Subgroups — #14439083

Order 1920.

1128 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×#91793	C₂	1	4	yes	maximal, complemented
2	#11756236	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	#11750816	C₂	1	4	yes	maximal, complemented
5	#11750813	C₂	1	4	yes	maximal, complemented
6	#10610230	—	3	1	no	maximal, Hall
7	C₂×#6817	C₂²	1	4	yes	—
8	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	#91819	—	2	4	no	—
11	C₂×Dic₆₀	C₂²	1	4	yes	—
12	#91793	—	2	16	no	—
13	C₂×D₁₂₀	C₂²	1	4	yes	—
14	C₂×#6706	C₂²	1	8	yes	—
15	#91764	C₂²	1	4	yes	—
16	#91763	C₂²	1	2	yes	—
17	C₄×D₆₀	—	2	4	no	—
18	#91758	C₂²	1	2	yes	—
19	C₃×C₅×#97	C₂²	1	4	yes	—
20	#78499	—	5	1	no	maximal, Hall
21	C₂×#4050	—	3	4	no	—
22	#65811	—	3	2	no	—
23	C₂×C₅×#97	D₃	1	1	yes	characteristic, complemented
24	#65004	—	3	4	no	—
25	#65001	—	3	4	no	—
26	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂×#1220	—	2	4	no	—
29	#6817	—	2	16	no	—
30	C₂×D₆₀	—	2	4	no	—
31	C₂×D₆₀	C₂³	1	4	yes	—
32	C₂×C₄×D₁₅	—	2	4	no	—
33	C₂×Dic₃₀	C₂³	1	4	yes	—
34	C₂×C₃×C₅×C₈	D₄	1	8	yes	complemented
35	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
36	C₃×C₄²×C₅	D₄	1	2	yes	—
37	C₃×C₄²×C₅	C₂³	1	2	yes	—
38	#6714	—	2	8	no	—
39	C₄⋊Dic₁₅	—	2	4	no	—
40	Dic₆₀	—	2	8	no	—
41	D₁₂₀	—	2	8	no	—
42	#6706	—	2	16	no	—
43	C₂×#904	—	5	4	no	—
44	#5477	—	5	2	no	—
45	C₂×C₃×#97	D₅	1	1	yes	characteristic, complemented
46	#4715	—	5	4	no	—
47	#4712	—	5	4	no	—
48	C₂×#709	—	3	4	no	—
49	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
50	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
51	#4076	—	6	4	no	—
52	C₂×Dic₂₀	—	3	4	no	—
53	#4050	—	6	16	no	—
54	C₂×D₄₀	—	3	4	no	—
55	C₂×#677	—	3	8	no	—
56	#4021	—	3	4	no	—
57	#4020	—	3	2	no	—
58	C₄×D₂₀	—	6	4	no	—
59	#4015	—	3	2	no	—
60	C₅×#97	C₂×D₃	1	4	yes	—
61	#3150	—	15	1	no	Sylow
62	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂²×D₁₅	—	2	4	no	—
64	C₂×C₃×C₄×C₅	—	4	2	no	—
65	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
66	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
68	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
69	#1220	—	4	8	no	—
70	C₂×Dic₁₅	—	2	4	no	—
71	D₆₀	—	2	8	no	—
72	D₆₀	—	4	8	no	—
73	C₄×D₁₅	—	4	8	no	—
74	Dic₃₀	—	2	8	no	—
75	C₃×C₅×C₈	C₂×D₄	1	8	yes	—
76	C₂×#235	—	5	4	no	—
77	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
78	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
79	#930	—	10	4	no	—
80	C₂×Dic₁₂	—	5	4	no	—
81	#904	—	10	16	no	—
82	C₂×D₂₄	—	5	4	no	—
83	C₂×#204	—	5	8	no	—
84	#875	—	5	4	no	—
85	#874	—	5	2	no	—
86	C₄×D₁₂	—	10	4	no	—
87	#869	—	5	2	no	—
88	C₃×#97	C₂×D₅	1	4	yes	—
89	C₂²×C₄×C₅	D₁₂	1	2	yes	—
90	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
91	C₂×C₂²⋊D₅	—	6	4	no	—
92	#709	—	6	16	no	—
93	C₂×D₂₀	—	3	4	no	—
94	C₂×D₂₀	—	6	4	no	—
95	C₂×C₄×D₅	—	6	4	no	—
96	C₂×Dic₁₀	—	3	4	no	—
97	C₂×C₅×C₈	D₁₂	1	8	yes	complemented
98	C₂×C₅×C₈	C₂²×D₃	1	4	yes	—
99	C₄²×C₅	D₁₂	1	2	yes	—
100	C₄²×C₅	C₂²×D₃	1	2	yes	—
101	#685	—	6	8	no	—
102	C₄⋊Dic₅	—	6	4	no	—
103	Dic₂₀	—	6	8	no	—
104	D₄₀	—	6	8	no	—
105	#677	—	6	16	no	—
106	C₂×#135	—	15	4	no	—
107	#532	—	15	2	no	—
108	#497	—	15	4	no	—
109	#496	—	15	4	no	—
110	C₂×#97	D₁₅	1	1	yes	characteristic, complemented
111	C₂²×C₃×C₅	—	2	2	no	—
112	C₂²×C₃×C₅	#127	1	2	yes	—
113	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
114	C₂×D₁₅	—	4	4	no	—
115	C₂×D₁₅	—	4	8	no	—
116	C₃×C₄×C₅	—	2	4	no	—
117	C₃×C₄×C₅	#127	1	2	yes	—
118	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
119	Dic₁₅	—	4	4	no	—
120	C₂²×C₃×C₄	D₂₀	1	2	yes	—
121	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
122	C₂×#67	—	10	4	no	—
123	#235	—	10	16	no	—
124	C₂×D₁₂	—	5	4	no	—
125	C₂×D₁₂	—	10	4	no	—
126	C₂×C₄×D₃	—	10	4	no	—
127	C₂×Dic₆	—	5	4	no	—
128	C₂×C₃×C₈	D₂₀	1	8	yes	complemented
129	C₂×C₃×C₈	C₂²×D₅	1	4	yes	—
130	C₃×C₄²	D₂₀	1	2	yes	—
131	C₃×C₄²	C₂²×D₅	1	2	yes	—
132	(C₂×D₃)⋊C₄	—	10	8	no	—
133	C₄⋊Dic₃	—	10	4	no	—
134	Dic₁₂	—	10	8	no	—
135	D₂₄	—	10	8	no	—
136	#204	—	10	16	no	—
137	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
138	C₂²×D₅	—	6	4	no	—
139	C₂×C₄×C₅	—	4	2	no	—
140	C₂×C₄×C₅	C₂×D₁₂	1	4	yes	—
141	C₂×C₄×C₅	C₂×D₁₂	1	4	yes	—
142	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
144	C₂²⋊D₅	—	12	8	no	—
145	C₂×Dic₅	—	6	4	no	—
146	D₂₀	—	6	8	no	—
147	D₂₀	—	12	8	no	—
148	C₄×D₅	—	12	8	no	—
149	Dic₁₀	—	6	8	no	—
150	C₅×C₈	C₂×D₁₂	1	8	yes	—
151	C₂×#41	—	15	4	no	—
152	#135	—	30	16	no	—
153	C₂×Q₁₆	—	15	4	no	—
154	C₂×SD₁₆	—	15	8	no	—
155	C₂×D₈	—	15	4	no	—
156	C₂²×C₈	C₂×D₁₅	1	2	yes	—
157	C₄⋊Q₈	—	15	2	no	—
158	#127	—	15	2	no	—
159	#124	—	15	4	no	—
160	#121	—	30	4	no	—
161	C₄×D₄	—	30	4	no	—
162	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
163	#97	C₂×D₁₅	1	4	yes	—
164	C₂×C₃×C₅	—	2	2	no	—
165	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
166	C₂×C₃×C₅	#575	1	2	yes	—
167	D₁₅	—	8	4	no	—
168	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
169	C₂²×D₃	—	10	4	no	—
170	C₂×C₃×C₄	—	4	2	no	—
171	C₂×C₃×C₄	C₂×D₂₀	1	4	yes	—
172	C₂×C₃×C₄	C₂×D₂₀	1	4	yes	—
173	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
175	#67	—	20	8	no	—
176	C₂×Dic₃	—	10	4	no	—
177	D₁₂	—	10	8	no	—
178	D₁₂	—	20	8	no	—
179	C₄×D₃	—	20	8	no	—
180	Dic₆	—	10	8	no	—
181	C₃×C₈	C₂×D₂₀	1	8	yes	—
182	C₂²×C₅	—	2	2	no	—
183	C₂²×C₅	#874	1	2	yes	—
184	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
185	C₂×D₅	—	12	4	no	—
186	C₂×D₅	—	12	8	no	—
187	C₄×C₅	—	2	4	no	—
188	C₄×C₅	#874	1	2	yes	—
189	C₄×C₅	C₂²×D₁₂	1	2	yes	—
190	Dic₅	—	12	4	no	—
191	#41	—	30	16	no	—
192	C₂×Q₈	—	15	4	no	—
193	C₂×D₄	—	15	4	no	—
194	C₂×D₄	—	30	4	no	—
195	C₂×D₄	—	30	4	no	—
196	C₂²×C₄	—	30	4	no	—
197	C₂²×C₄	D₆₀	1	2	yes	—
198	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
199	Q₁₆	—	30	8	no	—
200	SD₁₆	—	30	16	no	—
201	D₈	—	30	8	no	—
202	C₂×C₈	D₆₀	1	8	yes	complemented
203	C₂×C₈	C₂²×D₁₅	1	4	yes	—
204	C₄⋊C₄	—	30	4	no	—
205	C₂²⋊C₄	—	30	8	no	—
206	C₄²	D₆₀	1	2	yes	—
207	C₄²	C₂²×D₁₅	1	2	yes	—
208	C₃×C₅	#3150	1	1	yes	characteristic, complemented, Hall
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	#4020	1	2	yes	—
211	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
212	C₂×D₃	—	20	4	no	—
213	C₂×D₃	—	20	8	no	—
214	C₃×C₄	—	2	4	no	—
215	C₃×C₄	#4020	1	2	yes	—
216	C₃×C₄	C₂²×D₂₀	1	2	yes	—
217	Dic₃	—	20	4	no	—
218	C₂×C₅	—	2	2	no	—
219	C₂×C₅	C₂×#874	1	1	yes	characteristic
220	C₂×C₅	#5748	1	2	yes	—
221	D₅	—	24	4	no	—
222	C₂³	—	30	4	no	—
223	C₂³	C₂×D₆₀	1	1	yes	characteristic
224	Q₈	—	30	8	no	—
225	D₄	—	30	8	no	—
226	D₄	—	60	8	no	—
227	D₄	—	60	8	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	30	4	no	—
230	C₂×C₄	—	60	8	no	—
231	C₂×C₄	C₂×D₆₀	1	4	yes	—
232	C₂×C₄	C₂×D₆₀	1	4	yes	—
233	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
234	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
235	C₈	C₂×D₆₀	1	8	yes	—
236	C₂×C₃	—	2	2	no	—
237	C₂×C₃	C₂×#4020	1	1	yes	characteristic
238	C₂×C₃	#66082	1	2	yes	—
239	D₃	—	40	4	no	—
240	C₅	#78499	1	1	yes	characteristic, complemented, Sylow
241	C₂²	—	2	2	no	—
242	C₂²	—	60	4	no	—
243	C₂²	—	60	8	no	—
244	C₂²	#91763	1	2	yes	—
245	C₂²	C₂²×D₆₀	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	60	4	no	—
248	C₄	#91763	1	2	yes	—
249	C₄	C₂²×D₆₀	1	2	yes	—
250	C₃	#10610230	1	1	yes	characteristic, complemented, Sylow
251	C₂	—	2	2	no	—
252	C₂	—	120	4	no	—
253	C₂	C₂×#91763	1	1	yes	characteristic
254	C₂	#11756507	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₆₀
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅
Fitting	C₂×C₃×C₅×#97	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₂×#97	D₁₅
hypocenter	C₃×C₅	#3150
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
1. C₂×#91793	C₂	semidirect
2. #11756236	C₂	semidirect
3. C₂×C₃×C₅×#97	C₂	semidirect
4. #11750816	C₂	semidirect
5. #11750813	C₂	semidirect
23. C₂×C₅×#97	D₃	semidirect
34. C₂×C₃×C₅×C₈	D₄	semidirect
45. C₂×C₃×#97	D₅	semidirect
97. C₂×C₅×C₈	D₁₂	semidirect
110. C₂×#97	D₁₅	semidirect
128. C₂×C₃×C₈	D₂₀	semidirect
202. C₂×C₈	D₆₀	semidirect
208. C₃×C₅	#3150	semidirect
240. C₅	#78499	semidirect
250. C₃	#10610230	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#91793	2
#11756236	2
C₂×C₃×C₅×#97	2
#11750816	2
#11750813	2
#10610230	3
#78499	5