Subgroups — C₂×#11750800 #14438804

Order 1920.

1104 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₆₀	C₂	1	4	yes	maximal
2	C₂×#91758	C₂	1	2	yes	maximal
3	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
4	#11750800	C₂	1	8	yes	maximal, direct product factor
5	C₂×#64988	—	3	1	no	maximal, Hall
6	C₂²×Dic₃₀	C₂²	1	4	yes	—
7	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₅	—	2	4	no	—
10	C₂×Dic₆₀	—	2	16	no	—
11	C₂×Dic₆₀	C₂²	1	16	yes	—
12	#91758	—	2	4	no	—
13	#91758	C₂²	1	8	yes	—
14	C₃×C₄×C₅×C₈	C₂²	1	4	yes	—
15	C₂×#4699	—	5	1	no	maximal, Hall
16	C₂²×Dic₂₀	—	3	4	no	—
17	C₂×#4015	—	3	2	no	—
18	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic
19	#64988	—	3	8	no	—
20	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
21	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₂²×Dic₁₅	—	2	4	no	—
23	C₂×Dic₃₀	—	2	16	no	—
24	C₂×Dic₃₀	—	2	4	no	—
25	C₂×Dic₃₀	C₂³	1	8	yes	—
26	C₂×C₃×C₅×C₈	D₄	1	8	yes	—
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	C₄⋊Dic₁₅	—	2	16	no	—
31	Dic₆₀	—	2	32	no	—
32	C₂²×Dic₁₂	—	5	4	no	—
33	C₂×#869	—	5	2	no	—
34	C₂×C₃×C₄×C₈	D₅	1	1	yes	characteristic
35	#4699	—	5	8	no	—
36	C₂²×Dic₁₀	—	3	4	no	—
37	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
38	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
39	C₂×C₄⋊Dic₅	—	6	4	no	—
40	C₂×Dic₂₀	—	3	16	no	—
41	C₂×Dic₂₀	—	6	16	no	—
42	#4015	—	3	8	no	—
43	#4015	—	6	4	no	—
44	C₄×C₅×C₈	C₂×D₃	1	4	yes	—
45	C₂×#494	—	15	1	no	Sylow
46	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	Q₁₆	1	8	yes	complemented
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×D₄	1	4	yes	—
51	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
52	C₂×Dic₁₅	—	2	8	no	—
53	C₂×Dic₁₅	—	4	8	no	—
54	Dic₃₀	—	2	16	no	—
55	Dic₃₀	—	4	8	no	—
56	C₃×C₅×C₈	C₂×D₄	1	8	yes	—
57	C₂²×Dic₆	—	5	4	no	—
58	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
59	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
60	C₂×C₄⋊Dic₃	—	10	4	no	—
61	C₂×Dic₁₂	—	5	16	no	—
62	C₂×Dic₁₂	—	10	16	no	—
63	#869	—	5	8	no	—
64	#869	—	10	4	no	—
65	C₃×C₄×C₈	C₂×D₅	1	4	yes	—
66	C₂²×C₄×C₅	D₁₂	1	2	yes	—
67	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
68	C₂²×Dic₅	—	6	4	no	—
69	C₂×Dic₁₀	—	3	8	no	—
70	C₂×Dic₁₀	—	6	16	no	—
71	C₂×Dic₁₀	—	6	4	no	—
72	C₂×C₅×C₈	D₁₂	1	8	yes	—
73	C₂×C₅×C₈	C₂²×D₃	1	4	yes	—
74	C₄²×C₅	D₁₂	1	2	yes	—
75	C₄²×C₅	C₂²×D₃	1	2	yes	—
76	C₄⋊Dic₅	—	6	16	no	—
77	Dic₂₀	—	6	32	no	—
78	C₂²×Q₁₆	—	15	4	no	—
79	C₂×C₄⋊Q₈	—	15	2	no	—
80	#494	—	15	8	no	—
81	C₂×C₄×C₈	D₁₅	1	1	yes	characteristic
82	C₂²×C₃×C₅	#127	1	2	yes	—
83	C₂²×C₃×C₅	C₂×Q₁₆	1	4	yes	—
84	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
85	C₃×C₄×C₅	#127	1	2	yes	—
86	C₃×C₄×C₅	C₂×Q₁₆	1	8	yes	complemented
87	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
88	Dic₁₅	—	4	8	no	—
89	C₂²×C₃×C₄	D₂₀	1	2	yes	—
90	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
91	C₂²×Dic₃	—	10	4	no	—
92	C₂×Dic₆	—	5	8	no	—
93	C₂×Dic₆	—	10	16	no	—
94	C₂×Dic₆	—	10	4	no	—
95	C₂×C₃×C₈	D₂₀	1	8	yes	—
96	C₂×C₃×C₈	C₂²×D₅	1	4	yes	—
97	C₃×C₄²	D₂₀	1	2	yes	—
98	C₃×C₄²	C₂²×D₅	1	2	yes	—
99	C₄⋊Dic₃	—	10	16	no	—
100	Dic₁₂	—	10	32	no	—
101	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
102	C₂×C₄×C₅	Dic₁₂	1	8	yes	complemented
103	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
104	C₂×C₄×C₅	C₂×D₁₂	1	4	yes	—
105	C₂×C₄×C₅	C₂×D₁₂	1	4	yes	—
106	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
107	C₂×Dic₅	—	6	8	no	—
108	C₂×Dic₅	—	12	8	no	—
109	Dic₁₀	—	6	16	no	—
110	Dic₁₀	—	12	8	no	—
111	C₅×C₈	C₂×D₁₂	1	8	yes	—
112	C₂²×Q₈	—	15	4	no	—
113	C₂×Q₁₆	—	15	16	no	—
114	C₂×Q₁₆	—	30	16	no	—
115	C₂²×C₈	C₂×D₁₅	1	2	yes	—
116	C₄⋊Q₈	—	15	8	no	—
117	C₄⋊Q₈	—	30	4	no	—
118	C₂×C₄⋊C₄	—	30	4	no	—
119	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
120	C₄×C₈	C₂×D₁₅	1	4	yes	—
121	C₂×C₃×C₅	#494	1	4	yes	complemented
122	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
123	C₂×C₃×C₅	C₂²×Q₁₆	1	2	yes	—
124	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
125	C₂×C₃×C₄	Dic₂₀	1	8	yes	complemented
126	C₂×C₃×C₄	C₂×D₂₀	1	4	yes	—
127	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂×D₂₀	1	4	yes	—
129	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
130	C₂×Dic₃	—	10	8	no	—
131	C₂×Dic₃	—	20	8	no	—
132	Dic₆	—	10	16	no	—
133	Dic₆	—	20	8	no	—
134	C₃×C₈	C₂×D₂₀	1	8	yes	—
135	C₂²×C₅	#874	1	2	yes	—
136	C₂²×C₅	C₂×Dic₁₂	1	4	yes	—
137	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
138	C₄×C₅	#874	1	2	yes	—
139	C₄×C₅	C₂×Dic₁₂	1	8	yes	complemented
140	C₄×C₅	C₂²×D₁₂	1	2	yes	—
141	Dic₅	—	12	8	no	—
142	C₂×Q₈	—	15	8	no	—
143	C₂×Q₈	—	30	4	no	—
144	C₂×Q₈	—	30	16	no	—
145	C₂²×C₄	—	30	4	no	—
146	C₂²×C₄	D₆₀	1	2	yes	—
147	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
148	Q₁₆	—	30	32	no	—
149	C₂×C₈	D₆₀	1	8	yes	—
150	C₂×C₈	C₂²×D₁₅	1	4	yes	—
151	C₄⋊C₄	—	30	16	no	—
152	C₄²	D₆₀	1	2	yes	—
153	C₄²	C₂²×D₁₅	1	2	yes	—
154	C₃×C₅	C₂×#494	1	1	yes	characteristic, complemented, Hall
155	C₂²×C₃	#4020	1	2	yes	—
156	C₂²×C₃	C₂×Dic₂₀	1	4	yes	—
157	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
158	C₃×C₄	#4020	1	2	yes	—
159	C₃×C₄	C₂×Dic₂₀	1	8	yes	complemented
160	C₃×C₄	C₂²×D₂₀	1	2	yes	—
161	Dic₃	—	20	8	no	—
162	C₂×C₅	#4699	1	4	yes	complemented
163	C₂×C₅	C₂×#874	1	1	yes	characteristic
164	C₂×C₅	C₂²×Dic₁₂	1	2	yes	—
165	C₂³	C₂×D₆₀	1	1	yes	characteristic
166	Q₈	—	30	16	no	—
167	Q₈	—	60	8	no	—
168	C₂×C₄	—	30	8	no	—
169	C₂×C₄	—	60	8	no	—
170	C₂×C₄	Dic₆₀	1	8	yes	complemented
171	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
172	C₂×C₄	C₂×D₆₀	1	4	yes	—
173	C₂×C₄	C₂×D₆₀	1	4	yes	—
174	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
175	C₈	C₂×D₆₀	1	8	yes	—
176	C₂×C₃	#64988	1	4	yes	complemented
177	C₂×C₃	C₂×#4020	1	1	yes	characteristic
178	C₂×C₃	C₂²×Dic₂₀	1	2	yes	—
179	C₅	C₂×#4699	1	1	yes	characteristic, complemented, Sylow
180	C₂²	#91763	1	2	yes	—
181	C₂²	C₂×Dic₆₀	1	4	yes	—
182	C₂²	C₂²×D₆₀	1	1	yes	characteristic
183	C₄	—	60	8	no	—
184	C₄	#91763	1	2	yes	—
185	C₄	C₂×Dic₆₀	1	8	yes	complemented
186	C₄	C₂²×D₆₀	1	2	yes	—
187	C₃	C₂×#64988	1	1	yes	characteristic, complemented, Sylow
188	C₂	#11750800	1	4	yes	direct product factor
189	C₂	C₂×#91763	1	1	yes	characteristic
190	C₂	C₂²×Dic₆₀	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₆₀
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
hypercenter	C₂×C₄×C₈	D₁₅
hypocenter	C₃×C₅	C₂×#494
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #11750800	C₂	direct (N × H)
47. C₂×C₃×C₄×C₅	Q₁₆	semidirect
86. C₃×C₄×C₅	C₂×Q₁₆	semidirect
102. C₂×C₄×C₅	Dic₁₂	semidirect
121. C₂×C₃×C₅	#494	semidirect
125. C₂×C₃×C₄	Dic₂₀	semidirect
139. C₄×C₅	C₂×Dic₁₂	semidirect
154. C₃×C₅	C₂×#494	semidirect
159. C₃×C₄	C₂×Dic₂₀	semidirect
162. C₂×C₅	#4699	semidirect
170. C₂×C₄	Dic₆₀	semidirect
176. C₂×C₃	#64988	semidirect
179. C₅	C₂×#4699	semidirect
185. C₄	C₂×Dic₆₀	semidirect
187. C₃	C₂×#64988	semidirect
188. C₂	#11750800	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×Dic₆₀	2
C₂×#91758	2
C₂×C₃×C₄×C₅×C₈	2
#11750800	2
C₂×#64988	3
C₂×#4699	5

Subgroups — C2×#11750800 #14438804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11750800 #14438804