacteristic
142C2×C53#182512yes—
143D53—42no—
144C2×#117—532no—
145C22×C42—531no—
146C2×C4⋊C8—532no—
147C2×C4⋊C8C2×D5312yes—
148C2×C22⋊C8—1062no—
149#384—531no—
150#384C2×D5311yescharacteristic
151#382—532no—
152#380—1062no—
153C4×C4⋊C4—1062no—
154#336—1062no—
155C53#872811yescharacteristic, complemented, Sylow
156C23×C4—531no—
157C23×C4—532no—
158C22×C8—22no—
159C22×C8—1062no—
160#117—2124no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—532no—
163C2×C4⋊C4—1062no—
164C2×C4⋊C4C4×D5312yes—
165C2×C22⋊C4—1062no—
166C2×C42—531no—
167C2×C42—532no—
168C2×C42—1062no—
169C2×C42—1061no—
170C2×C42—1061no—
171C2×C42—1062no—
172C2×C42C22×D5311yescharacteristic
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—1062no—
176C4⋊C8—1062no—
177C22⋊C8—2124no—
178#95—1064no—
179C24—531no—
180C22×C4—22no—
181C22×C4—531no—
182C22×C4—532no—
183C22×C4—1062no—
184C22×C4—1062no—
185C22×C4—1061no—
186C22×C4—1061no—
187C22×C4—1062no—
188C22×C4—1062no—
189C22×C4—1062no—
190C22×C4—1062no—
191C22×C4—1062no—
192C22×C4—1062no—
193C22×C4—1062no—
194C22×C4C2×C4×D5311yescharacteristic
195C22×C4D4×D5312yes—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—2122no—
200C2×C8—2122no—
201C2×C8—2122no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—2122no—
206C4⋊C4—2122no—
207C4⋊C4—2122no—
208C22⋊C4—2124no—
209C42—1061no—
210C42—1061no—
211C42—1062no—
212C42—1062no—
213C42—2124no—
214C42#1171994512yes—
215C42#1171994711yescharacteristic
216C42#1171994911yescharacteristic
217C23—532no—
218C23—1061no—
219C23—1062no—
220C23—1062no—
221C23—1061no—
222C23C22⋊C4×D5311yescharacteristic
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—42no—
226C2×C4—44no—
227C2×C4—532no—
228C2×C4—1064no—
229C2×C4—1064no—
230C2×C4—1061no—
231C2×C4—1061no—
232C2×C4—1062no—
233C2×C4—1062no—
234C2×C4—1062no—
235C2×C4—2122no—
236C2×C4—2124no—
237C2×C4—2122no—
238C2×C4—2122no—
239C2×C4—2122no—
240C2×C4—2122no—
241C2×C4#1313317411yescharacteristic
242C2×C4#1313317511yescharacteristic
243C2×C4#1313319912yes—
244C2×C4#1313320212yes—
245C2×C4#1313323812yes—
246C2×C4#1313324012yes—
247C2×C4#1313324112yes—
248C2×C4#1313324612yes—
249C8—82no—
250C8—4242no—
251C22—1064no—
252C22—1062no—
253C22—2121no—
254C22—2122no—
255C22—2121no—
256C22#1681048511yescharacteristic
257C22#100×D5311yescharacteristic
258C22#101×D5311yescharacteristic
259C22#1681061812yes—
260C22#1681062112yes—
261C4—21no—
262C4—21no—
263C4—82no—
264C4—1062no—
265C4—4242no—
266C4—2122no—
267C4#1681094612yes—
268C4#1681095012yes—
269C4#1681096312yes—
270C4#1681096712yes—
271C2—2122no—
272C2#2027337611yescharacteristic
273C2#2027341412yes—
274C2#2027344711yescharacteristic
275C2#2027479411yescharacteristic
276C2#2027480912yes—
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1296C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1296D53
hypocenterC53#8728
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
3. #20272599C2semidirect
4. #20272598C2semidirect
5. C53×#1296C2semidirect
137. #1296D53semidirect
155. C53#8728semidirect

Maximal subgroups

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

subgroupindex
#202771202
#202746922
#202725992
#202725982
C53×#12962
#872853