Subgroups — #14305803

Order 1920.

872 conjugacy classes of subgroups.

The subgroup lattice has 292 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×#384	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91093	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
5	#10600373	—	3	1	no	maximal, Hall
6	C₂²×D₆₀	—	2	2	no	—
7	C₂²×D₆₀	C₂²	1	2	yes	—
8	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂²×C₃⋊#168	C₂²	1	2	yes	—
12	#91763	—	2	2	no	—
13	#91763	C₄	1	4	yes	complemented
14	#91093	—	2	8	no	—
15	C₄×C₃⋊#168	—	2	2	no	—
16	#69580	—	5	1	no	maximal, Hall
17	C₂×#4020	—	3	1	no	—
18	C₅×#384	D₃	1	1	yes	characteristic, complemented
19	C₂×#3941	—	3	4	no	—
20	C₂×C₄×#168	—	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	4	no	—
30	C₂×D₆₀	—	4	4	no	—
31	C₂×D₆₀	C₂×C₄	1	4	yes	—
32	C₃×C₅×C₄⋊C₄	—	2	4	no	—
33	C₃×C₅×C₄⋊C₄	—	4	4	no	—
34	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
35	C₃×C₄²×C₅	D₄	1	2	yes	—
36	C₂×C₃⋊#168	—	2	2	no	—
37	C₂×C₃⋊#168	—	2	2	no	—
38	C₂×C₃⋊#168	D₄	1	4	yes	complemented
39	C₂×#874	—	5	1	no	—
40	C₃×#384	D₅	1	1	yes	characteristic, complemented
41	C₂×#809	—	5	4	no	—
42	C₂×C₄×C₃⋊C₈	—	5	1	no	—
43	C₂²×D₂₀	—	3	2	no	—
44	C₂²×D₂₀	—	6	2	no	—
45	C₂×C₅×C₄⋊C₄	—	2	2	no	—
46	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
47	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
48	C₂²×#168	—	3	2	no	—
49	#4020	—	3	4	no	—
50	#4020	—	6	2	no	—
51	#3941	—	6	8	no	—
52	C₄×#168	—	6	2	no	—
53	#1957	—	15	1	no	Sylow
54	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂²×D₁₅	—	2	4	no	—
56	C₂²×D₁₅	—	4	2	no	—
57	C₂²×D₁₅	—	4	2	no	—
58	C₂²×D₁₅	—	4	4	no	—
59	C₂²×D₁₅	—	4	4	no	—
60	C₂×C₃×C₄×C₅	—	4	2	no	—
61	C₂×C₃×C₄×C₅	—	4	4	no	—
62	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
63	C₂×C₃×C₄×C₅	D₈	1	4	yes	—
64	C₂×C₃×C₄×C₅	SD₁₆	1	4	yes	—
65	C₂×C₃×C₄×C₅	C₂²×C₄	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	—	2	8	no	—
70	D₆₀	—	4	16	no	—
71	D₆₀	—	4	4	no	—
72	C₃⋊#168	—	2	4	no	—
73	C₂²×D₁₂	—	5	2	no	—
74	C₂²×D₁₂	—	10	2	no	—
75	C₂×C₃×C₄⋊C₄	—	2	2	no	—
76	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
77	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
78	C₂²×C₃⋊C₈	—	5	2	no	—
79	#874	—	5	4	no	—
80	#874	—	10	2	no	—
81	#809	—	10	8	no	—
82	C₄×C₃⋊C₈	—	10	2	no	—
83	C₂³×D₅	—	6	2	no	—
84	C₂²×C₄×C₅	—	2	2	no	—
85	C₂²×C₄×C₅	#67	1	2	yes	—
86	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
87	C₂×D₂₀	—	3	4	no	—
88	C₂×D₂₀	—	6	4	no	—
89	C₂×D₂₀	—	6	4	no	—
90	C₂×D₂₀	—	6	8	no	—
91	C₂×D₂₀	—	6	2	no	—
92	C₂×D₂₀	—	12	4	no	—
93	C₂×D₂₀	—	12	4	no	—
94	C₅×C₄⋊C₄	—	2	4	no	—
95	C₅×C₄⋊C₄	—	4	4	no	—
96	C₄²×C₅	C₄×D₃	1	2	yes	—
97	C₄²×C₅	D₁₂	1	2	yes	—
98	C₂×#168	—	3	4	no	—
99	C₂×#168	—	6	2	no	—
100	C₂×#168	—	6	2	no	—
101	C₂×#127	—	15	1	no	—
102	C₂×#102	—	15	4	no	—
103	C₂×C₄×C₈	—	15	1	no	—
104	#384	D₁₅	1	1	yes	characteristic, complemented
105	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂²×C₃×C₅	#124	1	1	yes	characteristic
107	C₂²×C₃×C₅	#127	1	1	yes	characteristic
108	C₂²×C₃×C₅	C₂×D₈	1	2	yes	—
109	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
110	C₂×D₁₅	—	4	4	no	—
111	C₂×D₁₅	—	4	8	no	—
112	C₂×D₁₅	—	8	4	no	—
113	C₂×D₁₅	—	8	2	no	—
114	C₂×D₁₅	—	8	2	no	—
115	C₃×C₄×C₅	—	8	2	no	—
116	C₃×C₄×C₅	#102	1	8	yes	complemented
117	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
118	C₃×C₄×C₅	#121	1	2	yes	—
119	C₂³×D₃	—	10	2	no	—
120	C₂²×C₃×C₄	—	2	2	no	—
121	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
122	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
123	C₂×D₁₂	—	5	4	no	—
124	C₂×D₁₂	—	10	4	no	—
125	C₂×D₁₂	—	10	4	no	—
126	C₂×D₁₂	—	10	8	no	—
127	C₂×D₁₂	—	10	2	no	—
128	C₂×D₁₂	—	20	4	no	—
129	C₂×D₁₂	—	20	4	no	—
130	C₃×C₄⋊C₄	—	2	4	no	—
131	C₃×C₄⋊C₄	—	4	4	no	—
132	C₃×C₄²	C₄×D₅	1	2	yes	—
133	C₃×C₄²	D₂₀	1	2	yes	—
134	C₂×C₃⋊C₈	—	5	4	no	—
135	C₂×C₃⋊C₈	—	10	2	no	—
136	C₂×C₃⋊C₈	—	10	2	no	—
137	C₂³×C₅	C₂×#67	1	1	yes	characteristic
138	C₂²×D₅	—	6	4	no	—
139	C₂²×D₅	—	12	4	no	—
140	C₂²×D₅	—	12	2	no	—
141	C₂²×D₅	—	12	4	no	—
142	C₂²×D₅	—	12	2	no	—
143	C₂×C₄×C₅	—	4	4	no	—
144	C₂×C₄×C₅	—	4	2	no	—
145	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	4	yes	—
146	C₂×C₄×C₅	#213	1	4	yes	—
147	C₂×C₄×C₅	#215	1	4	yes	—
148	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
150	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
151	C₂×C₄×C₅	#239	1	2	yes	—
152	D₂₀	—	6	8	no	—
153	D₂₀	—	12	16	no	—
154	D₂₀	—	12	4	no	—
155	#168	—	6	4	no	—
156	C₂²×D₄	—	15	2	no	—
157	C₂²×D₄	—	30	2	no	—
158	C₂²×C₈	—	15	2	no	—
159	#127	—	15	4	no	—
160	#127	—	30	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	C₂×D₁₅	1	2	yes	—
163	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
164	#102	—	30	8	no	—
165	C₄×C₈	—	30	2	no	—
166	C₂×C₃×C₅	#390	1	1	yes	characteristic
167	C₂×C₃×C₅	C₂×#102	1	2	yes	—
168	C₂×C₃×C₅	#486	1	2	yes	—
169	C₂×C₃×C₅	#492	1	1	yes	characteristic
170	C₂×C₃×C₅	#493	1	1	yes	characteristic
171	D₁₅	—	8	4	no	—
172	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
173	C₂²×D₃	—	10	4	no	—
174	C₂²×D₃	—	20	4	no	—
175	C₂²×D₃	—	20	2	no	—
176	C₂²×D₃	—	20	4	no	—
177	C₂²×D₃	—	20	2	no	—
178	C₂×C₃×C₄	—	4	4	no	—
179	C₂×C₃×C₄	—	4	2	no	—
180	C₂×C₃×C₄	#685	1	4	yes	—
181	C₂×C₃×C₄	#686	1	4	yes	—
182	C₂×C₃×C₄	#688	1	4	yes	—
183	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
185	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
186	C₂×C₃×C₄	#713	1	2	yes	—
187	D₁₂	—	10	8	no	—
188	D₁₂	—	20	16	no	—
189	D₁₂	—	20	4	no	—
190	C₃⋊C₈	—	10	4	no	—
191	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
192	C₂²×C₅	C₂×#213	1	2	yes	—
193	C₂²×C₅	#940	1	1	yes	characteristic
194	C₂²×C₅	C₂×#215	1	2	yes	—
195	C₂²×C₅	#947	1	1	yes	characteristic
196	C₂×D₅	—	12	8	no	—
197	C₂×D₅	—	12	4	no	—
198	C₂×D₅	—	24	2	no	—
199	C₂×D₅	—	24	2	no	—
200	C₂×D₅	—	24	4	no	—
201	C₄×C₅	—	8	2	no	—
202	C₄×C₅	#809	1	8	yes	complemented
203	C₄×C₅	#893	1	2	yes	—
204	C₄×C₅	#895	1	2	yes	—
205	C₂⁴	—	30	2	no	—
206	C₂×D₄	—	15	4	no	—
207	C₂×D₄	—	30	8	no	—
208	C₂×D₄	—	30	2	no	—
209	C₂×D₄	—	30	4	no	—
210	C₂×D₄	—	30	4	no	—
211	C₂×D₄	—	60	4	no	—
212	C₂×D₄	—	60	4	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	#1220	1	2	yes	—
215	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
216	C₂×C₈	—	15	4	no	—
217	C₂×C₈	—	30	2	no	—
218	C₂×C₈	—	30	2	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄²	C₄×D₁₅	1	2	yes	—
222	C₄²	D₆₀	1	2	yes	—
223	C₃×C₅	#1957	1	1	yes	characteristic, complemented, Hall
224	C₂²×C₃	C₂×#685	1	1	yes	characteristic
225	C₂²×C₃	C₂×#686	1	2	yes	—
226	C₂²×C₃	#4086	1	1	yes	characteristic
227	C₂²×C₃	C₂×#688	1	2	yes	—
228	C₂²×C₃	#4093	1	1	yes	characteristic
229	C₂×D₃	—	20	8	no	—
230	C₂×D₃	—	20	4	no	—
231	C₂×D₃	—	40	2	no	—
232	C₂×D₃	—	40	2	no	—
233	C₂×D₃	—	40	4	no	—
234	C₃×C₄	—	8	2	no	—
235	C₃×C₄	#3941	1	8	yes	complemented
236	C₃×C₄	#4039	1	2	yes	—
237	C₃×C₄	#4041	1	2	yes	—
238	C₂×C₅	C₂×#809	1	2	yes	—
239	C₂×C₅	#4988	1	1	yes	characteristic
240	C₂×C₅	#5073	1	1	yes	characteristic
241	C₂×C₅	#5085	1	1	yes	characteristic
242	C₂×C₅	#5088	1	2	yes	—
243	D₅	—	24	4	no	—
244	C₂³	—	30	4	no	—
245	C₂³	—	60	4	no	—
246	C₂³	—	60	2	no	—
247	C₂³	—	60	4	no	—
248	C₂³	—	60	2	no	—
249	C₂³	C₂×#1220	1	1	yes	characteristic
250	D₄	—	30	8	no	—
251	D₄	—	60	4	no	—
252	D₄	—	60	16	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	#6714	1	4	yes	—
256	C₂×C₄	#6715	1	4	yes	—
257	C₂×C₄	#6717	1	4	yes	—
258	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
259	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
260	C₂×C₄	D₄×D₁₅	1	2	yes	—
261	C₂×C₄	#6821	1	2	yes	—
262	C₈	—	30	4	no	—
263	C₂×C₃	C₂×#3941	1	2	yes	—
264	C₂×C₃	#65277	1	1	yes	characteristic
265	C₂×C₃	#65362	1	1	yes	characteristic
266	C₂×C₃	#65374	1	1	yes	characteristic
267	C₂×C₃	#65377	1	2	yes	—
268	D₃	—	40	4	no	—
269	C₅	#69580	1	1	yes	characteristic, complemented, Sylow
270	C₂²	—	60	8	no	—
271	C₂²	—	60	4	no	—
272	C₂²	—	120	2	no	—
273	C₂²	—	120	2	no	—
274	C₂²	—	120	4	no	—
275	C₂²	C₂×#6714	1	1	yes	characteristic
276	C₂²	C₂×#6715	1	2	yes	—
277	C₂²	#91829	1	1	yes	characteristic
278	C₂²	C₂×#6717	1	2	yes	—
279	C₂²	#91836	1	1	yes	characteristic
280	C₄	—	8	2	no	—
281	C₄	#91093	1	8	yes	complemented
282	C₄	#91782	1	2	yes	—
283	C₄	#91784	1	2	yes	—
284	C₃	#10600373	1	1	yes	characteristic, complemented, Sylow
285	C₂	—	120	4	no	—
286	C₂	C₂×#91093	1	2	yes	—
287	C₂	#11751089	1	1	yes	characteristic
288	C₂	#11751174	1	1	yes	characteristic
289	C₂	#11751186	1	1	yes	characteristic
290	C₂	#11751189	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×#384	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂×#91763	C₂
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	#384	D₁₅
hypocenter	C₃×C₅	#1957
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#384	C₂	semidirect
3. C₂×#91093	C₂	semidirect
4. C₂×C₄×C₃⋊#168	C₂	semidirect
13. #91763	C₄	semidirect
18. C₅×#384	D₃	semidirect
38. C₂×C₃⋊#168	D₄	semidirect
40. C₃×#384	D₅	semidirect
104. #384	D₁₅	semidirect
116. C₃×C₄×C₅	#102	semidirect
202. C₄×C₅	#809	semidirect
223. C₃×C₅	#1957	semidirect
235. C₃×C₄	#3941	semidirect
269. C₅	#69580	semidirect
281. C₄	#91093	semidirect
284. C₃	#10600373	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×#384	2
C₂×#91093	2
C₂×C₄×C₃⋊#168	2
#10600373	3
#69580	5