Finite Group Databasebeta

← #27277504

Subgroups — #27277504

Order 13568.

526 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 107 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#20277121C211yesmaximal, characteristic
2#20274478C212yesmaximal
3#20272597C212yesmaximal, complemented
4#20272596C211yesmaximal, characteristic, complemented
5C53×#1295C211yesmaximal, characteristic, complemented
6C2×#13133175C412yes—
7C2×C42×D53C2211yescharacteristic
8C2×#13133129—22no—
9C2×#13133128C2212yes—
10#16810492C412yes—
11#16810489C2211yescharacteristic
12#16810482—22no—
13#16810474—22no—
14#16810377—22no—
15C2×#97×C53C2212yes—
16C53×#382C2211yescharacteristic
17C22×C4×D53C2×C411yescharacteristic
18C22×C4×D53D412yes—
19C22×#88322—22no—
20#13133175—44no—
21C2×C4⋊Dic53C2×C412yes—
22C2×#11719919—22no—
23C2×#11719919—22no—
24C2×#11719921—22no—
25C2×C4×Dic53—22no—
26C2×C4×Dic53C2×C411yescharacteristic
27C2×C4×Dic53D412yes—
28C42×D53—22no—
29C42×D53—21no—
30C42×D53—21no—
31#13133129—28no—
32#13133128—22no—
33#13133128—22no—
34#13133124—22no—
35C22×C8×C53—22no—
36C2×C4⋊C4×C53C2×C412yes—
37C2×C42×C53C2311yescharacteristic
38#97×C53—22no—
39#97×C53—22no—
40#95×C53—22no—
41C23×D53C22⋊C411yescharacteristic
42C22×C4×C53—22no—
43C22×C4×C53C22×C411yescharacteristic
44C22×C4×C53C2×D412yes—
45C22×Dic53—22no—
46C22×Dic53C22⋊C411yescharacteristic
47C22×Dic53C22⋊C412yes—
48C2×C4×D53—22no—
49C2×C4×D53—21no—
50C2×C4×D53—21no—
51C2×C4×D53—22no—
52C2×C4×D53—22no—
53C2×C4×D53—22no—
54C2×C4×D53—24no—
55C2×C4×D53—42no—
56C2×C8×C53—24no—
57C2×C8×C53—42no—
58C2×C8×C53—42no—
59C4⋊C4×C53—42no—
60C42×C53#4112yes—
61C42×C53#4111yescharacteristic
62C42×C53#4111yescharacteristic
63#11719921—44no—
64C4⋊Dic53—42no—
65#11719919—44no—
66#11719919—44no—
67C4×Dic53—21no—
68C4×Dic53—21no—
69C4×Dic53—22no—
70C4×Dic53—22no—
71C4×Dic53—44no—
72C2×#88322—24no—
73C2×#88322—42no—
74C2×#88322—42no—
75C23×C53C2×C22⋊C411yescharacteristic
76C22×D53—21no—
77C22×D53—21no—
78C22×D53—22no—
79C22×D53—22no—
80C22×D53#10012yes—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53#11711yescharacteristic
89C2×C4×C53#11711yescharacteristic
90C2×C4×C53#12312yes—
91C2×C4×C53#12312yes—
92C2×Dic53—21no—
93C2×Dic53—22no—
94C2×Dic53—21no—
95C2×Dic53—22no—
96C2×Dic53—42no—
97C2×Dic53—42no—
98C2×Dic53—42no—
99C2×Dic53#10112yes—
100C2×Dic53#10414yes—
101C4×D53—24no—
102C4×D53—24no—
103C4×D53—44no—
104C4×D53—44no—
105C8×C53—44no—
106#88322—44no—
107#8722—531nomaximal, Sylow
108C22×C53#38111yescharacteristic
109C22×C53C2×#10011yescharacteristic
110C22×C53C2×#10111yescharacteristic
111C22×C53C2×#10412yes—
112C22×C53#42112yes—
113C2×D53—22no—
114C2×D53—24no—
115C2×D53—41no—
116C2×D53—42no—
117C2×D53—41no—
118C4×C53—21no—
119C4×C53—21no—
120C4×C53—22no—
121C4×C53—22no—
122C4×C53—82no—
123Dic53—22no—
124Dic53—24no—
125Dic53—82no—
126#2285—531no—
127#1749—532no—
128#1295—532no—
129#1295—531no—
130#1295D5311yescharacteristic, complemented
131C2×C53#152611yescharacteristic
132C2×C53#153012yes—
133C2×C53#153711yescharacteristic
134C2×C53#182111yescharacteristic
135C2×C53#182612yes—
136D53—42no—
137C2×#117—532no—
138C22×C42—531no—
139C2×C22⋊C8—1062no—
140C2×#97—532no—
141C2×#97C2×D5312yes—
142#384—532no—
143#382—531no—
144#382C2×D5311yescharacteristic
145#380—1062no—
146C4×C4⋊C4—1062no—
147#336—1062no—
148C53#872211yescharacteristic, complemented, Sylow
149C23×C4—531no—
150C23×C4—532no—
151C22×C8—22no—
152C22×C8—1062no—
153#117—2124no—
154C2×C4⋊C4—532no—
155C2×C4⋊C4—1062no—
156C2×C4⋊C4—1062no—
157C2×C4⋊C4C4×D5312yes—
158C2×C22⋊C4—1062no—
159C2×C42—531no—
160C2×C42—532no—
161C2×C42—1062no—
162C2×C42—1061no—
163C2×C42—1061no—
164C2×C42—1062no—
165C2×C42C22×D5311yescharacteristic
166C22⋊C8—1068no—
167#97—22no—
168#97—22no—
169#97—1062no—
170#97—1062no—
171#95—22no—
172#95—1062no—
173C24—531no—
174C22×C4—22no—
175C22×C4—531no—
176C22×C4—532no—
177C22×C4—1062no—
178C22×C4—1064no—
179C22×C4—1062no—
180C22×C4—1061no—
181C22×C4—1061no—
182C22×C4—1062no—
183C22×C4—1062no—
184C22×C4—1062no—
185C22×C4—2122no—
186C22×C4C2×C4×D5311yescharacteristic
187C22×C4D4×D5312yes—
188C2×C8—24no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—1064no—
192C2×C8—2122no—
193C2×C8—2122no—
194C4⋊C4—42no—
195C4⋊C4—2122no—
196C4⋊C4—2124no—
197C4⋊C4—2124no—
198C22⋊C4—2124no—
199C42—1061no—
200C42—1061no—
201C42—1062no—
202C42—1062no—
203C42—2124no—
204C42#1171994512yes—
205C42#1171994711yescharacteristic
206C42#1171994911yescharacteristic
207C23—532no—
208C23—1061no—
209C23—1062no—
210C23—1062no—
211C23—1061no—
212C23C22⋊C4×D5311yescharacteristic
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—532no—
221C2×C4—534no—
222C2×C4—1064no—
223C2×C4—1064no—
224C2×C4—1061no—
225C2×C4—1062no—
226C2×C4—1061no—
227C2×C4—1062no—
228C2×C4—2124no—
229C2×C4—2124no—
230C2×C4—2122no—
231C2×C4—2122no—
232C2×C4—2122no—
233C2×C4#1313317411yescharacteristic
234C2×C4#1313317511yescharacteristic
235C2×C4#1313319912yes—
236C2×C4#1313320212yes—
237C8—44no—
238C8—2124no—
239C22—1062no—
240C22—1064no—
241C22—2122no—
242C22—2121no—
243C22—2121no—
244C22#1681048511yescharacteristic
245C22#100×D5311yescharacteristic
246C22#101×D5311yescharacteristic
247C22#104×D5312yes—
248C22#1681063812yes—
249C4—21no—
250C4—21no—
251C4—22no—
252C4—22no—
253C4—82no—
254C4—1062no—
255C4—1064no—
256C4—4242no—
257C2—2122no—
258C2#2027330811yescharacteristic
259C2#2027332412yes—
260C2#2027335111yescharacteristic
261C2#2027479411yescharacteristic
262C2#2027481312yes—
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1295C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1295D53
hypocenterC53#8722
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
3. #20272597C2semidirect
4. #20272596C2semidirect
5. C53×#1295C2semidirect
130. #1295D53semidirect
148. C53#8722semidirect

Maximal subgroups

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

subgroupindex
#202771212
#202744782
#202725972
#202725962
C53×#12952
#872253