Subgroups — C₂³×C₅×#213 #14509639

Order 1920.

3952 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×#213	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₅×D₈	—	3	1	no	maximal, Hall
6	C₂⁵×C₃×C₅	—	2	1	no	—
7	C₂⁴×C₅×D₃	—	2	1	no	—
8	C₂²×C₃×C₅×D₄	—	2	7	no	—
9	C₂²×C₃×C₅×D₄	C₂²	1	14	yes	complemented
10	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂²×C₅×D₁₂	—	2	7	no	—
12	C₂²×C₅×D₁₂	C₂²	1	14	yes	complemented
13	C₂×C₅×#213	C₂²	1	112	yes	direct product factor
14	C₂²×C₅×C₃⋊C₈	C₂²	1	14	yes	complemented
15	C₂³×#213	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂³×C₅×D₄	—	3	1	no	—
17	C₂³×C₅×D₄	D₃	1	1	yes	characteristic, complemented
18	C₂²×C₅×D₈	—	3	28	no	—
19	C₂³×C₅×C₈	—	3	1	no	—
20	C₂⁴×C₃×C₅	—	2	14	no	—
21	C₂⁴×C₃×C₅	—	4	8	no	—
22	C₂⁴×C₃×C₅	D₄	1	1	yes	characteristic
23	C₂³×C₅×D₃	—	2	14	no	—
24	C₂³×C₅×D₃	—	4	8	no	—
25	C₂×C₃×C₅×D₄	—	2	42	no	—
26	C₂×C₃×C₅×D₄	C₂³	1	28	yes	complemented
27	C₂²×C₃×C₄×C₅	D₄	1	7	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	7	yes	—
29	C₂×C₅×D₁₂	—	2	42	no	—
30	C₂×C₅×D₁₂	C₂³	1	28	yes	complemented
31	C₅×#213	C₂³	1	64	yes	direct product factor
32	C₂×C₅×C₃⋊C₈	C₂³	1	28	yes	complemented
33	C₂³×C₃×D₄	C₂×C₅	1	1	yes	characteristic, complemented
34	C₂³×D₁₂	C₂×C₅	1	1	yes	characteristic, complemented
35	C₂²×#213	C₂×C₅	1	28	yes	direct product factor
36	C₂³×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
37	C₂⁵×C₅	—	2	1	no	—
38	C₂⁵×C₅	—	6	1	no	—
39	C₂²×C₅×D₄	—	2	7	no	—
40	C₂²×C₅×D₄	—	3	14	no	—
41	C₂²×C₅×D₄	—	6	7	no	—
42	C₂²×C₅×D₄	C₂×D₃	1	14	yes	complemented
43	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
44	C₂×C₅×D₈	—	3	112	no	—
45	C₂²×C₅×C₈	—	3	14	no	—
46	C₂³×D₈	—	3	1	no	Sylow
47	C₂³×C₃×C₅	—	2	28	no	—
48	C₂³×C₃×C₅	—	4	56	no	—
49	C₂³×C₃×C₅	D₈	1	8	yes	complemented
50	C₂³×C₃×C₅	C₂×D₄	1	7	yes	—
51	C₂²×C₅×D₃	—	2	28	no	—
52	C₂²×C₅×D₃	—	4	56	no	—
53	C₃×C₅×D₄	—	2	28	no	—
54	C₃×C₅×D₄	C₂⁴	1	8	yes	complemented
55	C₂×C₃×C₄×C₅	C₂×D₄	1	21	yes	—
56	C₂×C₃×C₄×C₅	C₂⁴	1	7	yes	—
57	C₅×D₁₂	—	2	28	no	—
58	C₅×D₁₂	C₂⁴	1	8	yes	complemented
59	C₅×C₃⋊C₈	C₂⁴	1	8	yes	complemented
60	C₂⁵×C₃	—	2	1	no	—
61	C₂⁴×D₃	—	2	1	no	—
62	C₂²×C₃×D₄	—	2	7	no	—
63	C₂²×C₃×D₄	C₂²×C₅	1	14	yes	complemented
64	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
65	C₂²×D₁₂	—	2	7	no	—
66	C₂²×D₁₂	C₂²×C₅	1	14	yes	complemented
67	C₂×#213	C₂²×C₅	1	112	yes	direct product factor
68	C₂²×C₃⋊C₈	C₂²×C₅	1	14	yes	complemented
69	C₂⁴×C₅	—	2	14	no	—
70	C₂⁴×C₅	—	4	8	no	—
71	C₂⁴×C₅	—	6	14	no	—
72	C₂⁴×C₅	—	12	8	no	—
73	C₂⁴×C₅	#67	1	1	yes	characteristic
74	C₂×C₅×D₄	—	2	42	no	—
75	C₂×C₅×D₄	—	3	28	no	—
76	C₂×C₅×D₄	—	6	42	no	—
77	C₂×C₅×D₄	C₂²×D₃	1	28	yes	complemented
78	C₂²×C₄×C₅	#67	1	7	yes	—
79	C₂²×C₄×C₅	C₂²×D₃	1	7	yes	—
80	C₅×D₈	—	3	64	no	—
81	C₂×C₅×C₈	—	3	28	no	—
82	C₂³×D₄	—	3	1	no	—
83	C₂³×D₄	C₅×D₃	1	1	yes	characteristic, complemented
84	C₂²×D₈	—	3	28	no	—
85	C₂³×C₈	—	3	1	no	—
86	C₂²×C₃×C₅	—	2	8	no	—
87	C₂²×C₃×C₅	—	4	56	no	—
88	C₂²×C₃×C₅	C₂×D₈	1	28	yes	complemented
89	C₂²×C₃×C₅	C₂²×D₄	1	7	yes	—
90	C₂×C₅×D₃	—	2	8	no	—
91	C₂×C₅×D₃	—	4	56	no	—
92	C₃×C₄×C₅	C₂²×D₄	1	7	yes	—
93	C₃×C₄×C₅	C₂⁵	1	1	yes	characteristic
94	C₂⁴×C₃	—	2	14	no	—
95	C₂⁴×C₃	—	4	8	no	—
96	C₂⁴×C₃	C₅×D₄	1	1	yes	characteristic
97	C₂³×D₃	—	2	14	no	—
98	C₂³×D₃	—	4	8	no	—
99	C₂×C₃×D₄	—	2	42	no	—
100	C₂×C₃×D₄	C₂³×C₅	1	28	yes	complemented
101	C₂²×C₃×C₄	C₅×D₄	1	7	yes	—
102	C₂²×C₃×C₄	C₂³×C₅	1	7	yes	—
103	C₂×D₁₂	—	2	42	no	—
104	C₂×D₁₂	C₂³×C₅	1	28	yes	complemented
105	#213	C₂³×C₅	1	64	yes	direct product factor
106	C₂×C₃⋊C₈	C₂³×C₅	1	28	yes	complemented
107	C₂³×C₅	—	2	28	no	—
108	C₂³×C₅	—	4	56	no	—
109	C₂³×C₅	—	6	28	no	—
110	C₂³×C₅	—	12	56	no	—
111	C₂³×C₅	#213	1	8	yes	direct product factor
112	C₂³×C₅	C₂×#67	1	7	yes	—
113	C₅×D₄	—	2	28	no	—
114	C₅×D₄	—	3	8	no	—
115	C₅×D₄	—	6	28	no	—
116	C₅×D₄	C₂³×D₃	1	8	yes	complemented
117	C₂×C₄×C₅	C₂×#67	1	21	yes	—
118	C₂×C₄×C₅	C₂³×D₃	1	7	yes	—
119	C₅×C₈	—	3	8	no	—
120	C₂⁵	—	2	1	no	—
121	C₂⁵	—	6	1	no	—
122	C₂²×D₄	—	2	7	no	—
123	C₂²×D₄	—	3	14	no	—
124	C₂²×D₄	—	6	7	no	—
125	C₂²×D₄	C₂×C₅×D₃	1	14	yes	complemented
126	C₂³×C₄	C₂×C₅×D₃	1	1	yes	characteristic
127	C₂×D₈	—	3	112	no	—
128	C₂²×C₈	—	3	14	no	—
129	C₂×C₃×C₅	—	4	8	no	—
130	C₂×C₃×C₅	C₂²×D₈	1	14	yes	complemented
131	C₂×C₃×C₅	C₂³×D₄	1	1	yes	characteristic
132	C₅×D₃	—	4	8	no	—
133	C₂³×C₃	—	2	28	no	—
134	C₂³×C₃	—	4	56	no	—
135	C₂³×C₃	C₅×D₈	1	8	yes	complemented
136	C₂³×C₃	C₂×C₅×D₄	1	7	yes	—
137	C₂²×D₃	—	2	28	no	—
138	C₂²×D₃	—	4	56	no	—
139	C₃×D₄	—	2	28	no	—
140	C₃×D₄	C₂⁴×C₅	1	8	yes	complemented
141	C₂×C₃×C₄	C₂×C₅×D₄	1	21	yes	—
142	C₂×C₃×C₄	C₂⁴×C₅	1	7	yes	—
143	D₁₂	—	2	28	no	—
144	D₁₂	C₂⁴×C₅	1	8	yes	complemented
145	C₃⋊C₈	C₂⁴×C₅	1	8	yes	complemented
146	C₂²×C₅	—	2	8	no	—
147	C₂²×C₅	—	4	56	no	—
148	C₂²×C₅	—	6	8	no	—
149	C₂²×C₅	—	12	56	no	—
150	C₂²×C₅	C₂×#213	1	28	yes	direct product factor
151	C₂²×C₅	C₂²×#67	1	7	yes	—
152	C₄×C₅	C₂²×#67	1	7	yes	—
153	C₄×C₅	C₂⁴×D₃	1	1	yes	characteristic
154	C₂⁴	—	2	14	no	—
155	C₂⁴	—	4	8	no	—
156	C₂⁴	—	6	14	no	—
157	C₂⁴	—	12	8	no	—
158	C₂⁴	C₅×#67	1	1	yes	characteristic
159	C₂×D₄	—	2	42	no	—
160	C₂×D₄	—	3	28	no	—
161	C₂×D₄	—	6	42	no	—
162	C₂×D₄	C₂²×C₅×D₃	1	28	yes	complemented
163	C₂²×C₄	C₅×#67	1	7	yes	—
164	C₂²×C₄	C₂²×C₅×D₃	1	7	yes	—
165	D₈	—	3	64	no	—
166	C₂×C₈	—	3	28	no	—
167	C₃×C₅	C₂³×D₈	1	1	yes	characteristic, complemented, Hall
168	C₂²×C₃	—	2	8	no	—
169	C₂²×C₃	—	4	56	no	—
170	C₂²×C₃	C₂×C₅×D₈	1	28	yes	complemented
171	C₂²×C₃	C₂²×C₅×D₄	1	7	yes	—
172	C₂×D₃	—	2	8	no	—
173	C₂×D₃	—	4	56	no	—
174	C₃×C₄	C₂²×C₅×D₄	1	7	yes	—
175	C₃×C₄	C₂⁵×C₅	1	1	yes	characteristic
176	C₂×C₅	—	4	8	no	—
177	C₂×C₅	—	12	8	no	—
178	C₂×C₅	C₂²×#213	1	14	yes	direct product factor
179	C₂×C₅	C₂³×#67	1	1	yes	characteristic
180	C₂³	—	2	28	no	—
181	C₂³	—	4	56	no	—
182	C₂³	—	6	28	no	—
183	C₂³	—	12	56	no	—
184	C₂³	C₅×#213	1	8	yes	direct product factor
185	C₂³	C₂×C₅×#67	1	7	yes	—
186	D₄	—	2	28	no	—
187	D₄	—	3	8	no	—
188	D₄	—	6	28	no	—
189	D₄	C₂³×C₅×D₃	1	8	yes	complemented
190	C₂×C₄	C₂×C₅×#67	1	21	yes	—
191	C₂×C₄	C₂³×C₅×D₃	1	7	yes	—
192	C₈	—	3	8	no	—
193	C₂×C₃	—	4	8	no	—
194	C₂×C₃	C₂²×C₅×D₈	1	14	yes	complemented
195	C₂×C₃	C₂³×C₅×D₄	1	1	yes	characteristic
196	D₃	—	4	8	no	—
197	C₅	C₂³×#213	1	1	yes	characteristic, direct product factor, Sylow
198	C₂²	—	2	8	no	—
199	C₂²	—	4	56	no	—
200	C₂²	—	6	8	no	—
201	C₂²	—	12	56	no	—
202	C₂²	C₂×C₅×#213	1	28	yes	direct product factor
203	C₂²	C₂²×C₅×#67	1	7	yes	—
204	C₄	C₂²×C₅×#67	1	7	yes	—
205	C₄	C₂⁴×C₅×D₃	1	1	yes	characteristic
206	C₃	C₂³×C₅×D₈	1	1	yes	characteristic, complemented, Sylow
207	C₂	—	4	8	no	—
208	C₂	—	12	8	no	—
209	C₂	C₂²×C₅×#213	1	14	yes	direct product factor
210	C₂	C₂³×C₅×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₅	#67
commutator, cosocle	C₃×C₄	C₂⁵×C₅
Frattini	C₄	C₂⁴×C₅×D₃
Fitting	C₂³×C₃×C₅×D₄	C₂
socle	C₂⁴×C₃×C₅	D₄
hypercenter	C₂³×C₅×D₄	D₃
hypocenter	C₃	C₂³×C₅×D₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₅×D₄	C₂	semidirect
2. C₂³×C₅×D₁₂	C₂	semidirect
3. C₂²×C₅×#213	C₂	direct (N × H)
4. C₂³×C₅×C₃⋊C₈	C₂	semidirect
9. C₂²×C₃×C₅×D₄	C₂²	semidirect
12. C₂²×C₅×D₁₂	C₂²	semidirect
13. C₂×C₅×#213	C₂²	direct (N × H)
14. C₂²×C₅×C₃⋊C₈	C₂²	semidirect
15. C₂³×#213	C₅	direct (N × H)
17. C₂³×C₅×D₄	D₃	semidirect
26. C₂×C₃×C₅×D₄	C₂³	semidirect
30. C₂×C₅×D₁₂	C₂³	semidirect
31. C₅×#213	C₂³	direct (N × H)
32. C₂×C₅×C₃⋊C₈	C₂³	semidirect
33. C₂³×C₃×D₄	C₂×C₅	semidirect
34. C₂³×D₁₂	C₂×C₅	semidirect
35. C₂²×#213	C₂×C₅	direct (N × H)
36. C₂³×C₃⋊C₈	C₂×C₅	semidirect
42. C₂²×C₅×D₄	C₂×D₃	semidirect
49. C₂³×C₃×C₅	D₈	semidirect
54. C₃×C₅×D₄	C₂⁴	semidirect
58. C₅×D₁₂	C₂⁴	semidirect
59. C₅×C₃⋊C₈	C₂⁴	semidirect
63. C₂²×C₃×D₄	C₂²×C₅	semidirect
66. C₂²×D₁₂	C₂²×C₅	semidirect
67. C₂×#213	C₂²×C₅	direct (N × H)
68. C₂²×C₃⋊C₈	C₂²×C₅	semidirect
77. C₂×C₅×D₄	C₂²×D₃	semidirect
83. C₂³×D₄	C₅×D₃	semidirect
88. C₂²×C₃×C₅	C₂×D₈	semidirect
100. C₂×C₃×D₄	C₂³×C₅	semidirect
104. C₂×D₁₂	C₂³×C₅	semidirect
105. #213	C₂³×C₅	direct (N × H)
106. C₂×C₃⋊C₈	C₂³×C₅	semidirect
111. C₂³×C₅	#213	direct (N × H)
116. C₅×D₄	C₂³×D₃	semidirect
125. C₂²×D₄	C₂×C₅×D₃	semidirect
130. C₂×C₃×C₅	C₂²×D₈	semidirect
135. C₂³×C₃	C₅×D₈	semidirect
140. C₃×D₄	C₂⁴×C₅	semidirect
144. D₁₂	C₂⁴×C₅	semidirect
145. C₃⋊C₈	C₂⁴×C₅	semidirect
150. C₂²×C₅	C₂×#213	direct (N × H)
162. C₂×D₄	C₂²×C₅×D₃	semidirect
167. C₃×C₅	C₂³×D₈	semidirect
170. C₂²×C₃	C₂×C₅×D₈	semidirect
178. C₂×C₅	C₂²×#213	direct (N × H)
184. C₂³	C₅×#213	direct (N × H)
189. D₄	C₂³×C₅×D₃	semidirect
194. C₂×C₃	C₂²×C₅×D₈	semidirect
197. C₅	C₂³×#213	direct (N × H)
202. C₂²	C₂×C₅×#213	direct (N × H)
206. C₃	C₂³×C₅×D₈	semidirect
209. C₂	C₂²×C₅×#213	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×C₅×D₄	2
C₂³×C₅×D₁₂	2
C₂²×C₅×#213	2
C₂³×C₅×C₃⋊C₈	2
C₂³×C₅×D₈	3
C₂³×#213	5

Subgroups — C23×C5×#213 #14509639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₅×#213 #14509639