Finite Group Databasebeta

← #27272353

Subgroups — #27272353

Order 13568.

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

normal subgroups only

This group has 520 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#20274568C212yesmaximal, complemented
2#20272541C212yesmaximal, complemented
3#20272536C212yesmaximal
4C4×#124×C53C211yesmaximal, characteristic
5C2×C4⋊#88322—22no—
6C2×#13133128—22no—
7#16810378C2214yes—
8C2×#124×C53C411yescharacteristic
9C2×C4×Q8×C53C411yescharacteristic
10C2×C4×D4×C53C411yescharacteristic
11C53×#391C2211yescharacteristic
12C53×#390C2211yescharacteristic
13C53×#388—22no—
14C53×#382C411yescharacteristic
15C4×C22⋊C4×C53—22no—
16C43×C53C2211yescharacteristic
17C22×#88322—24no—
18C4⋊#88322—28no—
19#13133128—44no—
20C22×Q8×C53C2×C411yescharacteristic
21C22×D4×C53C2×C411yescharacteristic
22C23×C4×C53—21no—
23#124×C53—22no—
24#124×C53—22no—
25C4×Q8×C53—42no—
26C4×D4×C53—42no—
27C2×C4⋊C4×C53—21no—
28C2×C4⋊C4×C53C2×C411yescharacteristic
29C2×C4⋊C4×C53C2×C411yescharacteristic
30C2×C22⋊C4×C53—22no—
31C2×C22⋊C4×C53—21no—
32C2×C22⋊C4×C53—22no—
33C2×C42×C53—21no—
34C2×C42×C53C2×C411yescharacteristic
35C2×C42×C53C2×C411yescharacteristic
36C2×C42×C53D412yes—
37C2×C42×C53D412yes—
38C2×C42×C53C2311yescharacteristic
39#95×C53—22no—
40#95×C53—22no—
41C24×C53—21no—
42C2×Q8×C53—21no—
43C2×Q8×C53—41no—
44C2×Q8×C53—41no—
45C2×Q8×C53M1612yes—
46C2×D4×C53—21no—
47C2×D4×C53—41no—
48C2×D4×C53—41no—
49C2×D4×C53M1612yes—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—42no—
54C22×C4×C53—44no—
55C22×C4×C53C22⋊C412yes—
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22×C411yescharacteristic
59C22×C4×C53C2×D412yes—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—41no—
62C4⋊C4×C53—41no—
63C22⋊C4×C53—28no—
64C22⋊C4×C53—42no—
65C22⋊C4×C53—42no—
66C22⋊C4×C53—42no—
67C42×C53—22no—
68C42×C53—24no—
69C42×C53—21no—
70C42×C53—21no—
71C42×C53—22no—
72C42×C53—22no—
73C42×C53—41no—
74C42×C53—42no—
75C2×#88322—28no—
76C2×#88322—44no—
77C2×#88322—44no—
78C23×C53—22no—
79C23×C53—41no—
80C23×C53—41no—
81C23×C53—42no—
82C23×C53—42no—
83C23×C53C2×C22⋊C411yescharacteristic
84Q8×C53—42no—
85Q8×C53—42no—
86D4×C53—42no—
87D4×C53—42no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
90C2×C4×C53—21no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—21no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—84no—
105C2×C4×C53#10012yes—
106C2×C4×C53#10414yes—
107C2×C4×C53C2×M1611yescharacteristic
108C2×C4×C53C2×M1611yescharacteristic
109C2×C4×C53AGL(1,ℤ/8)12yes—
110C2×C4×C53#13712yes—
111#88322—48no—
112#7635—531nomaximal, Sylow
113C22×C53—42no—
114C22×C53—44no—
115C22×C53—81no—
116C22×C53—82no—
117C22×C53—81no—
118C22×C53#40711yescharacteristic
119C22×C53#41011yescharacteristic
120C22×C53C2×#10011yescharacteristic
121C22×C53#41712yes—
122C22×C53C2×#10412yes—
123C4×C53—21no—
124C4×C53—21no—
125C4×C53—22no—
126C4×C53—22no—
127C4×C53—41no—
128C4×C53—42no—
129C4×C53—42no—
130C4×C53—81no—
131C4×C53—81no—
132C4×C53#32914yes—
133C4×#124D5311yescharacteristic
134#1767—532no—
135#1272—532no—
136#1270—532no—
137C2×C53—82no—
138C2×C53#146611yescharacteristic
139C2×C53#148912yes—
140C2×C53#150912yes—
141C2×C53C2×#32911yescharacteristic
142C2×C53#156011yescharacteristic
143C2×#124Dic5311yescharacteristic
144C2×C4×Q8Dic5311yescharacteristic
145C2×C4×D4Dic5311yescharacteristic
146C2×C4⋊C8—1062no—
147C2×#97—1062no—
148#391C2×D5311yescharacteristic
149#390C2×D5311yescharacteristic
150#388—22no—
151#382Dic5311yescharacteristic
152C4×C22⋊C4—22no—
153C43C2×D5311yescharacteristic
154#336—534no—
155C53#763511yescharacteristic, complemented, Sylow
156C22×Q8C2×Dic5311yescharacteristic
157C22×D4C2×Dic5311yescharacteristic
158C23×C4—21no—
159C22×C8—1064no—
160#124—22no—
161#124—22no—
162C4×Q8—42no—
163C4×D4—42no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4C2×Dic5311yescharacteristic
166C2×C4⋊C4C2×Dic5311yescharacteristic
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—21no—
170C2×C42—21no—
171C2×C42C2×Dic5311yescharacteristic
172C2×C42C2×Dic5311yescharacteristic
173C2×C42C22⋊D5312yes—
174C2×C42C22⋊D5312yes—
175C2×C42C22×D5311yescharacteristic
176C4⋊C8—1068no—
177#97—2124no—
178#95—22no—
179#95—22no—
180C24—21no—
181C2×Q8—21no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8#1171991712yes—
185C2×D4—21no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4#1171991712yes—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—42no—
193C22×C4—44no—
194C22×C4C22⋊Dic5312yes—
195C22×C4C22⋊Dic5311yescharacteristic
196C22×C4C22⋊Dic5311yescharacteristic
197C22×C4C22×Dic5311yescharacteristic
198C22×C4C2×C22⋊D5312yes—
199C2×C8—1068no—
200C2×C8—2124no—
201C2×C8—2124no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—42no—
205C22⋊C4—28no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C42—22no—
210C42—24no—
211C42—21no—
212C42—21no—
213C42—22no—
214C42—22no—
215C42—41no—
216C42—42no—
217C23—22no—
218C23—41no—
219C23—41no—
220C23—42no—
221C23—42no—
222C23C2×C22⋊Dic5311yescharacteristic
223Q8—42no—
224Q8—42no—
225D4—42no—
226D4—42no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—84no—
244C2×C4#1313313412yes—
245C2×C4#1313314414yes—
246C2×C4C2×#1171991711yescharacteristic
247C2×C4C2×#1171991711yescharacteristic
248C2×C4#1313324212yes—
249C2×C4#1313324712yes—
250C8—2128no—
251C22—44no—
252C22—42no—
253C22—81no—
254C22—82no—
255C22—81no—
256C22#1681058711yescharacteristic
257C22#1681059711yescharacteristic
258C22C2×#1313313411yescharacteristic
259C22#1681062412yes—
260C22C2×#1313314412yes—
261C4—22no—
262C4—22no—
263C4—21no—
264C4—21no—
265C4—42no—
266C4—41no—
267C4—42no—
268C4—81no—
269C4—81no—
270C4#1681036314yes—
271C2—82no—
272C2#2027305611yescharacteristic
273C2#2027315412yes—
274C2#2027324012yes—
275C2C2×#1681036311yescharacteristic
276C2#2027345011yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC4×#124×C53C2
socleC23×C53C2×C22⋊C4
prime radicalC22×D4×C53C2×C4
cosocleC2×C42×C53C23
hypercenterC4×#124D53
hypocenterC53#7635
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C43 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
1. #20274568C2semidirect
2. #20272541C2semidirect
155. C53#7635semidirect

Maximal subgroups

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

subgroupindex
#202745682
#202725412
#202725362
C4×#124×C532
#763553