Finite Group Databasebeta

← #18011032

Subgroups — #18011032

Order 4736.

912 conjugacy classes of subgroups; only the 290 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 290 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
1C22×C22⋊#3866C212yesmaximal
2C22⋊C4×#3866C218yesmaximal, complemented
3C4×C22⋊#3866C214yesmaximal, complemented
4C2×C22⋊C4×D37C211yesmaximal, characteristic
5C23×#3866C2214yes—
6C2×C22⋊#3866C4116yescomplemented
7C2×C22⋊#3866C2214yes—
8C2×C22⋊#3866C2214yes—
9C2×C4×#3866C2218yescomplemented
10(C2×#3866)⋊C4C2218yescomplemented
11C24×D37C2211yescharacteristic
12C22×C4×D37C2212yes—
13C2×C22⋊Dic37C411yescharacteristic, complemented
14C2×#10596938C412yescomplemented
15C22⋊C4×D37C414yescomplemented
16C22⋊C4×D37C2214yes—
17C2×C22⋊C4×C37C411yescharacteristic, complemented
18C24×C37C2×C411yescharacteristic
19C23×D37—44no—
20C23×D37C2×C412yes—
21C23×D37C2×C411yescharacteristic
22C23×D37C2×C418yes—
23C23×D37C2312yes—
24C23×D37C2311yescharacteristic
25C22×#3866—48no—
26C22×#3866C2×C4116yescomplemented
27C22×#3866C2318yes—
28C22×C4×C37C2×C412yescomplemented
29C22×Dic37C2×C412yescomplemented
30C2×C4×D37—44no—
31C2×C4×D37C2×C414yescomplemented
32C2×C4×D37C2314yes—
33C22⋊#3866C2×C4116yescomplemented
34C22⋊#3866C2×C4116yescomplemented
35C4×#3866—48no—
36C22⋊C4×C37C2×C414yescomplemented
37C22⋊Dic37C2×C414yescomplemented
38#10596938C2×C418yescomplemented
39C23×C37—42no—
40C23×C37C4214yescomplemented
41C23×C37C22×C412yes—
42C23×C37C22×C411yescharacteristic
43C22×D37—24no—
44C22×D37—28no—
45C22×D37—416no—
46C22×D37—44no—
47C22×D37—42no—
48C22×D37C4218yescomplemented
49C22×D37C4214yescomplemented
50C22×D37C22×C412yes—
51C22×D37C22×C411yescharacteristic
52C22×D37C22×C414yes—
53C22×D37C22×C414yes—
54C22×D37C22×C414yes—
55C22×D37C2411yescharacteristic
56C2×#3866—216no—
57C2×#3866—416no—
58C2×#3866C22×C4116yes—
59C2×C4×C37—42no—
60C2×C4×C37C22×C414yes—
61C2×Dic37—42no—
62C2×Dic37C22×C414yes—
63C4×D37—28no—
64C4×D37—44no—
65C22×C37—22no—
66C22×C37—24no—
67C22×C37—44no—
68C22×C37C2×C4212yes—
69C22×C37C2×C4214yes—
70C22×C37C23×C411yescharacteristic
71C2×D37—22no—
72C2×D37—24no—
73C2×D37—28no—
74C2×D37—28no—
75C2×D37—28no—
76C2×D37—48no—
77C2×D37—44no—
78C2×D37C2×C4214yes—
79C2×D37C2×C4212yes—
80C2×D37C23×C412yes—
81C2×D37H(5,𝔽2)14yes—
82#3866—216no—
83C4×C37—24no—
84Dic37—24no—
85#2273—371nomaximal, Sylow
86C2×C37—24no—
87C2×C37—22no—
88C2×C37C22×C4211yescharacteristic
89C2×C37#51812yes—
90D37—22no—
91D37—24no—
92D37#51814yes—
93C22×C22⋊C4—372no—
94C22×C22⋊C4—371no—
95C4×C22⋊C4—374no—
96C4×C22⋊C4—378no—
97C37#227311yescharacteristic, complemented, Sylow
98C25—371no—
99C23×C4—374no—
100C23×C4—372no—
101C2×C22⋊C4—374no—
102C2×C22⋊C4—374no—
103C2×C22⋊C4—3716no—
104C2×C22⋊C4—371no—
105C2×C22⋊C4—372no—
106C2×C22⋊C4—374no—
107C2×C22⋊C4—374no—
108C2×C22⋊C4#386611yescharacteristic, complemented
109C2×C42—378no—
110#95—378no—
111C24—371no—
112C24—372no—
113C24—378no—
114C24—371no—
115C24—372no—
116C24—1484no—
117C24C2×#386611yescharacteristic
118C22×C4—378no—
119C22×C4—3716no—
120C22×C4—372no—
121C22×C4—374no—
122C22×C4—374no—
123C22×C4—1484no—
124C22×C4—1488no—
125C22×C4C2×#386612yescomplemented
126C22⋊C4—3716no—
127C22⋊C4—3716no—
128C22⋊C4—374no—
129C22⋊C4—378no—
130C22⋊C4C2×#386614yescomplemented
131C42—1488no—
132C23—42no—
133C23—371no—
134C23—374no—
135C23—374no—
136C23—374no—
137C23—372no—
138C23—371no—
139C23—378no—
140C23—374no—
141C23—744no—
142C23—748no—
143C23—1482no—
144C23—1484no—
145C23—14816no—
146C23C4×#386614yescomplemented
147C23C22×#386612yes—
148C23C22×#386611yescharacteristic
149C2×C4—42no—
150C2×C4—3716no—
151C2×C4—374no—
152C2×C4—748no—
153C2×C4—7416no—
154C2×C4—1484no—
155C2×C4—1482no—
156C2×C4—14816no—
157C2×C4C22×#386614yes—
158C22—22no—
159C22—24no—
160C22—44no—
161C22—372no—
162C22—374no—
163C22—372no—
164C22—374no—
165C22—748no—
166C22—748no—
167C22—748no—
168C22—742no—
169C22—744no—
170C22—1484no—
171C22—1488no—
172C22C2×C4×#386612yes—
173C22C2×C4×#386614yes—
174C22C23×#386611yescharacteristic
175C4—24no—
176C4—7416no—
177C4—744no—
178C2—22no—
179C2—24no—
180C2—374no—
181C2—744no—
182C2—742no—
183C2C22×C4×#386611yescharacteristic
184C2#1455860012yes—
subgroupG/H
center, FrattiniC22C23×#3866
commutatorC2×C37C22×C42
FittingC2×C22⋊C4×C37C4
socleC22×C37C23×C4
prime radicalC24×D37C22
cosocleC22×D37C24
hypercenterC2×C22⋊C4#3866
hypocenterC37#2273
LCSG ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22⋊C4×#3866C2semidirect
3. C4×C22⋊#3866C2semidirect
6. C2×C22⋊#3866C4semidirect
9. C2×C4×#3866C22semidirect
10. (C2×#3866)⋊C4C22semidirect
13. C2×C22⋊Dic37C4semidirect
14. C2×#10596938C4semidirect
15. C22⋊C4×D37C4semidirect
17. C2×C22⋊C4×C37C4semidirect
26. C22×#3866C2×C4semidirect
28. C22×C4×C37C2×C4semidirect
29. C22×Dic37C2×C4semidirect
31. C2×C4×D37C2×C4semidirect
33. C22⋊#3866C2×C4semidirect
34. C22⋊#3866C2×C4semidirect
36. C22⋊C4×C37C2×C4semidirect
37. C22⋊Dic37C2×C4semidirect
38. #10596938C2×C4semidirect
40. C23×C37C42semidirect
48. C22×D37C42semidirect
49. C22×D37C42semidirect
97. C37#2273semidirect
108. C2×C22⋊C4#3866semidirect
125. C22×C4C2×#3866semidirect
130. C22⋊C4C2×#3866semidirect
146. C23C4×#3866semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊#38662
C22⋊C4×#38662
C4×C22⋊#38662
C2×C22⋊C4×D372
#227337