Finite Group Databasebeta

← C23×#13133286 #28329318

Subgroups — C23×#13133286 #28329318

Order 13568.

3614 conjugacy classes of subgroups; only the 1464 normal subgroups are shown.

normal subgroups only

This group has 3614 conjugacy classes of subgroups — too many to draw. Only its 1464 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
1C23×C4⋊#6157C212yesmaximal
2C22×#13133286C2128yesmaximal, direct product factor
3C23×C8×D53C211yesmaximal, characteristic
4C24×#6157—22no—
5C22×C4⋊#6157—214no—
6C22×C4⋊#6157C22128yes—
7C2×#13133286C221112yesdirect product factor
8C23×C4×D53C2211yescharacteristic
9C23×#88322C411yescharacteristic, complemented
10C22×C8×D53C4114yescomplemented
11C22×C8×D53C22114yes—
12C23×C8×C53C411yescharacteristic, complemented
13C23×#6157—228no—
14C23×#6157—416no—
15C2×C4⋊#6157—284no—
16C2×C4⋊#6157C23156yes—
17#13133286C23164yesdirect product factor
18C24×D53D411yescharacteristic
19C23×Dic53Q811yescharacteristic
20C22×C4×D53C2×C417yes—
21C22×C4×D53D417yes—
22C22×C4×D53Q817yes—
23C22×C4×D53C2317yes—
24C22×#88322C2×C4114yescomplemented
25C2×C8×D53C2×C4184yescomplemented
26C2×C8×D53C23128yes—
27C23×C4×C53C2×C411yescharacteristic
28C22×C8×C53C2×C4114yescomplemented
29C24×C53C4⋊C411yescharacteristic
30C23×D53C4⋊C417yes—
31C23×D53D818yes—
32C23×D53Q1618yes—
33C23×D53C2×D417yes—
34C22×#6157—256no—
35C22×#6157—4112no—
36C22×C4×C53C4⋊C417yes—
37C22×C4×C53C22×C417yes—
38C22×Dic53C4⋊C417yes—
39C22×Dic53C2×Q817yes—
40C2×C4×D53C4⋊C4142yes—
41C2×C4×D53C22×C4121yes—
42C2×C4×D53C2×D4121yes—
43C2×C4×D53C2×Q8121yes—
44C2×C4×D53C2417yes—
45C4⋊#6157—256no—
46C4⋊#6157C24116yes—
47C2×C8×C53C22×C4128yescomplemented
48C2×#88322C22×C4128yescomplemented
49C8×D53C22×C4156yescomplemented
50C8×D53C2418yes—
51C23×C53#10718yescomplemented
52C23×C53C2×C4⋊C417yes—
53C22×D53#107156yescomplemented
54C22×D53C2×C4⋊C4121yes—
55C22×D53C2×D8128yes—
56C22×D53C2×Q16128yes—
57C22×D53C22×D417yes—
58C2×#6157—216no—
59C2×#6157—4112no—
60C2×C4×C53C2×C4⋊C4121yes—
61C2×C4×C53C23×C417yes—
62C2×Dic53C2×C4⋊C4121yes—
63C2×Dic53C22×Q817yes—
64C4×D53C2×C4⋊C4142yes—
65C4×D53C23×C417yes—
66C4×D53C22×D417yes—
67C4×D53C22×Q817yes—
68C4×D53C2511yescharacteristic
69C8×C53C23×C418yescomplemented
70#88322C23×C418yescomplemented
71C23×#107—531nomaximal, Sylow
72C22×C53C2×#107128yescomplemented
73C22×C53C22×C4⋊C417yes—
74C2×D53C2×#107184yescomplemented
75C2×D53C22×C4⋊C417yes—
76C2×D53C22×D8114yes—
77C2×D53C22×Q16114yes—
78C2×D53C23×D411yescharacteristic
79#6157—416no—
80C4×C53C22×C4⋊C417yes—
81C4×C53C24×C411yescharacteristic
82Dic53C22×C4⋊C417yes—
83Dic53C23×Q811yescharacteristic
84C24×C8—531no—
85C23×C4⋊C4—532no—
86C22×#107—5328no—
87C2×C53C22×#107114yescomplemented
88C2×C53C23×C4⋊C411yescharacteristic
89D53C22×#107114yescomplemented
90D53C23×D811yescharacteristic
91D53C23×Q1611yescharacteristic
92C24×C4—531no—
93C24×C4—1062no—
94C23×C8—531no—
95C23×C8—5314no—
96C23×C8—5314no—
97C23×C8#615711yescharacteristic, complemented
98C22×C4⋊C4—5328no—
99C22×C4⋊C4—10614no—
100C2×#107—53112no—
101C53C23×#10711yescharacteristic, complemented, Sylow
102C25—531no—
103C23×C4—531no—
104C23×C4—537no—
105C23×C4—537no—
106C23×C4—537no—
107C23×C4—537no—
108C23×C4—10628no—
109C23×C4—21216no—
110C23×C4C2×#615711yescharacteristic
111C22×C8—5314no—
112C22×C8—5384no—
113C22×C8—5328no—
114C22×C8C2×#6157114yescomplemented
115C2×C4⋊C4—5356no—
116C2×C4⋊C4—10684no—
117#107—5364no—
118C24—538no—
119C24—538no—
120C24—537no—
121C24—537no—
122C24C4⋊#615711yescharacteristic
123C22×C4—537no—
124C22×C4—537no—
125C22×C4—5342no—
126C22×C4—5321no—
127C22×C4—5321no—
128C22×C4—5321no—
129C22×C4—537no—
130C22×C4—10656no—
131C22×C4—212112no—
132C22×C4C4⋊#615717yes—
133C22×C4C22×#615717yes—
134C2×C8—5328no—
135C2×C8—538no—
136C2×C8—5356no—
137C2×C8C22×#6157128yescomplemented
138C4⋊C4—5316no—
139C4⋊C4—10656no—
140C23—5356no—
141C23—5328no—
142C23—5328no—
143C23—5321no—
144C23—537no—
145C23#1313328618yesdirect product factor
146C23C2×C4⋊#615717yes—
147C2×C4—5321no—
148C2×C4—537no—
149C2×C4—5342no—
150C2×C4—537no—
151C2×C4—537no—
152C2×C4—537no—
153C2×C4—531no—
154C2×C4—10616no—
155C2×C4—212112no—
156C2×C4C2×C4⋊#6157121yes—
157C2×C4C23×#615717yes—
158C8—538no—
159C8C23×#615718yescomplemented
160C22—5314no—
161C22—5314no—
162C22—5384no—
163C22—531no—
164C22—537no—
165C22C2×#13133286128yesdirect product factor
166C22C22×C4⋊#615717yes—
167C4—537no—
168C4—531no—
169C4—21216no—
170C4C22×C4⋊#615717yes—
171C4C24×#615711yescharacteristic
172C2—531no—
173C2—5314no—
174C2—531no—
175C2C22×#13133286114yesdirect product factor
176C2C23×C4⋊#615711yescharacteristic
subgroupG/H
centerC24C4⋊#6157
commutatorC4×C53C24×C4
FrattiniC4C24×#6157
FittingC23×C8×C53C4
socleC24×C53C4⋊C4
prime radicalC24×D53D4
cosocleC4×D53C25
hypercenterC23×C8#6157
hypocenterC53C23×#107
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C24 ◁ C23×C4 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
2. C22×#13133286C2direct (N × H)
7. C2×#13133286C22direct (N × H)
9. C23×#88322C4semidirect
10. C22×C8×D53C4semidirect
12. C23×C8×C53C4semidirect
17. #13133286C23direct (N × H)
24. C22×#88322C2×C4semidirect
25. C2×C8×D53C2×C4semidirect
28. C22×C8×C53C2×C4semidirect
47. C2×C8×C53C22×C4semidirect
48. C2×#88322C22×C4semidirect
49. C8×D53C22×C4semidirect
51. C23×C53#107semidirect
53. C22×D53#107semidirect
69. C8×C53C23×C4semidirect
70. #88322C23×C4semidirect
72. C22×C53C2×#107semidirect
74. C2×D53C2×#107semidirect
87. C2×C53C22×#107semidirect
89. D53C22×#107semidirect
97. C23×C8#6157semidirect
101. C53C23×#107semidirect
114. C22×C8C2×#6157semidirect
137. C2×C8C22×#6157semidirect
145. C23#13133286direct (N × H)
159. C8C23×#6157semidirect
165. C22C2×#13133286direct (N × H)
175. C2C22×#13133286direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4⋊#61572
C22×#131332862
C23×C8×D532
C23×#10753