Subgroups — #14313954

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 252 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	1	yes	maximal, characteristic
2	C₂×C₅×#812	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65395	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745895	C₂	1	4	yes	maximal, complemented
5	#10602233	—	3	1	no	maximal, Hall
6	C₂²×D₆₀	—	2	1	no	—
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×D₁₂₀	—	2	1	no	—
9	C₂×D₁₂₀	—	2	2	no	—
10	C₂×D₁₂₀	C₄	1	4	yes	complemented
11	C₂×D₁₂₀	C₂²	1	2	yes	—
12	C₃×#4072	—	2	1	no	—
13	C₅×#812	—	2	2	no	—
14	C₂×C₅×C₃⋊C₁₆	C₂²	1	2	yes	complemented
15	C₃×#3950	C₂²	1	2	yes	complemented
16	C₃×#3949	—	2	1	no	—
17	#70652	—	5	1	no	maximal, Hall
18	C₂²×D₄₀	—	3	1	no	—
19	C₂×C₅×M₃₂	—	3	1	no	—
20	#65395	D₃	1	1	yes	characteristic, complemented
21	#64729	—	3	4	no	—
22	C₂³×D₁₅	—	4	1	no	—
23	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
24	C₂×D₆₀	—	2	4	no	—
25	C₂×D₆₀	—	2	2	no	—
26	C₂×D₆₀	—	4	1	no	—
27	C₂×D₆₀	—	4	2	no	—
28	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
31	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
32	D₁₂₀	—	2	4	no	—
33	D₁₂₀	—	2	2	no	—
34	D₁₂₀	C₂×C₄	1	4	yes	—
35	C₃×C₂²⋊Dic₅	—	4	1	no	—
36	C₃×C₄⋊Dic₅	—	2	2	no	—
37	C₃×C₄×Dic₅	—	4	1	no	—
38	C₅×C₃⋊C₁₆	—	2	2	no	—
39	C₂²×D₂₄	—	5	1	no	—
40	C₃×#427	—	5	1	no	—
41	C₂×#812	D₅	1	1	yes	characteristic, complemented
42	#4491	—	5	4	no	—
43	C₂²×D₂₀	—	6	1	no	—
44	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
45	#4072	—	2	1	no	—
46	C₂×D₄₀	—	3	4	no	—
47	C₂×D₄₀	—	3	2	no	—
48	C₂×D₄₀	—	6	2	no	—
49	C₂×D₄₀	—	6	1	no	—
50	C₅×M₃₂	—	6	2	no	—
51	C₂×C₅×C₁₆	—	3	2	no	—
52	#3950	C₂×D₃	1	2	yes	complemented
53	#3949	—	2	1	no	—
54	#2140	—	15	1	no	Sylow
55	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
56	C₂²×D₁₅	—	4	4	no	—
57	C₂²×D₁₅	—	4	2	no	—
58	C₂²×D₁₅	—	8	4	no	—
59	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
60	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
62	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	D₆₀	—	2	4	no	—
64	D₆₀	—	4	4	no	—
65	D₆₀	—	4	2	no	—
66	C₂×C₃×Dic₅	—	4	2	no	—
67	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
68	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂²×D₁₂	—	10	1	no	—
71	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
72	C₃×#117	—	10	1	no	—
73	C₂×D₂₄	—	5	4	no	—
74	C₂×D₂₄	—	5	2	no	—
75	C₂×D₂₄	—	10	1	no	—
76	C₂×D₂₄	—	10	2	no	—
77	C₃×#107	—	5	2	no	—
78	C₃×#106	—	10	1	no	—
79	#812	—	2	2	no	—
80	C₂×C₃⋊C₁₆	C₂×D₅	1	2	yes	complemented
81	C₂³×D₅	—	12	1	no	—
82	C₂²×C₄×C₅	#67	1	1	yes	characteristic
83	C₂×D₂₀	—	6	4	no	—
84	C₂×D₂₀	—	6	2	no	—
85	C₂×D₂₀	—	12	1	no	—
86	C₂×D₂₀	—	12	2	no	—
87	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
88	C₂×C₅×C₈	D₁₂	1	2	yes	—
89	C₂×C₅×C₈	#67	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
91	C₂²⋊Dic₅	—	4	1	no	—
92	C₄⋊Dic₅	—	2	2	no	—
93	C₄×Dic₅	—	4	1	no	—
94	D₄₀	—	3	4	no	—
95	D₄₀	—	6	4	no	—
96	D₄₀	—	6	2	no	—
97	C₅×C₁₆	—	6	2	no	—
98	C₂²×D₈	—	15	1	no	—
99	C₂×M₃₂	—	3	1	no	—
100	#427	—	5	1	no	—
101	#357	—	15	4	no	—
102	C₂²×C₃×C₅	—	2	2	no	—
103	C₂²×C₃×C₅	#102	1	2	yes	—
104	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
105	C₂×D₁₅	—	4	4	no	—
106	C₂×D₁₅	—	8	8	no	—
107	C₂×D₁₅	—	8	4	no	—
108	C₃×C₄×C₅	#102	1	2	yes	—
109	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
111	C₃×Dic₅	—	8	2	no	—
112	C₂³×D₃	—	20	1	no	—
113	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
114	C₂×D₁₂	—	10	4	no	—
115	C₂×D₁₂	—	10	2	no	—
116	C₂×D₁₂	—	20	2	no	—
117	C₂×D₁₂	—	20	1	no	—
118	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
119	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
120	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
121	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
122	C₃×C₄⋊C₄	—	10	2	no	—
123	C₃×C₂²⋊C₄	—	20	1	no	—
124	C₃×C₄²	—	20	1	no	—
125	D₂₄	—	5	4	no	—
126	D₂₄	—	10	4	no	—
127	D₂₄	—	10	2	no	—
128	C₃⋊C₁₆	—	2	2	no	—
129	C₂³×C₅	#213	1	1	yes	characteristic
130	C₂²×D₅	—	12	4	no	—
131	C₂²×D₅	—	12	2	no	—
132	C₂²×D₅	—	24	4	no	—
133	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
134	C₂×C₄×C₅	#213	1	1	yes	characteristic
135	C₂×C₄×C₅	#215	1	2	yes	—
136	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
137	C₂×Dic₅	—	4	2	no	—
138	D₂₀	—	6	4	no	—
139	D₂₀	—	12	4	no	—
140	D₂₀	—	12	2	no	—
141	C₅×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
142	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
143	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
144	C₂²×D₄	—	30	1	no	—
145	C₂×D₈	—	15	4	no	—
146	C₂×D₈	—	15	2	no	—
147	C₂×D₈	—	30	1	no	—
148	C₂×D₈	—	30	2	no	—
149	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
150	#117	—	10	1	no	—
151	M₃₂	—	6	2	no	—
152	C₂×C₁₆	—	3	2	no	—
153	#107	—	5	2	no	—
154	#106	—	10	1	no	—
155	C₂×C₃×C₅	—	2	2	no	—
156	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
157	C₂×C₃×C₅	#509	1	2	yes	—
158	D₁₅	—	8	4	no	—
159	C₂³×C₃	D₄₀	1	1	yes	characteristic
160	C₂²×D₃	—	20	4	no	—
161	C₂²×D₃	—	20	2	no	—
162	C₂²×D₃	—	40	4	no	—
163	C₂×C₃×C₄	—	20	2	no	—
164	C₂×C₃×C₄	#677	1	2	yes	—
165	C₂×C₃×C₄	D₄₀	1	1	yes	characteristic
166	C₂×C₃×C₄	#685	1	2	yes	—
167	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
168	D₁₂	—	10	4	no	—
169	D₁₂	—	20	4	no	—
170	D₁₂	—	20	2	no	—
171	C₃×C₈	#685	1	2	yes	—
172	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
173	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	#809	1	2	yes	—
176	C₂²×C₅	C₂×#213	1	1	yes	characteristic
177	C₂×D₅	—	12	4	no	—
178	C₂×D₅	—	24	8	no	—
179	C₂×D₅	—	24	4	no	—
180	C₄×C₅	#809	1	2	yes	—
181	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
182	C₄×C₅	C₂×#215	1	1	yes	characteristic
183	Dic₅	—	8	2	no	—
184	C₂⁴	—	60	1	no	—
185	C₂×D₄	—	30	4	no	—
186	C₂×D₄	—	30	2	no	—
187	C₂×D₄	—	60	2	no	—
188	C₂×D₄	—	60	1	no	—
189	C₂²×C₄	#1202	1	1	yes	characteristic
190	D₈	—	15	4	no	—
191	D₈	—	30	4	no	—
192	D₈	—	30	2	no	—
193	C₂×C₈	#1200	1	2	yes	—
194	C₂×C₈	#1202	1	1	yes	characteristic
195	C₂×C₈	#1203	1	2	yes	—
196	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
197	C₄⋊C₄	—	10	2	no	—
198	C₂²⋊C₄	—	20	1	no	—
199	C₄²	—	20	1	no	—
200	C₁₆	—	6	2	no	—
201	C₃×C₅	#2140	1	1	yes	characteristic, complemented, Hall
202	C₂²×C₃	—	2	2	no	—
203	C₂²×C₃	#3953	1	2	yes	—
204	C₂²×C₃	C₂×D₄₀	1	1	yes	characteristic
205	C₂×D₃	—	20	4	no	—
206	C₂×D₃	—	40	8	no	—
207	C₂×D₃	—	40	4	no	—
208	C₃×C₄	—	40	2	no	—
209	C₃×C₄	#3953	1	2	yes	—
210	C₃×C₄	C₂×#677	1	1	yes	characteristic
211	C₃×C₄	C₂×#685	1	1	yes	characteristic
212	C₂×C₅	—	2	2	no	—
213	C₂×C₅	C₂×#809	1	1	yes	characteristic
214	C₂×C₅	#5193	1	2	yes	—
215	D₅	—	24	4	no	—
216	C₂³	—	60	4	no	—
217	C₂³	—	60	2	no	—
218	C₂³	—	120	4	no	—
219	C₂³	#6653	1	1	yes	characteristic
220	D₄	—	30	4	no	—
221	D₄	—	60	4	no	—
222	D₄	—	60	2	no	—
223	C₂×C₄	—	20	2	no	—
224	C₂×C₄	#6653	1	1	yes	characteristic
225	C₂×C₄	#6658	1	2	yes	—
226	C₂×C₄	#6667	1	2	yes	—
227	C₂×C₄	C₂×#1202	1	1	yes	characteristic
228	C₈	#6667	1	2	yes	—
229	C₈	C₂×#1200	1	1	yes	characteristic
230	C₈	C₂×#1203	1	1	yes	characteristic
231	C₂×C₃	—	2	2	no	—
232	C₂×C₃	#65197	1	2	yes	—
233	C₂×C₃	C₂×#3953	1	1	yes	characteristic
234	D₃	—	40	4	no	—
235	C₅	#70652	1	1	yes	characteristic, complemented, Sylow
236	C₂²	—	2	2	no	—
237	C₂²	—	60	4	no	—
238	C₂²	—	120	8	no	—
239	C₂²	—	120	4	no	—
240	C₂²	#90968	1	2	yes	—
241	C₂²	C₂×#6653	1	1	yes	characteristic
242	C₄	—	40	2	no	—
243	C₄	#90968	1	2	yes	—
244	C₄	C₂×#6658	1	1	yes	characteristic
245	C₄	C₂×#6667	1	1	yes	characteristic
246	C₃	#10602233	1	1	yes	characteristic, complemented, Sylow
247	C₂	—	2	2	no	—
248	C₂	—	120	4	no	—
249	C₂	#11747544	1	2	yes	—
250	C₂	C₂×#90968	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6653
commutator	C₃×C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂×D₃×D₅
Fitting	C₂²×C₃×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂²×D₁₂₀	C₂
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	#2140
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#812	C₂	semidirect
3. C₃×#65395	C₂	semidirect
4. #11745895	C₂	semidirect
10. C₂×D₁₂₀	C₄	semidirect
14. C₂×C₅×C₃⋊C₁₆	C₂²	semidirect
15. C₃×#3950	C₂²	semidirect
20. #65395	D₃	semidirect
41. C₂×#812	D₅	semidirect
52. #3950	C₂×D₃	semidirect
80. C₂×C₃⋊C₁₆	C₂×D₅	semidirect
201. C₃×C₅	#2140	semidirect
235. C₅	#70652	semidirect
246. C₃	#10602233	semidirect

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×C₅×#812	2
C₃×#65395	2
#11745895	2
#10602233	3
#70652	5