Subgroups — C₂×#11751168 #14437919

Order 1920.

976 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#91763	C₂	1	1	yes	maximal, characteristic
3	#11751168	C₂	1	8	yes	maximal, direct product factor
4	C₂×#91093	C₂	1	4	yes	maximal, complemented
5	C₂×#91087	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#65356	—	3	1	no	maximal, Hall
7	C₂²×D₆₀	—	2	2	no	—
8	C₂²×D₆₀	C₂²	1	2	yes	—
9	C₃×C₅×#125	—	2	2	no	—
10	C₃×C₅×#125	C₂²	1	4	yes	complemented
11	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₂²×C₃⋊#168	C₂²	1	2	yes	—
15	#91763	—	2	2	no	—
16	#91763	C₂²	1	4	yes	—
17	#91093	C₂²	1	16	yes	complemented
18	#91087	C₂²	1	4	yes	complemented
19	C₂×#5067	—	5	1	no	maximal, Hall
20	C₂×C₅×#125	D₃	1	1	yes	characteristic, complemented
21	C₂×#4020	—	3	1	no	—
22	#65356	—	3	8	no	—
23	C₂×#3941	—	3	4	no	—
24	C₂×#3935	—	3	1	no	—
25	C₂³×D₁₅	—	2	2	no	—
26	C₂²×C₃×C₄×C₅	—	2	2	no	—
27	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂×D₆₀	—	2	8	no	—
30	C₂×D₆₀	—	2	2	no	—
31	C₂×D₆₀	—	2	8	no	—
32	C₂×D₆₀	—	4	8	no	—
33	C₂×D₆₀	C₂³	1	4	yes	—
34	C₃×C₅×C₄⋊C₄	—	2	8	no	—
35	C₃×C₅×C₄⋊C₄	—	2	2	no	—
36	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
37	C₃×C₄²×C₅	D₄	1	2	yes	—
38	C₃×C₄²×C₅	C₂³	1	2	yes	—
39	C₂×C₃⋊#168	—	2	4	no	—
40	C₂×C₃⋊#168	C₂³	1	4	yes	—
41	C₂×C₃×#125	D₅	1	1	yes	characteristic, complemented
42	C₂×#874	—	5	1	no	—
43	#5067	—	5	8	no	—
44	C₂×#809	—	5	4	no	—
45	C₂×#803	—	5	1	no	—
46	C₂²×D₂₀	—	3	2	no	—
47	C₂²×D₂₀	—	6	2	no	—
48	C₅×#125	—	2	2	no	—
49	C₅×#125	C₂×D₃	1	4	yes	complemented
50	C₂×C₅×C₄⋊C₄	—	2	2	no	—
51	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
52	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
53	C₂²×#168	—	3	2	no	—
54	#4020	—	3	4	no	—
55	#4020	—	6	2	no	—
56	#3941	—	3	16	no	—
57	#3935	—	3	4	no	—
58	C₂×#490	—	15	1	no	Sylow
59	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂²×D₁₅	—	2	4	no	—
61	C₂²×D₁₅	—	4	4	no	—
62	C₂²×D₁₅	—	4	8	no	—
63	C₂×C₃×C₄×C₅	—	2	4	no	—
64	C₂×C₃×C₄×C₅	—	2	4	no	—
65	C₂×C₃×C₄×C₅	—	4	4	no	—
66	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
67	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	#41	1	4	yes	—
69	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
70	D₆₀	—	2	8	no	—
71	D₆₀	—	4	16	no	—
72	D₆₀	—	4	4	no	—
73	C₃⋊#168	—	2	4	no	—
74	C₂²×D₁₂	—	5	2	no	—
75	C₂²×D₁₂	—	10	2	no	—
76	C₃×#125	—	2	2	no	—
77	C₃×#125	C₂×D₅	1	4	yes	complemented
78	C₂×C₃×C₄⋊C₄	—	2	2	no	—
79	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
80	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
81	C₂²×C₃⋊C₈	—	5	2	no	—
82	#874	—	5	4	no	—
83	#874	—	10	2	no	—
84	#809	—	5	16	no	—
85	#803	—	5	4	no	—
86	C₂³×D₅	—	6	2	no	—
87	C₂²×C₄×C₅	—	2	2	no	—
88	C₂²×C₄×C₅	#67	1	2	yes	—
89	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
90	C₂×D₂₀	—	3	4	no	—
91	C₂×D₂₀	—	6	8	no	—
92	C₂×D₂₀	—	6	2	no	—
93	C₂×D₂₀	—	6	8	no	—
94	C₂×D₂₀	—	12	8	no	—
95	C₅×C₄⋊C₄	—	2	8	no	—
96	C₅×C₄⋊C₄	—	2	2	no	—
97	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
98	C₄²×C₅	#67	1	2	yes	—
99	C₄²×C₅	C₂²×D₃	1	2	yes	—
100	C₂×#168	—	3	4	no	—
101	C₂×#168	—	6	4	no	—
102	C₂×#127	—	15	1	no	—
103	C₂×#125	D₁₅	1	1	yes	characteristic, complemented
104	#490	—	15	8	no	—
105	C₂×#102	—	15	4	no	—
106	C₂×#97	—	15	1	no	—
107	C₂²×C₃×C₅	#124	1	2	yes	—
108	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	4	yes	—
109	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
110	C₂×D₁₅	—	4	4	no	—
111	C₂×D₁₅	—	4	8	no	—
112	C₂×D₁₅	—	8	8	no	—
113	C₃×C₄×C₅	—	2	4	no	—
114	C₃×C₄×C₅	—	4	4	no	—
115	C₃×C₄×C₅	#124	1	2	yes	—
116	C₃×C₄×C₅	C₂×#41	1	2	yes	—
117	C₂³×D₃	—	10	2	no	—
118	C₂²×C₃×C₄	—	2	2	no	—
119	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
120	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
121	C₂×D₁₂	—	5	4	no	—
122	C₂×D₁₂	—	10	8	no	—
123	C₂×D₁₂	—	10	8	no	—
124	C₂×D₁₂	—	10	2	no	—
125	C₂×D₁₂	—	20	8	no	—
126	C₃×C₄⋊C₄	—	2	8	no	—
127	C₃×C₄⋊C₄	—	2	2	no	—
128	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
129	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
130	C₃×C₄²	C₂²×D₅	1	2	yes	—
131	C₂×C₃⋊C₈	—	5	4	no	—
132	C₂×C₃⋊C₈	—	10	4	no	—
133	C₂³×C₅	C₂×#67	1	1	yes	characteristic
134	C₂²×D₅	—	6	4	no	—
135	C₂²×D₅	—	12	4	no	—
136	C₂²×D₅	—	12	8	no	—
137	C₂×C₄×C₅	—	2	4	no	—
138	C₂×C₄×C₅	—	2	4	no	—
139	C₂×C₄×C₅	—	4	4	no	—
140	C₂×C₄×C₅	#239	1	4	yes	—
141	C₂×C₄×C₅	C₂×#67	1	4	yes	—
142	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
144	D₂₀	—	6	8	no	—
145	D₂₀	—	12	16	no	—
146	D₂₀	—	12	4	no	—
147	#168	—	6	4	no	—
148	C₂²×D₄	—	15	2	no	—
149	C₂²×D₄	—	30	2	no	—
150	C₂²×C₈	—	15	2	no	—
151	#127	—	15	4	no	—
152	#127	—	30	2	no	—
153	#125	—	2	2	no	—
154	#125	C₂×D₁₅	1	4	yes	complemented
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	C₂×D₁₅	1	2	yes	—
157	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
158	#102	—	15	16	no	—
159	#97	—	15	4	no	—
160	C₂×C₃×C₅	#490	1	4	yes	complemented
161	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
162	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
163	D₁₅	—	8	4	no	—
164	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
165	C₂²×D₃	—	10	4	no	—
166	C₂²×D₃	—	20	4	no	—
167	C₂²×D₃	—	20	8	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	2	4	no	—
170	C₂×C₃×C₄	—	4	4	no	—
171	C₂×C₃×C₄	#713	1	4	yes	—
172	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	4	yes	—
173	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
175	D₁₂	—	10	8	no	—
176	D₁₂	—	20	16	no	—
177	D₁₂	—	20	4	no	—
178	C₃⋊C₈	—	10	4	no	—
179	C₂²×C₅	#947	1	2	yes	—
180	C₂²×C₅	#949	1	4	yes	—
181	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
182	C₂×D₅	—	12	4	no	—
183	C₂×D₅	—	12	8	no	—
184	C₂×D₅	—	24	8	no	—
185	C₄×C₅	—	2	4	no	—
186	C₄×C₅	—	4	4	no	—
187	C₄×C₅	#947	1	2	yes	—
188	C₄×C₅	C₂×#239	1	2	yes	—
189	C₂⁴	—	30	2	no	—
190	C₂×D₄	—	15	4	no	—
191	C₂×D₄	—	30	8	no	—
192	C₂×D₄	—	30	8	no	—
193	C₂×D₄	—	30	2	no	—
194	C₂×D₄	—	60	8	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	#1220	1	2	yes	—
197	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
198	C₂×C₈	—	15	4	no	—
199	C₂×C₈	—	30	4	no	—
200	C₄⋊C₄	—	2	8	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	C₂²×D₁₅	1	4	yes	—
203	C₄²	#1220	1	2	yes	—
204	C₄²	C₂²×D₁₅	1	2	yes	—
205	C₃×C₅	C₂×#490	1	1	yes	characteristic, complemented, Hall
206	C₂²×C₃	#4093	1	2	yes	—
207	C₂²×C₃	#4095	1	4	yes	—
208	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
209	C₂×D₃	—	20	4	no	—
210	C₂×D₃	—	20	8	no	—
211	C₂×D₃	—	40	8	no	—
212	C₃×C₄	—	2	4	no	—
213	C₃×C₄	—	4	4	no	—
214	C₃×C₄	#4093	1	2	yes	—
215	C₃×C₄	C₂×#713	1	2	yes	—
216	C₂×C₅	#5067	1	4	yes	complemented
217	C₂×C₅	C₂×#947	1	1	yes	characteristic
218	C₂×C₅	C₂×#949	1	2	yes	—
219	D₅	—	24	4	no	—
220	C₂³	—	30	4	no	—
221	C₂³	—	60	4	no	—
222	C₂³	—	60	8	no	—
223	C₂³	C₂×#1220	1	1	yes	characteristic
224	D₄	—	30	8	no	—
225	D₄	—	60	16	no	—
226	D₄	—	60	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	#6821	1	4	yes	—
231	C₂×C₄	C₂×#1220	1	4	yes	—
232	C₂×C₄	C₂×#1220	1	1	yes	characteristic
233	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
234	C₈	—	30	4	no	—
235	C₂×C₃	#65356	1	4	yes	complemented
236	C₂×C₃	C₂×#4093	1	1	yes	characteristic
237	C₂×C₃	C₂×#4095	1	2	yes	—
238	D₃	—	40	4	no	—
239	C₅	C₂×#5067	1	1	yes	characteristic, complemented, Sylow
240	C₂²	—	60	4	no	—
241	C₂²	—	60	8	no	—
242	C₂²	—	120	8	no	—
243	C₂²	#91836	1	2	yes	—
244	C₂²	#91838	1	4	yes	—
245	C₂²	C₂²×#1220	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	4	4	no	—
248	C₄	#91836	1	2	yes	—
249	C₄	C₂×#6821	1	2	yes	—
250	C₃	C₂×#65356	1	1	yes	characteristic, complemented, Sylow
251	C₂	—	120	4	no	—
252	C₂	#11751168	1	4	yes	direct product factor
253	C₂	C₂×#91836	1	1	yes	characteristic
254	C₂	C₂×#91838	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#1220
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅
Fitting	C₂×C₃×C₅×#125	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂×#91763	C₂
hypercenter	C₂×#125	D₁₅
hypocenter	C₃×C₅	C₂×#490
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#125	C₂	semidirect
3. #11751168	C₂	direct (N × H)
4. C₂×#91093	C₂	semidirect
5. C₂×#91087	C₂	semidirect
10. C₃×C₅×#125	C₂²	semidirect
17. #91093	C₂²	semidirect
18. #91087	C₂²	semidirect
20. C₂×C₅×#125	D₃	semidirect
41. C₂×C₃×#125	D₅	semidirect
49. C₅×#125	C₂×D₃	semidirect
77. C₃×#125	C₂×D₅	semidirect
103. C₂×#125	D₁₅	semidirect
154. #125	C₂×D₁₅	semidirect
160. C₂×C₃×C₅	#490	semidirect
205. C₃×C₅	C₂×#490	semidirect
216. C₂×C₅	#5067	semidirect
235. C₂×C₃	#65356	semidirect
239. C₅	C₂×#5067	semidirect
250. C₃	C₂×#65356	semidirect
252. C₂	#11751168	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×C₃×C₅×#125	2
C₂×#91763	2
#11751168	2
C₂×#91093	2
C₂×#91087	2
C₂×#65356	3
C₂×#5067	5

Subgroups — C2×#11751168 #14437919

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11751168 #14437919