Subgroups — C₂²×C₃×#4021 #14479660

Order 1920.

3760 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×D₂₀	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₄²×C₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃×#685	C₂	1	4	yes	maximal, complemented
5	C₂×C₃×#4021	C₂	1	24	yes	maximal, direct product factor
6	C₂²×#4021	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂⁴×C₃×D₅	C₂²	1	2	yes	—
8	C₂³×C₃×C₄×C₅	C₂²	1	2	yes	—
9	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂³×C₃×Dic₅	C₂²	1	2	yes	—
11	C₂²×C₃×D₂₀	—	2	8	no	—
12	C₂²×C₃×D₂₀	C₂²	1	12	yes	—
13	C₂²×C₃×Dic₁₀	—	2	8	no	—
14	C₂²×C₃×Dic₁₀	C₂²	1	12	yes	complemented
15	C₂×C₃×C₄²×C₅	C₂²	1	12	yes	complemented
16	C₂×C₃×#685	C₂²	1	48	yes	complemented
17	C₃×#4021	C₂²	1	64	yes	direct product factor
18	C₂²×C₃×#124	—	5	1	no	maximal, Hall
19	C₂³×D₂₀	C₂×C₃	1	1	yes	characteristic
20	C₂³×Dic₁₀	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂²×C₄²×C₅	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂²×#685	C₂×C₃	1	4	yes	complemented
23	C₂×#4021	C₂×C₃	1	24	yes	direct product factor
24	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
25	C₂³×C₃×D₅	—	2	8	no	—
26	C₂³×C₃×D₅	—	2	16	no	—
27	C₂³×C₃×D₅	C₂³	1	12	yes	—
28	C₂²×C₃×C₄×C₅	—	2	8	no	—
29	C₂²×C₃×C₄×C₅	D₄	1	8	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	12	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
32	C₂²×C₃×Dic₅	—	2	8	no	—
33	C₂²×C₃×Dic₅	C₂³	1	12	yes	—
34	C₂×C₃×D₂₀	—	2	48	no	—
35	C₂×C₃×D₂₀	C₂³	1	16	yes	—
36	C₂×C₃×Dic₁₀	—	2	48	no	—
37	C₂×C₃×Dic₁₀	C₂³	1	16	yes	complemented
38	C₃×C₄²×C₅	C₂³	1	16	yes	complemented
39	C₃×#685	C₂³	1	64	yes	complemented
40	C₂³×C₃×Q₈	—	5	1	no	—
41	C₂³×C₃×D₄	—	5	1	no	—
42	C₂×C₃×#124	—	5	24	no	—
43	C₂²×C₃×C₂²⋊C₄	—	5	4	no	—
44	C₂²×C₃×C₄²	D₅	1	1	yes	characteristic, complemented
45	C₂⁴×D₅	C₂²×C₃	1	2	yes	—
46	C₂³×C₄×C₅	C₂²×C₃	1	2	yes	—
47	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
48	C₂³×Dic₅	C₂²×C₃	1	2	yes	—
49	C₂²×D₂₀	—	2	8	no	—
50	C₂²×D₂₀	C₂²×C₃	1	12	yes	—
51	C₂²×Dic₁₀	—	2	8	no	—
52	C₂²×Dic₁₀	C₂²×C₃	1	12	yes	complemented
53	C₂×C₄²×C₅	C₂²×C₃	1	12	yes	complemented
54	C₂×#685	C₂²×C₃	1	48	yes	complemented
55	#4021	C₂²×C₃	1	64	yes	direct product factor
56	C₂²×#124	—	5	1	no	Sylow
57	C₂³×C₃×C₅	C₂×D₄	1	4	yes	—
58	C₂³×C₃×C₅	#41	1	8	yes	—
59	C₂³×C₃×C₅	C₂⁴	1	3	yes	—
60	C₂²×C₃×D₅	—	2	24	no	—
61	C₂²×C₃×D₅	—	2	48	no	—
62	C₂²×C₃×D₅	—	4	32	no	—
63	C₂²×C₃×D₅	C₂⁴	1	8	yes	—
64	C₂×C₃×C₄×C₅	—	2	24	no	—
65	C₂×C₃×C₄×C₅	C₂×D₄	1	24	yes	—
66	C₂×C₃×C₄×C₅	C₂⁴	1	8	yes	—
67	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
68	C₂×C₃×Dic₅	—	2	24	no	—
69	C₂×C₃×Dic₅	C₂⁴	1	8	yes	—
70	C₃×D₂₀	—	2	32	no	—
71	C₃×Dic₁₀	—	2	32	no	—
72	C₂⁵×C₃	—	5	2	no	—
73	C₂²×C₃×Q₈	—	5	12	no	—
74	C₂²×C₃×Q₈	—	10	8	no	—
75	C₂²×C₃×D₄	—	5	12	no	—
76	C₂²×C₃×D₄	—	10	8	no	—
77	C₂³×C₃×C₄	—	5	2	no	—
78	C₂³×C₃×C₄	C₂×D₅	1	2	yes	—
79	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
80	C₃×#124	—	5	64	no	—
81	C₂×C₃×C₂²⋊C₄	—	5	48	no	—
82	C₂×C₃×C₄²	C₂×D₅	1	12	yes	complemented
83	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
84	C₂³×D₅	—	2	8	no	—
85	C₂³×D₅	—	2	16	no	—
86	C₂³×D₅	C₂³×C₃	1	12	yes	—
87	C₂²×C₄×C₅	—	2	8	no	—
88	C₂²×C₄×C₅	C₃×D₄	1	8	yes	—
89	C₂²×C₄×C₅	C₂³×C₃	1	12	yes	—
90	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
91	C₂²×Dic₅	—	2	8	no	—
92	C₂²×Dic₅	C₂³×C₃	1	12	yes	—
93	C₂×D₂₀	—	2	48	no	—
94	C₂×D₂₀	C₂³×C₃	1	16	yes	—
95	C₂×Dic₁₀	—	2	48	no	—
96	C₂×Dic₁₀	C₂³×C₃	1	16	yes	complemented
97	C₄²×C₅	C₂³×C₃	1	16	yes	complemented
98	#685	C₂³×C₃	1	64	yes	complemented
99	C₂³×Q₈	—	5	1	no	—
100	C₂³×D₄	—	5	1	no	—
101	C₂×#124	—	5	24	no	—
102	C₂²×C₂²⋊C₄	—	5	4	no	—
103	C₂²×C₄²	C₃×D₅	1	1	yes	characteristic, complemented
104	C₂²×C₃×C₅	#124	1	16	yes	complemented
105	C₂²×C₃×C₅	C₂²×D₄	1	6	yes	—
106	C₂²×C₃×C₅	C₂×#41	1	12	yes	—
107	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
108	C₂×C₃×D₅	—	2	8	no	—
109	C₂×C₃×D₅	—	2	16	no	—
110	C₂×C₃×D₅	—	4	48	no	—
111	C₃×C₄×C₅	—	2	8	no	—
112	C₃×C₄×C₅	C₂²×D₄	1	8	yes	—
113	C₃×Dic₅	—	2	8	no	—
114	C₂⁴×C₃	—	5	12	no	—
115	C₂⁴×C₃	—	10	8	no	—
116	C₂⁴×C₃	—	10	16	no	—
117	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
118	C₂×C₃×Q₈	—	5	16	no	—
119	C₂×C₃×Q₈	—	10	48	no	—
120	C₂×C₃×D₄	—	5	16	no	—
121	C₂×C₃×D₄	—	10	48	no	—
122	C₂²×C₃×C₄	—	2	8	no	—
123	C₂²×C₃×C₄	—	5	12	no	—
124	C₂²×C₃×C₄	—	10	8	no	—
125	C₂²×C₃×C₄	D₂₀	1	8	yes	—
126	C₂²×C₃×C₄	C₂²×D₅	1	12	yes	—
127	C₂²×C₃×C₄	C₂²×D₅	1	6	yes	—
128	C₃×C₂²⋊C₄	—	5	64	no	—
129	C₃×C₄²	C₂²×D₅	1	16	yes	complemented
130	C₂³×C₅	C₂×C₃×D₄	1	4	yes	—
131	C₂³×C₅	C₃×#41	1	8	yes	—
132	C₂³×C₅	C₂⁴×C₃	1	3	yes	—
133	C₂²×D₅	—	2	24	no	—
134	C₂²×D₅	—	2	48	no	—
135	C₂²×D₅	—	4	32	no	—
136	C₂²×D₅	C₂⁴×C₃	1	8	yes	—
137	C₂×C₄×C₅	—	2	24	no	—
138	C₂×C₄×C₅	C₂×C₃×D₄	1	24	yes	—
139	C₂×C₄×C₅	C₂⁴×C₃	1	8	yes	—
140	C₂×C₄×C₅	C₂⁴×C₃	1	4	yes	—
141	C₂×Dic₅	—	2	24	no	—
142	C₂×Dic₅	C₂⁴×C₃	1	8	yes	—
143	D₂₀	—	2	32	no	—
144	Dic₁₀	—	2	32	no	—
145	C₂⁵	—	5	2	no	—
146	C₂²×Q₈	—	5	12	no	—
147	C₂²×Q₈	—	10	8	no	—
148	C₂²×D₄	—	5	12	no	—
149	C₂²×D₄	—	10	8	no	—
150	C₂³×C₄	—	5	2	no	—
151	C₂³×C₄	C₂×C₃×D₅	1	2	yes	—
152	C₂³×C₄	C₂×C₃×D₅	1	1	yes	characteristic
153	#124	—	5	64	no	—
154	C₂×C₂²⋊C₄	—	5	48	no	—
155	C₂×C₄²	C₂×C₃×D₅	1	12	yes	complemented
156	C₂×C₃×C₅	C₂×#124	1	12	yes	complemented
157	C₂×C₃×C₅	C₂³×D₄	1	1	yes	characteristic
158	C₂×C₃×C₅	C₂²×#41	1	2	yes	—
159	C₃×D₅	—	4	8	no	—
160	C₂³×C₃	—	5	8	no	—
161	C₂³×C₃	—	10	24	no	—
162	C₂³×C₃	—	10	48	no	—
163	C₂³×C₃	—	20	32	no	—
164	C₂³×C₃	C₂×D₂₀	1	4	yes	—
165	C₂³×C₃	#709	1	8	yes	—
166	C₂³×C₃	C₂³×D₅	1	3	yes	—
167	C₃×Q₈	—	10	32	no	—
168	C₃×D₄	—	10	32	no	—
169	C₂×C₃×C₄	—	2	24	no	—
170	C₂×C₃×C₄	—	5	8	no	—
171	C₂×C₃×C₄	—	10	24	no	—
172	C₂×C₃×C₄	C₂×D₂₀	1	24	yes	—
173	C₂×C₃×C₄	C₂³×D₅	1	8	yes	—
174	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
175	C₂²×C₅	C₃×#124	1	16	yes	complemented
176	C₂²×C₅	C₂²×C₃×D₄	1	6	yes	—
177	C₂²×C₅	C₂×C₃×#41	1	12	yes	—
178	C₂²×C₅	C₂⁵×C₃	1	1	yes	characteristic
179	C₂×D₅	—	2	8	no	—
180	C₂×D₅	—	2	16	no	—
181	C₂×D₅	—	4	48	no	—
182	C₄×C₅	—	2	8	no	—
183	C₄×C₅	C₂²×C₃×D₄	1	8	yes	—
184	Dic₅	—	2	8	no	—
185	C₂⁴	—	5	12	no	—
186	C₂⁴	—	10	8	no	—
187	C₂⁴	—	10	16	no	—
188	C₂⁴	C₂²×C₃×D₅	1	1	yes	characteristic
189	C₂×Q₈	—	5	16	no	—
190	C₂×Q₈	—	10	48	no	—
191	C₂×D₄	—	5	16	no	—
192	C₂×D₄	—	10	48	no	—
193	C₂²×C₄	—	2	8	no	—
194	C₂²×C₄	—	5	12	no	—
195	C₂²×C₄	—	10	8	no	—
196	C₂²×C₄	C₃×D₂₀	1	8	yes	—
197	C₂²×C₄	C₂²×C₃×D₅	1	12	yes	—
198	C₂²×C₄	C₂²×C₃×D₅	1	6	yes	—
199	C₂²⋊C₄	—	5	64	no	—
200	C₄²	C₂²×C₃×D₅	1	16	yes	complemented
201	C₃×C₅	C₂²×#124	1	1	yes	characteristic, complemented, Hall
202	C₂²×C₃	—	10	8	no	—
203	C₂²×C₃	—	10	16	no	—
204	C₂²×C₃	—	20	48	no	—
205	C₂²×C₃	#4021	1	16	yes	direct product factor
206	C₂²×C₃	C₂²×D₂₀	1	6	yes	—
207	C₂²×C₃	C₂×#709	1	12	yes	—
208	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
209	C₃×C₄	—	2	8	no	—
210	C₃×C₄	—	10	8	no	—
211	C₃×C₄	C₂²×D₂₀	1	8	yes	—
212	C₂×C₅	C₂×C₃×#124	1	12	yes	complemented
213	C₂×C₅	C₂³×C₃×D₄	1	1	yes	characteristic
214	C₂×C₅	C₂²×C₃×#41	1	2	yes	—
215	D₅	—	4	8	no	—
216	C₂³	—	5	8	no	—
217	C₂³	—	10	24	no	—
218	C₂³	—	10	48	no	—
219	C₂³	—	20	32	no	—
220	C₂³	C₂×C₃×D₂₀	1	4	yes	—
221	C₂³	C₃×#709	1	8	yes	—
222	C₂³	C₂³×C₃×D₅	1	3	yes	—
223	Q₈	—	10	32	no	—
224	D₄	—	10	32	no	—
225	C₂×C₄	—	2	24	no	—
226	C₂×C₄	—	5	8	no	—
227	C₂×C₄	—	10	24	no	—
228	C₂×C₄	C₂×C₃×D₂₀	1	24	yes	—
229	C₂×C₄	C₂³×C₃×D₅	1	8	yes	—
230	C₂×C₄	C₂³×C₃×D₅	1	4	yes	—
231	C₂×C₃	—	20	8	no	—
232	C₂×C₃	C₂×#4021	1	12	yes	direct product factor
233	C₂×C₃	C₂³×D₂₀	1	1	yes	characteristic
234	C₂×C₃	C₂²×#709	1	2	yes	—
235	C₅	C₂²×C₃×#124	1	1	yes	characteristic, complemented, Sylow
236	C₂²	—	10	8	no	—
237	C₂²	—	10	16	no	—
238	C₂²	—	20	48	no	—
239	C₂²	C₃×#4021	1	16	yes	direct product factor
240	C₂²	C₂²×C₃×D₂₀	1	6	yes	—
241	C₂²	C₂×C₃×#709	1	12	yes	—
242	C₂²	C₂⁴×C₃×D₅	1	1	yes	characteristic
243	C₄	—	2	8	no	—
244	C₄	—	10	8	no	—
245	C₄	C₂²×C₃×D₂₀	1	8	yes	—
246	C₃	C₂²×#4021	1	1	yes	characteristic, direct product factor, Sylow
247	C₂	—	20	8	no	—
248	C₂	C₂×C₃×#4021	1	12	yes	direct product factor
249	C₂	C₂³×C₃×D₂₀	1	1	yes	characteristic
250	C₂	C₂²×C₃×#709	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₃	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂⁵×C₃
Frattini	C₂²	C₂⁴×C₃×D₅
Fitting	C₂²×C₃×C₄²×C₅	C₂
socle	C₂⁴×C₃×C₅	C₂³
prime radical	C₂³×C₃×D₂₀	C₂
hypercenter	C₂²×C₃×C₄²	D₅
hypocenter	C₅	C₂²×C₃×#124
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂²×C₃×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃×Dic₁₀	C₂	semidirect
3. C₂²×C₃×C₄²×C₅	C₂	semidirect
4. C₂²×C₃×#685	C₂	semidirect
5. C₂×C₃×#4021	C₂	direct (N × H)
6. C₂²×#4021	C₃	direct (N × H)
14. C₂²×C₃×Dic₁₀	C₂²	semidirect
15. C₂×C₃×C₄²×C₅	C₂²	semidirect
16. C₂×C₃×#685	C₂²	semidirect
17. C₃×#4021	C₂²	direct (N × H)
20. C₂³×Dic₁₀	C₂×C₃	semidirect
21. C₂²×C₄²×C₅	C₂×C₃	semidirect
22. C₂²×#685	C₂×C₃	semidirect
23. C₂×#4021	C₂×C₃	direct (N × H)
37. C₂×C₃×Dic₁₀	C₂³	semidirect
38. C₃×C₄²×C₅	C₂³	semidirect
39. C₃×#685	C₂³	semidirect
44. C₂²×C₃×C₄²	D₅	semidirect
52. C₂²×Dic₁₀	C₂²×C₃	semidirect
53. C₂×C₄²×C₅	C₂²×C₃	semidirect
54. C₂×#685	C₂²×C₃	semidirect
55. #4021	C₂²×C₃	direct (N × H)
82. C₂×C₃×C₄²	C₂×D₅	semidirect
96. C₂×Dic₁₀	C₂³×C₃	semidirect
97. C₄²×C₅	C₂³×C₃	semidirect
98. #685	C₂³×C₃	semidirect
103. C₂²×C₄²	C₃×D₅	semidirect
104. C₂²×C₃×C₅	#124	semidirect
129. C₃×C₄²	C₂²×D₅	semidirect
155. C₂×C₄²	C₂×C₃×D₅	semidirect
156. C₂×C₃×C₅	C₂×#124	semidirect
175. C₂²×C₅	C₃×#124	semidirect
200. C₄²	C₂²×C₃×D₅	semidirect
201. C₃×C₅	C₂²×#124	semidirect
205. C₂²×C₃	#4021	direct (N × H)
212. C₂×C₅	C₂×C₃×#124	semidirect
232. C₂×C₃	C₂×#4021	direct (N × H)
235. C₅	C₂²×C₃×#124	semidirect
239. C₂²	C₃×#4021	direct (N × H)
246. C₃	C₂²×#4021	direct (N × H)
248. C₂	C₂×C₃×#4021	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×D₂₀	2
C₂³×C₃×Dic₁₀	2
C₂²×C₃×C₄²×C₅	2
C₂²×C₃×#685	2
C₂×C₃×#4021	2
C₂²×#4021	3
C₂²×C₃×#124	5

Subgroups — C22×C3×#4021 #14479660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#4021 #14479660