Subgroups — C₂×#11750784 #14439773

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91759	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#91102	C₂	1	2	yes	maximal
4	C₂×#91101	C₂	1	2	yes	maximal
5	#11750784	C₂	1	8	yes	maximal, direct product factor
6	C₂×#64972	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₅	C₂²	1	4	yes	—
11	C₂×#6712	—	2	4	no	—
12	#91759	—	2	4	no	—
13	#91759	C₂²	1	8	yes	—
14	C₃×C₄×C₅×C₈	C₂²	1	4	yes	—
15	#91102	C₂²	1	8	yes	—
16	#91101	C₂²	1	8	yes	—
17	C₂×#4683	—	5	1	no	maximal, Hall
18	C₂×#4016	—	3	2	no	—
19	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic
20	C₂×#3950	—	3	2	no	—
21	C₂×#3949	—	3	2	no	—
22	#64972	—	3	8	no	—
23	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂²×Dic₁₅	—	2	4	no	—
26	C₂×C₃×C₅×C₈	Q₈	1	4	yes	—
27	C₂×C₃×C₅×C₈	Q₈	1	4	yes	—
28	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
29	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
30	C₃×C₄²×C₅	D₄	1	2	yes	—
31	C₃×C₄²×C₅	C₂³	1	2	yes	—
32	C₄⋊Dic₁₅	—	2	4	no	—
33	C₄⋊Dic₁₅	C₂³	1	8	yes	—
34	#6712	—	2	16	no	—
35	C₂×#870	—	5	2	no	—
36	C₂×C₃×C₄×C₈	D₅	1	1	yes	characteristic
37	C₂×#818	—	5	2	no	—
38	C₂×#817	—	5	2	no	—
39	#4683	—	5	8	no	—
40	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
42	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
43	C₂×C₄⋊Dic₅	—	3	4	no	—
44	C₂×#683	—	6	4	no	—
45	#4016	—	3	8	no	—
46	#4016	—	6	4	no	—
47	C₄×C₅×C₈	C₂×D₃	1	4	yes	—
48	#3950	—	3	8	no	—
49	#3949	—	3	8	no	—
50	C₂×#499	—	15	1	no	Sylow
51	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	—	2	4	no	—
53	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
54	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
56	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
57	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₁₅	—	2	8	no	—
59	C₂×Dic₁₅	—	4	8	no	—
60	C₃×C₅×C₈	C₂×Q₈	1	4	yes	—
61	C₃×C₅×C₈	C₂×Q₈	1	4	yes	—
62	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
63	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
64	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
65	C₂×C₄⋊Dic₃	—	5	4	no	—
66	C₂×Dic₃⋊C₄	—	10	4	no	—
67	#870	—	5	8	no	—
68	#870	—	10	4	no	—
69	C₃×C₄×C₈	C₂×D₅	1	4	yes	—
70	#818	—	5	8	no	—
71	#817	—	5	8	no	—
72	C₂²×C₄×C₅	D₁₂	1	2	yes	—
73	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
74	C₂²×Dic₅	—	6	4	no	—
75	C₂×C₅×C₈	Dic₆	1	4	yes	—
76	C₂×C₅×C₈	Dic₆	1	4	yes	—
77	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
78	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
79	C₄²×C₅	D₁₂	1	2	yes	—
80	C₄²×C₅	C₂²×D₃	1	2	yes	—
81	C₄⋊Dic₅	—	3	8	no	—
82	C₄⋊Dic₅	—	6	4	no	—
83	#683	—	6	16	no	—
84	C₂×#125	—	15	2	no	—
85	#499	—	15	8	no	—
86	C₂×#107	—	15	2	no	—
87	C₂×#106	—	15	2	no	—
88	C₂×C₄×C₈	D₁₅	1	1	yes	characteristic
89	C₂²×C₃×C₅	C₄⋊Q₈	1	2	yes	—
90	C₂²×C₃×C₅	#135	1	4	yes	—
91	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
92	C₃×C₄×C₅	—	2	4	no	—
93	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
94	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
95	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
96	Dic₁₅	—	4	8	no	—
97	C₂²×C₃×C₄	D₂₀	1	2	yes	—
98	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
99	C₂²×Dic₃	—	10	4	no	—
100	C₂×C₃×C₈	Dic₁₀	1	4	yes	—
101	C₂×C₃×C₈	Dic₁₀	1	4	yes	—
102	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
103	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
104	C₃×C₄²	D₂₀	1	2	yes	—
105	C₃×C₄²	C₂²×D₅	1	2	yes	—
106	C₄⋊Dic₃	—	5	8	no	—
107	C₄⋊Dic₃	—	10	4	no	—
108	Dic₃⋊C₄	—	10	16	no	—
109	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
110	C₂×C₄×C₅	—	2	4	no	—
111	C₂×C₄×C₅	C₂×Dic₆	1	2	yes	—
112	C₂×C₄×C₅	C₂×Dic₆	1	2	yes	—
113	C₂×C₄×C₅	C₂×D₁₂	1	4	yes	—
114	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
116	C₂×Dic₅	—	6	8	no	—
117	C₂×Dic₅	—	12	8	no	—
118	C₅×C₈	C₂×Dic₆	1	4	yes	—
119	C₅×C₈	C₂×Dic₆	1	4	yes	—
120	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
121	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
122	#125	—	15	8	no	—
123	#125	—	30	4	no	—
124	C₂×C₄⋊C₄	—	15	4	no	—
125	C₂×C₄⋊C₄	—	30	4	no	—
126	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
127	#107	—	15	8	no	—
128	#106	—	15	8	no	—
129	C₄×C₈	C₂×D₁₅	1	4	yes	—
130	C₂×C₃×C₅	#499	1	4	yes	complemented
131	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
132	C₂×C₃×C₅	C₂×#135	1	2	yes	—
133	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
134	C₂×C₃×C₄	—	2	4	no	—
135	C₂×C₃×C₄	C₂×Dic₁₀	1	2	yes	—
136	C₂×C₃×C₄	C₂×Dic₁₀	1	2	yes	—
137	C₂×C₃×C₄	C₂×D₂₀	1	4	yes	—
138	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
140	C₂×Dic₃	—	10	8	no	—
141	C₂×Dic₃	—	20	8	no	—
142	C₃×C₈	C₂×Dic₁₀	1	4	yes	—
143	C₃×C₈	C₂×Dic₁₀	1	4	yes	—
144	C₂²×C₅	#869	1	2	yes	—
145	C₂²×C₅	#904	1	4	yes	—
146	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
147	C₄×C₅	—	2	4	no	—
148	C₄×C₅	#869	1	2	yes	—
149	C₄×C₅	C₂²×Dic₆	1	1	yes	characteristic
150	C₄×C₅	C₂²×Dic₆	1	1	yes	characteristic
151	Dic₅	—	12	8	no	—
152	C₂²×C₄	—	30	4	no	—
153	C₂²×C₄	D₆₀	1	2	yes	—
154	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
155	C₂×C₈	Dic₃₀	1	4	yes	—
156	C₂×C₈	Dic₃₀	1	4	yes	—
157	C₂×C₈	C₂²×D₁₅	1	2	yes	—
158	C₂×C₈	C₂²×D₁₅	1	2	yes	—
159	C₄⋊C₄	—	15	8	no	—
160	C₄⋊C₄	—	30	4	no	—
161	C₄⋊C₄	—	30	16	no	—
162	C₄²	D₆₀	1	2	yes	—
163	C₄²	C₂²×D₁₅	1	2	yes	—
164	C₃×C₅	C₂×#499	1	1	yes	characteristic, complemented, Hall
165	C₂²×C₃	#4015	1	2	yes	—
166	C₂²×C₃	#4050	1	4	yes	—
167	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
168	C₃×C₄	—	2	4	no	—
169	C₃×C₄	#4015	1	2	yes	—
170	C₃×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
171	C₃×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
172	Dic₃	—	20	8	no	—
173	C₂×C₅	#4683	1	4	yes	complemented
174	C₂×C₅	C₂×#869	1	1	yes	characteristic
175	C₂×C₅	C₂×#904	1	2	yes	—
176	C₂³	C₂×D₆₀	1	1	yes	characteristic
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	30	8	no	—
179	C₂×C₄	—	60	8	no	—
180	C₂×C₄	C₂×Dic₃₀	1	2	yes	—
181	C₂×C₄	C₂×Dic₃₀	1	2	yes	—
182	C₂×C₄	C₂×D₆₀	1	4	yes	—
183	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
184	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
185	C₈	C₂×Dic₃₀	1	4	yes	—
186	C₈	C₂×Dic₃₀	1	4	yes	—
187	C₂×C₃	#64972	1	4	yes	complemented
188	C₂×C₃	C₂×#4015	1	1	yes	characteristic
189	C₂×C₃	C₂×#4050	1	2	yes	—
190	C₅	C₂×#4683	1	1	yes	characteristic, complemented, Sylow
191	C₂²	#91758	1	2	yes	—
192	C₂²	#91793	1	4	yes	—
193	C₂²	C₂²×D₆₀	1	1	yes	characteristic
194	C₄	—	2	4	no	—
195	C₄	—	60	8	no	—
196	C₄	#91758	1	2	yes	—
197	C₄	C₂²×Dic₃₀	1	1	yes	characteristic
198	C₄	C₂²×Dic₃₀	1	1	yes	characteristic
199	C₃	C₂×#64972	1	1	yes	characteristic, complemented, Sylow
200	C₂	#11750784	1	4	yes	direct product factor
201	C₂	C₂×#91758	1	1	yes	characteristic
202	C₂	C₂×#91793	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×#499
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
5. #11750784	C₂	direct (N × H)
130. C₂×C₃×C₅	#499	semidirect
164. C₃×C₅	C₂×#499	semidirect
173. C₂×C₅	#4683	semidirect
187. C₂×C₃	#64972	semidirect
190. C₅	C₂×#4683	semidirect
199. C₃	C₂×#64972	semidirect
200. C₂	#11750784	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×#91759	2
C₂×C₃×C₄×C₅×C₈	2
C₂×#91102	2
C₂×#91101	2
#11750784	2
C₂×#64972	3
C₂×#4683	5

Subgroups — C2×#11750784 #14439773

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11750784 #14439773