Subgroups — C₂×C₅×#4461 #14312545

Order 1920.

400 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×C₃⋊C₁₆	C₂	1	2	yes	maximal
3	C₅×#4461	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅×#346	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₅×C₈	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₄²×C₅	C₄	1	1	yes	characteristic
8	C₃×C₄×C₅×C₈	C₄	1	2	yes	—
9	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
10	C₂×C₅×C₃⋊C₁₆	C₄	1	8	yes	complemented
11	C₂×C₅×C₃⋊C₁₆	C₂²	1	4	yes	—
12	C₂×#4461	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂²×C₅×C₁₆	—	3	2	no	—
14	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic
15	C₅×#346	—	3	4	no	—
16	C₂²×C₃×C₄×C₅	C₈	1	2	yes	—
17	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃×C₅×C₈	C₈	1	4	yes	—
19	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
21	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
22	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
23	C₃×C₄²×C₅	C₈	1	2	yes	—
24	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
25	C₅×C₃⋊C₁₆	C₂×C₄	1	8	yes	complemented
26	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
27	C₂²×C₃⋊C₁₆	C₂×C₅	1	2	yes	—
28	#4461	C₂×C₅	1	4	yes	direct product factor
29	C₂²×C₅×C₈	Dic₃	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
31	C₂×C₄²×C₅	Dic₃	1	1	yes	characteristic
32	C₂×C₅×C₁₆	—	3	8	no	—
33	C₂×C₅×C₁₆	—	3	4	no	—
34	C₄×C₅×C₈	Dic₃	1	2	yes	—
35	C₄×C₅×C₈	C₂×D₃	1	2	yes	—
36	C₂×#346	—	3	1	no	Sylow
37	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
38	C₂×C₃×C₄×C₅	—	2	4	no	—
39	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
40	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
41	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
43	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₃×C₅×C₈	C₄²	1	2	yes	—
45	C₃×C₅×C₈	C₂×C₈	1	4	yes	—
46	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
47	C₂²×C₃×C₈	C₄×C₅	1	1	yes	characteristic
48	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
49	C₂×C₃×C₄²	C₄×C₅	1	1	yes	characteristic
50	C₃×C₄×C₈	C₄×C₅	1	2	yes	—
51	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
52	C₂×C₃⋊C₁₆	C₄×C₅	1	8	yes	complemented
53	C₂×C₃⋊C₁₆	C₂²×C₅	1	4	yes	—
54	C₂²×C₄×C₅	C₃⋊C₈	1	2	yes	—
55	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
56	C₂×C₅×C₈	C₃⋊C₈	1	4	yes	—
57	C₂×C₅×C₈	C₄×D₃	1	4	yes	—
58	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
59	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
60	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
61	C₄²×C₅	C₃⋊C₈	1	2	yes	—
62	C₄²×C₅	C₂×Dic₃	1	2	yes	—
63	C₅×C₁₆	—	3	8	no	—
64	C₂²×C₁₆	—	3	2	no	—
65	C₂×C₄×C₈	C₅×D₃	1	1	yes	characteristic
66	#346	—	3	4	no	—
67	C₂²×C₃×C₅	C₄×C₈	1	2	yes	—
68	C₂²×C₃×C₅	M₃₂	1	4	yes	—
69	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
70	C₃×C₄×C₅	—	2	4	no	—
71	C₃×C₄×C₅	C₄×C₈	1	2	yes	—
72	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
73	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
74	C₂²×C₃×C₄	C₅×C₈	1	2	yes	—
75	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
76	C₂×C₃×C₈	C₅×C₈	1	4	yes	—
77	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	—
78	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
79	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
80	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
81	C₃×C₄²	C₅×C₈	1	2	yes	—
82	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
83	C₃⋊C₁₆	C₂×C₄×C₅	1	8	yes	complemented
84	C₂³×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
85	C₂×C₄×C₅	—	2	4	no	—
86	C₂×C₄×C₅	C₂×C₃⋊C₈	1	4	yes	—
87	C₂×C₄×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₅	C₂×C₃⋊C₈	1	2	yes	—
89	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
90	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
91	C₅×C₈	C₂×C₃⋊C₈	1	4	yes	—
92	C₅×C₈	C₄×Dic₃	1	2	yes	—
93	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
94	C₂²×C₈	C₅×Dic₃	1	1	yes	characteristic
95	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
96	C₂×C₄²	C₅×Dic₃	1	1	yes	characteristic
97	C₂×C₁₆	—	3	8	no	—
98	C₂×C₁₆	—	3	4	no	—
99	C₄×C₈	C₅×Dic₃	1	2	yes	—
100	C₄×C₈	C₂×C₅×D₃	1	2	yes	—
101	C₂×C₃×C₅	#346	1	4	yes	complemented
102	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
103	C₂×C₃×C₅	C₂×M₃₂	1	2	yes	—
104	C₂³×C₃	C₂×C₅×C₈	1	1	yes	characteristic
105	C₂×C₃×C₄	—	2	4	no	—
106	C₂×C₃×C₄	C₄²×C₅	1	2	yes	—
107	C₂×C₃×C₄	C₂×C₅×C₈	1	4	yes	—
108	C₂×C₃×C₄	C₂×C₅×C₈	1	2	yes	—
109	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
110	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
111	C₃×C₈	C₄²×C₅	1	2	yes	—
112	C₃×C₈	C₂×C₅×C₈	1	4	yes	—
113	C₃×C₈	C₂²×C₄×C₅	1	2	yes	—
114	C₂²×C₅	C₄×C₃⋊C₈	1	2	yes	—
115	C₂²×C₅	#812	1	4	yes	—
116	C₂²×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
117	C₄×C₅	—	2	4	no	—
118	C₄×C₅	C₄×C₃⋊C₈	1	2	yes	—
119	C₄×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
120	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
121	C₂²×C₄	C₅×C₃⋊C₈	1	2	yes	—
122	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
123	C₂×C₈	C₅×C₃⋊C₈	1	4	yes	—
124	C₂×C₈	C₄×C₅×D₃	1	4	yes	—
125	C₂×C₈	C₂×C₅×Dic₃	1	2	yes	—
126	C₂×C₈	C₂×C₅×Dic₃	1	1	yes	characteristic
127	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
128	C₄²	C₅×C₃⋊C₈	1	2	yes	—
129	C₄²	C₂×C₅×Dic₃	1	2	yes	—
130	C₁₆	—	3	8	no	—
131	C₃×C₅	C₂×#346	1	1	yes	characteristic, complemented, Hall
132	C₂²×C₃	C₄×C₅×C₈	1	2	yes	—
133	C₂²×C₃	C₅×M₃₂	1	4	yes	—
134	C₂²×C₃	C₂²×C₅×C₈	1	1	yes	characteristic
135	C₃×C₄	—	2	4	no	—
136	C₃×C₄	C₄×C₅×C₈	1	2	yes	—
137	C₃×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
138	C₃×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
139	C₂×C₅	#4461	1	4	yes	direct product factor
140	C₂×C₅	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
141	C₂×C₅	C₂×#812	1	2	yes	—
142	C₂³	C₂×C₅×C₃⋊C₈	1	1	yes	characteristic
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	C₂×C₅×C₃⋊C₈	1	4	yes	—
145	C₂×C₄	C₂×C₅×C₃⋊C₈	1	2	yes	—
146	C₂×C₄	C₂×C₅×C₃⋊C₈	1	1	yes	characteristic
147	C₂×C₄	C₄×C₅×Dic₃	1	2	yes	—
148	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
149	C₈	C₂×C₅×C₃⋊C₈	1	4	yes	—
150	C₈	C₄×C₅×Dic₃	1	2	yes	—
151	C₈	C₂×C₄×C₅×D₃	1	2	yes	—
152	C₂×C₃	C₅×#346	1	4	yes	complemented
153	C₂×C₃	C₂×C₄×C₅×C₈	1	1	yes	characteristic
154	C₂×C₃	C₂×C₅×M₃₂	1	2	yes	—
155	C₅	C₂×#4461	1	1	yes	characteristic, direct product factor, Sylow
156	C₂²	C₄×C₅×C₃⋊C₈	1	2	yes	—
157	C₂²	C₅×#812	1	4	yes	—
158	C₂²	C₂²×C₅×C₃⋊C₈	1	1	yes	characteristic
159	C₄	—	2	4	no	—
160	C₄	C₄×C₅×C₃⋊C₈	1	2	yes	—
161	C₄	C₂²×C₅×C₃⋊C₈	1	1	yes	characteristic
162	C₄	C₂×C₄×C₅×Dic₃	1	1	yes	characteristic
163	C₃	C₂×C₅×#346	1	1	yes	characteristic, complemented, Sylow
164	C₂	C₅×#4461	1	4	yes	direct product factor
165	C₂	C₂×C₄×C₅×C₃⋊C₈	1	1	yes	characteristic
166	C₂	C₂×C₅×#812	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅×C₈	C₂×D₃
commutator	C₂×C₃	C₂×C₄×C₅×C₈
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×C₈
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₂×C₄×C₅×C₈	D₃
hypocenter	C₃	C₂×C₅×#346
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅×C₈ ◁ C₂×C₄×C₅×C₈

Complements

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

N	H ≅ G/N	type
3. C₅×#4461	C₂	direct (N × H)
10. C₂×C₅×C₃⋊C₁₆	C₄	semidirect
12. C₂×#4461	C₅	direct (N × H)
25. C₅×C₃⋊C₁₆	C₂×C₄	semidirect
28. #4461	C₂×C₅	direct (N × H)
52. C₂×C₃⋊C₁₆	C₄×C₅	semidirect
83. C₃⋊C₁₆	C₂×C₄×C₅	semidirect
101. C₂×C₃×C₅	#346	semidirect
131. C₃×C₅	C₂×#346	semidirect
139. C₂×C₅	#4461	direct (N × H)
152. C₂×C₃	C₅×#346	semidirect
155. C₅	C₂×#4461	direct (N × H)
163. C₃	C₂×C₅×#346	semidirect
164. C₂	C₅×#4461	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×C₃×C₄×C₅×C₈	2
C₂²×C₅×C₃⋊C₁₆	2
C₅×#4461	2
C₂×C₅×#346	3
C₂×#4461	5

Subgroups — C2×C5×#4461 #14312545

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4461 #14312545