Subgroups — C₂×C₅×#4680 #14439738

Order 1920.

848 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#869	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#817	C₂	1	4	yes	maximal
4	C₅×#4680	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#498	—	3	1	no	maximal, Hall
6	C₂²×C₅×Dic₆	—	2	2	no	—
7	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₄⋊Dic₃	—	2	2	no	—
10	C₂×C₅×C₄⋊Dic₃	C₂²	1	4	yes	—
11	C₅×#869	—	2	4	no	—
12	C₅×#869	C₂²	1	8	yes	—
13	C₃×C₄×C₅×C₈	C₂²	1	4	yes	—
14	C₅×#817	C₂²	1	16	yes	—
15	C₂×#4680	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₅×C₄⋊Q₈	—	3	2	no	—
17	C₅×#498	—	3	8	no	—
18	C₂×C₅×#106	—	3	4	no	—
19	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic
20	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
21	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₂²×C₅×Dic₃	—	2	4	no	—
23	C₂×C₅×Dic₆	—	2	8	no	—
24	C₂×C₅×Dic₆	—	4	8	no	—
25	C₂×C₃×C₅×C₈	Q₈	1	8	yes	complemented
26	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
27	C₃×C₄²×C₅	D₄	1	2	yes	—
28	C₃×C₄²×C₅	C₂³	1	2	yes	—
29	C₅×C₄⋊Dic₃	—	2	4	no	—
30	C₅×C₄⋊Dic₃	—	2	8	no	—
31	C₅×C₄⋊Dic₃	C₂³	1	8	yes	—
32	C₂×#869	C₂×C₅	1	2	yes	—
33	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
34	C₂×#817	C₂×C₅	1	4	yes	—
35	#4680	C₂×C₅	1	8	yes	direct product factor
36	C₂²×C₅×Q₈	—	6	2	no	—
37	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
38	C₅×C₄⋊Q₈	—	3	8	no	—
39	C₅×C₄⋊Q₈	—	6	4	no	—
40	C₂×C₅×C₄⋊C₄	—	3	4	no	—
41	C₂×C₅×C₄⋊C₄	—	6	2	no	—
42	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
43	C₅×#106	—	3	16	no	—
44	C₄×C₅×C₈	C₂×D₃	1	4	yes	—
45	C₂×#498	—	3	1	no	Sylow
46	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	SD₁₆	1	8	yes	—
48	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
50	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
51	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
52	C₂×C₅×Dic₃	—	2	8	no	—
53	C₂×C₅×Dic₃	—	4	8	no	—
54	C₅×Dic₆	—	4	16	no	—
55	C₃×C₅×C₈	C₂×Q₈	1	8	yes	complemented
56	C₂²×Dic₆	—	2	2	no	—
57	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
58	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
59	C₂×C₄⋊Dic₃	—	2	2	no	—
60	C₂×C₄⋊Dic₃	C₂²×C₅	1	4	yes	—
61	#869	—	2	4	no	—
62	#869	C₂²×C₅	1	8	yes	—
63	C₃×C₄×C₈	C₂²×C₅	1	4	yes	—
64	#817	C₂²×C₅	1	16	yes	—
65	C₂×C₅×Q₈	—	6	8	no	—
66	C₂×C₅×Q₈	—	12	8	no	—
67	C₂²×C₄×C₅	—	6	4	no	—
68	C₂²×C₄×C₅	D₁₂	1	2	yes	—
69	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
70	C₂×C₅×C₈	Dic₆	1	8	yes	complemented
71	C₂×C₅×C₈	C₂²×D₃	1	4	yes	—
72	C₅×C₄⋊C₄	—	3	8	no	—
73	C₅×C₄⋊C₄	—	6	4	no	—
74	C₅×C₄⋊C₄	—	6	8	no	—
75	C₄²×C₅	D₁₂	1	2	yes	—
76	C₄²×C₅	C₂²×D₃	1	2	yes	—
77	C₂×C₄⋊Q₈	—	3	2	no	—
78	#498	—	3	8	no	—
79	C₂×#106	—	3	4	no	—
80	C₂×C₄×C₈	C₅×D₃	1	1	yes	characteristic
81	C₂²×C₃×C₅	C₄⋊Q₈	1	2	yes	—
82	C₂²×C₃×C₅	C₂×SD₁₆	1	4	yes	—
83	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
84	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
85	C₃×C₄×C₅	C₂×SD₁₆	1	8	yes	—
86	C₃×C₄×C₅	C₂²×Q₈	1	2	yes	—
87	C₅×Dic₃	—	4	8	no	—
88	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
89	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
90	C₂²×Dic₃	—	2	4	no	—
91	C₂×Dic₆	—	2	8	no	—
92	C₂×Dic₆	—	4	8	no	—
93	C₂×C₃×C₈	C₅×Q₈	1	8	yes	complemented
94	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
95	C₃×C₄²	C₅×D₄	1	2	yes	—
96	C₃×C₄²	C₂³×C₅	1	2	yes	—
97	C₄⋊Dic₃	—	2	4	no	—
98	C₄⋊Dic₃	—	2	8	no	—
99	C₄⋊Dic₃	C₂³×C₅	1	8	yes	—
100	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
101	C₅×Q₈	—	12	16	no	—
102	C₂×C₄×C₅	—	6	8	no	—
103	C₂×C₄×C₅	—	12	8	no	—
104	C₂×C₄×C₅	#204	1	8	yes	—
105	C₂×C₄×C₅	C₂×Dic₆	1	4	yes	—
106	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×D₁₂	1	4	yes	—
108	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
109	C₅×C₈	C₂×Dic₆	1	8	yes	complemented
110	C₂²×Q₈	—	6	2	no	—
111	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
112	C₄⋊Q₈	—	3	8	no	—
113	C₄⋊Q₈	—	6	4	no	—
114	C₂×C₄⋊C₄	—	3	4	no	—
115	C₂×C₄⋊C₄	—	6	2	no	—
116	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
117	#106	—	3	16	no	—
118	C₄×C₈	C₂×C₅×D₃	1	4	yes	—
119	C₂×C₃×C₅	#498	1	4	yes	complemented
120	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
121	C₂×C₃×C₅	C₂²×SD₁₆	1	2	yes	—
122	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
123	C₂×C₃×C₄	C₅×SD₁₆	1	8	yes	—
124	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
126	C₂×C₃×C₄	C₂×C₅×Q₈	1	4	yes	—
127	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
128	C₂×Dic₃	—	2	8	no	—
129	C₂×Dic₃	—	4	8	no	—
130	Dic₆	—	4	16	no	—
131	C₃×C₈	C₂×C₅×Q₈	1	8	yes	complemented
132	C₂²×C₅	#869	1	2	yes	—
133	C₂²×C₅	C₂×#204	1	4	yes	—
134	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
135	C₄×C₅	—	12	8	no	—
136	C₄×C₅	#869	1	2	yes	—
137	C₄×C₅	C₂×#204	1	8	yes	—
138	C₄×C₅	C₂²×Dic₆	1	2	yes	—
139	C₂×Q₈	—	6	8	no	—
140	C₂×Q₈	—	12	8	no	—
141	C₂²×C₄	—	6	4	no	—
142	C₂²×C₄	C₅×D₁₂	1	2	yes	—
143	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
144	C₂×C₈	C₅×Dic₆	1	8	yes	complemented
145	C₂×C₈	C₂²×C₅×D₃	1	4	yes	—
146	C₄⋊C₄	—	3	8	no	—
147	C₄⋊C₄	—	6	4	no	—
148	C₄⋊C₄	—	6	8	no	—
149	C₄²	C₅×D₁₂	1	2	yes	—
150	C₄²	C₂²×C₅×D₃	1	2	yes	—
151	C₃×C₅	C₂×#498	1	1	yes	characteristic, complemented, Hall
152	C₂²×C₃	C₅×C₄⋊Q₈	1	2	yes	—
153	C₂²×C₃	C₂×C₅×SD₁₆	1	4	yes	—
154	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
155	C₃×C₄	C₅×C₄⋊Q₈	1	2	yes	—
156	C₃×C₄	C₂×C₅×SD₁₆	1	8	yes	—
157	C₃×C₄	C₂²×C₅×Q₈	1	2	yes	—
158	Dic₃	—	4	8	no	—
159	C₂×C₅	#4680	1	4	yes	direct product factor
160	C₂×C₅	C₂×#869	1	1	yes	characteristic
161	C₂×C₅	C₂²×#204	1	2	yes	—
162	C₂³	C₂×C₅×D₁₂	1	1	yes	characteristic
163	Q₈	—	12	16	no	—
164	C₂×C₄	—	6	8	no	—
165	C₂×C₄	—	12	8	no	—
166	C₂×C₄	C₅×#204	1	8	yes	—
167	C₂×C₄	C₂×C₅×Dic₆	1	4	yes	—
168	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
169	C₂×C₄	C₂×C₅×D₁₂	1	4	yes	—
170	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
171	C₈	C₂×C₅×Dic₆	1	8	yes	complemented
172	C₂×C₃	C₅×#498	1	4	yes	complemented
173	C₂×C₃	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
174	C₂×C₃	C₂²×C₅×SD₁₆	1	2	yes	—
175	C₅	C₂×#4680	1	1	yes	characteristic, direct product factor, Sylow
176	C₂²	C₅×#869	1	2	yes	—
177	C₂²	C₂×C₅×#204	1	4	yes	—
178	C₂²	C₂²×C₅×D₁₂	1	1	yes	characteristic
179	C₄	—	12	8	no	—
180	C₄	C₅×#869	1	2	yes	—
181	C₄	C₂×C₅×#204	1	8	yes	—
182	C₄	C₂²×C₅×Dic₆	1	2	yes	—
183	C₃	C₂×C₅×#498	1	1	yes	characteristic, complemented, Sylow
184	C₂	C₅×#4680	1	4	yes	direct product factor
185	C₂	C₂×C₅×#869	1	1	yes	characteristic
186	C₂	C₂²×C₅×#204	1	2	yes	—

	subgroup	G/H
center	C₂³×C₅	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
hypercenter	C₂×C₄×C₅×C₈	D₃
hypocenter	C₃	C₂×C₅×#498
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈

Complements

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

N	H ≅ G/N	type
4. C₅×#4680	C₂	direct (N × H)
15. C₂×#4680	C₅	direct (N × H)
25. C₂×C₃×C₅×C₈	Q₈	semidirect
35. #4680	C₂×C₅	direct (N × H)
55. C₃×C₅×C₈	C₂×Q₈	semidirect
70. C₂×C₅×C₈	Dic₆	semidirect
93. C₂×C₃×C₈	C₅×Q₈	semidirect
109. C₅×C₈	C₂×Dic₆	semidirect
119. C₂×C₃×C₅	#498	semidirect
131. C₃×C₈	C₂×C₅×Q₈	semidirect
144. C₂×C₈	C₅×Dic₆	semidirect
151. C₃×C₅	C₂×#498	semidirect
159. C₂×C₅	#4680	direct (N × H)
171. C₈	C₂×C₅×Dic₆	semidirect
172. C₂×C₃	C₅×#498	semidirect
175. C₅	C₂×#4680	direct (N × H)
183. C₃	C₂×C₅×#498	semidirect
184. C₂	C₅×#4680	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×C₅×#869	2
C₂×C₃×C₄×C₅×C₈	2
C₂×C₅×#817	2
C₅×#4680	2
C₂×C₅×#498	3
C₂×#4680	5

Subgroups — C2×C5×#4680 #14439738

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4680 #14439738