Subgroups — C₃²×C₄×C₅×D₄ #13088227

Order 1440.

564 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₅×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃²×C₄×C₅	C₂	1	2	yes	maximal, complemented
3	C₃²×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₅×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₃²×C₄²×C₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₄×C₅×D₄	C₃	1	4	yes	maximal, direct product factor
7	C₂³×C₃²×C₅	C₂²	1	2	yes	—
8	C₃²×C₅×D₄	C₄	1	4	yes	direct product factor
9	C₂×C₃²×C₄×C₅	—	2	4	no	—
10	C₂×C₃²×C₄×C₅	C₂²	1	2	yes	—
11	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₃²×C₄×D₄	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂×C₃×C₅×D₄	C₂×C₃	1	4	yes	—
16	C₂²×C₃×C₄×C₅	C₂×C₃	1	8	yes	complemented
17	C₃×C₅×C₄⋊C₄	C₂×C₃	1	4	yes	complemented
18	C₃×C₅×C₂²⋊C₄	C₂×C₃	1	8	yes	complemented
19	C₃×C₄²×C₅	C₂×C₃	1	4	yes	complemented
20	C₂²×C₃²×C₅	—	2	2	no	—
21	C₂²×C₃²×C₅	—	2	2	no	—
22	C₂²×C₃²×C₅	C₂×C₄	1	4	yes	complemented
23	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
24	C₃²×C₄×C₅	—	2	2	no	—
25	C₃²×C₄×C₅	—	2	1	no	—
26	C₃²×C₄×C₅	C₂×C₄	1	2	yes	complemented
27	C₃²×C₄×C₅	D₄	1	2	yes	direct product factor
28	C₄×C₅×D₄	C₃²	1	1	yes	characteristic, direct product factor, Hall
29	C₂×C₃²×D₄	C₂×C₅	1	1	yes	characteristic
30	C₂²×C₃²×C₄	C₂×C₅	1	2	yes	complemented
31	C₃²×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
32	C₃²×C₂²⋊C₄	C₂×C₅	1	2	yes	complemented
33	C₃²×C₄²	C₂×C₅	1	1	yes	characteristic, complemented
34	C₂³×C₃×C₅	C₂²×C₃	1	8	yes	—
35	C₃×C₅×D₄	C₃×C₄	1	16	yes	direct product factor
36	C₂×C₃×C₄×C₅	—	2	16	no	—
37	C₂×C₃×C₄×C₅	C₂²×C₃	1	4	yes	—
38	C₂×C₃×C₄×C₅	C₂²×C₃	1	8	yes	—
39	C₂×C₃×C₄×C₅	C₂²×C₃	1	4	yes	—
40	C₂×C₃×C₄×C₅	C₂²×C₃	1	4	yes	—
41	C₃×C₄×D₄	C₃×C₅	1	4	yes	direct product factor
42	C₂×C₃²×C₅	—	2	4	no	—
43	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
45	C₂×C₃²×C₅	#41	1	1	yes	characteristic
46	C₂×C₅×D₄	C₂×C₃²	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂×C₃²	1	2	yes	complemented
48	C₅×C₄⋊C₄	C₂×C₃²	1	1	yes	characteristic, complemented
49	C₅×C₂²⋊C₄	C₂×C₃²	1	2	yes	complemented
50	C₄²×C₅	C₂×C₃²	1	1	yes	characteristic, complemented
51	C₂³×C₃²	C₂²×C₅	1	2	yes	—
52	C₃²×D₄	C₄×C₅	1	4	yes	direct product factor
53	C₂×C₃²×C₄	—	2	4	no	—
54	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
55	C₂×C₃²×C₄	C₂²×C₅	1	2	yes	—
56	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
57	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
58	C₂²×C₃×C₅	—	2	8	no	—
59	C₂²×C₃×C₅	—	2	8	no	—
60	C₂²×C₃×C₅	C₂×C₃×C₄	1	16	yes	complemented
61	C₂²×C₃×C₅	C₂³×C₃	1	4	yes	—
62	C₃×C₄×C₅	—	2	8	no	—
63	C₃×C₄×C₅	—	2	4	no	—
64	C₃×C₄×C₅	C₂×C₃×C₄	1	8	yes	complemented
65	C₃×C₄×C₅	C₃×D₄	1	8	yes	direct product factor
66	C₂×C₃×D₄	C₂×C₃×C₅	1	4	yes	—
67	C₂²×C₃×C₄	C₂×C₃×C₅	1	8	yes	complemented
68	C₃×C₄⋊C₄	C₂×C₃×C₅	1	4	yes	complemented
69	C₃×C₂²⋊C₄	C₂×C₃×C₅	1	8	yes	complemented
70	C₃×C₄²	C₂×C₃×C₅	1	4	yes	complemented
71	C₃²×C₅	C₄×D₄	1	1	yes	characteristic, direct product factor, Hall
72	C₂³×C₅	C₂²×C₃²	1	2	yes	—
73	C₅×D₄	C₃²×C₄	1	4	yes	direct product factor
74	C₂×C₄×C₅	—	2	4	no	—
75	C₂×C₄×C₅	C₂²×C₃²	1	1	yes	characteristic
76	C₂×C₄×C₅	C₂²×C₃²	1	2	yes	—
77	C₂×C₄×C₅	C₂²×C₃²	1	1	yes	characteristic
78	C₂×C₄×C₅	C₂²×C₃²	1	1	yes	characteristic
79	C₂²×C₃²	—	2	2	no	—
80	C₂²×C₃²	—	2	2	no	—
81	C₂²×C₃²	C₂×C₄×C₅	1	4	yes	complemented
82	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
83	C₃²×C₄	—	2	2	no	—
84	C₃²×C₄	—	2	1	no	—
85	C₃²×C₄	C₂×C₄×C₅	1	2	yes	complemented
86	C₃²×C₄	C₅×D₄	1	2	yes	direct product factor
87	C₄×D₄	C₃²×C₅	1	1	yes	characteristic, direct product factor, Sylow
88	C₂×C₃×C₅	—	2	16	no	—
89	C₂×C₃×C₅	C₂²×C₃×C₄	1	4	yes	—
90	C₂×C₃×C₅	C₂×C₃×D₄	1	4	yes	—
91	C₂×C₃×C₅	C₃×#41	1	4	yes	—
92	C₂³×C₃	C₂²×C₃×C₅	1	8	yes	—
93	C₃×D₄	C₃×C₄×C₅	1	16	yes	direct product factor
94	C₂×C₃×C₄	—	2	16	no	—
95	C₂×C₃×C₄	C₂²×C₃×C₅	1	4	yes	—
96	C₂×C₃×C₄	C₂²×C₃×C₅	1	8	yes	—
97	C₂×C₃×C₄	C₂²×C₃×C₅	1	4	yes	—
98	C₂×C₃×C₄	C₂²×C₃×C₅	1	4	yes	—
99	C₂²×C₅	—	2	2	no	—
100	C₂²×C₅	—	2	2	no	—
101	C₂²×C₅	C₂×C₃²×C₄	1	4	yes	complemented
102	C₂²×C₅	C₂³×C₃²	1	1	yes	characteristic
103	C₄×C₅	—	2	2	no	—
104	C₄×C₅	—	2	1	no	—
105	C₄×C₅	C₂×C₃²×C₄	1	2	yes	complemented
106	C₄×C₅	C₃²×D₄	1	2	yes	direct product factor
107	C₂×C₃²	—	2	4	no	—
108	C₂×C₃²	C₂²×C₄×C₅	1	1	yes	characteristic
109	C₂×C₃²	C₂×C₅×D₄	1	1	yes	characteristic
110	C₂×C₃²	C₅×#41	1	1	yes	characteristic
111	C₂×D₄	C₂×C₃²×C₅	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₃²×C₅	1	2	yes	complemented
113	C₄⋊C₄	C₂×C₃²×C₅	1	1	yes	characteristic, complemented
114	C₂²⋊C₄	C₂×C₃²×C₅	1	2	yes	complemented
115	C₄²	C₂×C₃²×C₅	1	1	yes	characteristic, complemented
116	C₃×C₅	C₃×C₄×D₄	1	4	yes	direct product factor
117	C₂²×C₃	—	2	8	no	—
118	C₂²×C₃	—	2	8	no	—
119	C₂²×C₃	C₂×C₃×C₄×C₅	1	16	yes	complemented
120	C₂²×C₃	C₂³×C₃×C₅	1	4	yes	—
121	C₃×C₄	—	2	8	no	—
122	C₃×C₄	—	2	4	no	—
123	C₃×C₄	C₂×C₃×C₄×C₅	1	8	yes	complemented
124	C₃×C₄	C₃×C₅×D₄	1	8	yes	direct product factor
125	C₂×C₅	—	2	4	no	—
126	C₂×C₅	C₂²×C₃²×C₄	1	1	yes	characteristic
127	C₂×C₅	C₂×C₃²×D₄	1	1	yes	characteristic
128	C₂×C₅	C₃²×#41	1	1	yes	characteristic
129	C₃²	C₄×C₅×D₄	1	1	yes	characteristic, direct product factor, Sylow
130	C₂³	C₂²×C₃²×C₅	1	2	yes	—
131	D₄	C₃²×C₄×C₅	1	4	yes	direct product factor
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	C₂²×C₃²×C₅	1	1	yes	characteristic
134	C₂×C₄	C₂²×C₃²×C₅	1	2	yes	—
135	C₂×C₄	C₂²×C₃²×C₅	1	1	yes	characteristic
136	C₂×C₄	C₂²×C₃²×C₅	1	1	yes	characteristic
137	C₂×C₃	—	2	16	no	—
138	C₂×C₃	C₂²×C₃×C₄×C₅	1	4	yes	—
139	C₂×C₃	C₂×C₃×C₅×D₄	1	4	yes	—
140	C₂×C₃	C₃×C₅×#41	1	4	yes	—
141	C₅	C₃²×C₄×D₄	1	1	yes	characteristic, direct product factor, Sylow
142	C₂²	—	2	2	no	—
143	C₂²	—	2	2	no	—
144	C₂²	C₂×C₃²×C₄×C₅	1	4	yes	complemented
145	C₂²	C₂³×C₃²×C₅	1	1	yes	characteristic
146	C₄	—	2	2	no	—
147	C₄	—	2	1	no	—
148	C₄	C₂×C₃²×C₄×C₅	1	2	yes	complemented
149	C₄	C₃²×C₅×D₄	1	2	yes	direct product factor
150	C₃	C₃×C₄×C₅×D₄	1	4	yes	direct product factor
151	C₂	—	2	4	no	—
152	C₂	C₂²×C₃²×C₄×C₅	1	1	yes	characteristic
153	C₂	C₂×C₃²×C₅×D₄	1	1	yes	characteristic
154	C₂	C₃²×C₅×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²×C₄×C₅	C₂²
commutator	C₂	C₂²×C₃²×C₄×C₅
Frattini, cosocle	C₂²	C₂³×C₃²×C₅
socle	C₂²×C₃²×C₅	C₂³
prime radical	C₂×C₃²×C₅×D₄	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃²×C₄×C₅	C₂	semidirect
3. C₃²×C₅×C₄⋊C₄	C₂	semidirect
4. C₃²×C₅×C₂²⋊C₄	C₂	semidirect
5. C₃²×C₄²×C₅	C₂	semidirect
6. C₃×C₄×C₅×D₄	C₃	direct (N × H)
8. C₃²×C₅×D₄	C₄	direct (N × H)
14. C₃²×C₄×D₄	C₅	direct (N × H)
16. C₂²×C₃×C₄×C₅	C₂×C₃	semidirect
17. C₃×C₅×C₄⋊C₄	C₂×C₃	semidirect
18. C₃×C₅×C₂²⋊C₄	C₂×C₃	semidirect
19. C₃×C₄²×C₅	C₂×C₃	semidirect
22. C₂²×C₃²×C₅	C₂×C₄	semidirect
26. C₃²×C₄×C₅	C₂×C₄	semidirect
27. C₃²×C₄×C₅	D₄	direct (N × H)
28. C₄×C₅×D₄	C₃²	direct (N × H)
30. C₂²×C₃²×C₄	C₂×C₅	semidirect
31. C₃²×C₄⋊C₄	C₂×C₅	semidirect
32. C₃²×C₂²⋊C₄	C₂×C₅	semidirect
33. C₃²×C₄²	C₂×C₅	semidirect
35. C₃×C₅×D₄	C₃×C₄	direct (N × H)
41. C₃×C₄×D₄	C₃×C₅	direct (N × H)
47. C₂²×C₄×C₅	C₂×C₃²	semidirect
48. C₅×C₄⋊C₄	C₂×C₃²	semidirect
49. C₅×C₂²⋊C₄	C₂×C₃²	semidirect
50. C₄²×C₅	C₂×C₃²	semidirect
52. C₃²×D₄	C₄×C₅	direct (N × H)
60. C₂²×C₃×C₅	C₂×C₃×C₄	semidirect
64. C₃×C₄×C₅	C₂×C₃×C₄	semidirect
65. C₃×C₄×C₅	C₃×D₄	direct (N × H)
67. C₂²×C₃×C₄	C₂×C₃×C₅	semidirect
68. C₃×C₄⋊C₄	C₂×C₃×C₅	semidirect
69. C₃×C₂²⋊C₄	C₂×C₃×C₅	semidirect
70. C₃×C₄²	C₂×C₃×C₅	semidirect
71. C₃²×C₅	C₄×D₄	direct (N × H)
73. C₅×D₄	C₃²×C₄	direct (N × H)
81. C₂²×C₃²	C₂×C₄×C₅	semidirect
85. C₃²×C₄	C₂×C₄×C₅	semidirect
86. C₃²×C₄	C₅×D₄	direct (N × H)
87. C₄×D₄	C₃²×C₅	direct (N × H)
93. C₃×D₄	C₃×C₄×C₅	direct (N × H)
101. C₂²×C₅	C₂×C₃²×C₄	semidirect
105. C₄×C₅	C₂×C₃²×C₄	semidirect
106. C₄×C₅	C₃²×D₄	direct (N × H)
112. C₂²×C₄	C₂×C₃²×C₅	semidirect
113. C₄⋊C₄	C₂×C₃²×C₅	semidirect
114. C₂²⋊C₄	C₂×C₃²×C₅	semidirect
115. C₄²	C₂×C₃²×C₅	semidirect
116. C₃×C₅	C₃×C₄×D₄	direct (N × H)
119. C₂²×C₃	C₂×C₃×C₄×C₅	semidirect
123. C₃×C₄	C₂×C₃×C₄×C₅	semidirect
124. C₃×C₄	C₃×C₅×D₄	direct (N × H)
129. C₃²	C₄×C₅×D₄	direct (N × H)
131. D₄	C₃²×C₄×C₅	direct (N × H)
141. C₅	C₃²×C₄×D₄	direct (N × H)
144. C₂²	C₂×C₃²×C₄×C₅	semidirect
148. C₄	C₂×C₃²×C₄×C₅	semidirect
149. C₄	C₃²×C₅×D₄	direct (N × H)
150. C₃	C₃×C₄×C₅×D₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₅×D₄	2
C₂²×C₃²×C₄×C₅	2
C₃²×C₅×C₄⋊C₄	2
C₃²×C₅×C₂²⋊C₄	2
C₃²×C₄²×C₅	2
C₃×C₄×C₅×D₄	3
C₃²×C₄×D₄	5

Subgroups — C32×C4×C5×D4 #13088227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄×C₅×D₄ #13088227