Subgroups — #14305957

Order 1920.

824 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91763	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#382	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91093	C₂	1	4	yes	maximal, complemented
4	C₂×#91087	C₂	1	1	yes	maximal, characteristic, complemented
5	#10600346	—	3	1	no	maximal, Hall
6	C₂²×D₆₀	—	2	2	no	—
7	C₂²×D₆₀	C₂²	1	2	yes	—
8	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂²×C₃⋊#168	C₂²	1	2	yes	—
11	#91763	—	2	2	no	—
12	#91763	C₄	1	4	yes	complemented
13	C₃×C₅×#95	—	2	2	no	—
14	#91093	—	2	8	no	—
15	#91087	—	2	2	no	—
16	#69553	—	5	1	no	maximal, Hall
17	C₂×#4020	—	3	1	no	—
18	C₅×#382	D₃	1	1	yes	characteristic, complemented
19	C₂×#3941	—	3	4	no	—
20	C₂×#3935	—	3	1	no	—
21	C₂³×D₁₅	—	2	2	no	—
22	C₂²×C₃×C₄×C₅	—	2	2	no	—
23	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂×D₆₀	—	2	4	no	—
26	C₂×D₆₀	—	2	4	no	—
27	C₂×D₆₀	—	2	8	no	—
28	C₂×D₆₀	—	2	2	no	—
29	C₂×D₆₀	—	4	8	no	—
30	C₂×D₆₀	C₂×C₄	1	4	yes	—
31	C₃×C₅×C₄⋊C₄	—	2	4	no	—
32	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
33	C₃×C₄²×C₅	D₄	1	2	yes	—
34	C₂×C₃⋊#168	—	2	2	no	—
35	C₂×C₃⋊#168	—	2	2	no	—
36	C₂×C₃⋊#168	D₄	1	4	yes	complemented
37	C₂×#874	—	5	1	no	—
38	C₃×#382	D₅	1	1	yes	characteristic, complemented
39	C₂×#809	—	5	4	no	—
40	C₂×#803	—	5	1	no	—
41	C₂²×D₂₀	—	3	2	no	—
42	C₂²×D₂₀	—	6	2	no	—
43	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
44	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
45	C₂²×#168	—	3	2	no	—
46	#4020	—	3	4	no	—
47	#4020	—	6	2	no	—
48	C₅×#95	—	2	2	no	—
49	#3941	—	6	8	no	—
50	#3935	—	6	2	no	—
51	#1961	—	15	1	no	Sylow
52	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂²×D₁₅	—	2	4	no	—
54	C₂²×D₁₅	—	4	2	no	—
55	C₂²×D₁₅	—	4	2	no	—
56	C₂²×D₁₅	—	4	8	no	—
57	C₂×C₃×C₄×C₅	—	2	4	no	—
58	C₂×C₃×C₄×C₅	—	4	2	no	—
59	C₂×C₃×C₄×C₅	—	4	4	no	—
60	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
61	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
63	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	#41	1	2	yes	—
65	D₆₀	—	2	8	no	—
66	D₆₀	—	4	16	no	—
67	D₆₀	—	4	4	no	—
68	C₃⋊#168	—	2	4	no	—
69	C₂²×D₁₂	—	5	2	no	—
70	C₂²×D₁₂	—	10	2	no	—
71	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
72	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
73	C₂²×C₃⋊C₈	—	5	2	no	—
74	#874	—	5	4	no	—
75	#874	—	10	2	no	—
76	C₃×#95	—	2	2	no	—
77	#809	—	10	8	no	—
78	#803	—	10	2	no	—
79	C₂³×D₅	—	6	2	no	—
80	C₂²×C₄×C₅	—	2	2	no	—
81	C₂²×C₄×C₅	#67	1	2	yes	—
82	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
83	C₂×D₂₀	—	3	4	no	—
84	C₂×D₂₀	—	6	4	no	—
85	C₂×D₂₀	—	6	4	no	—
86	C₂×D₂₀	—	6	8	no	—
87	C₂×D₂₀	—	6	2	no	—
88	C₂×D₂₀	—	12	8	no	—
89	C₅×C₄⋊C₄	—	2	4	no	—
90	C₄²×C₅	C₄×D₃	1	2	yes	—
91	C₄²×C₅	D₁₂	1	2	yes	—
92	C₂×#168	—	3	4	no	—
93	C₂×#168	—	6	2	no	—
94	C₂×#168	—	6	2	no	—
95	C₂×#127	—	15	1	no	—
96	C₂×#102	—	15	4	no	—
97	C₂×#97	—	15	1	no	—
98	#382	D₁₅	1	1	yes	characteristic, complemented
99	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂²×C₃×C₅	#124	1	1	yes	characteristic
101	C₂²×C₃×C₅	#127	1	1	yes	characteristic
102	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	4	yes	—
103	C₂×D₁₅	—	4	4	no	—
104	C₂×D₁₅	—	4	8	no	—
105	C₂×D₁₅	—	8	4	no	—
106	C₂×D₁₅	—	8	4	no	—
107	C₃×C₄×C₅	—	2	4	no	—
108	C₃×C₄×C₅	—	8	2	no	—
109	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
110	C₃×C₄×C₅	#121	1	2	yes	—
111	C₂³×D₃	—	10	2	no	—
112	C₂²×C₃×C₄	—	2	2	no	—
113	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
114	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
115	C₂×D₁₂	—	5	4	no	—
116	C₂×D₁₂	—	10	4	no	—
117	C₂×D₁₂	—	10	4	no	—
118	C₂×D₁₂	—	10	8	no	—
119	C₂×D₁₂	—	10	2	no	—
120	C₂×D₁₂	—	20	8	no	—
121	C₃×C₄⋊C₄	—	2	4	no	—
122	C₃×C₄²	C₄×D₅	1	2	yes	—
123	C₃×C₄²	D₂₀	1	2	yes	—
124	C₂×C₃⋊C₈	—	5	4	no	—
125	C₂×C₃⋊C₈	—	10	2	no	—
126	C₂×C₃⋊C₈	—	10	2	no	—
127	C₂³×C₅	C₂×#67	1	1	yes	characteristic
128	C₂²×D₅	—	6	4	no	—
129	C₂²×D₅	—	12	2	no	—
130	C₂²×D₅	—	12	2	no	—
131	C₂²×D₅	—	12	8	no	—
132	C₂×C₄×C₅	—	2	4	no	—
133	C₂×C₄×C₅	—	4	2	no	—
134	C₂×C₄×C₅	—	4	4	no	—
135	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	4	yes	—
136	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
138	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
139	C₂×C₄×C₅	#239	1	2	yes	—
140	D₂₀	—	6	8	no	—
141	D₂₀	—	12	16	no	—
142	D₂₀	—	12	4	no	—
143	#168	—	6	4	no	—
144	C₂²×D₄	—	15	2	no	—
145	C₂²×D₄	—	30	2	no	—
146	C₂²×C₈	—	15	2	no	—
147	#127	—	15	4	no	—
148	#127	—	30	2	no	—
149	C₂×C₄⋊C₄	C₂×D₁₅	1	2	yes	—
150	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
151	#102	—	30	8	no	—
152	#97	—	30	2	no	—
153	#95	—	2	2	no	—
154	C₂×C₃×C₅	#390	1	1	yes	characteristic
155	C₂×C₃×C₅	#418	1	2	yes	—
156	C₂×C₃×C₅	#490	1	2	yes	—
157	C₂×C₃×C₅	#496	1	2	yes	—
158	D₁₅	—	8	4	no	—
159	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
160	C₂²×D₃	—	10	4	no	—
161	C₂²×D₃	—	20	2	no	—
162	C₂²×D₃	—	20	2	no	—
163	C₂²×D₃	—	20	8	no	—
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	—	4	2	no	—
166	C₂×C₃×C₄	—	4	4	no	—
167	C₂×C₃×C₄	#685	1	4	yes	—
168	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
170	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
171	C₂×C₃×C₄	#713	1	2	yes	—
172	D₁₂	—	10	8	no	—
173	D₁₂	—	20	16	no	—
174	D₁₂	—	20	4	no	—
175	C₃⋊C₈	—	10	4	no	—
176	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
177	C₂²×C₅	#940	1	1	yes	characteristic
178	C₂²×C₅	#947	1	1	yes	characteristic
179	C₂²×C₅	#949	1	4	yes	—
180	C₂×D₅	—	12	4	no	—
181	C₂×D₅	—	12	8	no	—
182	C₂×D₅	—	24	4	no	—
183	C₂×D₅	—	24	4	no	—
184	C₄×C₅	—	2	4	no	—
185	C₄×C₅	—	8	2	no	—
186	C₄×C₅	#893	1	2	yes	—
187	C₄×C₅	#895	1	2	yes	—
188	C₂⁴	—	30	2	no	—
189	C₂×D₄	—	15	4	no	—
190	C₂×D₄	—	30	8	no	—
191	C₂×D₄	—	30	2	no	—
192	C₂×D₄	—	30	4	no	—
193	C₂×D₄	—	30	4	no	—
194	C₂×D₄	—	60	8	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	#1220	1	2	yes	—
197	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
198	C₂×C₈	—	15	4	no	—
199	C₂×C₈	—	30	2	no	—
200	C₂×C₈	—	30	2	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄²	C₄×D₁₅	1	2	yes	—
203	C₄²	D₆₀	1	2	yes	—
204	C₃×C₅	#1961	1	1	yes	characteristic, complemented, Hall
205	C₂²×C₃	C₂×#685	1	1	yes	characteristic
206	C₂²×C₃	#4086	1	1	yes	characteristic
207	C₂²×C₃	#4093	1	1	yes	characteristic
208	C₂²×C₃	#4095	1	4	yes	—
209	C₂×D₃	—	20	8	no	—
210	C₂×D₃	—	20	4	no	—
211	C₂×D₃	—	40	4	no	—
212	C₂×D₃	—	40	4	no	—
213	C₃×C₄	—	2	4	no	—
214	C₃×C₄	—	8	2	no	—
215	C₃×C₄	#4039	1	2	yes	—
216	C₃×C₄	#4041	1	2	yes	—
217	C₂×C₅	#4988	1	1	yes	characteristic
218	C₂×C₅	#5001	1	2	yes	—
219	C₂×C₅	#5058	1	2	yes	—
220	C₂×C₅	#5067	1	2	yes	—
221	D₅	—	24	4	no	—
222	C₂³	—	30	4	no	—
223	C₂³	—	60	2	no	—
224	C₂³	—	60	2	no	—
225	C₂³	—	60	8	no	—
226	C₂³	C₂×#1220	1	1	yes	characteristic
227	D₄	—	30	8	no	—
228	D₄	—	60	16	no	—
229	D₄	—	60	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	#6714	1	4	yes	—
234	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
235	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
236	C₂×C₄	D₄×D₁₅	1	2	yes	—
237	C₂×C₄	#6821	1	2	yes	—
238	C₈	—	30	4	no	—
239	C₂×C₃	#65277	1	1	yes	characteristic
240	C₂×C₃	#65290	1	2	yes	—
241	C₂×C₃	#65347	1	2	yes	—
242	C₂×C₃	#65356	1	2	yes	—
243	D₃	—	40	4	no	—
244	C₅	#69553	1	1	yes	characteristic, complemented, Sylow
245	C₂²	—	60	4	no	—
246	C₂²	—	60	8	no	—
247	C₂²	—	120	4	no	—
248	C₂²	—	120	4	no	—
249	C₂²	C₂×#6714	1	1	yes	characteristic
250	C₂²	#91829	1	1	yes	characteristic
251	C₂²	#91836	1	1	yes	characteristic
252	C₂²	#91838	1	4	yes	—
253	C₄	—	2	4	no	—
254	C₄	—	8	2	no	—
255	C₄	#91782	1	2	yes	—
256	C₄	#91784	1	2	yes	—
257	C₃	#10600346	1	1	yes	characteristic, complemented, Sylow
258	C₂	—	120	4	no	—
259	C₂	#11751089	1	1	yes	characteristic
260	C₂	#11751102	1	2	yes	—
261	C₂	#11751159	1	2	yes	—
262	C₂	#11751168	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#1220
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₅×#382	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂×#91763	C₂
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	#382	D₁₅
hypocenter	C₃×C₅	#1961
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#382	C₂	semidirect
3. C₂×#91093	C₂	semidirect
4. C₂×#91087	C₂	semidirect
12. #91763	C₄	semidirect
18. C₅×#382	D₃	semidirect
36. C₂×C₃⋊#168	D₄	semidirect
38. C₃×#382	D₅	semidirect
98. #382	D₁₅	semidirect
204. C₃×C₅	#1961	semidirect
244. C₅	#69553	semidirect
257. C₃	#10600346	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91763	2
C₃×C₅×#382	2
C₂×#91093	2
C₂×#91087	2
#10600346	3
#69553	5