Subgroups — #14306118

Order 1920.

792 conjugacy classes of subgroups.

The subgroup lattice has 276 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×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91107	C₂	1	4	yes	maximal, complemented
4	C₄×#6710	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601314	—	3	1	no	maximal, Hall
6	C₂²×D₆₀	C₄	1	4	yes	—
7	C₂²×D₆₀	C₂²	1	2	yes	—
8	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×#6710	C₂²	1	2	yes	—
11	#91763	—	2	1	no	—
12	#91763	—	2	1	no	—
13	#91763	—	2	2	no	—
14	C₃×C₅×C₄⋊C₈	—	2	2	no	—
15	#91107	—	2	8	no	—
16	#91087	—	2	1	no	—
17	C₄×C₃⋊#168	—	2	1	no	—
18	#70521	—	5	1	no	maximal, Hall
19	C₂×#4020	—	3	1	no	—
20	C₅×#423	D₃	1	1	yes	characteristic, complemented
21	C₂×#3955	—	3	4	no	—
22	C₄×#681	—	3	1	no	—
23	C₂³×D₁₅	C₂×C₄	1	4	yes	—
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂×D₆₀	—	2	4	no	—
27	C₂×D₆₀	—	2	4	no	—
28	C₂×D₆₀	—	2	2	no	—
29	C₂×D₆₀	—	2	2	no	—
30	C₂×D₆₀	—	2	4	no	—
31	C₂×D₆₀	—	2	4	no	—
32	C₂×D₆₀	—	4	8	no	—
33	C₃×C₅×M₁₆	—	2	4	no	—
34	C₂×C₃×C₅×C₈	—	2	2	no	—
35	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
36	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	D₄	1	2	yes	—
38	#6710	—	2	2	no	—
39	#6710	D₄	1	4	yes	complemented
40	C₂×C₃⋊#168	—	2	2	no	—
41	C₂×#874	—	5	1	no	—
42	C₃×#423	D₅	1	1	yes	characteristic, complemented
43	C₂×#823	—	5	4	no	—
44	C₄×#208	—	5	1	no	—
45	C₂²×D₂₀	—	3	2	no	—
46	C₂²×D₂₀	—	3	4	no	—
47	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
48	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
49	C₂×#681	—	3	2	no	—
50	#4020	—	6	1	no	—
51	#4020	—	6	1	no	—
52	#4020	—	6	2	no	—
53	C₅×C₄⋊C₈	—	2	2	no	—
54	#3955	—	6	8	no	—
55	#3935	—	6	1	no	—
56	C₄×#168	—	6	1	no	—
57	#1965	—	15	1	no	Sylow
58	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂²×D₁₅	—	2	4	no	—
60	C₂²×D₁₅	—	2	8	no	—
61	C₂²×D₁₅	—	4	8	no	—
62	C₂×C₃×C₄×C₅	—	2	4	no	—
63	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
65	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
67	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	#41	1	2	yes	—
69	D₆₀	—	4	8	no	—
70	D₆₀	—	4	8	no	—
71	D₆₀	—	4	4	no	—
72	D₆₀	—	4	2	no	—
73	D₆₀	—	4	2	no	—
74	C₃×C₅×C₈	—	4	2	no	—
75	C₃⋊#168	—	2	4	no	—
76	C₂²×D₁₂	—	5	2	no	—
77	C₂²×D₁₂	—	5	4	no	—
78	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
79	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
80	C₂×#208	—	5	2	no	—
81	#874	—	10	1	no	—
82	#874	—	10	1	no	—
83	#874	—	10	2	no	—
84	C₃×C₄⋊C₈	—	2	2	no	—
85	#823	—	10	8	no	—
86	#803	—	10	1	no	—
87	C₄×C₃⋊C₈	—	10	1	no	—
88	C₂³×D₅	—	3	4	no	—
89	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
90	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
91	C₂×D₂₀	—	6	4	no	—
92	C₂×D₂₀	—	6	4	no	—
93	C₂×D₂₀	—	6	2	no	—
94	C₂×D₂₀	—	6	4	no	—
95	C₂×D₂₀	—	6	4	no	—
96	C₂×D₂₀	—	6	2	no	—
97	C₂×D₂₀	—	12	8	no	—
98	C₅×M₁₆	—	2	4	no	—
99	C₂×C₅×C₈	—	2	2	no	—
100	C₄²×C₅	D₁₂	1	1	yes	characteristic
101	C₄²×C₅	D₁₂	1	1	yes	characteristic
102	C₄²×C₅	#67	1	2	yes	—
103	#681	—	3	4	no	—
104	#681	—	6	2	no	—
105	C₂×#168	—	6	2	no	—
106	C₂×#127	—	15	1	no	—
107	#423	D₁₅	1	1	yes	characteristic, complemented
108	C₂×#100	—	15	4	no	—
109	C₄×M₁₆	—	15	1	no	—
110	C₂²×C₃×C₅	—	2	2	no	—
111	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
113	C₂×D₁₅	—	4	4	no	—
114	C₂×D₁₅	—	4	8	no	—
115	C₂×D₁₅	—	8	8	no	—
116	C₃×C₄×C₅	—	2	2	no	—
117	C₃×C₄×C₅	#100	1	4	yes	complemented
118	C₃×C₄×C₅	#121	1	2	yes	—
119	C₃×C₄×C₅	#124	1	1	yes	characteristic
120	C₃×C₄×C₅	#127	1	1	yes	characteristic
121	C₂³×D₃	—	5	4	no	—
122	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
123	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
124	C₂×D₁₂	—	10	4	no	—
125	C₂×D₁₂	—	10	4	no	—
126	C₂×D₁₂	—	10	2	no	—
127	C₂×D₁₂	—	10	4	no	—
128	C₂×D₁₂	—	10	4	no	—
129	C₂×D₁₂	—	10	2	no	—
130	C₂×D₁₂	—	20	8	no	—
131	C₃×M₁₆	—	2	4	no	—
132	C₂×C₃×C₈	—	2	2	no	—
133	C₃×C₄²	D₂₀	1	1	yes	characteristic
134	C₃×C₄²	D₂₀	1	1	yes	characteristic
135	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
136	#208	—	5	4	no	—
137	#208	—	10	2	no	—
138	C₂×C₃⋊C₈	—	10	2	no	—
139	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
140	C₂²×D₅	—	6	4	no	—
141	C₂²×D₅	—	6	8	no	—
142	C₂²×D₅	—	12	8	no	—
143	C₂×C₄×C₅	—	2	4	no	—
144	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
145	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
146	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
147	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
148	C₂×C₄×C₅	#239	1	2	yes	—
149	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
150	D₂₀	—	12	8	no	—
151	D₂₀	—	12	8	no	—
152	D₂₀	—	12	4	no	—
153	D₂₀	—	12	2	no	—
154	D₂₀	—	12	2	no	—
155	C₅×C₈	—	4	2	no	—
156	#168	—	6	4	no	—
157	C₂²×D₄	—	15	4	no	—
158	C₂²×D₄	—	15	2	no	—
159	C₂×M₁₆	—	15	2	no	—
160	C₂×M₁₆	C₂×D₁₅	1	2	yes	—
161	#127	—	30	1	no	—
162	#127	—	30	1	no	—
163	#127	—	30	2	no	—
164	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
165	C₄⋊C₈	—	2	2	no	—
166	#100	—	30	8	no	—
167	#97	—	30	1	no	—
168	C₄×C₈	—	30	1	no	—
169	C₂×C₃×C₅	—	2	2	no	—
170	C₂×C₃×C₅	#390	1	1	yes	characteristic
171	C₂×C₃×C₅	C₂×#100	1	2	yes	—
172	D₁₅	—	8	4	no	—
173	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
174	C₂²×D₃	—	10	4	no	—
175	C₂²×D₃	—	10	8	no	—
176	C₂²×D₃	—	20	8	no	—
177	C₂×C₃×C₄	—	2	4	no	—
178	C₂×C₃×C₄	#685	1	2	yes	—
179	C₂×C₃×C₄	#685	1	2	yes	—
180	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
181	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
182	C₂×C₃×C₄	#713	1	2	yes	—
183	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
184	D₁₂	—	20	8	no	—
185	D₁₂	—	20	8	no	—
186	D₁₂	—	20	4	no	—
187	D₁₂	—	20	2	no	—
188	D₁₂	—	20	2	no	—
189	C₃×C₈	—	4	2	no	—
190	C₃⋊C₈	—	10	4	no	—
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	#893	1	2	yes	—
193	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
194	C₂×D₅	—	12	8	no	—
195	C₂×D₅	—	12	4	no	—
196	C₂×D₅	—	24	8	no	—
197	C₄×C₅	—	2	2	no	—
198	C₄×C₅	#823	1	4	yes	complemented
199	C₄×C₅	#895	1	2	yes	—
200	C₄×C₅	#940	1	1	yes	characteristic
201	C₄×C₅	#947	1	1	yes	characteristic
202	C₂⁴	—	15	4	no	—
203	C₂×D₄	—	30	4	no	—
204	C₂×D₄	—	30	4	no	—
205	C₂×D₄	—	30	2	no	—
206	C₂×D₄	—	30	4	no	—
207	C₂×D₄	—	30	4	no	—
208	C₂×D₄	—	30	2	no	—
209	C₂×D₄	—	60	8	no	—
210	C₂²×C₄	C₄×D₁₅	1	2	yes	—
211	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
212	M₁₆	—	2	4	no	—
213	M₁₆	—	15	4	no	—
214	M₁₆	—	30	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	30	2	no	—
217	C₄²	D₆₀	1	1	yes	characteristic
218	C₄²	D₆₀	1	1	yes	characteristic
219	C₄²	#1220	1	2	yes	—
220	C₃×C₅	#1965	1	1	yes	characteristic, complemented, Hall
221	C₂²×C₃	—	2	2	no	—
222	C₂²×C₃	#4039	1	2	yes	—
223	C₂²×C₃	C₂×#685	1	1	yes	characteristic
224	C₂×D₃	—	20	8	no	—
225	C₂×D₃	—	20	4	no	—
226	C₂×D₃	—	40	8	no	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	#3955	1	4	yes	complemented
229	C₃×C₄	#4041	1	2	yes	—
230	C₃×C₄	#4086	1	1	yes	characteristic
231	C₃×C₄	#4093	1	1	yes	characteristic
232	C₂×C₅	—	2	2	no	—
233	C₂×C₅	#4988	1	1	yes	characteristic
234	C₂×C₅	C₂×#823	1	2	yes	—
235	D₅	—	24	4	no	—
236	C₂³	—	30	4	no	—
237	C₂³	—	30	8	no	—
238	C₂³	—	60	8	no	—
239	C₂³	C₂×C₄×D₁₅	1	1	yes	characteristic
240	D₄	—	60	8	no	—
241	D₄	—	60	4	no	—
242	D₄	—	60	2	no	—
243	D₄	—	60	2	no	—
244	D₄	—	60	8	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	#6714	1	2	yes	—
247	C₂×C₄	#6714	1	2	yes	—
248	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
249	C₂×C₄	D₄×D₁₅	1	2	yes	—
250	C₂×C₄	#6821	1	2	yes	—
251	C₂×C₄	C₂×#1220	1	1	yes	characteristic
252	C₈	—	4	2	no	—
253	C₈	—	30	4	no	—
254	C₂×C₃	—	2	2	no	—
255	C₂×C₃	#65277	1	1	yes	characteristic
256	C₂×C₃	C₂×#3955	1	2	yes	—
257	D₃	—	40	4	no	—
258	C₅	#70521	1	1	yes	characteristic, complemented, Sylow
259	C₂²	—	2	2	no	—
260	C₂²	—	60	8	no	—
261	C₂²	—	60	4	no	—
262	C₂²	—	120	8	no	—
263	C₂²	#91782	1	2	yes	—
264	C₂²	C₂×#6714	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	#91107	1	4	yes	complemented
267	C₄	#91784	1	2	yes	—
268	C₄	#91829	1	1	yes	characteristic
269	C₄	#91836	1	1	yes	characteristic
270	C₃	#10601314	1	1	yes	characteristic, complemented, Sylow
271	C₂	—	2	2	no	—
272	C₂	—	120	4	no	—
273	C₂	#11751089	1	1	yes	characteristic
274	C₂	C₂×#91107	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#6714
commutator	C₂³×C₃×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₅×#423	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂×#91763	C₂
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	#423	D₁₅
hypocenter	C₃×C₅	#1965
LCS	G ▷ C₂³×C₃×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#423	C₂	semidirect
3. C₂×#91107	C₂	semidirect
4. C₄×#6710	C₂	semidirect
20. C₅×#423	D₃	semidirect
39. #6710	D₄	semidirect
42. C₃×#423	D₅	semidirect
107. #423	D₁₅	semidirect
117. C₃×C₄×C₅	#100	semidirect
198. C₄×C₅	#823	semidirect
220. C₃×C₅	#1965	semidirect
228. C₃×C₄	#3955	semidirect
258. C₅	#70521	semidirect
266. C₄	#91107	semidirect
270. C₃	#10601314	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×#423	2
C₂×#91107	2
C₄×#6710	2
#10601314	3
#70521	5