Subgroups — C₂³×C₃×#686 #14509640

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×C₃×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×#686	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₃×#168	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×#686	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂⁵×C₃×C₅	—	2	1	no	—
7	C₂⁴×C₃×D₅	—	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×C₃×D₂₀	—	2	7	no	—
12	C₂²×C₃×D₂₀	C₂²	1	14	yes	complemented
13	C₂×C₃×#686	C₂²	1	112	yes	direct product factor
14	C₂²×C₃×#168	C₂²	1	14	yes	complemented
15	C₂³×C₃×D₈	—	5	1	no	maximal, Hall
16	C₂³×C₅×D₄	C₂×C₃	1	1	yes	characteristic, complemented
17	C₂³×D₂₀	C₂×C₃	1	1	yes	characteristic, complemented
18	C₂²×#686	C₂×C₃	1	28	yes	direct product factor
19	C₂³×#168	C₂×C₃	1	1	yes	characteristic, complemented
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×C₃×D₅	—	2	14	no	—
24	C₂³×C₃×D₅	—	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×C₃×D₂₀	—	2	42	no	—
30	C₂×C₃×D₂₀	C₂³	1	28	yes	complemented
31	C₃×#686	C₂³	1	64	yes	direct product factor
32	C₂×C₃×#168	C₂³	1	28	yes	complemented
33	C₂³×C₃×D₄	—	5	1	no	—
34	C₂³×C₃×D₄	D₅	1	1	yes	characteristic, complemented
35	C₂²×C₃×D₈	—	5	28	no	—
36	C₂³×C₃×C₈	—	5	1	no	—
37	C₂⁵×C₅	—	2	1	no	—
38	C₂⁴×D₅	—	2	1	no	—
39	C₂²×C₅×D₄	—	2	7	no	—
40	C₂²×C₅×D₄	C₂²×C₃	1	14	yes	complemented
41	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
42	C₂²×D₂₀	—	2	7	no	—
43	C₂²×D₂₀	C₂²×C₃	1	14	yes	complemented
44	C₂×#686	C₂²×C₃	1	112	yes	direct product factor
45	C₂²×#168	C₂²×C₃	1	14	yes	complemented
46	C₂³×D₈	—	5	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×C₃×D₅	—	2	28	no	—
52	C₂²×C₃×D₅	—	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	C₃×D₂₀	—	2	28	no	—
58	C₃×D₂₀	C₂⁴	1	8	yes	complemented
59	C₃×#168	C₂⁴	1	8	yes	complemented
60	C₂⁵×C₃	—	2	1	no	—
61	C₂⁵×C₃	—	10	1	no	—
62	C₂²×C₃×D₄	—	2	7	no	—
63	C₂²×C₃×D₄	—	5	14	no	—
64	C₂²×C₃×D₄	—	10	7	no	—
65	C₂²×C₃×D₄	C₂×D₅	1	14	yes	complemented
66	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
67	C₂×C₃×D₈	—	5	112	no	—
68	C₂²×C₃×C₈	—	5	14	no	—
69	C₂⁴×C₅	—	2	14	no	—
70	C₂⁴×C₅	—	4	8	no	—
71	C₂⁴×C₅	C₃×D₄	1	1	yes	characteristic
72	C₂³×D₅	—	2	14	no	—
73	C₂³×D₅	—	4	8	no	—
74	C₂×C₅×D₄	—	2	42	no	—
75	C₂×C₅×D₄	C₂³×C₃	1	28	yes	complemented
76	C₂²×C₄×C₅	C₃×D₄	1	7	yes	—
77	C₂²×C₄×C₅	C₂³×C₃	1	7	yes	—
78	C₂×D₂₀	—	2	42	no	—
79	C₂×D₂₀	C₂³×C₃	1	28	yes	complemented
80	#686	C₂³×C₃	1	64	yes	direct product factor
81	C₂×#168	C₂³×C₃	1	28	yes	complemented
82	C₂³×D₄	—	5	1	no	—
83	C₂³×D₄	C₃×D₅	1	1	yes	characteristic, complemented
84	C₂²×D₈	—	5	28	no	—
85	C₂³×C₈	—	5	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×C₃×D₅	—	2	8	no	—
91	C₂×C₃×D₅	—	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	—	10	14	no	—
97	C₂⁴×C₃	—	20	8	no	—
98	C₂⁴×C₃	C₂²⋊D₅	1	1	yes	characteristic
99	C₂×C₃×D₄	—	2	42	no	—
100	C₂×C₃×D₄	—	5	28	no	—
101	C₂×C₃×D₄	—	10	42	no	—
102	C₂×C₃×D₄	C₂²×D₅	1	28	yes	complemented
103	C₂²×C₃×C₄	C₂²⋊D₅	1	7	yes	—
104	C₂²×C₃×C₄	C₂²×D₅	1	7	yes	—
105	C₃×D₈	—	5	64	no	—
106	C₂×C₃×C₈	—	5	28	no	—
107	C₂³×C₅	—	2	28	no	—
108	C₂³×C₅	—	4	56	no	—
109	C₂³×C₅	C₃×D₈	1	8	yes	complemented
110	C₂³×C₅	C₂×C₃×D₄	1	7	yes	—
111	C₂²×D₅	—	2	28	no	—
112	C₂²×D₅	—	4	56	no	—
113	C₅×D₄	—	2	28	no	—
114	C₅×D₄	C₂⁴×C₃	1	8	yes	complemented
115	C₂×C₄×C₅	C₂×C₃×D₄	1	21	yes	—
116	C₂×C₄×C₅	C₂⁴×C₃	1	7	yes	—
117	D₂₀	—	2	28	no	—
118	D₂₀	C₂⁴×C₃	1	8	yes	complemented
119	#168	C₂⁴×C₃	1	8	yes	complemented
120	C₂⁵	—	2	1	no	—
121	C₂⁵	—	10	1	no	—
122	C₂²×D₄	—	2	7	no	—
123	C₂²×D₄	—	5	14	no	—
124	C₂²×D₄	—	10	7	no	—
125	C₂²×D₄	C₂×C₃×D₅	1	14	yes	complemented
126	C₂³×C₄	C₂×C₃×D₅	1	1	yes	characteristic
127	C₂×D₈	—	5	112	no	—
128	C₂²×C₈	—	5	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	C₃×D₅	—	4	8	no	—
133	C₂³×C₃	—	2	28	no	—
134	C₂³×C₃	—	4	56	no	—
135	C₂³×C₃	—	10	28	no	—
136	C₂³×C₃	—	20	56	no	—
137	C₂³×C₃	#686	1	8	yes	direct product factor
138	C₂³×C₃	C₂×C₂²⋊D₅	1	7	yes	—
139	C₃×D₄	—	2	28	no	—
140	C₃×D₄	—	5	8	no	—
141	C₃×D₄	—	10	28	no	—
142	C₃×D₄	C₂³×D₅	1	8	yes	complemented
143	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	21	yes	—
144	C₂×C₃×C₄	C₂³×D₅	1	7	yes	—
145	C₃×C₈	—	5	8	no	—
146	C₂²×C₅	—	2	8	no	—
147	C₂²×C₅	—	4	56	no	—
148	C₂²×C₅	C₂×C₃×D₈	1	28	yes	complemented
149	C₂²×C₅	C₂²×C₃×D₄	1	7	yes	—
150	C₂×D₅	—	2	8	no	—
151	C₂×D₅	—	4	56	no	—
152	C₄×C₅	C₂²×C₃×D₄	1	7	yes	—
153	C₄×C₅	C₂⁵×C₃	1	1	yes	characteristic
154	C₂⁴	—	2	14	no	—
155	C₂⁴	—	4	8	no	—
156	C₂⁴	—	10	14	no	—
157	C₂⁴	—	20	8	no	—
158	C₂⁴	C₃×C₂²⋊D₅	1	1	yes	characteristic
159	C₂×D₄	—	2	42	no	—
160	C₂×D₄	—	5	28	no	—
161	C₂×D₄	—	10	42	no	—
162	C₂×D₄	C₂²×C₃×D₅	1	28	yes	complemented
163	C₂²×C₄	C₃×C₂²⋊D₅	1	7	yes	—
164	C₂²×C₄	C₂²×C₃×D₅	1	7	yes	—
165	D₈	—	5	64	no	—
166	C₂×C₈	—	5	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	—	10	8	no	—
171	C₂²×C₃	—	20	56	no	—
172	C₂²×C₃	C₂×#686	1	28	yes	direct product factor
173	C₂²×C₃	C₂²×C₂²⋊D₅	1	7	yes	—
174	C₃×C₄	C₂²×C₂²⋊D₅	1	7	yes	—
175	C₃×C₄	C₂⁴×D₅	1	1	yes	characteristic
176	C₂×C₅	—	4	8	no	—
177	C₂×C₅	C₂²×C₃×D₈	1	14	yes	complemented
178	C₂×C₅	C₂³×C₃×D₄	1	1	yes	characteristic
179	D₅	—	4	8	no	—
180	C₂³	—	2	28	no	—
181	C₂³	—	4	56	no	—
182	C₂³	—	10	28	no	—
183	C₂³	—	20	56	no	—
184	C₂³	C₃×#686	1	8	yes	direct product factor
185	C₂³	C₂×C₃×C₂²⋊D₅	1	7	yes	—
186	D₄	—	2	28	no	—
187	D₄	—	5	8	no	—
188	D₄	—	10	28	no	—
189	D₄	C₂³×C₃×D₅	1	8	yes	complemented
190	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	21	yes	—
191	C₂×C₄	C₂³×C₃×D₅	1	7	yes	—
192	C₈	—	5	8	no	—
193	C₂×C₃	—	4	8	no	—
194	C₂×C₃	—	20	8	no	—
195	C₂×C₃	C₂²×#686	1	14	yes	direct product factor
196	C₂×C₃	C₂³×C₂²⋊D₅	1	1	yes	characteristic
197	C₅	C₂³×C₃×D₈	1	1	yes	characteristic, complemented, Sylow
198	C₂²	—	2	8	no	—
199	C₂²	—	4	56	no	—
200	C₂²	—	10	8	no	—
201	C₂²	—	20	56	no	—
202	C₂²	C₂×C₃×#686	1	28	yes	direct product factor
203	C₂²	C₂²×C₃×C₂²⋊D₅	1	7	yes	—
204	C₄	C₂²×C₃×C₂²⋊D₅	1	7	yes	—
205	C₄	C₂⁴×C₃×D₅	1	1	yes	characteristic
206	C₃	C₂³×#686	1	1	yes	characteristic, direct product factor, Sylow
207	C₂	—	4	8	no	—
208	C₂	—	20	8	no	—
209	C₂	C₂²×C₃×#686	1	14	yes	direct product factor
210	C₂	C₂³×C₃×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₃	C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂⁵×C₃
Frattini	C₄	C₂⁴×C₃×D₅
Fitting	C₂³×C₃×C₅×D₄	C₂
socle	C₂⁴×C₃×C₅	D₄
hypercenter	C₂³×C₃×D₄	D₅
hypocenter	C₅	C₂³×C₃×D₈
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂³×C₃×C₄ ◁ C₂³×C₃×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×C₃×D₂₀	C₂	semidirect
3. C₂²×C₃×#686	C₂	direct (N × H)
4. C₂³×C₃×#168	C₂	semidirect
5. C₂³×#686	C₃	direct (N × H)
9. C₂²×C₃×C₅×D₄	C₂²	semidirect
12. C₂²×C₃×D₂₀	C₂²	semidirect
13. C₂×C₃×#686	C₂²	direct (N × H)
14. C₂²×C₃×#168	C₂²	semidirect
16. C₂³×C₅×D₄	C₂×C₃	semidirect
17. C₂³×D₂₀	C₂×C₃	semidirect
18. C₂²×#686	C₂×C₃	direct (N × H)
19. C₂³×#168	C₂×C₃	semidirect
26. C₂×C₃×C₅×D₄	C₂³	semidirect
30. C₂×C₃×D₂₀	C₂³	semidirect
31. C₃×#686	C₂³	direct (N × H)
32. C₂×C₃×#168	C₂³	semidirect
34. C₂³×C₃×D₄	D₅	semidirect
40. C₂²×C₅×D₄	C₂²×C₃	semidirect
43. C₂²×D₂₀	C₂²×C₃	semidirect
44. C₂×#686	C₂²×C₃	direct (N × H)
45. C₂²×#168	C₂²×C₃	semidirect
49. C₂³×C₃×C₅	D₈	semidirect
54. C₃×C₅×D₄	C₂⁴	semidirect
58. C₃×D₂₀	C₂⁴	semidirect
59. C₃×#168	C₂⁴	semidirect
65. C₂²×C₃×D₄	C₂×D₅	semidirect
75. C₂×C₅×D₄	C₂³×C₃	semidirect
79. C₂×D₂₀	C₂³×C₃	semidirect
80. #686	C₂³×C₃	direct (N × H)
81. C₂×#168	C₂³×C₃	semidirect
83. C₂³×D₄	C₃×D₅	semidirect
88. C₂²×C₃×C₅	C₂×D₈	semidirect
102. C₂×C₃×D₄	C₂²×D₅	semidirect
109. C₂³×C₅	C₃×D₈	semidirect
114. C₅×D₄	C₂⁴×C₃	semidirect
118. D₂₀	C₂⁴×C₃	semidirect
119. #168	C₂⁴×C₃	semidirect
125. C₂²×D₄	C₂×C₃×D₅	semidirect
130. C₂×C₃×C₅	C₂²×D₈	semidirect
137. C₂³×C₃	#686	direct (N × H)
142. C₃×D₄	C₂³×D₅	semidirect
148. C₂²×C₅	C₂×C₃×D₈	semidirect
162. C₂×D₄	C₂²×C₃×D₅	semidirect
167. C₃×C₅	C₂³×D₈	semidirect
172. C₂²×C₃	C₂×#686	direct (N × H)
177. C₂×C₅	C₂²×C₃×D₈	semidirect
184. C₂³	C₃×#686	direct (N × H)
189. D₄	C₂³×C₃×D₅	semidirect
195. C₂×C₃	C₂²×#686	direct (N × H)
197. C₅	C₂³×C₃×D₈	semidirect
202. C₂²	C₂×C₃×#686	direct (N × H)
206. C₃	C₂³×#686	direct (N × H)
209. C₂	C₂²×C₃×#686	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×C₃×D₂₀	2
C₂²×C₃×#686	2
C₂³×C₃×#168	2
C₂³×#686	3
C₂³×C₃×D₈	5

Subgroups — C23×C3×#686 #14509640

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×#686 #14509640