Finite Group Databasebeta

← #28313697

Subgroups — #28313697

Order 13568.

1182 conjugacy classes of subgroups; only the 272 normal subgroups are shown.

normal subgroups only

This group has 1182 conjugacy classes of subgroups — too many to draw. Only its 272 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
1C2×#16811569C211yesmaximal, characteristic
2C2×#16811568C211yesmaximal, characteristic
3#20289903C216yesmaximal
4#20289901C216yesmaximal
5#383×D53C211yesmaximal, characteristic
6C22×C4⋊#6157C2216yes—
7C22×C4⋊#6157C2214yes—
8#16811569—24no—
9#16811568—24no—
10C2×(C2×#6157)⋊C4C2213yes—
11C2×(C2×#6157)⋊C4C2213yes—
12C2×(C2×#6157)⋊C4C22112yes—
13C2×C42×D53C2211yescharacteristic
14#16810491C411yescharacteristic, complemented
15#16810490C416yescomplemented
16#95×D53C2216yes—
17C53×#383C411yescharacteristic, complemented
18C23×#6157C2313yes—
19C23×#6157C2311yescharacteristic
20C23×#6157C2314yes—
21C2×C4⋊#6157—212no—
22C2×C4⋊#6157—412no—
23C2×C4⋊#6157—412no—
24(C2×#6157)⋊C4—26no—
25(C2×#6157)⋊C4—26no—
26(C2×#6157)⋊C4—412no—
27C22×C4×D53C2314yes—
28C22×C4×D53C2313yes—
29C2×C4×Dic53C2×C413yes—
30C42×D53D414yes—
31C42×D53Q814yes—
32#13133124C2×C4112yes—
33#13133124C2×C416yes—
34C2×C42×C53C2×C411yescharacteristic
35#95×C53C2×C416yes—
36C23×D53C2411yescharacteristic
37C22×#6157—21no—
38C22×#6157—23no—
39C22×#6157—24no—
40C22×#6157—26no—
41C22×#6157—212no—
42C22×#6157—212no—
43C22×#6157—26no—
44C22×#6157—212no—
45C22×C4×C53C22×C414yes—
46C22×C4×C53C22×C413yes—
47C22×Dic53C22×C414yes—
48C22×Dic53C22×C413yes—
49C2×C4×D53—26no—
50C2×C4×D53—26no—
51C2×C4×D53—212no—
52C2×C4×D53—212no—
53C2×C4×D53C2×D416yes—
54C2×C4×D53C2×Q816yes—
55C4⋊#6157—424no—
56C4⋊#6157—416no—
57C42×C53C4⋊C414yescomplemented
58C4×Dic53C4⋊C4112yescomplemented
59C23×C53C23×C411yescharacteristic
60C22×D53C22×D411yescharacteristic
61C22×D53C22×Q811yescharacteristic
62C22×D53H(5,𝔽2)16yes—
63C22×D532−1+416yes—
64C2×#6157—412no—
65C2×#6157—43no—
66C2×#6157—43no—
67C2×#6157—46no—
68C2×#6157—44no—
69C2×#6157—44no—
70C2×#6157—412no—
71C2×#6157—412no—
72C2×C4×C53—26no—
73C2×C4×C53—212no—
74C2×C4×C53C2×C4⋊C416yes—
75C2×Dic53—26no—
76C2×Dic53—212no—
77C2×Dic53C2×C4⋊C416yes—
78C4×D53—26no—
79C4×D53—26no—
80C4×D53—212no—
81C4×D53—44no—
82C4×D53—412no—
83#14678—531nomaximal, Sylow
84C22×C53C22×C4⋊C411yescharacteristic
85C22×C53#52016yes—
86C2×D53#53513yes—
87C2×D53#53613yes—
88C2×D53#54416yes—
89C2×D53#56014yes—
90C2×D53#563112yes—
91#6157—84no—
92#6157—81no—
93#6157—83no—
94C4×C53—26no—
95C4×C53—44no—
96Dic53—26no—
97Dic53—44no—
98C2×#384—531no—
99C2×#383—531no—
100C2×#383—531no—
101#1444—536no—
102#1442—536no—
103C2×C53#231713yes—
104C2×C53#238314yes—
105D53#281811yescharacteristic
106D53#282113yes—
107D53#284314yes—
108C22×C4⋊C4—536no—
109C22×C4⋊C4—534no—
110C22×C42—531no—
111#384—1064no—
112#383—531no—
113#383—536no—
114#383—1064no—
115#383#615711yescharacteristic, complemented
116C2×#95—536no—
117C2×#95—533no—
118C2×#95—533no—
119C2×#95—5312no—
120C53#1467811yescharacteristic, complemented, Sylow
121C23×C4—533no—
122C23×C4—534no—
123C23×C4—531no—
124C23×C4—534no—
125C23×C4—533no—
126C2×C4⋊C4—10612no—
127C2×C4⋊C4—21212no—
128C2×C4⋊C4—21212no—
129C2×C42—533no—
130C2×C42—534no—
131C2×C42—534no—
132C2×C42C2×#615711yescharacteristic
133#95—5312no—
134#95—536no—
135#95—1066no—
136#95—1066no—
137#95—21212no—
138#95C2×#615716yes—
139C24—531no—
140C22×C4—533no—
141C22×C4—536no—
142C22×C4—536no—
143C22×C4—534no—
144C22×C4—1066no—
145C22×C4—1066no—
146C22×C4—10612no—
147C22×C4—10612no—
148C22×C4—1061no—
149C22×C4—1063no—
150C22×C4—1066no—
151C22×C4—10612no—
152C22×C4—1064no—
153C22×C4—10612no—
154C22×C4—1066no—
155C22×C4—10612no—
156C22×C4C22×#615714yes—
157C22×C4C22×#615713yes—
158C4⋊C4—21224no—
159C4⋊C4—21216no—
160C42—5312no—
161C42C4⋊#615714yescomplemented
162C23—531no—
163C23—531no—
164C23—536no—
165C23—536no—
166C23C23×#615711yescharacteristic
167C2×C4—26no—
168C2×C4—212no—
169C2×C4—536no—
170C2×C4—1066no—
171C2×C4—1066no—
172C2×C4—10612no—
173C2×C4—1066no—
174C2×C4—10612no—
175C2×C4—2124no—
176C2×C4—21212no—
177C2×C4—21212no—
178C2×C4—2124no—
179C2×C4—2124no—
180C2×C4—2123no—
181C2×C4—2123no—
182C2×C4—2126no—
183C2×C4—21212no—
184C2×C4—21212no—
185C2×C4C2×C4⋊#615716yes—
186C22—533no—
187C22—533no—
188C22—536no—
189C22—534no—
190C22—5312no—
191C22C22×C4⋊#615711yescharacteristic
192C22#1681169216yes—
193C4—26no—
194C4—44no—
195C4—1066no—
196C4—4244no—
197C4—4241no—
198C4—4243no—
199C4—2124no—
200C2—531no—
201C2—533no—
202C2—534no—
203C2#2029119613yes—
204C2#2029126214yes—
subgroupG/H
center, FrattiniC23C23×#6157
commutator, socleC23×C53C23×C4
FittingC53×#383C4
prime radical, cosocleC23×D53C24
hypercenter#383#6157
hypocenterC53#14678
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #383

Complements

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

NH ≅ G/Ntype
14. #16810491C4semidirect
15. #16810490C4semidirect
17. C53×#383C4semidirect
57. C42×C53C4⋊C4semidirect
58. C4×Dic53C4⋊C4semidirect
115. #383#6157semidirect
120. C53#14678semidirect
161. C42C4⋊#6157semidirect

Maximal subgroups

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

subgroupindex
C2×#168115692
C2×#168115682
#202899032
#202899012
#383×D532
#1467853