Subgroups — #14287263

Order 1920.

360 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₁₂₀	C₂	1	1	yes	maximal, characteristic
2	#11750789	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746280	C₂	1	2	yes	maximal, complemented
5	#11746279	C₂	1	1	yes	maximal, characteristic, complemented
6	#10598845	—	3	1	no	maximal, Hall
7	C₂×D₁₂₀	C₄	1	1	yes	characteristic
8	C₂×#6705	—	2	2	no	—
9	C₄×D₆₀	C₂²	1	2	yes	—
10	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	#91105	C₄	1	2	yes	—
14	#91104	—	2	2	no	—
15	#91102	C₄	1	1	yes	characteristic
16	C₄⋊(C₃⋊#168)	C₂²	1	2	yes	—
17	#68052	—	5	1	no	maximal, Hall
18	C₄×D₄₀	—	3	1	no	—
19	#64977	—	3	2	no	—
20	C₅×C₈²	D₃	1	1	yes	characteristic, complemented
21	#64679	—	3	2	no	—
22	#64678	—	3	1	no	—
23	C₂×D₆₀	C₂×C₄	1	2	yes	—
24	C₂×C₄×D₁₅	—	2	2	no	—
25	C₂×C₃×C₅×C₈	—	2	1	no	—
26	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
30	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
31	#6714	—	2	2	no	—
32	C₄⋊Dic₁₅	C₂×C₄	1	2	yes	—
33	C₂×C₃⋊#168	—	2	2	no	—
34	D₁₂₀	—	2	2	no	—
35	#6705	—	4	4	no	—
36	C₄×D₂₄	—	5	1	no	—
37	#4688	—	5	2	no	—
38	C₃×C₈²	D₅	1	1	yes	characteristic, complemented
39	#4459	—	5	2	no	—
40	#4458	—	5	1	no	—
41	C₂×D₄₀	—	3	1	no	—
42	C₂×#676	—	6	2	no	—
43	C₄×D₂₀	—	3	2	no	—
44	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
45	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
46	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
47	#3953	—	3	2	no	—
48	#3952	—	6	2	no	—
49	#3950	—	3	1	no	—
50	C₄⋊#168	—	3	2	no	—
51	#1589	—	15	1	no	Sylow
52	C₂²×D₁₅	—	2	2	no	—
53	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
56	C₂×Dic₁₅	—	2	2	no	—
57	D₆₀	—	2	2	no	—
58	D₆₀	—	2	2	no	—
59	C₄×D₁₅	—	4	2	no	—
60	C₄×D₁₅	—	4	2	no	—
61	C₃×C₅×C₈	—	2	2	no	—
62	C₃×C₅×C₈	—	2	1	no	—
63	C₃×C₅×C₈	—	2	1	no	—
64	C₃×C₅×C₈	M₁₆	1	2	yes	complemented
65	C₃×C₅×C₈	D₈	1	2	yes	complemented
66	C₃⋊#168	—	4	2	no	—
67	C₂×D₂₄	—	5	1	no	—
68	C₂×ΓL(1,𝔽₂₅)	—	10	2	no	—
69	C₄×D₁₂	—	5	2	no	—
70	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
71	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
72	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
73	#821	—	5	2	no	—
74	(C₂×D₃)⋊C₈	—	10	2	no	—
75	#818	—	5	1	no	—
76	C₄⋊(C₃⋊C₈)	—	5	2	no	—
77	C₂×D₂₀	—	3	2	no	—
78	C₂×C₄×D₅	—	6	2	no	—
79	C₂×C₅×C₈	—	2	1	no	—
80	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
81	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
82	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
83	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
84	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
85	#685	—	6	2	no	—
86	C₄⋊Dic₅	—	3	2	no	—
87	C₂×#168	—	6	2	no	—
88	D₄₀	—	6	2	no	—
89	#676	—	12	4	no	—
90	C₄×D₈	—	15	1	no	—
91	#436	—	15	2	no	—
92	#335	—	15	1	no	—
93	D₄⋊C₈	—	15	2	no	—
94	C₈²	D₁₅	1	1	yes	characteristic, complemented
95	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
96	C₂×D₁₅	—	4	2	no	—
97	C₂×D₁₅	—	4	2	no	—
98	C₂×D₁₅	—	4	2	no	—
99	C₃×C₄×C₅	—	2	1	no	—
100	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
101	C₃×C₄×C₅	#131	1	1	yes	characteristic
102	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
103	C₃×C₄×C₅	#135	1	1	yes	characteristic
104	Dic₁₅	—	4	2	no	—
105	C₂×D₁₂	—	5	2	no	—
106	C₂×C₄×D₃	—	10	2	no	—
107	C₂×C₃×C₈	—	2	1	no	—
108	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
109	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
110	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
111	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
112	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
113	(C₂×D₃)⋊C₄	—	10	2	no	—
114	C₄⋊Dic₃	—	5	2	no	—
115	C₂×C₃⋊C₈	—	10	2	no	—
116	D₂₄	—	10	2	no	—
117	ΓL(1,𝔽₂₅)	—	20	4	no	—
118	C₂²×D₅	—	6	2	no	—
119	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
121	C₂×C₄×C₅	#235	1	1	yes	characteristic
122	C₂×Dic₅	—	6	2	no	—
123	D₂₀	—	6	2	no	—
124	D₂₀	—	6	2	no	—
125	C₄×D₅	—	12	2	no	—
126	C₄×D₅	—	12	2	no	—
127	C₅×C₈	—	2	2	no	—
128	C₅×C₈	—	2	1	no	—
129	C₅×C₈	—	2	1	no	—
130	C₅×C₈	ΓL(1,𝔽₂₅)	1	2	yes	complemented
131	C₅×C₈	D₂₄	1	2	yes	complemented
132	#168	—	12	2	no	—
133	C₂×D₈	—	15	1	no	—
134	C₂×M₁₆	—	30	2	no	—
135	C₄×D₄	—	15	2	no	—
136	#107	—	15	1	no	—
137	C₄⋊C₈	—	15	2	no	—
138	#102	—	15	2	no	—
139	C₂²⋊C₈	—	30	2	no	—
140	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
141	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
142	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
143	C₂×C₃×C₅	#436	1	1	yes	characteristic
144	C₂×C₃×C₅	C₄×D₈	1	1	yes	characteristic
145	C₂×C₃×C₅	#443	1	1	yes	characteristic
146	D₁₅	—	8	2	no	—
147	C₂²×D₃	—	10	2	no	—
148	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
150	C₂×C₃×C₄	#709	1	1	yes	characteristic
151	C₂×Dic₃	—	10	2	no	—
152	D₁₂	—	10	2	no	—
153	D₁₂	—	10	2	no	—
154	C₄×D₃	—	20	2	no	—
155	C₄×D₃	—	20	2	no	—
156	C₃×C₈	—	2	2	no	—
157	C₃×C₈	—	2	1	no	—
158	C₃×C₈	—	2	1	no	—
159	C₃×C₈	#676	1	2	yes	complemented
160	C₃×C₈	D₄₀	1	2	yes	complemented
161	C₃⋊C₈	—	20	2	no	—
162	C₂²×C₅	C₄×D₁₂	1	1	yes	characteristic
163	C₂×D₅	—	12	2	no	—
164	C₂×D₅	—	12	2	no	—
165	C₂×D₅	—	12	2	no	—
166	C₄×C₅	—	2	1	no	—
167	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
168	C₄×C₅	#901	1	1	yes	characteristic
169	C₄×C₅	C₂×D₂₄	1	1	yes	characteristic
170	C₄×C₅	#904	1	1	yes	characteristic
171	Dic₅	—	12	2	no	—
172	C₂×D₄	—	15	2	no	—
173	C₂²×C₄	—	30	2	no	—
174	D₈	—	30	2	no	—
175	M₁₆	—	60	4	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	30	2	no	—
178	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
179	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
180	C₂×C₈	D₆₀	1	1	yes	characteristic
181	C₂×C₈	D₆₀	1	1	yes	characteristic
182	C₄⋊C₄	—	15	2	no	—
183	C₂²⋊C₄	—	30	2	no	—
184	C₄²	C₂²×D₁₅	1	1	yes	characteristic
185	C₃×C₅	#1589	1	1	yes	characteristic, complemented, Hall
186	C₂²×C₃	C₄×D₂₀	1	1	yes	characteristic
187	C₂×D₃	—	20	2	no	—
188	C₂×D₃	—	20	2	no	—
189	C₂×D₃	—	20	2	no	—
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	C₂×#676	1	1	yes	characteristic
192	C₃×C₄	#4047	1	1	yes	characteristic
193	C₃×C₄	C₂×D₄₀	1	1	yes	characteristic
194	C₃×C₄	#4050	1	1	yes	characteristic
195	Dic₃	—	20	2	no	—
196	C₂×C₅	#4688	1	1	yes	characteristic
197	C₂×C₅	C₄×D₂₄	1	1	yes	characteristic
198	C₂×C₅	#4700	1	1	yes	characteristic
199	D₅	—	24	2	no	—
200	C₂³	—	30	2	no	—
201	D₄	—	30	2	no	—
202	D₄	—	30	2	no	—
203	C₂×C₄	—	30	2	no	—
204	C₂×C₄	—	60	2	no	—
205	C₂×C₄	—	60	2	no	—
206	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
207	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
208	C₂×C₄	#6817	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	60	2	no	—
213	C₈	#6705	1	2	yes	complemented
214	C₈	D₁₂₀	1	2	yes	complemented
215	C₂×C₃	#64977	1	1	yes	characteristic
216	C₂×C₃	C₄×D₄₀	1	1	yes	characteristic
217	C₂×C₃	#64989	1	1	yes	characteristic
218	D₃	—	40	2	no	—
219	C₅	#68052	1	1	yes	characteristic, complemented, Sylow
220	C₂²	—	60	2	no	—
221	C₂²	—	60	2	no	—
222	C₂²	—	60	2	no	—
223	C₂²	C₄×D₆₀	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	60	2	no	—
226	C₄	C₂×#6705	1	1	yes	characteristic
227	C₄	#91790	1	1	yes	characteristic
228	C₄	C₂×D₁₂₀	1	1	yes	characteristic
229	C₄	#91793	1	1	yes	characteristic
230	C₃	#10598845	1	1	yes	characteristic, complemented, Sylow
231	C₂	—	120	2	no	—
232	C₂	#11750789	1	1	yes	characteristic
233	C₂	C₄×D₁₂₀	1	1	yes	characteristic
234	C₂	#11750801	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₆₀
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₅×C₈²	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×D₁₂₀	C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₈²	D₁₅
hypocenter	C₃×C₅	#1589
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. #11750789	C₂	semidirect
3. C₃×C₅×C₈²	C₂	semidirect
4. #11746280	C₂	semidirect
5. #11746279	C₂	semidirect
20. C₅×C₈²	D₃	semidirect
38. C₃×C₈²	D₅	semidirect
64. C₃×C₅×C₈	M₁₆	semidirect
65. C₃×C₅×C₈	D₈	semidirect
94. C₈²	D₁₅	semidirect
130. C₅×C₈	ΓL(1,𝔽₂₅)	semidirect
131. C₅×C₈	D₂₄	semidirect
159. C₃×C₈	#676	semidirect
160. C₃×C₈	D₄₀	semidirect
185. C₃×C₅	#1589	semidirect
213. C₈	#6705	semidirect
214. C₈	D₁₂₀	semidirect
219. C₅	#68052	semidirect
230. C₃	#10598845	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₁₂₀	2
#11750789	2
C₃×C₅×C₈²	2
#11746280	2
#11746279	2
#10598845	3
#68052	5