Subgroups — C₂×#11750796 #14438774

Order 1920.

1520 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₂₀	C₂	1	4	yes	maximal, complemented
2	C₂×#91763	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#11750796	C₂	1	8	yes	maximal, direct product factor
5	C₂×#64984	—	3	1	no	maximal, Hall
6	C₂²×D₆₀	—	2	4	no	—
7	C₂²×D₆₀	C₂²	1	4	yes	—
8	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×D₁₂₀	—	2	16	no	—
11	C₂×D₁₂₀	C₂²	1	16	yes	complemented
12	#91763	—	2	4	no	—
13	#91763	C₂²	1	8	yes	complemented
14	C₃×C₄×C₅×C₈	C₂²	1	4	yes	complemented
15	C₂×#4695	—	5	1	no	maximal, Hall
16	C₂²×D₄₀	—	3	4	no	—
17	C₂×#4020	—	3	2	no	—
18	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic, complemented
19	#64984	—	3	8	no	—
20	C₂³×D₁₅	—	2	4	no	—
21	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
22	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂×D₆₀	—	2	16	no	—
24	C₂×D₆₀	—	2	4	no	—
25	C₂×D₆₀	—	2	16	no	—
26	C₂×D₆₀	—	4	16	no	—
27	C₂×D₆₀	C₂³	1	8	yes	—
28	C₂×C₃×C₅×C₈	D₄	1	8	yes	complemented
29	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
30	C₃×C₄²×C₅	D₄	1	2	yes	—
31	C₃×C₄²×C₅	C₂³	1	2	yes	—
32	D₁₂₀	—	2	32	no	—
33	C₂²×D₂₄	—	5	4	no	—
34	C₂×#874	—	5	2	no	—
35	C₂×C₃×C₄×C₈	D₅	1	1	yes	characteristic, complemented
36	#4695	—	5	8	no	—
37	C₂²×D₂₀	—	3	4	no	—
38	C₂²×D₂₀	—	6	4	no	—
39	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
40	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
41	C₂×D₄₀	—	3	16	no	—
42	C₂×D₄₀	—	6	16	no	—
43	#4020	—	3	8	no	—
44	#4020	—	6	4	no	—
45	C₄×C₅×C₈	C₂×D₃	1	4	yes	complemented
46	C₂×#493	—	15	1	no	Sylow
47	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
48	C₂²×D₁₅	—	2	8	no	—
49	C₂²×D₁₅	—	4	8	no	—
50	C₂²×D₁₅	—	4	16	no	—
51	C₂×C₃×C₄×C₅	D₈	1	8	yes	complemented
52	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
54	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
55	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
56	D₆₀	—	2	16	no	—
57	D₆₀	—	4	8	no	—
58	D₆₀	—	4	32	no	—
59	C₃×C₅×C₈	C₂×D₄	1	8	yes	complemented
60	C₂²×D₁₂	—	5	4	no	—
61	C₂²×D₁₂	—	10	4	no	—
62	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
63	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
64	C₂×D₂₄	—	5	16	no	—
65	C₂×D₂₄	—	10	16	no	—
66	#874	—	5	8	no	—
67	#874	—	10	4	no	—
68	C₃×C₄×C₈	C₂×D₅	1	4	yes	complemented
69	C₂³×D₅	—	6	4	no	—
70	C₂²×C₄×C₅	D₁₂	1	2	yes	—
71	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
72	C₂×D₂₀	—	3	8	no	—
73	C₂×D₂₀	—	6	16	no	—
74	C₂×D₂₀	—	6	4	no	—
75	C₂×D₂₀	—	6	16	no	—
76	C₂×D₂₀	—	12	16	no	—
77	C₂×C₅×C₈	D₁₂	1	8	yes	complemented
78	C₂×C₅×C₈	C₂²×D₃	1	4	yes	—
79	C₄²×C₅	D₁₂	1	2	yes	—
80	C₄²×C₅	C₂²×D₃	1	2	yes	—
81	D₄₀	—	6	32	no	—
82	C₂²×D₈	—	15	4	no	—
83	C₂×#127	—	15	2	no	—
84	#493	—	15	8	no	—
85	C₂×C₄×C₈	D₁₅	1	1	yes	characteristic, complemented
86	C₂²×C₃×C₅	#127	1	2	yes	—
87	C₂²×C₃×C₅	C₂×D₈	1	4	yes	—
88	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
89	C₂×D₁₅	—	4	8	no	—
90	C₂×D₁₅	—	4	16	no	—
91	C₂×D₁₅	—	8	16	no	—
92	C₃×C₄×C₅	#127	1	2	yes	—
93	C₃×C₄×C₅	C₂×D₈	1	8	yes	complemented
94	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
95	C₂³×D₃	—	10	4	no	—
96	C₂²×C₃×C₄	D₂₀	1	2	yes	—
97	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
98	C₂×D₁₂	—	5	8	no	—
99	C₂×D₁₂	—	10	16	no	—
100	C₂×D₁₂	—	10	16	no	—
101	C₂×D₁₂	—	10	4	no	—
102	C₂×D₁₂	—	20	16	no	—
103	C₂×C₃×C₈	D₂₀	1	8	yes	complemented
104	C₂×C₃×C₈	C₂²×D₅	1	4	yes	—
105	C₃×C₄²	D₂₀	1	2	yes	—
106	C₃×C₄²	C₂²×D₅	1	2	yes	—
107	D₂₄	—	10	32	no	—
108	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
109	C₂²×D₅	—	6	8	no	—
110	C₂²×D₅	—	12	8	no	—
111	C₂²×D₅	—	12	16	no	—
112	C₂×C₄×C₅	D₂₄	1	8	yes	complemented
113	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×D₁₂	1	4	yes	—
115	C₂×C₄×C₅	C₂×D₁₂	1	4	yes	—
116	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
117	D₂₀	—	6	16	no	—
118	D₂₀	—	12	8	no	—
119	D₂₀	—	12	32	no	—
120	C₅×C₈	C₂×D₁₂	1	8	yes	complemented
121	C₂²×D₄	—	15	4	no	—
122	C₂²×D₄	—	30	4	no	—
123	C₂×D₈	—	15	16	no	—
124	C₂×D₈	—	30	16	no	—
125	C₂²×C₈	C₂×D₁₅	1	2	yes	—
126	#127	—	15	8	no	—
127	#127	—	30	4	no	—
128	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
129	C₄×C₈	C₂×D₁₅	1	4	yes	complemented
130	C₂×C₃×C₅	#493	1	4	yes	complemented
131	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
132	C₂×C₃×C₅	C₂²×D₈	1	2	yes	—
133	D₁₅	—	8	8	no	—
134	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
135	C₂²×D₃	—	10	8	no	—
136	C₂²×D₃	—	20	8	no	—
137	C₂²×D₃	—	20	16	no	—
138	C₂×C₃×C₄	D₄₀	1	8	yes	complemented
139	C₂×C₃×C₄	C₂×D₂₀	1	4	yes	—
140	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂×D₂₀	1	4	yes	—
142	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
143	D₁₂	—	10	16	no	—
144	D₁₂	—	20	8	no	—
145	D₁₂	—	20	32	no	—
146	C₃×C₈	C₂×D₂₀	1	8	yes	complemented
147	C₂²×C₅	#874	1	2	yes	—
148	C₂²×C₅	C₂×D₂₄	1	4	yes	—
149	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
150	C₂×D₅	—	12	8	no	—
151	C₂×D₅	—	12	16	no	—
152	C₂×D₅	—	24	16	no	—
153	C₄×C₅	#874	1	2	yes	—
154	C₄×C₅	C₂×D₂₄	1	8	yes	complemented
155	C₄×C₅	C₂²×D₁₂	1	2	yes	—
156	C₂⁴	—	30	4	no	—
157	C₂×D₄	—	15	8	no	—
158	C₂×D₄	—	30	16	no	—
159	C₂×D₄	—	30	16	no	—
160	C₂×D₄	—	30	4	no	—
161	C₂×D₄	—	60	16	no	—
162	C₂²×C₄	D₆₀	1	2	yes	—
163	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
164	D₈	—	30	32	no	—
165	C₂×C₈	D₆₀	1	8	yes	complemented
166	C₂×C₈	C₂²×D₁₅	1	4	yes	—
167	C₄²	D₆₀	1	2	yes	—
168	C₄²	C₂²×D₁₅	1	2	yes	—
169	C₃×C₅	C₂×#493	1	1	yes	characteristic, complemented, Hall
170	C₂²×C₃	#4020	1	2	yes	—
171	C₂²×C₃	C₂×D₄₀	1	4	yes	—
172	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
173	C₂×D₃	—	20	8	no	—
174	C₂×D₃	—	20	16	no	—
175	C₂×D₃	—	40	16	no	—
176	C₃×C₄	#4020	1	2	yes	—
177	C₃×C₄	C₂×D₄₀	1	8	yes	complemented
178	C₃×C₄	C₂²×D₂₀	1	2	yes	—
179	C₂×C₅	#4695	1	4	yes	complemented
180	C₂×C₅	C₂×#874	1	1	yes	characteristic
181	C₂×C₅	C₂²×D₂₄	1	2	yes	—
182	D₅	—	24	8	no	—
183	C₂³	—	30	8	no	—
184	C₂³	—	60	8	no	—
185	C₂³	—	60	16	no	—
186	C₂³	C₂×D₆₀	1	1	yes	characteristic
187	D₄	—	30	16	no	—
188	D₄	—	60	8	no	—
189	D₄	—	60	32	no	—
190	C₂×C₄	D₁₂₀	1	8	yes	complemented
191	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
192	C₂×C₄	C₂×D₆₀	1	4	yes	—
193	C₂×C₄	C₂×D₆₀	1	4	yes	—
194	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
195	C₈	C₂×D₆₀	1	8	yes	complemented
196	C₂×C₃	#64984	1	4	yes	complemented
197	C₂×C₃	C₂×#4020	1	1	yes	characteristic
198	C₂×C₃	C₂²×D₄₀	1	2	yes	—
199	D₃	—	40	8	no	—
200	C₅	C₂×#4695	1	1	yes	characteristic, complemented, Sylow
201	C₂²	—	60	8	no	—
202	C₂²	—	60	16	no	—
203	C₂²	—	120	16	no	—
204	C₂²	#91763	1	2	yes	—
205	C₂²	C₂×D₁₂₀	1	4	yes	—
206	C₂²	C₂²×D₆₀	1	1	yes	characteristic
207	C₄	#91763	1	2	yes	—
208	C₄	C₂×D₁₂₀	1	8	yes	complemented
209	C₄	C₂²×D₆₀	1	2	yes	—
210	C₃	C₂×#64984	1	1	yes	characteristic, complemented, Sylow
211	C₂	—	120	8	no	—
212	C₂	#11750796	1	4	yes	direct product factor
213	C₂	C₂×#91763	1	1	yes	characteristic
214	C₂	C₂²×D₁₂₀	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	C₂³×C₃×C₅	C₂×D₄
hypercenter	C₂×C₄×C₈	D₁₅
hypocenter	C₃×C₅	C₂×#493
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
1. C₂²×D₁₂₀	C₂	semidirect
2. C₂×#91763	C₂	semidirect
3. C₂×C₃×C₄×C₅×C₈	C₂	semidirect
4. #11750796	C₂	direct (N × H)
11. C₂×D₁₂₀	C₂²	semidirect
13. #91763	C₂²	semidirect
14. C₃×C₄×C₅×C₈	C₂²	semidirect
18. C₂×C₄×C₅×C₈	D₃	semidirect
28. C₂×C₃×C₅×C₈	D₄	semidirect
35. C₂×C₃×C₄×C₈	D₅	semidirect
45. C₄×C₅×C₈	C₂×D₃	semidirect
51. C₂×C₃×C₄×C₅	D₈	semidirect
59. C₃×C₅×C₈	C₂×D₄	semidirect
68. C₃×C₄×C₈	C₂×D₅	semidirect
77. C₂×C₅×C₈	D₁₂	semidirect
85. C₂×C₄×C₈	D₁₅	semidirect
93. C₃×C₄×C₅	C₂×D₈	semidirect
103. C₂×C₃×C₈	D₂₀	semidirect
112. C₂×C₄×C₅	D₂₄	semidirect
120. C₅×C₈	C₂×D₁₂	semidirect
129. C₄×C₈	C₂×D₁₅	semidirect
130. C₂×C₃×C₅	#493	semidirect
138. C₂×C₃×C₄	D₄₀	semidirect
146. C₃×C₈	C₂×D₂₀	semidirect
154. C₄×C₅	C₂×D₂₄	semidirect
165. C₂×C₈	D₆₀	semidirect
169. C₃×C₅	C₂×#493	semidirect
177. C₃×C₄	C₂×D₄₀	semidirect
179. C₂×C₅	#4695	semidirect
190. C₂×C₄	D₁₂₀	semidirect
195. C₈	C₂×D₆₀	semidirect
196. C₂×C₃	#64984	semidirect
200. C₅	C₂×#4695	semidirect
208. C₄	C₂×D₁₂₀	semidirect
210. C₃	C₂×#64984	semidirect
212. C₂	#11750796	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×D₁₂₀	2
C₂×#91763	2
C₂×C₃×C₄×C₅×C₈	2
#11750796	2
C₂×#64984	3
C₂×#4695	5

Subgroups — C2×#11750796 #14438774

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11750796 #14438774