Subgroups — #14315135

Order 1920.

324 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91082	C₂	1	2	yes	maximal, complemented
2	C₈×D₆₀	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746328	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₃⋊#672)	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599848	—	3	1	no	maximal, Hall
7	C₂×C₈×D₁₅	C₂²	1	2	yes	—
8	C₄×D₆₀	C₄	1	1	yes	characteristic
9	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	#91104	C₄	1	2	yes	—
13	C₂×C₃⋊#672	C₂²	1	2	yes	—
14	C₄⋊(C₃⋊#168)	C₄	1	1	yes	characteristic
15	#91082	—	2	4	no	—
16	#69055	—	5	1	no	maximal, Hall
17	C₂×#3930	—	3	2	no	—
18	C₈×D₂₀	—	3	1	no	—
19	C₄×C₅×C₁₆	D₃	1	1	yes	characteristic, complemented
20	#64727	—	3	2	no	—
21	C₄⋊#672	—	3	1	no	—
22	C₂×D₆₀	C₈	1	1	yes	characteristic
23	C₂×C₄×D₁₅	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
26	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	#6714	C₈	1	2	yes	—
28	C₄⋊Dic₁₅	C₈	1	1	yes	characteristic
29	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
30	C₈×D₁₅	—	2	2	no	—
31	C₈×D₁₅	—	2	2	no	—
32	C₃×C₅×C₁₆	—	2	1	no	—
33	C₃×C₅×C₁₆	D₄	1	2	yes	complemented
34	C₃⋊#672	—	2	2	no	—
35	C₂×#798	—	5	2	no	—
36	C₈×D₁₂	—	5	1	no	—
37	C₃×C₄×C₁₆	D₅	1	1	yes	characteristic, complemented
38	(C₂×D₃)⋊C₁₆	—	5	2	no	—
39	C₄⋊(C₃⋊C₁₆)	—	5	1	no	—
40	C₂×C₈×D₅	—	3	2	no	—
41	C₄×D₂₀	—	3	1	no	—
42	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
43	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
44	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
45	#3952	—	3	2	no	—
46	C₂×#672	—	3	2	no	—
47	C₄⋊#168	—	3	1	no	—
48	#3930	—	6	4	no	—
49	#2169	—	15	1	no	Sylow
50	C₂²×D₁₅	C₂×C₈	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂×Dic₁₅	C₂×C₈	1	2	yes	—
55	D₆₀	—	2	2	no	—
56	C₄×D₁₅	—	2	2	no	—
57	C₄×D₁₅	—	2	2	no	—
58	C₃×C₅×C₈	—	2	1	no	—
59	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
60	C₃×C₅×C₈	#41	1	1	yes	characteristic
61	C₃⋊#168	—	2	2	no	—
62	C₂×D₃×C₈	—	5	2	no	—
63	C₄×D₁₂	—	5	1	no	—
64	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
65	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
66	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
67	(C₂×D₃)⋊C₈	—	5	2	no	—
68	C₂×C₃⋊C₁₆	—	5	2	no	—
69	C₄⋊(C₃⋊C₈)	—	5	1	no	—
70	#798	—	10	4	no	—
71	C₂×D₂₀	—	3	1	no	—
72	C₂×C₄×D₅	—	3	2	no	—
73	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
74	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
75	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
76	#685	—	3	2	no	—
77	C₄⋊Dic₅	—	3	1	no	—
78	C₂×#168	—	3	2	no	—
79	C₈×D₅	—	6	2	no	—
80	C₈×D₅	—	6	2	no	—
81	C₅×C₁₆	—	2	1	no	—
82	C₅×C₁₆	D₁₂	1	2	yes	complemented
83	#672	—	6	2	no	—
84	C₂×M₃₂	—	15	2	no	—
85	C₈×D₄	—	15	1	no	—
86	C₄⋊C₁₆	—	15	1	no	—
87	C₂²⋊C₁₆	—	15	2	no	—
88	C₄×C₁₆	D₁₅	1	1	yes	characteristic, complemented
89	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
90	C₂×D₁₅	—	2	2	no	—
91	C₂×D₁₅	—	2	2	no	—
92	C₂×D₁₅	—	2	2	no	—
93	C₃×C₄×C₅	—	2	1	no	—
94	C₃×C₄×C₅	M₃₂	1	2	yes	complemented
95	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
96	C₃×C₄×C₅	#131	1	1	yes	characteristic
97	Dic₁₅	—	2	2	no	—
98	C₂×D₁₂	—	5	1	no	—
99	C₂×C₄×D₃	—	5	2	no	—
100	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
102	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
103	(C₂×D₃)⋊C₄	—	5	2	no	—
104	C₄⋊Dic₃	—	5	1	no	—
105	C₂×C₃⋊C₈	—	5	2	no	—
106	D₃×C₈	—	10	2	no	—
107	D₃×C₈	—	10	2	no	—
108	C₃×C₁₆	—	2	1	no	—
109	C₃×C₁₆	D₂₀	1	2	yes	complemented
110	C₃⋊C₁₆	—	10	2	no	—
111	C₂²×D₅	—	3	2	no	—
112	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
113	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
115	C₂×Dic₅	—	3	2	no	—
116	D₂₀	—	6	2	no	—
117	C₄×D₅	—	6	2	no	—
118	C₄×D₅	—	6	2	no	—
119	C₅×C₈	—	2	1	no	—
120	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
121	C₅×C₈	#235	1	1	yes	characteristic
122	#168	—	6	2	no	—
123	C₂²×C₈	—	15	2	no	—
124	C₄×D₄	—	15	1	no	—
125	M₃₂	—	30	4	no	—
126	C₂×C₁₆	—	15	2	no	—
127	C₂×C₁₆	C₂×D₁₅	1	1	yes	characteristic
128	C₂×C₁₆	C₂×D₁₅	1	1	yes	characteristic
129	C₄⋊C₈	—	15	1	no	—
130	C₂²⋊C₈	—	15	2	no	—
131	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
132	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
133	C₂×C₃×C₅	C₂×M₃₂	1	1	yes	characteristic
134	C₂×C₃×C₅	#504	1	1	yes	characteristic
135	D₁₅	—	4	2	no	—
136	C₂²×D₃	—	5	2	no	—
137	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
138	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
140	C₂×Dic₃	—	5	2	no	—
141	D₁₂	—	10	2	no	—
142	C₄×D₃	—	10	2	no	—
143	C₄×D₃	—	10	2	no	—
144	C₃×C₈	—	2	1	no	—
145	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
146	C₃×C₈	#709	1	1	yes	characteristic
147	C₃⋊C₈	—	10	2	no	—
148	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
149	C₂×D₅	—	6	2	no	—
150	C₂×D₅	—	6	2	no	—
151	C₂×D₅	—	6	2	no	—
152	C₄×C₅	—	2	1	no	—
153	C₄×C₅	#798	1	2	yes	complemented
154	C₄×C₅	C₄×D₁₂	1	1	yes	characteristic
155	C₄×C₅	#901	1	1	yes	characteristic
156	Dic₅	—	6	2	no	—
157	C₂×D₄	—	15	1	no	—
158	C₂²×C₄	—	15	2	no	—
159	C₂×C₈	—	15	2	no	—
160	C₂×C₈	—	30	2	no	—
161	C₂×C₈	—	30	2	no	—
162	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
163	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
164	C₄⋊C₄	—	15	1	no	—
165	C₂²⋊C₄	—	15	2	no	—
166	C₄²	C₄×D₁₅	1	1	yes	characteristic
167	C₁₆	—	2	1	no	—
168	C₁₆	—	30	2	no	—
169	C₁₆	D₆₀	1	2	yes	complemented
170	C₃×C₅	#2169	1	1	yes	characteristic, complemented, Hall
171	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
172	C₂×D₃	—	10	2	no	—
173	C₂×D₃	—	10	2	no	—
174	C₂×D₃	—	10	2	no	—
175	C₃×C₄	—	2	1	no	—
176	C₃×C₄	#3930	1	2	yes	complemented
177	C₃×C₄	C₄×D₂₀	1	1	yes	characteristic
178	C₃×C₄	#4047	1	1	yes	characteristic
179	Dic₃	—	10	2	no	—
180	C₂×C₅	C₈×D₁₂	1	1	yes	characteristic
181	C₂×C₅	C₂×#798	1	1	yes	characteristic
182	C₂×C₅	#4901	1	1	yes	characteristic
183	D₅	—	12	2	no	—
184	C₂³	—	15	2	no	—
185	D₄	—	30	2	no	—
186	C₂×C₄	—	15	2	no	—
187	C₂×C₄	—	30	2	no	—
188	C₂×C₄	—	30	2	no	—
189	C₂×C₄	C₈×D₁₅	1	1	yes	characteristic
190	C₂×C₄	C₈×D₁₅	1	1	yes	characteristic
191	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
192	C₈	—	2	1	no	—
193	C₈	—	30	2	no	—
194	C₈	C₂×D₆₀	1	1	yes	characteristic
195	C₈	#6817	1	1	yes	characteristic
196	C₂×C₃	C₈×D₂₀	1	1	yes	characteristic
197	C₂×C₃	C₂×#3930	1	1	yes	characteristic
198	C₂×C₃	#65190	1	1	yes	characteristic
199	D₃	—	20	2	no	—
200	C₅	#69055	1	1	yes	characteristic, complemented, Sylow
201	C₂²	—	30	2	no	—
202	C₂²	—	30	2	no	—
203	C₂²	—	30	2	no	—
204	C₂²	C₂×C₈×D₁₅	1	1	yes	characteristic
205	C₄	—	2	1	no	—
206	C₄	—	30	2	no	—
207	C₄	#91082	1	2	yes	complemented
208	C₄	C₄×D₆₀	1	1	yes	characteristic
209	C₄	#91790	1	1	yes	characteristic
210	C₃	#10599848	1	1	yes	characteristic, complemented, Sylow
211	C₂	—	60	2	no	—
212	C₂	C₈×D₆₀	1	1	yes	characteristic
213	C₂	C₂×#91082	1	1	yes	characteristic
214	C₂	#11751002	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₅
commutator, socle	C₂²×C₃×C₅	C₂²×C₈
Fitting	C₃×C₄×C₅×C₁₆	C₂
prime radical	C₂×D₆₀	C₈
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₄×C₁₆	D₁₅
hypocenter	C₃×C₅	#2169
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#91082	C₂	semidirect
3. C₃×C₄×C₅×C₁₆	C₂	semidirect
4. #11746328	C₂	semidirect
5. C₄⋊(C₃⋊#672)	C₂	semidirect
19. C₄×C₅×C₁₆	D₃	semidirect
33. C₃×C₅×C₁₆	D₄	semidirect
37. C₃×C₄×C₁₆	D₅	semidirect
82. C₅×C₁₆	D₁₂	semidirect
88. C₄×C₁₆	D₁₅	semidirect
94. C₃×C₄×C₅	M₃₂	semidirect
109. C₃×C₁₆	D₂₀	semidirect
153. C₄×C₅	#798	semidirect
169. C₁₆	D₆₀	semidirect
170. C₃×C₅	#2169	semidirect
176. C₃×C₄	#3930	semidirect
200. C₅	#69055	semidirect
207. C₄	#91082	semidirect
210. C₃	#10599848	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91082	2
C₈×D₆₀	2
C₃×C₄×C₅×C₁₆	2
#11746328	2
C₄⋊(C₃⋊#672)	2
#10599848	3
#69055	5