Subgroups — C₂×C₅×#3828 #13089491

Order 1440.

984 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₄×C₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#638	C₂	1	2	yes	maximal, complemented
3	C₅×#3828	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×C₅×C₃⋊D₃	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#634	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#235	—	3	4	no	maximal
8	C₂³×C₃²×C₅	C₂²	1	1	yes	characteristic
9	C₂²×C₅×C₃⋊D₃	C₂²	1	2	yes	—
10	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₃²×C₄×C₅	C₂²	1	4	yes	complemented
13	C₅×#638	C₂²	1	8	yes	complemented
14	C₂×C₅×C₃⋊Dic₃	C₂²	1	2	yes	—
15	C₅×C₄⋊(C₃⋊D₃)	C₂²	1	4	yes	complemented
16	C₄×C₅×C₃⋊D₃	C₂²	1	8	yes	complemented
17	C₅×#634	C₂²	1	4	yes	complemented
18	C₂×#3828	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂²×C₃×C₄×C₅	D₃	1	4	yes	complemented
20	C₂×C₅×#67	—	3	8	no	—
21	C₅×#235	—	3	32	no	—
22	C₂×C₅×D₁₂	—	3	4	no	—
23	C₂×C₄×C₅×D₃	—	3	8	no	—
24	C₂×C₅×Dic₆	—	3	4	no	—
25	C₂²×C₃²×C₅	—	2	2	no	—
26	C₂²×C₃²×C₅	C₂³	1	2	yes	—
27	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₅×C₃⋊D₃	—	2	4	no	—
29	C₂×C₅×C₃⋊D₃	C₂³	1	4	yes	—
30	C₃²×C₄×C₅	C₂³	1	2	yes	—
31	C₃²×C₄×C₅	C₂³	1	2	yes	—
32	C₅×C₃⋊Dic₃	C₂³	1	4	yes	—
33	C₂×C₅×#41	—	9	1	no	Hall
34	C₂²×C₃²×C₄	C₂×C₅	1	1	yes	characteristic, complemented
35	C₂×#638	C₂×C₅	1	2	yes	complemented
36	#3828	C₂×C₅	1	8	yes	direct product factor
37	C₂×C₄⋊(C₃⋊D₃)	C₂×C₅	1	1	yes	characteristic, complemented
38	C₂×C₄×C₃⋊D₃	C₂×C₅	1	2	yes	complemented
39	C₂×#634	C₂×C₅	1	1	yes	characteristic, complemented
40	C₂³×C₃×C₅	C₂×D₃	1	4	yes	—
41	C₂²×C₅×D₃	—	3	8	no	—
42	C₂×C₃×C₄×C₅	C₂×D₃	1	4	yes	—
43	C₂×C₃×C₄×C₅	C₂×D₃	1	4	yes	—
44	C₂×C₃×C₄×C₅	C₂×D₃	1	16	yes	complemented
45	C₅×#67	—	3	32	no	—
46	C₂×C₅×Dic₃	—	3	8	no	—
47	C₅×D₁₂	—	3	16	no	—
48	C₄×C₅×D₃	—	3	32	no	—
49	C₅×Dic₆	—	3	16	no	—
50	C₂×#235	—	3	4	no	—
51	C₂×C₃²×C₅	—	2	2	no	—
52	C₂×C₃²×C₅	#41	1	2	yes	complemented
53	C₂×C₃²×C₅	C₂⁴	1	1	yes	characteristic
54	C₅×C₃⋊D₃	—	2	4	no	—
55	C₅×#41	—	9	8	no	—
56	C₂×C₅×Q₈	—	9	1	no	—
57	C₂×C₅×D₄	—	9	2	no	—
58	C₂×C₅×D₄	—	9	1	no	—
59	C₂²×C₄×C₅	—	9	2	no	—
60	C₂²×C₄×C₅	C₃⋊D₃	1	1	yes	characteristic, complemented
61	C₂³×C₃²	C₂²×C₅	1	1	yes	characteristic
62	C₂²×C₃⋊D₃	C₂²×C₅	1	2	yes	—
63	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₂²×C₅	1	4	yes	complemented
66	#638	C₂²×C₅	1	8	yes	complemented
67	C₂×C₃⋊Dic₃	C₂²×C₅	1	2	yes	—
68	C₄⋊(C₃⋊D₃)	C₂²×C₅	1	4	yes	complemented
69	C₄×C₃⋊D₃	C₂²×C₅	1	8	yes	complemented
70	#634	C₂²×C₅	1	4	yes	complemented
71	C₂²×C₃×C₅	—	2	8	no	—
72	C₂²×C₃×C₅	C₂²×D₃	1	8	yes	—
73	C₂²×C₃×C₅	C₂²×D₃	1	4	yes	—
74	C₂×C₅×D₃	—	3	16	no	—
75	C₂×C₅×D₃	—	6	16	no	—
76	C₃×C₄×C₅	C₂²×D₃	1	8	yes	—
77	C₃×C₄×C₅	C₂²×D₃	1	8	yes	—
78	C₅×Dic₃	—	3	16	no	—
79	C₂²×C₃×C₄	C₅×D₃	1	4	yes	complemented
80	C₂×#67	—	3	8	no	—
81	#235	—	3	32	no	—
82	C₂×D₁₂	—	3	4	no	—
83	C₂×C₄×D₃	—	3	8	no	—
84	C₂×Dic₆	—	3	4	no	—
85	C₃²×C₅	C₂×#41	1	1	yes	characteristic, complemented, Hall
86	C₂³×C₅	—	9	2	no	—
87	C₂³×C₅	C₂×C₃⋊D₃	1	1	yes	characteristic
88	C₅×Q₈	—	9	4	no	—
89	C₅×D₄	—	9	8	no	—
90	C₅×D₄	—	9	4	no	—
91	C₂×C₄×C₅	—	9	2	no	—
92	C₂×C₄×C₅	—	9	8	no	—
93	C₂×C₄×C₅	C₂×C₃⋊D₃	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×C₃⋊D₃	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂×C₃⋊D₃	1	4	yes	complemented
96	C₂²×C₃²	—	2	2	no	—
97	C₂²×C₃²	C₂³×C₅	1	2	yes	—
98	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
99	C₂×C₃⋊D₃	—	2	4	no	—
100	C₂×C₃⋊D₃	C₂³×C₅	1	4	yes	—
101	C₃²×C₄	C₂³×C₅	1	2	yes	—
102	C₃²×C₄	C₂³×C₅	1	2	yes	—
103	C₃⋊Dic₃	C₂³×C₅	1	4	yes	—
104	C₂×#41	—	9	1	no	Sylow
105	C₂×C₃×C₅	—	2	8	no	—
106	C₂×C₃×C₅	#235	1	8	yes	complemented
107	C₂×C₃×C₅	C₂³×D₃	1	4	yes	—
108	C₅×D₃	—	6	16	no	—
109	C₂³×C₃	C₂×C₅×D₃	1	4	yes	—
110	C₂²×D₃	—	3	8	no	—
111	C₂×C₃×C₄	C₂×C₅×D₃	1	4	yes	—
112	C₂×C₃×C₄	C₂×C₅×D₃	1	4	yes	—
113	C₂×C₃×C₄	C₂×C₅×D₃	1	16	yes	complemented
114	#67	—	3	32	no	—
115	C₂×Dic₃	—	3	8	no	—
116	D₁₂	—	3	16	no	—
117	C₄×D₃	—	3	32	no	—
118	Dic₆	—	3	16	no	—
119	C₂²×C₅	—	2	2	no	—
120	C₂²×C₅	—	9	4	no	—
121	C₂²×C₅	—	18	4	no	—
122	C₂²×C₅	C₂²×C₃⋊D₃	1	2	yes	—
123	C₂²×C₅	C₂²×C₃⋊D₃	1	1	yes	characteristic
124	C₄×C₅	—	9	4	no	—
125	C₄×C₅	C₂²×C₃⋊D₃	1	2	yes	—
126	C₄×C₅	C₂²×C₃⋊D₃	1	2	yes	—
127	C₂×C₃²	—	2	2	no	—
128	C₂×C₃²	C₅×#41	1	2	yes	complemented
129	C₂×C₃²	C₂⁴×C₅	1	1	yes	characteristic
130	C₃⋊D₃	—	2	4	no	—
131	#41	—	9	8	no	—
132	C₂×Q₈	—	9	1	no	—
133	C₂×D₄	—	9	2	no	—
134	C₂×D₄	—	9	1	no	—
135	C₂²×C₄	—	9	2	no	—
136	C₂²×C₄	C₅×C₃⋊D₃	1	1	yes	characteristic, complemented
137	C₃×C₅	C₂×#235	1	4	yes	complemented
138	C₂²×C₃	—	2	8	no	—
139	C₂²×C₃	C₂²×C₅×D₃	1	8	yes	—
140	C₂²×C₃	C₂²×C₅×D₃	1	4	yes	—
141	C₂×D₃	—	3	16	no	—
142	C₂×D₃	—	6	16	no	—
143	C₃×C₄	C₂²×C₅×D₃	1	8	yes	—
144	C₃×C₄	C₂²×C₅×D₃	1	8	yes	—
145	Dic₃	—	3	16	no	—
146	C₂×C₅	—	2	2	no	—
147	C₂×C₅	—	18	4	no	—
148	C₂×C₅	#3828	1	2	yes	direct product factor
149	C₂×C₅	C₂³×C₃⋊D₃	1	1	yes	characteristic
150	C₃²	C₂×C₅×#41	1	1	yes	characteristic, complemented, Sylow
151	C₂³	—	9	2	no	—
152	C₂³	C₂×C₅×C₃⋊D₃	1	1	yes	characteristic
153	Q₈	—	9	4	no	—
154	D₄	—	9	8	no	—
155	D₄	—	9	4	no	—
156	C₂×C₄	—	9	2	no	—
157	C₂×C₄	—	9	8	no	—
158	C₂×C₄	C₂×C₅×C₃⋊D₃	1	1	yes	characteristic
159	C₂×C₄	C₂×C₅×C₃⋊D₃	1	1	yes	characteristic
160	C₂×C₄	C₂×C₅×C₃⋊D₃	1	4	yes	complemented
161	C₂×C₃	—	2	8	no	—
162	C₂×C₃	C₅×#235	1	8	yes	complemented
163	C₂×C₃	C₂³×C₅×D₃	1	4	yes	—
164	D₃	—	6	16	no	—
165	C₅	C₂×#3828	1	1	yes	characteristic, direct product factor, Sylow
166	C₂²	—	2	2	no	—
167	C₂²	—	9	4	no	—
168	C₂²	—	18	4	no	—
169	C₂²	C₂²×C₅×C₃⋊D₃	1	2	yes	—
170	C₂²	C₂²×C₅×C₃⋊D₃	1	1	yes	characteristic
171	C₄	—	9	4	no	—
172	C₄	C₂²×C₅×C₃⋊D₃	1	2	yes	—
173	C₄	C₂²×C₅×C₃⋊D₃	1	2	yes	—
174	C₃	C₂×C₅×#235	1	4	yes	complemented
175	C₂	—	2	2	no	—
176	C₂	—	18	4	no	—
177	C₂	C₅×#3828	1	2	yes	direct product factor
178	C₂	C₂³×C₅×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×C₃⋊D₃
commutator, cosocle	C₂×C₃²	C₂⁴×C₅
Frattini	C₂	C₂³×C₅×C₃⋊D₃
Fitting	C₂²×C₃²×C₄×C₅	C₂
socle	C₂²×C₃²×C₅	C₂³
hypercenter	C₂²×C₄×C₅	C₃⋊D₃
hypocenter	C₃²	C₂×C₅×#41
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×C₄×C₅	C₂	semidirect
2. C₂×C₅×#638	C₂	semidirect
3. C₅×#3828	C₂	direct (N × H)
4. C₂×C₅×C₄⋊(C₃⋊D₃)	C₂	semidirect
5. C₂×C₄×C₅×C₃⋊D₃	C₂	semidirect
6. C₂×C₅×#634	C₂	semidirect
12. C₂×C₃²×C₄×C₅	C₂²	semidirect
13. C₅×#638	C₂²	semidirect
15. C₅×C₄⋊(C₃⋊D₃)	C₂²	semidirect
16. C₄×C₅×C₃⋊D₃	C₂²	semidirect
17. C₅×#634	C₂²	semidirect
18. C₂×#3828	C₅	direct (N × H)
19. C₂²×C₃×C₄×C₅	D₃	semidirect
34. C₂²×C₃²×C₄	C₂×C₅	semidirect
35. C₂×#638	C₂×C₅	semidirect
36. #3828	C₂×C₅	direct (N × H)
37. C₂×C₄⋊(C₃⋊D₃)	C₂×C₅	semidirect
38. C₂×C₄×C₃⋊D₃	C₂×C₅	semidirect
39. C₂×#634	C₂×C₅	semidirect
44. C₂×C₃×C₄×C₅	C₂×D₃	semidirect
52. C₂×C₃²×C₅	#41	semidirect
60. C₂²×C₄×C₅	C₃⋊D₃	semidirect
65. C₂×C₃²×C₄	C₂²×C₅	semidirect
66. #638	C₂²×C₅	semidirect
68. C₄⋊(C₃⋊D₃)	C₂²×C₅	semidirect
69. C₄×C₃⋊D₃	C₂²×C₅	semidirect
70. #634	C₂²×C₅	semidirect
79. C₂²×C₃×C₄	C₅×D₃	semidirect
85. C₃²×C₅	C₂×#41	semidirect
95. C₂×C₄×C₅	C₂×C₃⋊D₃	semidirect
106. C₂×C₃×C₅	#235	semidirect
113. C₂×C₃×C₄	C₂×C₅×D₃	semidirect
128. C₂×C₃²	C₅×#41	semidirect
136. C₂²×C₄	C₅×C₃⋊D₃	semidirect
137. C₃×C₅	C₂×#235	semidirect
148. C₂×C₅	#3828	direct (N × H)
150. C₃²	C₂×C₅×#41	semidirect
160. C₂×C₄	C₂×C₅×C₃⋊D₃	semidirect
162. C₂×C₃	C₅×#235	semidirect
165. C₅	C₂×#3828	direct (N × H)
174. C₃	C₂×C₅×#235	semidirect
177. C₂	C₅×#3828	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₄×C₅	2
C₂×C₅×#638	2
C₅×#3828	2
C₂×C₅×C₄⋊(C₃⋊D₃)	2
C₂×C₄×C₅×C₃⋊D₃	2
C₂×C₅×#634	2
C₂×C₅×#235	3
C₂×#3828	5

Subgroups — C2×C5×#3828 #13089491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#3828 #13089491