Finite Group Databasebeta

← H(5,𝔽2)×C3⋊(C3⋊D3) #13181764

Subgroups — H(5,𝔽2)×C3⋊(C3⋊D3) #13181764

Order 1728.

12572 conjugacy classes of subgroups; only the 2213 normal subgroups are shown.

normal subgroups only

This group has 12572 conjugacy classes of subgroups — too many to draw. Only its 2213 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
1C33×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2#11724684C216yesmaximal, complemented
3#41×C3⋊(C3⋊D3)C216yesmaximal, complemented
4#11724679C219yesmaximal, complemented
5C2×D4×C3⋊(C3⋊D3)C219yesmaximal, complemented
6C3⋊D3×H(5,𝔽2)—313nomaximal
7C23×C3⋊(C3⋊D3)C2216yescomplemented
8C33×#41C2216yescomplemented
9C2×C33×D4C2219yescomplemented
10C2×#6312C22118yescomplemented
11#89080C2216yescomplemented
12Q8×C3⋊(C3⋊D3)C2212yescomplemented
13#89078C22118yescomplemented
14D4×C3⋊(C3⋊D3)C22118yescomplemented
15D4×C3⋊(C3⋊D3)C22136yescomplemented
16#89076C22118yescomplemented
17C2×C4⋊(C3⋊(C3⋊D3))C2219yescomplemented
18C2×C4×C3⋊(C3⋊D3)C2219yescomplemented
19C32×H(5,𝔽2)D3113yescomplemented
20#64366—378no—
21#41×C3⋊D3—378no—
22#64361—3117no—
23C2×D4×C3⋊D3—3117no—
24C23×C33C2316yescomplemented
25C22×C3⋊(C3⋊D3)—412no—
26C22×C3⋊(C3⋊D3)C2319yes—
27C22×C3⋊(C3⋊D3)C23118yescomplemented
28C33×Q8C2312yescomplemented
29C33×D4C23118yescomplemented
30C2×C33×C4C2319yescomplemented
31#6312C23136yescomplemented
32C2×C3⋊(C3⋊Dic3)C2319yescomplemented
33C4⋊(C3⋊(C3⋊D3))C23118yescomplemented
34C4×C3⋊(C3⋊D3)C2316yes—
35C4×C3⋊(C3⋊D3)C23118yescomplemented
36#6308C2316yescomplemented
37D3×H(5,𝔽2)—913no—
38C23×C3⋊D3—378no—
39C32×#41C2×D3178yescomplemented
40C2×C32×D4C2×D31117yescomplemented
41C2×#638—3234no—
42#3832—378no—
43Q8×C3⋊D3—326no—
44#3830—3234no—
45D4×C3⋊D3—3234no—
46D4×C3⋊D3—3468no—
47#3828—3234no—
48C2×C4⋊(C3⋊D3)—3117no—
49C2×C4×C3⋊D3—3117no—
50C22×C33—46no—
51C22×C33C2419yes—
52C2×C3⋊(C3⋊D3)—218no—
53C2×C3⋊(C3⋊D3)—418no—
54C2×C3⋊(C3⋊D3)C2419yes—
55C2×C3⋊(C3⋊D3)C2411yescharacteristic
56C33×C4C2416yes—
57C3⋊(C3⋊Dic3)C2416yes—
58C3×H(5,𝔽2)C3⋊D3113yescomplemented
59#1009—978no—
60D3×#41—978no—
61#1004—9117no—
62C2×D3×D4—9117no—
63C23×C32C22×D3178yescomplemented
64C22×C3⋊D3—3117no—
65C22×C3⋊D3—3234no—
66C22×C3⋊D3—12156no—
67C32×Q8C22×D3126yescomplemented
68C32×D4C22×D31234yescomplemented
69C2×C32×C4C22×D31117yescomplemented
70#638—3468no—
71C2×C3⋊Dic3—3117no—
72C4⋊(C3⋊D3)—3234no—
73C4×C3⋊D3—378no—
74C4×C3⋊D3—3234no—
75#634—378no—
76C2×H(5,𝔽2)—271noSylow
77C2×C33—29no—
78C2×C33C2511yescharacteristic
79C3⋊(C3⋊D3)—29no—
80C3⋊(C3⋊D3)H(5,𝔽2)12yesdirect product factor
81C23×D3—978no—
82C3×#41C2×C3⋊D3178yescomplemented
83C2×C3×D4C2×C3⋊D31117yescomplemented
84C2×#67—9234no—
85#239—978no—
86D3×Q8—926no—
87#237—9234no—
88D3×D4—9234no—
89D3×D4—9468no—
90#235—9234no—
91C2×D12—9117no—
92C2×C4×D3—9117no—
93C22×C32—478no—
94C22×C32C23×D31117yes—
95C2×C3⋊D3—313no—
96C2×C3⋊D3—3117no—
97C2×C3⋊D3—6234no—
98C2×C3⋊D3—12234no—
99C32×C4C23×D3178yes—
100C3⋊Dic3—378no—
101H(5,𝔽2)—276no—
102H(5,𝔽2)—279no—
103H(5,𝔽2)C3⋊(C3⋊D3)11yescharacteristic, direct product factor
104C2×#41—276no—
105C22×D4—279no—
106C33C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
107C23×C3C22×C3⋊D3178yescomplemented
108C22×D3—9117no—
109C22×D3—9234no—
110C22×D3—36156no—
111C3×Q8C22×C3⋊D3126yescomplemented
112C3×D4C22×C3⋊D31234yescomplemented
113C2×C3×C4C22×C3⋊D31117yescomplemented
114#67—9468no—
115C2×Dic3—9117no—
116D12—9234no—
117C4×D3—978no—
118C4×D3—9234no—
119Dic6—978no—
120C2×C32—2117no—
121C2×C32C24×D3113yes—
122C3⋊D3—326no—
123C3⋊D3—6117no—
124C24—276no—
125#41—276no—
126#41—2718no—
127#41—2718no—
128#41C2×C3⋊(C3⋊D3)16yescomplemented
129C2×Q8—272no—
130C2×D4—2718no—
131C2×D4—2718no—
132C2×D4—2736no—
133C2×D4—279no—
134C2×D4C2×C3⋊(C3⋊D3)19yescomplemented
135C22×C4—279no—
136C22×C3—478no—
137C22×C3C23×C3⋊D31117yes—
138C2×D3—913no—
139C2×D3—9117no—
140C2×D3—18234no—
141C2×D3—36234no—
142C3×C4C23×C3⋊D3178yes—
143Dic3—978no—
144C32D3×H(5,𝔽2)113yescomplemented
145C23—279no—
146C23—2718no—
147C23—10812no—
148C23C22×C3⋊(C3⋊D3)16yescomplemented
149Q8—276no—
150Q8C22×C3⋊(C3⋊D3)12yescomplemented
151D4—2736no—
152D4—2718no—
153D4C22×C3⋊(C3⋊D3)118yescomplemented
154C2×C4—279no—
155C2×C4—276no—
156C2×C4—2718no—
157C2×C4C22×C3⋊(C3⋊D3)19yescomplemented
158C2×C3—2117no—
159C2×C3C24×C3⋊D3113yes—
160D3—926no—
161D3—18117no—
162C22—46no—
163C22—271no—
164C22—279no—
165C22—5418no—
166C22—10818no—
167C22C23×C3⋊(C3⋊D3)19yes—
168C4—276no—
169C4C23×C3⋊(C3⋊D3)16yes—
170C3C3⋊D3×H(5,𝔽2)113yescomplemented
171C2—29no—
172C2—272no—
173C2—549no—
174C2C24×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
center, FrattiniC2C24×C3⋊(C3⋊D3)
commutator, socle, cosocleC2×C33C25
FittingC33×H(5,𝔽2)C2
hypercenterH(5,𝔽2)C3⋊(C3⋊D3)
hypocenterC33C2×H(5,𝔽2)
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C33×H(5,𝔽2)C2semidirect
2. #11724684C2semidirect
3. #41×C3⋊(C3⋊D3)C2semidirect
4. #11724679C2semidirect
5. C2×D4×C3⋊(C3⋊D3)C2semidirect
7. C23×C3⋊(C3⋊D3)C22semidirect
8. C33×#41C22semidirect
9. C2×C33×D4C22semidirect
10. C2×#6312C22semidirect
11. #89080C22semidirect
12. Q8×C3⋊(C3⋊D3)C22semidirect
13. #89078C22semidirect
14. D4×C3⋊(C3⋊D3)C22semidirect
15. D4×C3⋊(C3⋊D3)C22semidirect
16. #89076C22semidirect
17. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
18. C2×C4×C3⋊(C3⋊D3)C22semidirect
19. C32×H(5,𝔽2)D3semidirect
24. C23×C33C23semidirect
27. C22×C3⋊(C3⋊D3)C23semidirect
28. C33×Q8C23semidirect
29. C33×D4C23semidirect
30. C2×C33×C4C23semidirect
31. #6312C23semidirect
32. C2×C3⋊(C3⋊Dic3)C23semidirect
33. C4⋊(C3⋊(C3⋊D3))C23semidirect
35. C4×C3⋊(C3⋊D3)C23semidirect
36. #6308C23semidirect
39. C32×#41C2×D3semidirect
40. C2×C32×D4C2×D3semidirect
58. C3×H(5,𝔽2)C3⋊D3semidirect
63. C23×C32C22×D3semidirect
67. C32×Q8C22×D3semidirect
68. C32×D4C22×D3semidirect
69. C2×C32×C4C22×D3semidirect
80. C3⋊(C3⋊D3)H(5,𝔽2)direct (N × H)
82. C3×#41C2×C3⋊D3semidirect
83. C2×C3×D4C2×C3⋊D3semidirect
103. H(5,𝔽2)C3⋊(C3⋊D3)direct (N × H)
106. C33C2×H(5,𝔽2)semidirect
107. C23×C3C22×C3⋊D3semidirect
111. C3×Q8C22×C3⋊D3semidirect
112. C3×D4C22×C3⋊D3semidirect
113. C2×C3×C4C22×C3⋊D3semidirect
128. #41C2×C3⋊(C3⋊D3)semidirect
134. C2×D4C2×C3⋊(C3⋊D3)semidirect
144. C32D3×H(5,𝔽2)semidirect
148. C23C22×C3⋊(C3⋊D3)semidirect
150. Q8C22×C3⋊(C3⋊D3)semidirect
153. D4C22×C3⋊(C3⋊D3)semidirect
157. C2×C4C22×C3⋊(C3⋊D3)semidirect
170. C3C3⋊D3×H(5,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
C33×H(5,𝔽2)2
#117246842
#41×C3⋊(C3⋊D3)2
#117246792
C2×D4×C3⋊(C3⋊D3)2
C3⋊D3×H(5,𝔽2)3