Subgroups — C₂×C₅×#4682 #14439810

Order 1920.

848 conjugacy classes of subgroups.

The subgroup lattice has 212 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×#818	C₂	1	4	yes	maximal
4	C₅×#4682	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#500	—	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×#818	C₂²	1	16	yes	—
15	C₂×#4682	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₅×C₄⋊Q₈	—	3	2	no	—
17	C₅×#500	—	3	8	no	—
18	C₂×C₅×#107	—	3	4	no	—
19	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic
20	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
21	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
22	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂²×C₅×Dic₃	—	2	4	no	—
24	C₂×C₅×Dic₆	—	2	8	no	—
25	C₂×C₅×Dic₆	—	4	8	no	—
26	C₂×C₃×C₅×C₈	Q₈	1	8	yes	complemented
27	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
28	C₃×C₄²×C₅	D₄	1	2	yes	—
29	C₃×C₄²×C₅	C₂³	1	2	yes	—
30	C₅×C₄⋊Dic₃	—	2	4	no	—
31	C₅×C₄⋊Dic₃	—	2	8	no	—
32	C₅×C₄⋊Dic₃	C₂³	1	8	yes	—
33	C₂×#869	C₂×C₅	1	2	yes	—
34	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
35	C₂×#818	C₂×C₅	1	4	yes	—
36	#4682	C₂×C₅	1	8	yes	direct product factor
37	C₂²×C₅×Q₈	—	6	2	no	—
38	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
39	C₅×C₄⋊Q₈	—	3	8	no	—
40	C₅×C₄⋊Q₈	—	6	4	no	—
41	C₂×C₅×C₄⋊C₄	—	3	4	no	—
42	C₂×C₅×C₄⋊C₄	—	6	2	no	—
43	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
44	C₅×#107	—	3	16	no	—
45	C₄×C₅×C₈	C₂×D₃	1	4	yes	—
46	C₂×#500	—	3	1	no	Sylow
47	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
48	C₂×C₃×C₄×C₅	D₈	1	4	yes	—
49	C₂×C₃×C₄×C₅	Q₁₆	1	4	yes	—
50	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
52	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
53	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
54	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
55	C₂×C₅×Dic₃	—	2	8	no	—
56	C₂×C₅×Dic₃	—	4	8	no	—
57	C₅×Dic₆	—	4	16	no	—
58	C₃×C₅×C₈	C₂×Q₈	1	8	yes	complemented
59	C₂²×Dic₆	—	2	2	no	—
60	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
61	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
62	C₂×C₄⋊Dic₃	—	2	2	no	—
63	C₂×C₄⋊Dic₃	C₂²×C₅	1	4	yes	—
64	#869	—	2	4	no	—
65	#869	C₂²×C₅	1	8	yes	—
66	C₃×C₄×C₈	C₂²×C₅	1	4	yes	—
67	#818	C₂²×C₅	1	16	yes	—
68	C₂×C₅×Q₈	—	6	8	no	—
69	C₂×C₅×Q₈	—	12	8	no	—
70	C₂²×C₄×C₅	—	6	4	no	—
71	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
72	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
74	C₂×C₅×C₈	Dic₆	1	8	yes	complemented
75	C₂×C₅×C₈	C₂²×D₃	1	4	yes	—
76	C₅×C₄⋊C₄	—	3	8	no	—
77	C₅×C₄⋊C₄	—	6	4	no	—
78	C₅×C₄⋊C₄	—	6	8	no	—
79	C₄²×C₅	D₁₂	1	2	yes	—
80	C₄²×C₅	C₂²×D₃	1	2	yes	—
81	C₂×C₄⋊Q₈	—	3	2	no	—
82	#500	—	3	8	no	—
83	C₂×#107	—	3	4	no	—
84	C₂×C₄×C₈	C₅×D₃	1	1	yes	characteristic
85	C₂²×C₃×C₅	C₄⋊Q₈	1	2	yes	—
86	C₂²×C₃×C₅	C₂×D₈	1	2	yes	—
87	C₂²×C₃×C₅	C₂×Q₁₆	1	2	yes	—
88	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
89	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
90	C₃×C₄×C₅	C₂×D₈	1	4	yes	—
91	C₃×C₄×C₅	C₂×Q₁₆	1	4	yes	—
92	C₃×C₄×C₅	C₂²×Q₈	1	2	yes	—
93	C₅×Dic₃	—	4	8	no	—
94	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
95	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
96	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
97	C₂²×Dic₃	—	2	4	no	—
98	C₂×Dic₆	—	2	8	no	—
99	C₂×Dic₆	—	4	8	no	—
100	C₂×C₃×C₈	C₅×Q₈	1	8	yes	complemented
101	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
102	C₃×C₄²	C₅×D₄	1	2	yes	—
103	C₃×C₄²	C₂³×C₅	1	2	yes	—
104	C₄⋊Dic₃	—	2	4	no	—
105	C₄⋊Dic₃	—	2	8	no	—
106	C₄⋊Dic₃	C₂³×C₅	1	8	yes	—
107	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
108	C₅×Q₈	—	12	16	no	—
109	C₂×C₄×C₅	—	6	8	no	—
110	C₂×C₄×C₅	—	12	8	no	—
111	C₂×C₄×C₅	D₂₄	1	4	yes	—
112	C₂×C₄×C₅	Dic₁₂	1	4	yes	—
113	C₂×C₄×C₅	C₂×Dic₆	1	4	yes	—
114	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
116	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
117	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
118	C₅×C₈	C₂×Dic₆	1	8	yes	complemented
119	C₂²×Q₈	—	6	2	no	—
120	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
121	C₄⋊Q₈	—	3	8	no	—
122	C₄⋊Q₈	—	6	4	no	—
123	C₂×C₄⋊C₄	—	3	4	no	—
124	C₂×C₄⋊C₄	—	6	2	no	—
125	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
126	#107	—	3	16	no	—
127	C₄×C₈	C₂×C₅×D₃	1	4	yes	—
128	C₂×C₃×C₅	#500	1	4	yes	complemented
129	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
130	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
131	C₂×C₃×C₅	C₂²×Q₁₆	1	1	yes	characteristic
132	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
133	C₂×C₃×C₄	C₅×D₈	1	4	yes	—
134	C₂×C₃×C₄	C₅×Q₁₆	1	4	yes	—
135	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
137	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
138	C₂×C₃×C₄	C₂×C₅×Q₈	1	4	yes	—
139	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
140	C₂×Dic₃	—	2	8	no	—
141	C₂×Dic₃	—	4	8	no	—
142	Dic₆	—	4	16	no	—
143	C₃×C₈	C₂×C₅×Q₈	1	8	yes	complemented
144	C₂²×C₅	#869	1	2	yes	—
145	C₂²×C₅	C₂×D₂₄	1	2	yes	—
146	C₂²×C₅	C₂×Dic₁₂	1	2	yes	—
147	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
148	C₄×C₅	—	12	8	no	—
149	C₄×C₅	#869	1	2	yes	—
150	C₄×C₅	C₂×D₂₄	1	4	yes	—
151	C₄×C₅	C₂×Dic₁₂	1	4	yes	—
152	C₄×C₅	C₂²×Dic₆	1	2	yes	—
153	C₂×Q₈	—	6	8	no	—
154	C₂×Q₈	—	12	8	no	—
155	C₂²×C₄	—	6	4	no	—
156	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
157	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
158	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
159	C₂×C₈	C₅×Dic₆	1	8	yes	complemented
160	C₂×C₈	C₂²×C₅×D₃	1	4	yes	—
161	C₄⋊C₄	—	3	8	no	—
162	C₄⋊C₄	—	6	4	no	—
163	C₄⋊C₄	—	6	8	no	—
164	C₄²	C₅×D₁₂	1	2	yes	—
165	C₄²	C₂²×C₅×D₃	1	2	yes	—
166	C₃×C₅	C₂×#500	1	1	yes	characteristic, complemented, Hall
167	C₂²×C₃	C₅×C₄⋊Q₈	1	2	yes	—
168	C₂²×C₃	C₂×C₅×D₈	1	2	yes	—
169	C₂²×C₃	C₂×C₅×Q₁₆	1	2	yes	—
170	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
171	C₃×C₄	C₅×C₄⋊Q₈	1	2	yes	—
172	C₃×C₄	C₂×C₅×D₈	1	4	yes	—
173	C₃×C₄	C₂×C₅×Q₁₆	1	4	yes	—
174	C₃×C₄	C₂²×C₅×Q₈	1	2	yes	—
175	Dic₃	—	4	8	no	—
176	C₂×C₅	#4682	1	4	yes	direct product factor
177	C₂×C₅	C₂×#869	1	1	yes	characteristic
178	C₂×C₅	C₂²×D₂₄	1	1	yes	characteristic
179	C₂×C₅	C₂²×Dic₁₂	1	1	yes	characteristic
180	C₂³	C₂×C₅×D₁₂	1	1	yes	characteristic
181	Q₈	—	12	16	no	—
182	C₂×C₄	—	6	8	no	—
183	C₂×C₄	—	12	8	no	—
184	C₂×C₄	C₅×D₂₄	1	4	yes	—
185	C₂×C₄	C₅×Dic₁₂	1	4	yes	—
186	C₂×C₄	C₂×C₅×Dic₆	1	4	yes	—
187	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
188	C₂×C₄	C₂×C₅×D₁₂	1	2	yes	—
189	C₂×C₄	C₂×C₅×D₁₂	1	2	yes	—
190	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
191	C₈	C₂×C₅×Dic₆	1	8	yes	complemented
192	C₂×C₃	C₅×#500	1	4	yes	complemented
193	C₂×C₃	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
194	C₂×C₃	C₂²×C₅×D₈	1	1	yes	characteristic
195	C₂×C₃	C₂²×C₅×Q₁₆	1	1	yes	characteristic
196	C₅	C₂×#4682	1	1	yes	characteristic, direct product factor, Sylow
197	C₂²	C₅×#869	1	2	yes	—
198	C₂²	C₂×C₅×D₂₄	1	2	yes	—
199	C₂²	C₂×C₅×Dic₁₂	1	2	yes	—
200	C₂²	C₂²×C₅×D₁₂	1	1	yes	characteristic
201	C₄	—	12	8	no	—
202	C₄	C₅×#869	1	2	yes	—
203	C₄	C₂×C₅×D₂₄	1	4	yes	—
204	C₄	C₂×C₅×Dic₁₂	1	4	yes	—
205	C₄	C₂²×C₅×Dic₆	1	2	yes	—
206	C₃	C₂×C₅×#500	1	1	yes	characteristic, complemented, Sylow
207	C₂	C₅×#4682	1	4	yes	direct product factor
208	C₂	C₂×C₅×#869	1	1	yes	characteristic
209	C₂	C₂²×C₅×D₂₄	1	1	yes	characteristic
210	C₂	C₂²×C₅×Dic₁₂	1	1	yes	characteristic

	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×#500
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×#4682	C₂	direct (N × H)
15. C₂×#4682	C₅	direct (N × H)
26. C₂×C₃×C₅×C₈	Q₈	semidirect
36. #4682	C₂×C₅	direct (N × H)
58. C₃×C₅×C₈	C₂×Q₈	semidirect
74. C₂×C₅×C₈	Dic₆	semidirect
100. C₂×C₃×C₈	C₅×Q₈	semidirect
118. C₅×C₈	C₂×Dic₆	semidirect
128. C₂×C₃×C₅	#500	semidirect
143. C₃×C₈	C₂×C₅×Q₈	semidirect
159. C₂×C₈	C₅×Dic₆	semidirect
166. C₃×C₅	C₂×#500	semidirect
176. C₂×C₅	#4682	direct (N × H)
191. C₈	C₂×C₅×Dic₆	semidirect
192. C₂×C₃	C₅×#500	semidirect
196. C₅	C₂×#4682	direct (N × H)
206. C₃	C₂×C₅×#500	semidirect
207. C₂	C₅×#4682	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×#818	2
C₅×#4682	2
C₂×C₅×#500	3
C₂×#4682	5

Subgroups — C2×C5×#4682 #14439810

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4682 #14439810