Finite Group Databasebeta

← #27272072

Subgroups — #27272072

Order 13568.

516 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 143 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#20277284C211yesmaximal, characteristic
2#20274563C212yesmaximal
3#20272541C211yesmaximal, characteristic
4#20272540C212yesmaximal
5C53×#1272C211yesmaximal, characteristic
6C2×#13133215C411yescharacteristic
7C2×Q8×Dic53C411yescharacteristic
8C2×#13133182C411yescharacteristic
9C2×#13133146—22no—
10C2×C8×Dic53—22no—
11#16810529C2211yescharacteristic
12#16810520—22no—
13#16810492C411yescharacteristic
14#16810474—22no—
15C42×Dic53C2211yescharacteristic
16#16810378C2212yes—
17C53×#391C2211yescharacteristic
18C53×#336C2212yes—
19C22×Dic106C2×C411yescharacteristic
20C22×#88322—22no—
21#13133215—22no—
22#13133215C814yescomplemented
23Q8×Dic53—42no—
24#13133182—42no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719919—22no—
28C2×#11719919—21no—
29C2×#11719919—22no—
30C2×C4×Dic53—21no—
31C2×C4×Dic53—21no—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53C2×C411yescharacteristic
34C2×C4×Dic53D412yes—
35C2×C4×Dic53D412yes—
36#13133146—28no—
37C8×Dic53—44no—
38#13133124—22no—
39C22×Q8×C53C2×C411yescharacteristic
40C22×C8×C53—22no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C42×C53C2311yescharacteristic
43#95×C53—22no—
44C2×Q8×C53—21no—
45C2×Q8×C53—41no—
46C2×Q8×C53—41no—
47C2×Q8×C53C2×C812yes—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53C22×C411yescharacteristic
51C22×C4×C53C2×D412yes—
52C22×Dic53—21no—
53C22×Dic53—21no—
54C22×Dic53C22⋊C411yescharacteristic
55C22×Dic53C22⋊C412yes—
56C22×Dic53C22⋊C411yescharacteristic
57C2×Dic106—21no—
58C2×Dic106—41no—
59C2×Dic106—41no—
60C2×Dic106C2×C812yes—
61C2×C8×C53—24no—
62C2×C8×C53—42no—
63C2×C8×C53—42no—
64C4⋊C4×C53—41no—
65C42×C53—41no—
66C4⋊Dic53—42no—
67C4⋊Dic53—41no—
68#11719919—28no—
69#11719919—42no—
70#11719919—44no—
71C4×Dic53—22no—
72C4×Dic53—24no—
73C4×Dic53—42no—
74C4×Dic53—42no—
75C4×Dic53C2×C812yes—
76C4×Dic53M1612yes—
77C4×Dic53SD1614yes—
78C4×Dic53Q1614yes—
79C2×#88322—24no—
80C2×#88322—42no—
81C2×#88322—42no—
82C23×C53C2×C22⋊C411yescharacteristic
83Q8×C53—42no—
84Q8×C53—42no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—42no—
94C2×C4×C53C22×C811yescharacteristic
95C2×C4×C53C2×M1611yescharacteristic
96C2×C4×C53C2×SD1612yes—
97C2×C4×C53C2×Q1612yes—
98C2×Dic53—22no—
99C2×Dic53—21no—
100C2×Dic53—22no—
101C2×Dic53—41no—
102C2×Dic53—41no—
103C2×Dic53—41no—
104C2×Dic53—42no—
105C2×Dic53C22⋊C814yes—
106C2×Dic53#10012yes—
107C2×Dic53#10314yes—
108C2×Dic53#10314yes—
109C2×Dic53#10414yes—
110Dic106—42no—
111Dic106—42no—
112C8×C53—44no—
113#88322—44no—
114#7571—531nomaximal, Sylow
115C22×C53C2×C22⋊C811yescharacteristic
116C22×C53#41011yescharacteristic
117C22×C53C2×#10011yescharacteristic
118C22×C53C2×#10312yes—
119C22×C53C2×#10412yes—
120C4×C53—21no—
121C4×C53—21no—
122C4×C53—22no—
123C4×C53—22no—
124C4×C53—42no—
125C4×C53—81no—
126Dic53—42no—
127Dic53—41no—
128Dic53—81no—
129Dic53Q8⋊C818yes—
130Dic53#33314yes—
131C4×C4⋊Q8—531no—
132C4×C4⋊C8—532no—
133#1272—531no—
134#1272—532no—
135#1272D5311yescharacteristic
136C2×C53#146011yescharacteristic
137C2×C53C2×Q8⋊C812yes—
138C2×C53#150312yes—
139C2×C53#152511yescharacteristic
140C2×C53C2×#33311yescharacteristic
141C2×C4⋊Q8—531no—
142C2×C4×Q8—531no—
143C2×C4×Q8—531no—
144C2×C4⋊C8—1062no—
145C2×C4×C8—1062no—
146#391—531no—
147#391C2×D5311yescharacteristic
148#389—1062no—
149#384—531no—
150C4×C4⋊C4—1062no—
151C43—531no—
152#336—532no—
153#336C2×D5312yes—
154C53#757111yescharacteristic, complemented, Sylow
155C22×Q8—531no—
156C22×Q8C4×D5311yescharacteristic
157C22×C8—22no—
158C22×C8—1062no—
159C4⋊Q8—534no—
160C4⋊Q8—1062no—
161C4×Q8—2122no—
162C4×Q8—2122no—
163C2×C4⋊C4—531no—
164C2×C4⋊C4—1062no—
165C2×C4⋊C4—1061no—
166C2×C4⋊C4—1061no—
167C2×C4⋊C4—1062no—
168C2×C4⋊C4C4×D5311yescharacteristic
169C2×C42—531no—
170C2×C42—532no—
171C2×C42—531no—
172C2×C42—532no—
173C2×C42—1061no—
174C2×C42—1061no—
175C2×C42C22×D5311yescharacteristic
176C4⋊C8—1068no—
177C4×C8—2124no—
178#95—22no—
179#95—1062no—
180C2×Q8—21no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—532no—
184C2×Q8—1061no—
185C2×Q8—2121no—
186C2×Q8—2121no—
187C2×Q8C8×D5312yes—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—531no—
191C22×C4—532no—
192C22×C4—531no—
193C22×C4—1061no—
194C22×C4—1061no—
195C22×C4C2×C4×D5311yescharacteristic
196C22×C4D4×D5312yes—
197C2×C8—24no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—1064no—
201C2×C8—2122no—
202C2×C8—2122no—
203C4⋊C4—41no—
204C4⋊C4—1068no—
205C4⋊C4—2121no—
206C4⋊C4—2124no—
207C4⋊C4—2122no—
208C4⋊C4—2122no—
209C42—41no—
210C42—532no—
211C42—532no—
212C42—534no—
213C42—534no—
214C42—1064no—
215C42—1062no—
216C42—2122no—
217C42—2122no—
218C23C22⋊C4×D5311yescharacteristic
219Q8—42no—
220Q8—42no—
221Q8—2122no—
222Q8—2122no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—534no—
233C2×C4—532no—
234C2×C4—534no—
235C2×C4—534no—
236C2×C4—534no—
237C2×C4—1062no—
238C2×C4—1061no—
239C2×C4—1062no—
240C2×C4—2121no—
241C2×C4—2122no—
242C2×C4—2121no—
243C2×C4—2121no—
244C2×C4C2×C8×D5311yescharacteristic
245C2×C4M16×D5311yescharacteristic
246C2×C4SD16×D5312yes—
247C2×C4Q16×D5312yes—
248C8—44no—
249C8—2124no—
250C22C22⋊C8×D5311yescharacteristic
251C22#1681059511yescharacteristic
252C22#100×D5311yescharacteristic
253C22#103×D5312yes—
254C22#104×D5312yes—
255C4—22no—
256C4—22no—
257C4—21no—
258C4—21no—
259C4—42no—
260C4—81no—
261C4—538no—
262C4—534no—
263C4—4241no—
264C4—2122no—
265C4—2121no—
266C2#2027303411yescharacteristic
267C2Q8⋊C8×D5312yes—
268C2#2027321212yes—
269C2#2027330511yescharacteristic
270C2#333×D5311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC53×#1272C2
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1272D53
hypocenterC53#7571
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
22. #13133215C8semidirect
154. C53#7571semidirect

Maximal subgroups

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

subgroupindex
#202772842
#202745632
#202725412
#202725402
C53×#12722
#757153