Finite Group Databasebeta

← #27295596

Subgroups — #27295596

Order 13568.

300 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 83 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#20285299C211yesmaximal, characteristic
2#20276469C211yesmaximal, characteristic
3#20273477C211yesmaximal, characteristic
4#20272774C211yesmaximal, characteristic, complemented
5#20272772C211yesmaximal, characteristic, complemented
6#20272764C211yesmaximal, characteristic, complemented
7C53×#1362C211yesmaximal, characteristic, complemented
8#16811178—21no—
9C2×#13133156—21no—
10#16811028C2211yescharacteristic
11#16810973—21no—
12#16810861—21no—
13#16810861C412yes—
14Q16×Dic53—21no—
15Q16×Dic53C412yes—
16#16810679—21no—
17C4×C8×D53C2211yescharacteristic
18C4⋊#11719908C2211yescharacteristic
19#16810441—21no—
20#16810376C2211yescharacteristic
21#16810373—21no—
22C53×#494C2211yescharacteristic
23C53×C4⋊C16C2211yescharacteristic
24C53×#335C2211yescharacteristic
25C2×C8×D53C2×C411yescharacteristic
26C2×C8×D53D411yescharacteristic
27#13133217—21no—
28#13133208—41no—
29#13133197—41no—
30#13133193—22no—
31Q8×Dic53—22no—
32#13133182—41no—
33#13133175—41no—
34C42×D53D411yescharacteristic
35#13133156—42no—
36C2×#11719908—21no—
37#13133151—21no—
38#13133151C2×C412yes—
39C4⋊#88322—21no—
40#13133141—22no—
41#13133141—22no—
42#13133129—41no—
43#13133127—41no—
44C4×#88322D411yescharacteristic
45C8×Dic53C2×C411yescharacteristic
46C8×Dic53D411yescharacteristic
47C2×Q16×C53—21no—
48C2×Q16×C53C2×C412yes—
49C4⋊Q8×C53—21no—
50C2×C16×C53—21no—
51C4⋊C8×C53—21no—
52C4×C8×C53C2311yescharacteristic
53C2×Q8×C53—22no—
54C2×C4×D53C22⋊C411yescharacteristic
55C2×C4×D53D811yescharacteristic
56C2×C4×D53SD1611yescharacteristic
57C2×Dic106—41no—
58Q16×C53—42no—
59Q16×C53—41no—
60Q16×C53—41no—
61C2×C8×C53—41no—
62C2×C8×C53C22×C411yescharacteristic
63C2×C8×C53C2×D411yescharacteristic
64C4⋊C4×C53—41no—
65C4⋊C4×C53—41no—
66C42×C53C2×D411yescharacteristic
67#11719921—42no—
68C4⋊Dic53—22no—
69C4⋊Dic53—41no—
70C4⋊Dic53—42no—
71#11719919—42no—
72C4×Dic53—42no—
73C4×Dic53C22⋊C411yescharacteristic
74C4×Dic53D811yescharacteristic
75C4×Dic53SD1611yescharacteristic
76C2×#88322—41no—
77C2×#88322C22⋊C411yescharacteristic
78C2×#88322C22⋊C411yescharacteristic
79C8×D53—21no—
80C8×D53—21no—
81C8×D53—41no—
82C16×C53—41no—
83#11719908—41no—
84C22×D53#10211yescharacteristic
85Q8×C53—42no—
86Q8×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53C2×C22⋊C411yescharacteristic
89C2×C4×C53C2×D811yescharacteristic
90C2×C4×C53C2×SD1611yescharacteristic
91C2×Dic53—42no—
92C2×Dic53#10211yescharacteristic
93C2×Dic53#10211yescharacteristic
94C2×Dic53#10211yescharacteristic
95C4×D53—21no—
96C4×D53—21no—
97C4×D53—21no—
98C4×D53—21no—
99C4×D53—41no—
100Dic106—82no—
101C8×C53—21no—
102C8×C53—81no—
103C8×C53AGL(1,ℤ/8)11yescharacteristic
104C8×C53#13711yescharacteristic
105#88322—21no—
106#88322—21no—
107#88322—81no—
108#11987—531nomaximal, Sylow
109C22×C53C2×#10211yescharacteristic
110C2×D53—21no—
111C2×D53—21no—
112C2×D53—21no—
113C4×C53—21no—
114C4×C53—82no—
115C4×C53#41311yescharacteristic
116C4×C53#41711yescharacteristic
117C4×C53#51011yescharacteristic
118C4×C53#51011yescharacteristic
119Dic53—21no—
120Dic53—21no—
121Dic53—21no—
122Dic53—82no—
123#3170—531no—
124#2150—531no—
125#1568—531no—
126#1365—531no—
127#1365—531no—
128#1362—531no—
129#1362D5311yescharacteristic, complemented
130C2×C53#154711yescharacteristic
131C2×C53#214111yescharacteristic
132C2×C53#214811yescharacteristic
133D53—41no—
134#533—1061no—
135C2×M32—1061no—
136#500—531no—
137#494C2×D5311yescharacteristic
138#487—1061no—
139#467—532no—
140#467—1061no—
141C4×Q16—532no—
142C4×Q16—1061no—
143#432—1061no—
144C2×C4×C8—531no—
145C4⋊C16—531no—
146C4⋊C16—1061no—
147C4⋊C16C2×D5311yescharacteristic
148#335—531no—
149#335C2×D5311yescharacteristic
150#334—1061no—
151C53#1198711yescharacteristic, complemented, Sylow
152C2×Q16—21no—
153C2×Q16C4×D5312yes—
154C22×C8—531no—
155C22×C8—531no—
156C4⋊Q8—21no—
157C4⋊Q8—1061no—
158#125—2121no—
159C22⋊Q8—1062no—
160C22⋊Q8—2121no—
161C4×Q8—1062no—
162C4×Q8—2121no—
163#117—2121no—
164C2×C42—531no—
165M32—2122no—
166C2×C16—21no—
167C2×C16—1061no—
168#107—532no—
169#107—1061no—
170C4⋊C8—21no—
171C4⋊C8—1061no—
172#103—1062no—
173#103—1062no—
174C22⋊C8—2121no—
175#97—2121no—
176C4×C8—531no—
177C4×C8—531no—
178C4×C8—531no—
179C4×C8C22×D5311yescharacteristic
180C2×Q8—22no—
181C2×Q8—2121no—
182C22×C4—531no—
183C22×C4—531no—
184C22×C4—531no—
185Q16—42no—
186Q16—41no—
187Q16—41no—
188C2×C8—41no—
189C2×C8—531no—
190C2×C8—531no—
191C2×C8—1061no—
192C2×C8—1061no—
193C2×C8—2121no—
194C2×C8—2121no—
195C2×C8C2×C4×D5311yescharacteristic
196C2×C8D4×D5311yescharacteristic
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—1062no—
200C4⋊C4—2122no—
201C4⋊C4—2122no—
202C4⋊C4—2121no—
203C22⋊C4—2122no—
204C42—531no—
205C42—531no—
206C42—531no—
207C42—2122no—
208C42D4×D5311yescharacteristic
209C16—41no—
210C16—2121no—
211C23—531no—
212Q8—42no—
213Q8—42no—
214Q8—4242no—
215C2×C4—42no—
216C2×C4—531no—
217C2×C4—531no—
218C2×C4—531no—
219C2×C4—1061no—
220C2×C4—1061no—
221C2×C4—1061no—
222C2×C4—1061no—
223C2×C4—2121no—
224C2×C4—2122no—
225C2×C4C22⋊C4×D5311yescharacteristic
226C2×C4D8×D5311yescharacteristic
227C2×C4SD16×D5311yescharacteristic
228C8—21no—
229C8—81no—
230C8—1061no—
231C8—1061no—
232C8—4241no—
233C8#1313324111yescharacteristic
234C8#1313324611yescharacteristic
235C22—1061no—
236C22—1061no—
237C22—1061no—
238C22#102×D5311yescharacteristic
239C4—21no—
240C4—82no—
241C4—1061no—
242C4—1061no—
243C4—1061no—
244C4—4242no—
245C4#1681060411yescharacteristic
246C4#1681062111yescharacteristic
247C4#1681106211yescharacteristic
248C4#1681106511yescharacteristic
249C2—2121no—
250C2#2027338911yescharacteristic
251C2#2027643311yescharacteristic
252C2#2027646311yescharacteristic
subgroupG/H
centerC22#102×D53
commutatorC2×C8×C53C22×C4
FrattiniC4×C8C22×D53
FittingC53×#1362C2
socleC22×C53C2×#102
prime radicalC22×D53#102
cosocleC4×C8×C53C23
hypercenter#1362D53
hypocenterC53#11987
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
4. #20272774C2semidirect
5. #20272772C2semidirect
6. #20272764C2semidirect
7. C53×#1362C2semidirect
129. #1362D53semidirect
151. C53#11987semidirect

Maximal subgroups

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

subgroupindex
#202852992
#202764692
#202734772
#202727742
#202727722
#202727642
C53×#13622
#1198753