Subgroups — #14313951

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	C₅×#5106	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3944	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745896	C₂	1	4	yes	maximal, complemented
5	#10601445	—	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	C₅×#926	—	2	1	no	—
13	C₅×#818	C₂²	1	2	yes	complemented
14	C₅×#817	—	2	1	no	—
15	C₃×#3944	—	2	2	no	—
16	C₂×C₃×#672	C₂²	1	2	yes	complemented
17	#71440	—	5	1	no	maximal, Hall
18	C₂²×D₄₀	—	3	1	no	—
19	C₅×#427	—	3	1	no	—
20	C₂×#3944	D₃	1	1	yes	characteristic, complemented
21	#64711	—	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	C₅×#217	—	4	1	no	—
36	C₅×C₄⋊Dic₃	—	2	2	no	—
37	C₄×C₅×Dic₃	—	4	1	no	—
38	C₃×#672	—	2	2	no	—
39	C₂²×D₂₄	—	5	1	no	—
40	C₂×C₃×M₃₂	—	5	1	no	—
41	#5106	D₅	1	1	yes	characteristic, complemented
42	#4509	—	5	4	no	—
43	C₂²×D₂₀	—	6	1	no	—
44	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
45	C₅×#117	—	6	1	no	—
46	C₂×D₄₀	—	3	4	no	—
47	C₂×D₄₀	—	3	2	no	—
48	C₂×D₄₀	—	6	1	no	—
49	C₂×D₄₀	—	6	2	no	—
50	C₅×#107	—	3	2	no	—
51	C₅×#106	—	6	1	no	—
52	#3944	—	2	2	no	—
53	C₂×#672	C₂×D₃	1	2	yes	complemented
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×C₅×Dic₃	—	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	#926	—	2	1	no	—
73	C₂×D₂₄	—	5	4	no	—
74	C₂×D₂₄	—	5	2	no	—
75	C₂×D₂₄	—	10	2	no	—
76	C₂×D₂₄	—	10	1	no	—
77	C₃×M₃₂	—	10	2	no	—
78	C₂×C₃×C₁₆	—	5	2	no	—
79	#818	C₂×D₅	1	2	yes	complemented
80	#817	—	2	1	no	—
81	C₂³×D₅	—	12	1	no	—
82	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
83	C₂×D₂₀	—	6	4	no	—
84	C₂×D₂₀	—	6	2	no	—
85	C₂×D₂₀	—	12	2	no	—
86	C₂×D₂₀	—	12	1	no	—
87	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
88	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
89	C₂×C₅×C₈	#67	1	2	yes	—
90	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
91	C₅×C₄⋊C₄	—	6	2	no	—
92	C₅×C₂²⋊C₄	—	12	1	no	—
93	C₄²×C₅	—	12	1	no	—
94	D₄₀	—	3	4	no	—
95	D₄₀	—	6	4	no	—
96	D₄₀	—	6	2	no	—
97	#672	—	2	2	no	—
98	C₂²×D₈	—	15	1	no	—
99	C₂×M₃₂	—	5	1	no	—
100	#427	—	3	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	C₅×Dic₃	—	8	2	no	—
112	C₂³×D₃	—	20	1	no	—
113	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
114	C₂×D₁₂	—	10	4	no	—
115	C₂×D₁₂	—	10	2	no	—
116	C₂×D₁₂	—	20	1	no	—
117	C₂×D₁₂	—	20	2	no	—
118	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
119	C₂×C₃×C₈	D₂₀	1	2	yes	—
120	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
121	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
122	#217	—	4	1	no	—
123	C₄⋊Dic₃	—	2	2	no	—
124	C₄×Dic₃	—	4	1	no	—
125	D₂₄	—	5	4	no	—
126	D₂₄	—	10	4	no	—
127	D₂₄	—	10	2	no	—
128	C₃×C₁₆	—	10	2	no	—
129	C₂³×C₅	D₂₄	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	—	12	2	no	—
134	C₂×C₄×C₅	#204	1	2	yes	—
135	C₂×C₄×C₅	D₂₄	1	1	yes	characteristic
136	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
137	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
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×#67	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	2	no	—
148	C₂×D₈	—	30	1	no	—
149	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
150	#117	—	6	1	no	—
151	M₃₂	—	10	2	no	—
152	C₂×C₁₆	—	5	2	no	—
153	#107	—	3	2	no	—
154	#106	—	6	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	#686	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	#685	1	2	yes	—
164	C₂×C₃×C₄	#686	1	1	yes	characteristic
165	C₂×C₃×C₄	#688	1	2	yes	—
166	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
167	C₂×Dic₃	—	4	2	no	—
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×D₂₀	1	1	yes	characteristic
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	#821	1	2	yes	—
176	C₂²×C₅	C₂×D₂₄	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	—	24	2	no	—
181	C₄×C₅	#821	1	2	yes	—
182	C₄×C₅	C₂×#204	1	1	yes	characteristic
183	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
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	#1203	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	2	yes	—
195	C₂×C₈	#1203	1	1	yes	characteristic
196	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
197	C₄⋊C₄	—	6	2	no	—
198	C₂²⋊C₄	—	12	1	no	—
199	C₄²	—	12	1	no	—
200	C₁₆	—	10	2	no	—
201	C₃×C₅	#2140	1	1	yes	characteristic, complemented, Hall
202	C₂²×C₃	—	2	2	no	—
203	C₂²×C₃	#3941	1	2	yes	—
204	C₂²×C₃	C₂×#686	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	#3941	1	2	yes	—
209	C₃×C₄	C₂×#685	1	1	yes	characteristic
210	C₃×C₄	C₂×#688	1	1	yes	characteristic
211	Dic₃	—	8	2	no	—
212	C₂×C₅	—	2	2	no	—
213	C₂×C₅	#4908	1	2	yes	—
214	C₂×C₅	C₂×#821	1	1	yes	characteristic
215	D₅	—	24	4	no	—
216	C₂³	—	60	4	no	—
217	C₂³	—	60	2	no	—
218	C₂³	—	120	4	no	—
219	C₂³	#6654	1	1	yes	characteristic
220	D₄	—	30	4	no	—
221	D₄	—	60	4	no	—
222	D₄	—	60	2	no	—
223	C₂×C₄	—	12	2	no	—
224	C₂×C₄	#6654	1	1	yes	characteristic
225	C₂×C₄	#6660	1	2	yes	—
226	C₂×C₄	#6667	1	2	yes	—
227	C₂×C₄	C₂×#1203	1	1	yes	characteristic
228	C₈	#6667	1	2	yes	—
229	C₈	C₂×#1200	1	1	yes	characteristic
230	C₈	C₂×#1202	1	1	yes	characteristic
231	C₂×C₃	—	2	2	no	—
232	C₂×C₃	C₂×#3941	1	1	yes	characteristic
233	C₂×C₃	#65482	1	2	yes	—
234	D₃	—	40	4	no	—
235	C₅	#71440	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	#90969	1	2	yes	—
241	C₂²	C₂×#6654	1	1	yes	characteristic
242	C₄	—	24	2	no	—
243	C₄	#90969	1	2	yes	—
244	C₄	C₂×#6660	1	1	yes	characteristic
245	C₄	C₂×#6667	1	1	yes	characteristic
246	C₃	#10601445	1	1	yes	characteristic, complemented, Sylow
247	C₂	—	2	2	no	—
248	C₂	—	120	4	no	—
249	C₂	#11747548	1	2	yes	—
250	C₂	C₂×#90969	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6654
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. C₅×#5106	C₂	semidirect
3. C₂×C₃×#3944	C₂	semidirect
4. #11745896	C₂	semidirect
10. C₂×D₁₂₀	C₄	semidirect
13. C₅×#818	C₂²	semidirect
16. C₂×C₃×#672	C₂²	semidirect
20. C₂×#3944	D₃	semidirect
41. #5106	D₅	semidirect
53. C₂×#672	C₂×D₃	semidirect
79. #818	C₂×D₅	semidirect
201. C₃×C₅	#2140	semidirect
235. C₅	#71440	semidirect
246. C₃	#10601445	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
C₅×#5106	2
C₂×C₃×#3944	2
#11745896	2
#10601445	3
#71440	5