Subgroups — C₂×C₃×#65362 #14438770

Order 1920.

1520 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×#686	C₂	1	4	yes	maximal, complemented
3	C₂×C₃×#4020	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65362	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#65362	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×D₄	—	2	2	no	—
8	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
9	C₂²×C₃×D₂₀	—	2	2	no	—
10	C₂²×C₃×D₂₀	C₂²	1	2	yes	—
11	C₃×C₅×#127	—	2	2	no	—
12	C₃×C₅×#127	C₂²	1	4	yes	complemented
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×#686	—	2	16	no	—
15	C₂×C₃×#686	C₂²	1	16	yes	complemented
16	C₂²×C₃×#168	C₂²	1	2	yes	—
17	C₃×#4020	—	2	2	no	—
18	C₃×#4020	C₂²	1	4	yes	complemented
19	C₃×C₄×#168	C₂²	1	4	yes	complemented
20	C₂×C₃×#493	—	5	1	no	maximal, Hall
21	C₂×C₅×#127	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂²×#686	C₂×C₃	1	4	yes	complemented
23	C₂×#4020	C₂×C₃	1	1	yes	characteristic, complemented
24	#65362	C₂×C₃	1	8	yes	direct product factor
25	C₂×C₄×#168	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂⁴×C₃×C₅	—	2	2	no	—
27	C₂³×C₃×D₅	—	2	2	no	—
28	C₂×C₃×C₅×D₄	—	2	8	no	—
29	C₂×C₃×C₅×D₄	—	2	8	no	—
30	C₂×C₃×C₅×D₄	—	2	2	no	—
31	C₂×C₃×C₅×D₄	—	4	8	no	—
32	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
33	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂×C₃×D₂₀	—	2	8	no	—
36	C₂×C₃×D₂₀	—	2	8	no	—
37	C₂×C₃×D₂₀	—	2	2	no	—
38	C₂×C₃×D₂₀	—	4	8	no	—
39	C₂×C₃×D₂₀	C₂³	1	4	yes	—
40	C₃×C₄²×C₅	D₄	1	2	yes	—
41	C₃×C₄²×C₅	C₂³	1	2	yes	—
42	C₃×#686	—	2	32	no	—
43	C₂×C₃×#168	D₄	1	8	yes	complemented
44	C₂×C₃×#168	C₂³	1	4	yes	—
45	C₂²×C₃×D₈	—	5	4	no	—
46	C₂×C₃×#127	—	5	1	no	—
47	C₂×C₃×#127	D₅	1	1	yes	characteristic, complemented
48	C₃×#493	—	5	8	no	—
49	C₂×C₃×C₄×C₈	—	5	1	no	—
50	C₂²×C₅×D₄	—	2	2	no	—
51	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
52	C₂²×D₂₀	—	2	2	no	—
53	C₂²×D₂₀	C₂²×C₃	1	2	yes	—
54	C₅×#127	—	2	2	no	—
55	C₅×#127	C₂²×C₃	1	4	yes	complemented
56	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
57	C₂×#686	—	2	16	no	—
58	C₂×#686	C₂²×C₃	1	16	yes	complemented
59	C₂²×#168	C₂²×C₃	1	2	yes	—
60	#4020	—	2	2	no	—
61	#4020	C₂²×C₃	1	4	yes	complemented
62	C₄×#168	C₂²×C₃	1	4	yes	complemented
63	C₂×#493	—	5	1	no	Sylow
64	C₂³×C₃×C₅	—	2	4	no	—
65	C₂³×C₃×C₅	—	4	4	no	—
66	C₂³×C₃×C₅	—	4	8	no	—
67	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₃×D₅	—	2	4	no	—
69	C₂²×C₃×D₅	—	4	4	no	—
70	C₂²×C₃×D₅	—	4	8	no	—
71	C₃×C₅×D₄	—	2	8	no	—
72	C₃×C₅×D₄	—	4	4	no	—
73	C₃×C₅×D₄	—	4	16	no	—
74	C₂×C₃×C₄×C₅	D₈	1	8	yes	complemented
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
77	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
78	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
79	C₃×D₂₀	—	2	8	no	—
80	C₃×D₂₀	—	4	4	no	—
81	C₃×D₂₀	—	4	16	no	—
82	C₃×#168	C₂×D₄	1	8	yes	complemented
83	C₂²×C₃×D₄	—	2	2	no	—
84	C₂²×C₃×D₄	—	5	2	no	—
85	C₂²×C₃×D₄	—	10	2	no	—
86	C₂²×C₃×D₄	C₂×D₅	1	2	yes	—
87	C₂×C₃×D₈	—	5	16	no	—
88	C₂×C₃×D₈	—	10	16	no	—
89	C₂²×C₃×C₈	—	5	2	no	—
90	C₃×#127	—	2	2	no	—
91	C₃×#127	—	5	4	no	—
92	C₃×#127	—	10	2	no	—
93	C₃×#127	C₂×D₅	1	4	yes	complemented
94	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
95	C₃×C₄×C₈	—	5	4	no	—
96	C₂⁴×C₅	—	2	2	no	—
97	C₂³×D₅	—	2	2	no	—
98	C₂×C₅×D₄	—	2	8	no	—
99	C₂×C₅×D₄	—	2	2	no	—
100	C₂×C₅×D₄	—	2	8	no	—
101	C₂×C₅×D₄	—	4	8	no	—
102	C₂×C₅×D₄	C₂³×C₃	1	4	yes	—
103	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
104	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
105	C₂×D₂₀	—	2	8	no	—
106	C₂×D₂₀	—	2	8	no	—
107	C₂×D₂₀	—	2	2	no	—
108	C₂×D₂₀	—	4	8	no	—
109	C₂×D₂₀	C₂³×C₃	1	4	yes	—
110	C₄²×C₅	C₃×D₄	1	2	yes	—
111	C₄²×C₅	C₂³×C₃	1	2	yes	—
112	#686	—	2	32	no	—
113	C₂×#168	C₃×D₄	1	8	yes	complemented
114	C₂×#168	C₂³×C₃	1	4	yes	—
115	C₂²×D₈	—	5	4	no	—
116	C₂×#127	—	5	1	no	—
117	C₂×#127	C₃×D₅	1	1	yes	characteristic, complemented
118	#493	—	5	8	no	—
119	C₂×C₄×C₈	—	5	1	no	—
120	C₂²×C₃×C₅	—	4	4	no	—
121	C₂²×C₃×C₅	—	4	8	no	—
122	C₂²×C₃×C₅	—	8	8	no	—
123	C₂²×C₃×C₅	#127	1	2	yes	—
124	C₂²×C₃×C₅	C₂×D₈	1	4	yes	—
125	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₃×D₅	—	4	4	no	—
127	C₂×C₃×D₅	—	4	8	no	—
128	C₂×C₃×D₅	—	8	8	no	—
129	C₃×C₄×C₅	#127	1	2	yes	—
130	C₃×C₄×C₅	C₂×D₈	1	8	yes	complemented
131	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
132	C₂⁴×C₃	—	2	2	no	—
133	C₂⁴×C₃	—	10	2	no	—
134	C₂×C₃×D₄	—	2	8	no	—
135	C₂×C₃×D₄	—	2	2	no	—
136	C₂×C₃×D₄	—	2	8	no	—
137	C₂×C₃×D₄	—	4	8	no	—
138	C₂×C₃×D₄	—	5	4	no	—
139	C₂×C₃×D₄	—	10	8	no	—
140	C₂×C₃×D₄	—	10	8	no	—
141	C₂×C₃×D₄	—	10	2	no	—
142	C₂×C₃×D₄	—	20	8	no	—
143	C₂×C₃×D₄	C₂²×D₅	1	4	yes	—
144	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
145	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
146	C₃×D₈	—	10	32	no	—
147	C₂×C₃×C₈	—	5	8	no	—
148	C₂×C₃×C₈	—	5	4	no	—
149	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
150	C₃×C₄²	C₂²×D₅	1	2	yes	—
151	C₂³×C₅	—	2	4	no	—
152	C₂³×C₅	—	4	4	no	—
153	C₂³×C₅	—	4	8	no	—
154	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
155	C₂²×D₅	—	2	4	no	—
156	C₂²×D₅	—	4	4	no	—
157	C₂²×D₅	—	4	8	no	—
158	C₅×D₄	—	2	8	no	—
159	C₅×D₄	—	4	4	no	—
160	C₅×D₄	—	4	16	no	—
161	C₂×C₄×C₅	C₃×D₈	1	8	yes	complemented
162	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
163	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
165	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
166	D₂₀	—	2	8	no	—
167	D₂₀	—	4	4	no	—
168	D₂₀	—	4	16	no	—
169	#168	C₂×C₃×D₄	1	8	yes	complemented
170	C₂²×D₄	—	2	2	no	—
171	C₂²×D₄	—	5	2	no	—
172	C₂²×D₄	—	10	2	no	—
173	C₂²×D₄	C₂×C₃×D₅	1	2	yes	—
174	C₂×D₈	—	5	16	no	—
175	C₂×D₈	—	10	16	no	—
176	C₂²×C₈	—	5	2	no	—
177	#127	—	2	2	no	—
178	#127	—	5	4	no	—
179	#127	—	10	2	no	—
180	#127	C₂×C₃×D₅	1	4	yes	complemented
181	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
182	C₄×C₈	—	5	4	no	—
183	C₂×C₃×C₅	—	8	4	no	—
184	C₂×C₃×C₅	#493	1	4	yes	complemented
185	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
186	C₂×C₃×C₅	C₂²×D₈	1	2	yes	—
187	C₃×D₅	—	8	4	no	—
188	C₂³×C₃	—	2	4	no	—
189	C₂³×C₃	—	4	8	no	—
190	C₂³×C₃	—	4	4	no	—
191	C₂³×C₃	—	10	4	no	—
192	C₂³×C₃	—	20	8	no	—
193	C₂³×C₃	—	20	4	no	—
194	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
195	C₃×D₄	—	2	8	no	—
196	C₃×D₄	—	4	16	no	—
197	C₃×D₄	—	4	4	no	—
198	C₃×D₄	—	10	8	no	—
199	C₃×D₄	—	20	16	no	—
200	C₃×D₄	—	20	4	no	—
201	C₂×C₃×C₄	#686	1	8	yes	complemented
202	C₂×C₃×C₄	D₄×D₅	1	4	yes	—
203	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	4	yes	—
204	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
206	C₃×C₈	—	5	8	no	—
207	C₂²×C₅	—	4	8	no	—
208	C₂²×C₅	—	4	4	no	—
209	C₂²×C₅	—	8	8	no	—
210	C₂²×C₅	C₃×#127	1	2	yes	—
211	C₂²×C₅	C₂×C₃×D₈	1	4	yes	—
212	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
213	C₂×D₅	—	4	8	no	—
214	C₂×D₅	—	4	4	no	—
215	C₂×D₅	—	8	8	no	—
216	C₄×C₅	C₃×#127	1	2	yes	—
217	C₄×C₅	C₂×C₃×D₈	1	8	yes	complemented
218	C₄×C₅	C₂²×C₃×D₄	1	2	yes	—
219	C₂⁴	—	2	2	no	—
220	C₂⁴	—	10	2	no	—
221	C₂×D₄	—	2	8	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	2	8	no	—
224	C₂×D₄	—	4	8	no	—
225	C₂×D₄	—	5	4	no	—
226	C₂×D₄	—	10	8	no	—
227	C₂×D₄	—	10	8	no	—
228	C₂×D₄	—	10	2	no	—
229	C₂×D₄	—	20	8	no	—
230	C₂×D₄	C₂²×C₃×D₅	1	4	yes	—
231	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
232	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
233	D₈	—	10	32	no	—
234	C₂×C₈	—	5	8	no	—
235	C₂×C₈	—	5	4	no	—
236	C₄²	C₃×C₂²⋊D₅	1	2	yes	—
237	C₄²	C₂²×C₃×D₅	1	2	yes	—
238	C₃×C₅	C₂×#493	1	1	yes	characteristic, complemented, Hall
239	C₂²×C₃	—	4	8	no	—
240	C₂²×C₃	—	4	4	no	—
241	C₂²×C₃	—	8	8	no	—
242	C₂²×C₃	—	20	8	no	—
243	C₂²×C₃	—	20	4	no	—
244	C₂²×C₃	—	40	8	no	—
245	C₂²×C₃	C₂×#686	1	4	yes	—
246	C₂²×C₃	#4086	1	2	yes	—
247	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
248	C₃×C₄	C₂×#686	1	8	yes	complemented
249	C₃×C₄	#4086	1	2	yes	—
250	C₃×C₄	C₂×D₄×D₅	1	2	yes	—
251	C₂×C₅	—	8	4	no	—
252	C₂×C₅	C₃×#493	1	4	yes	complemented
253	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
254	C₂×C₅	C₂²×C₃×D₈	1	2	yes	—
255	D₅	—	8	4	no	—
256	C₂³	—	2	4	no	—
257	C₂³	—	4	8	no	—
258	C₂³	—	4	4	no	—
259	C₂³	—	10	4	no	—
260	C₂³	—	20	8	no	—
261	C₂³	—	20	4	no	—
262	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
263	D₄	—	2	8	no	—
264	D₄	—	4	16	no	—
265	D₄	—	4	4	no	—
266	D₄	—	10	8	no	—
267	D₄	—	20	16	no	—
268	D₄	—	20	4	no	—
269	C₂×C₄	C₃×#686	1	8	yes	complemented
270	C₂×C₄	C₃×D₄×D₅	1	4	yes	—
271	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	4	yes	—
272	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
273	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
274	C₈	—	5	8	no	—
275	C₂×C₃	—	8	4	no	—
276	C₂×C₃	—	40	4	no	—
277	C₂×C₃	#65362	1	4	yes	direct product factor
278	C₂×C₃	C₂²×#686	1	2	yes	—
279	C₂×C₃	C₂×#4086	1	1	yes	characteristic
280	C₅	C₂×C₃×#493	1	1	yes	characteristic, complemented, Sylow
281	C₂²	—	4	8	no	—
282	C₂²	—	4	4	no	—
283	C₂²	—	8	8	no	—
284	C₂²	—	20	8	no	—
285	C₂²	—	20	4	no	—
286	C₂²	—	40	8	no	—
287	C₂²	C₂×C₃×#686	1	4	yes	—
288	C₂²	C₃×#4086	1	2	yes	—
289	C₂²	C₂²×C₃×C₂²⋊D₅	1	1	yes	characteristic
290	C₄	C₂×C₃×#686	1	8	yes	complemented
291	C₄	C₃×#4086	1	2	yes	—
292	C₄	C₂×C₃×D₄×D₅	1	2	yes	—
293	C₃	C₂×#65362	1	1	yes	characteristic, direct product factor, Sylow
294	C₂	—	8	4	no	—
295	C₂	—	40	4	no	—
296	C₂	C₃×#65362	1	4	yes	direct product factor
297	C₂	C₂²×C₃×#686	1	2	yes	—
298	C₂	C₂×C₃×#4086	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₂×C₃×C₅×#127	C₂
socle	C₂³×C₃×C₅	C₂×D₄
hypercenter	C₂×C₃×#127	D₅
hypocenter	C₅	C₂×C₃×#493
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×#127

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#127	C₂	semidirect
2. C₂²×C₃×#686	C₂	semidirect
3. C₂×C₃×#4020	C₂	semidirect
4. C₃×#65362	C₂	direct (N × H)
5. C₂×C₃×C₄×#168	C₂	semidirect
6. C₂×#65362	C₃	direct (N × H)
12. C₃×C₅×#127	C₂²	semidirect
15. C₂×C₃×#686	C₂²	semidirect
18. C₃×#4020	C₂²	semidirect
19. C₃×C₄×#168	C₂²	semidirect
21. C₂×C₅×#127	C₂×C₃	semidirect
22. C₂²×#686	C₂×C₃	semidirect
23. C₂×#4020	C₂×C₃	semidirect
24. #65362	C₂×C₃	direct (N × H)
25. C₂×C₄×#168	C₂×C₃	semidirect
43. C₂×C₃×#168	D₄	semidirect
47. C₂×C₃×#127	D₅	semidirect
55. C₅×#127	C₂²×C₃	semidirect
58. C₂×#686	C₂²×C₃	semidirect
61. #4020	C₂²×C₃	semidirect
62. C₄×#168	C₂²×C₃	semidirect
74. C₂×C₃×C₄×C₅	D₈	semidirect
82. C₃×#168	C₂×D₄	semidirect
93. C₃×#127	C₂×D₅	semidirect
113. C₂×#168	C₃×D₄	semidirect
117. C₂×#127	C₃×D₅	semidirect
130. C₃×C₄×C₅	C₂×D₈	semidirect
161. C₂×C₄×C₅	C₃×D₈	semidirect
169. #168	C₂×C₃×D₄	semidirect
180. #127	C₂×C₃×D₅	semidirect
184. C₂×C₃×C₅	#493	semidirect
201. C₂×C₃×C₄	#686	semidirect
217. C₄×C₅	C₂×C₃×D₈	semidirect
238. C₃×C₅	C₂×#493	semidirect
248. C₃×C₄	C₂×#686	semidirect
252. C₂×C₅	C₃×#493	semidirect
269. C₂×C₄	C₃×#686	semidirect
277. C₂×C₃	#65362	direct (N × H)
280. C₅	C₂×C₃×#493	semidirect
290. C₄	C₂×C₃×#686	semidirect
293. C₃	C₂×#65362	direct (N × H)
296. C₂	C₃×#65362	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#127	2
C₂²×C₃×#686	2
C₂×C₃×#4020	2
C₃×#65362	2
C₂×C₃×C₄×#168	2
C₂×#65362	3
C₂×C₃×#493	5

Subgroups — C2×C3×#65362 #14438770

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#65362 #14438770