Subgroups — C₄×C₅×#815 #14294203

Order 1920.

720 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	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#815	C₂	1	4	yes	maximal
3	C₄²×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₄×C₅×#97	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅×Dic₃	C₄	1	4	yes	—
9	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
10	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
11	C₃×C₄×C₅×C₈	C₄	1	4	yes	complemented
12	C₅×#815	C₄	1	16	yes	direct product factor
13	C₄×C₅×C₃⋊C₈	C₄	1	4	yes	complemented
14	C₄×#815	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂×C₅×#97	—	3	4	no	—
16	C₂×C₄×C₅×C₈	—	3	1	no	—
17	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic
18	C₄³×C₅	—	3	1	no	—
19	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
20	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂²×C₅×Dic₃	C₂×C₄	1	4	yes	—
22	C₂×C₃×C₅×C₈	C₂×C₄	1	8	yes	—
23	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
24	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
25	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
26	C₄×C₅×Dic₃	—	2	8	no	—
27	C₄×C₅×Dic₃	C₂×C₄	1	8	yes	—
28	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
29	C₂×C₅×C₃⋊C₈	C₂×C₄	1	8	yes	—
30	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
31	C₂×#815	C₂×C₅	1	4	yes	—
32	C₄²×Dic₃	C₂×C₅	1	1	yes	characteristic
33	C₂×C₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic
34	C₂²×C₅×C₈	—	3	2	no	—
35	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
36	C₂×C₄²×C₅	—	3	2	no	—
37	C₂×C₄²×C₅	—	3	4	no	—
38	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
39	C₅×#97	—	3	16	no	—
40	C₄×C₅×C₈	—	3	4	no	—
41	C₄×C₅×C₈	Dic₃	1	4	yes	complemented
42	C₄×#97	—	3	1	no	Sylow
43	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₃×C₄×C₅	C₄²	1	4	yes	—
45	C₂×C₃×C₄×C₅	M₁₆	1	8	yes	—
46	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
48	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₅×Dic₃	—	2	8	no	—
50	C₂×C₅×Dic₃	C₄²	1	8	yes	—
51	C₃×C₅×C₈	C₄²	1	8	yes	complemented
52	C₅×C₃⋊C₈	C₄²	1	8	yes	complemented
53	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
54	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
55	C₂×C₄×Dic₃	C₄×C₅	1	4	yes	—
56	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
57	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
58	C₃×C₄×C₈	C₄×C₅	1	4	yes	complemented
59	#815	C₄×C₅	1	16	yes	direct product factor
60	C₄×C₃⋊C₈	C₄×C₅	1	4	yes	complemented
61	C₂²×C₄×C₅	—	3	4	no	—
62	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
63	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
64	C₂×C₅×C₈	—	3	4	no	—
65	C₂×C₅×C₈	—	3	8	no	—
66	C₂×C₅×C₈	C₄×D₃	1	8	yes	—
67	C₂×C₅×C₈	C₂×Dic₃	1	4	yes	—
68	C₄²×C₅	—	3	8	no	—
69	C₄²×C₅	—	6	8	no	—
70	C₄²×C₅	C₄×D₃	1	2	yes	—
71	C₄²×C₅	C₂×Dic₃	1	2	yes	—
72	C₂×#97	—	3	4	no	—
73	C₂×C₄×C₈	—	3	1	no	—
74	C₂×C₄×C₈	C₅×D₃	1	1	yes	characteristic
75	C₄³	—	3	1	no	—
76	C₂²×C₃×C₅	C₂×C₄²	1	2	yes	—
77	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
78	C₂²×C₃×C₅	C₂×M₁₆	1	4	yes	—
79	C₃×C₄×C₅	#97	1	8	yes	complemented
80	C₃×C₄×C₅	C₂×C₄²	1	2	yes	—
81	C₃×C₄×C₅	C₂×C₄²	1	2	yes	—
82	C₅×Dic₃	—	2	8	no	—
83	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
84	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
85	C₂²×Dic₃	C₂×C₄×C₅	1	4	yes	—
86	C₂×C₃×C₈	C₂×C₄×C₅	1	8	yes	—
87	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	—
88	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
89	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
90	C₄×Dic₃	—	2	8	no	—
91	C₄×Dic₃	C₂×C₄×C₅	1	8	yes	—
92	C₂×C₃⋊C₈	C₂×C₄×C₅	1	4	yes	—
93	C₂×C₃⋊C₈	C₂×C₄×C₅	1	8	yes	—
94	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
95	C₂×C₄×C₅	—	3	8	no	—
96	C₂×C₄×C₅	—	6	8	no	—
97	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	8	yes	—
98	C₂×C₄×C₅	C₄×Dic₃	1	4	yes	—
99	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
100	C₂×C₄×C₅	C₂×C₄×D₃	1	4	yes	—
101	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
102	C₅×C₈	—	3	8	no	—
103	C₅×C₈	C₄×Dic₃	1	8	yes	complemented
104	C₂²×C₈	—	3	2	no	—
105	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
106	C₂×C₄²	—	3	2	no	—
107	C₂×C₄²	—	3	4	no	—
108	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
109	#97	—	3	16	no	—
110	C₄×C₈	—	3	4	no	—
111	C₄×C₈	C₅×Dic₃	1	4	yes	complemented
112	C₂×C₃×C₅	C₄³	1	1	yes	characteristic
113	C₂×C₃×C₅	C₂×#97	1	2	yes	—
114	C₂×C₃×C₅	C₄×M₁₆	1	4	yes	—
115	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
116	C₂×C₃×C₄	C₄²×C₅	1	4	yes	—
117	C₂×C₃×C₄	C₅×M₁₆	1	8	yes	—
118	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂²×C₄×C₅	1	4	yes	—
120	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
121	C₂×Dic₃	—	2	8	no	—
122	C₂×Dic₃	C₄²×C₅	1	8	yes	—
123	C₃×C₈	C₄²×C₅	1	8	yes	complemented
124	C₃⋊C₈	C₄²×C₅	1	8	yes	complemented
125	C₂²×C₅	C₄²×D₃	1	2	yes	—
126	C₂²×C₅	C₂×ΓL(1,𝔽₂₅)	1	4	yes	—
127	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
128	C₄×C₅	—	6	8	no	—
129	C₄×C₅	#815	1	8	yes	direct product factor
130	C₄×C₅	C₄²×D₃	1	2	yes	—
131	C₄×C₅	C₂×C₄×Dic₃	1	2	yes	—
132	C₂²×C₄	—	3	4	no	—
133	C₂²×C₄	C₄×C₅×D₃	1	2	yes	—
134	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
135	C₂×C₈	—	3	4	no	—
136	C₂×C₈	—	3	8	no	—
137	C₂×C₈	C₄×C₅×D₃	1	8	yes	—
138	C₂×C₈	C₂×C₅×Dic₃	1	4	yes	—
139	C₄²	—	3	8	no	—
140	C₄²	—	6	8	no	—
141	C₄²	C₄×C₅×D₃	1	2	yes	—
142	C₄²	C₂×C₅×Dic₃	1	2	yes	—
143	C₃×C₅	C₄×#97	1	1	yes	characteristic, complemented, Hall
144	C₂²×C₃	C₂×C₄²×C₅	1	2	yes	—
145	C₂²×C₃	C₂×C₄²×C₅	1	1	yes	characteristic
146	C₂²×C₃	C₂×C₅×M₁₆	1	4	yes	—
147	C₃×C₄	C₅×#97	1	8	yes	complemented
148	C₃×C₄	C₂×C₄²×C₅	1	2	yes	—
149	C₃×C₄	C₂×C₄²×C₅	1	2	yes	—
150	Dic₃	—	2	8	no	—
151	C₂×C₅	C₄×ΓL(1,𝔽₂₅)	1	4	yes	—
152	C₂×C₅	C₄²×Dic₃	1	1	yes	characteristic
153	C₂×C₅	C₂×#815	1	2	yes	—
154	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
155	C₂×C₄	—	3	8	no	—
156	C₂×C₄	—	6	8	no	—
157	C₂×C₄	C₅×ΓL(1,𝔽₂₅)	1	8	yes	—
158	C₂×C₄	C₄×C₅×Dic₃	1	4	yes	—
159	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
160	C₂×C₄	C₂×C₄×C₅×D₃	1	4	yes	—
161	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
162	C₈	—	3	8	no	—
163	C₈	C₄×C₅×Dic₃	1	8	yes	complemented
164	C₂×C₃	C₄³×C₅	1	1	yes	characteristic
165	C₂×C₃	C₂×C₅×#97	1	2	yes	—
166	C₂×C₃	C₄×C₅×M₁₆	1	4	yes	—
167	C₅	C₄×#815	1	1	yes	characteristic, direct product factor, Sylow
168	C₂²	C₄²×C₅×D₃	1	2	yes	—
169	C₂²	C₂×C₅×ΓL(1,𝔽₂₅)	1	4	yes	—
170	C₂²	C₂×C₄×C₅×Dic₃	1	1	yes	characteristic
171	C₄	—	6	8	no	—
172	C₄	C₅×#815	1	8	yes	direct product factor
173	C₄	C₄²×C₅×D₃	1	2	yes	—
174	C₄	C₂×C₄×C₅×Dic₃	1	2	yes	—
175	C₃	C₄×C₅×#97	1	1	yes	characteristic, complemented, Sylow
176	C₂	C₄×C₅×ΓL(1,𝔽₂₅)	1	4	yes	—
177	C₂	C₄²×C₅×Dic₃	1	1	yes	characteristic
178	C₂	C₂×C₅×#815	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²×C₅	C₂×D₃
commutator	C₂×C₃	C₄³×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₄×C₅×C₈	D₃
hypocenter	C₃	C₄×C₅×#97
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈

Complements

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

N	H ≅ G/N	type
11. C₃×C₄×C₅×C₈	C₄	semidirect
12. C₅×#815	C₄	direct (N × H)
13. C₄×C₅×C₃⋊C₈	C₄	semidirect
14. C₄×#815	C₅	direct (N × H)
41. C₄×C₅×C₈	Dic₃	semidirect
51. C₃×C₅×C₈	C₄²	semidirect
52. C₅×C₃⋊C₈	C₄²	semidirect
58. C₃×C₄×C₈	C₄×C₅	semidirect
59. #815	C₄×C₅	direct (N × H)
60. C₄×C₃⋊C₈	C₄×C₅	semidirect
79. C₃×C₄×C₅	#97	semidirect
103. C₅×C₈	C₄×Dic₃	semidirect
111. C₄×C₈	C₅×Dic₃	semidirect
123. C₃×C₈	C₄²×C₅	semidirect
124. C₃⋊C₈	C₄²×C₅	semidirect
129. C₄×C₅	#815	direct (N × H)
143. C₃×C₅	C₄×#97	semidirect
147. C₃×C₄	C₅×#97	semidirect
163. C₈	C₄×C₅×Dic₃	semidirect
167. C₅	C₄×#815	direct (N × H)
172. C₄	C₅×#815	direct (N × H)
175. C₃	C₄×C₅×#97	semidirect

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
C₂×C₅×#815	2
C₄²×C₅×Dic₃	2
C₂×C₄×C₅×C₃⋊C₈	2
C₄×C₅×#97	3
C₄×#815	5

Subgroups — C4×C5×#815 #14294203

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×#815 #14294203