Subgroups — C₂×C₃³×#123 #13179979

Order 1728.

4788 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃³×C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃³×#123	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃³×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₂×C₃³×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₂×C₃³×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃²×#123	C₃	1	13	yes	maximal, direct product factor
8	C₂⁴×C₃³	C₂²	1	1	yes	characteristic
9	C₂⁴×C₃³	C₂²	1	1	yes	characteristic
10	C₂×C₃³×D₄	—	2	4	no	—
11	C₂×C₃³×D₄	C₂²	1	4	yes	—
12	C₂²×C₃³×C₄	—	2	4	no	—
13	C₂²×C₃³×C₄	C₂²	1	2	yes	—
14	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
15	C₂²×C₃³×C₄	C₂²	1	2	yes	—
16	C₂²×C₃³×C₄	C₂²	1	4	yes	complemented
17	C₃³×C₄⋊C₄	C₂²	1	8	yes	complemented
18	C₃³×C₂²⋊C₄	C₂²	1	4	yes	complemented
19	C₃³×C₂²⋊C₄	C₂²	1	8	yes	complemented
20	C₂²×C₃²×D₄	C₂×C₃	1	13	yes	—
21	C₂³×C₃²×C₄	C₂×C₃	1	13	yes	complemented
22	C₃²×#123	C₂×C₃	1	104	yes	direct product factor
23	C₂×C₃²×C₄⋊C₄	C₂×C₃	1	26	yes	complemented
24	C₂×C₃²×C₂²⋊C₄	C₂×C₃	1	26	yes	complemented
25	C₂×C₃²×C₂²⋊C₄	C₂×C₃	1	13	yes	complemented
26	C₂³×C₃³	—	2	4	no	—
27	C₂³×C₃³	—	2	2	no	—
28	C₂³×C₃³	—	2	4	no	—
29	C₂³×C₃³	D₄	1	4	yes	—
30	C₂³×C₃³	C₂³	1	2	yes	—
31	C₂³×C₃³	C₂³	1	2	yes	—
32	C₂³×C₃³	C₂³	1	1	yes	characteristic
33	C₃³×D₄	—	2	8	no	—
34	C₂×C₃³×C₄	—	2	4	no	—
35	C₂×C₃³×C₄	—	2	8	no	—
36	C₂×C₃³×C₄	—	2	2	no	—
37	C₂×C₃³×C₄	—	2	4	no	—
38	C₂×C₃³×C₄	C₂³	1	4	yes	—
39	C₂×C₃³×C₄	C₂³	1	4	yes	—
40	C₂×C₃³×C₄	C₂³	1	2	yes	—
41	C₂×C₃×#123	C₃²	1	13	yes	direct product factor
42	C₂⁴×C₃²	C₂²×C₃	1	13	yes	—
43	C₂⁴×C₃²	C₂²×C₃	1	13	yes	—
44	C₂×C₃²×D₄	—	2	52	no	—
45	C₂×C₃²×D₄	C₂²×C₃	1	52	yes	—
46	C₂²×C₃²×C₄	—	2	52	no	—
47	C₂²×C₃²×C₄	C₂²×C₃	1	26	yes	—
48	C₂²×C₃²×C₄	C₂²×C₃	1	13	yes	—
49	C₂²×C₃²×C₄	C₂²×C₃	1	26	yes	—
50	C₂²×C₃²×C₄	C₂²×C₃	1	52	yes	complemented
51	C₃²×C₄⋊C₄	C₂²×C₃	1	104	yes	complemented
52	C₃²×C₂²⋊C₄	C₂²×C₃	1	52	yes	complemented
53	C₃²×C₂²⋊C₄	C₂²×C₃	1	104	yes	complemented
54	C₂²×C₃³	—	2	8	no	—
55	C₂²×C₃³	—	2	4	no	—
56	C₂²×C₃³	—	2	2	no	—
57	C₂²×C₃³	—	2	4	no	—
58	C₂²×C₃³	—	4	4	no	—
59	C₂²×C₃³	C₂×D₄	1	4	yes	—
60	C₂²×C₃³	C₂×D₄	1	2	yes	—
61	C₂²×C₃³	#41	1	4	yes	—
62	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
63	C₃³×C₄	—	2	4	no	—
64	C₃³×C₄	—	2	2	no	—
65	C₃³×C₄	—	2	4	no	—
66	C₂²×C₃×D₄	C₂×C₃²	1	13	yes	—
67	C₂³×C₃×C₄	C₂×C₃²	1	13	yes	complemented
68	C₃×#123	C₂×C₃²	1	104	yes	direct product factor
69	C₂×C₃×C₄⋊C₄	C₂×C₃²	1	26	yes	complemented
70	C₂×C₃×C₂²⋊C₄	C₂×C₃²	1	13	yes	complemented
71	C₂×C₃×C₂²⋊C₄	C₂×C₃²	1	26	yes	complemented
72	C₂³×C₃²	—	2	52	no	—
73	C₂³×C₃²	—	2	26	no	—
74	C₂³×C₃²	—	2	52	no	—
75	C₂³×C₃²	C₃×D₄	1	52	yes	—
76	C₂³×C₃²	C₂³×C₃	1	26	yes	—
77	C₂³×C₃²	C₂³×C₃	1	26	yes	—
78	C₂³×C₃²	C₂³×C₃	1	13	yes	—
79	C₃²×D₄	—	2	104	no	—
80	C₂×C₃²×C₄	—	2	52	no	—
81	C₂×C₃²×C₄	—	2	104	no	—
82	C₂×C₃²×C₄	—	2	26	no	—
83	C₂×C₃²×C₄	—	2	52	no	—
84	C₂×C₃²×C₄	C₂³×C₃	1	52	yes	—
85	C₂×C₃²×C₄	C₂³×C₃	1	52	yes	—
86	C₂×C₃²×C₄	C₂³×C₃	1	26	yes	—
87	C₂×#123	C₃³	1	1	yes	characteristic, direct product factor, Sylow
88	C₂×C₃³	—	2	4	no	—
89	C₂×C₃³	—	4	2	no	—
90	C₂×C₃³	#123	1	4	yes	direct product factor
91	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₃³	C₂×#41	1	2	yes	—
93	C₂⁴×C₃	C₂²×C₃²	1	13	yes	—
94	C₂⁴×C₃	C₂²×C₃²	1	13	yes	—
95	C₂×C₃×D₄	—	2	52	no	—
96	C₂×C₃×D₄	C₂²×C₃²	1	52	yes	—
97	C₂²×C₃×C₄	—	2	52	no	—
98	C₂²×C₃×C₄	C₂²×C₃²	1	26	yes	—
99	C₂²×C₃×C₄	C₂²×C₃²	1	26	yes	—
100	C₂²×C₃×C₄	C₂²×C₃²	1	13	yes	—
101	C₂²×C₃×C₄	C₂²×C₃²	1	52	yes	complemented
102	C₃×C₄⋊C₄	C₂²×C₃²	1	104	yes	complemented
103	C₃×C₂²⋊C₄	C₂²×C₃²	1	52	yes	complemented
104	C₃×C₂²⋊C₄	C₂²×C₃²	1	104	yes	complemented
105	C₂²×C₃²	—	2	104	no	—
106	C₂²×C₃²	—	2	52	no	—
107	C₂²×C₃²	—	2	26	no	—
108	C₂²×C₃²	—	2	52	no	—
109	C₂²×C₃²	—	4	52	no	—
110	C₂²×C₃²	C₂×C₃×D₄	1	52	yes	—
111	C₂²×C₃²	C₂×C₃×D₄	1	26	yes	—
112	C₂²×C₃²	C₃×#41	1	52	yes	—
113	C₂²×C₃²	C₂⁴×C₃	1	13	yes	—
114	C₃²×C₄	—	2	52	no	—
115	C₃²×C₄	—	2	26	no	—
116	C₃²×C₄	—	2	52	no	—
117	C₂²×D₄	C₂×C₃³	1	1	yes	characteristic
118	C₂³×C₄	C₂×C₃³	1	1	yes	characteristic, complemented
119	#123	C₂×C₃³	1	8	yes	direct product factor
120	C₂×C₄⋊C₄	C₂×C₃³	1	2	yes	complemented
121	C₂×C₂²⋊C₄	C₂×C₃³	1	1	yes	characteristic, complemented
122	C₂×C₂²⋊C₄	C₂×C₃³	1	2	yes	complemented
123	C₃³	C₂×#123	1	1	yes	characteristic, direct product factor, Sylow
124	C₂³×C₃	—	2	52	no	—
125	C₂³×C₃	—	2	26	no	—
126	C₂³×C₃	—	2	52	no	—
127	C₂³×C₃	C₃²×D₄	1	52	yes	—
128	C₂³×C₃	C₂³×C₃²	1	26	yes	—
129	C₂³×C₃	C₂³×C₃²	1	26	yes	—
130	C₂³×C₃	C₂³×C₃²	1	13	yes	—
131	C₃×D₄	—	2	104	no	—
132	C₂×C₃×C₄	—	2	52	no	—
133	C₂×C₃×C₄	—	2	104	no	—
134	C₂×C₃×C₄	—	2	26	no	—
135	C₂×C₃×C₄	—	2	52	no	—
136	C₂×C₃×C₄	C₂³×C₃²	1	52	yes	—
137	C₂×C₃×C₄	C₂³×C₃²	1	52	yes	—
138	C₂×C₃×C₄	C₂³×C₃²	1	26	yes	—
139	C₂×C₃²	—	2	52	no	—
140	C₂×C₃²	—	4	26	no	—
141	C₂×C₃²	C₃×#123	1	52	yes	direct product factor
142	C₂×C₃²	C₂²×C₃×D₄	1	13	yes	—
143	C₂×C₃²	C₂×C₃×#41	1	26	yes	—
144	C₂⁴	C₂²×C₃³	1	1	yes	characteristic
145	C₂⁴	C₂²×C₃³	1	1	yes	characteristic
146	C₂×D₄	—	2	4	no	—
147	C₂×D₄	C₂²×C₃³	1	4	yes	—
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	C₂²×C₃³	1	2	yes	—
150	C₂²×C₄	C₂²×C₃³	1	2	yes	—
151	C₂²×C₄	C₂²×C₃³	1	1	yes	characteristic
152	C₂²×C₄	C₂²×C₃³	1	4	yes	complemented
153	C₄⋊C₄	C₂²×C₃³	1	8	yes	complemented
154	C₂²⋊C₄	C₂²×C₃³	1	4	yes	complemented
155	C₂²⋊C₄	C₂²×C₃³	1	8	yes	complemented
156	C₂²×C₃	—	2	52	no	—
157	C₂²×C₃	—	2	104	no	—
158	C₂²×C₃	—	2	26	no	—
159	C₂²×C₃	—	2	52	no	—
160	C₂²×C₃	—	4	52	no	—
161	C₂²×C₃	C₂×C₃²×D₄	1	52	yes	—
162	C₂²×C₃	C₂×C₃²×D₄	1	26	yes	—
163	C₂²×C₃	C₃²×#41	1	52	yes	—
164	C₂²×C₃	C₂⁴×C₃²	1	13	yes	—
165	C₃×C₄	—	2	52	no	—
166	C₃×C₄	—	2	26	no	—
167	C₃×C₄	—	2	52	no	—
168	C₃²	C₂×C₃×#123	1	13	yes	direct product factor
169	C₂³	—	2	4	no	—
170	C₂³	—	2	2	no	—
171	C₂³	—	2	4	no	—
172	C₂³	C₃³×D₄	1	4	yes	—
173	C₂³	C₂³×C₃³	1	2	yes	—
174	C₂³	C₂³×C₃³	1	2	yes	—
175	C₂³	C₂³×C₃³	1	1	yes	characteristic
176	D₄	—	2	8	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	C₂³×C₃³	1	4	yes	—
182	C₂×C₄	C₂³×C₃³	1	4	yes	—
183	C₂×C₄	C₂³×C₃³	1	2	yes	—
184	C₂×C₃	—	2	52	no	—
185	C₂×C₃	—	4	26	no	—
186	C₂×C₃	C₃²×#123	1	52	yes	direct product factor
187	C₂×C₃	C₂²×C₃²×D₄	1	13	yes	—
188	C₂×C₃	C₂×C₃²×#41	1	26	yes	—
189	C₂²	—	2	4	no	—
190	C₂²	—	2	8	no	—
191	C₂²	—	2	2	no	—
192	C₂²	—	2	4	no	—
193	C₂²	—	4	4	no	—
194	C₂²	C₂×C₃³×D₄	1	4	yes	—
195	C₂²	C₂×C₃³×D₄	1	2	yes	—
196	C₂²	C₃³×#41	1	4	yes	—
197	C₂²	C₂⁴×C₃³	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	2	2	no	—
200	C₄	—	2	4	no	—
201	C₃	C₂×C₃²×#123	1	13	yes	direct product factor
202	C₂	—	2	4	no	—
203	C₂	—	4	2	no	—
204	C₂	C₃³×#123	1	4	yes	direct product factor
205	C₂	C₂²×C₃³×D₄	1	1	yes	characteristic
206	C₂	C₂×C₃³×#41	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃³	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃³
prime radical	C₂²×C₃³×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃³ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃³×C₄	C₂	semidirect
3. C₃³×#123	C₂	direct (N × H)
4. C₂×C₃³×C₄⋊C₄	C₂	semidirect
5. C₂×C₃³×C₂²⋊C₄	C₂	semidirect
6. C₂×C₃³×C₂²⋊C₄	C₂	semidirect
7. C₂×C₃²×#123	C₃	direct (N × H)
16. C₂²×C₃³×C₄	C₂²	semidirect
17. C₃³×C₄⋊C₄	C₂²	semidirect
18. C₃³×C₂²⋊C₄	C₂²	semidirect
19. C₃³×C₂²⋊C₄	C₂²	semidirect
21. C₂³×C₃²×C₄	C₂×C₃	semidirect
22. C₃²×#123	C₂×C₃	direct (N × H)
23. C₂×C₃²×C₄⋊C₄	C₂×C₃	semidirect
24. C₂×C₃²×C₂²⋊C₄	C₂×C₃	semidirect
25. C₂×C₃²×C₂²⋊C₄	C₂×C₃	semidirect
41. C₂×C₃×#123	C₃²	direct (N × H)
50. C₂²×C₃²×C₄	C₂²×C₃	semidirect
51. C₃²×C₄⋊C₄	C₂²×C₃	semidirect
52. C₃²×C₂²⋊C₄	C₂²×C₃	semidirect
53. C₃²×C₂²⋊C₄	C₂²×C₃	semidirect
67. C₂³×C₃×C₄	C₂×C₃²	semidirect
68. C₃×#123	C₂×C₃²	direct (N × H)
69. C₂×C₃×C₄⋊C₄	C₂×C₃²	semidirect
70. C₂×C₃×C₂²⋊C₄	C₂×C₃²	semidirect
71. C₂×C₃×C₂²⋊C₄	C₂×C₃²	semidirect
87. C₂×#123	C₃³	direct (N × H)
90. C₂×C₃³	#123	direct (N × H)
101. C₂²×C₃×C₄	C₂²×C₃²	semidirect
102. C₃×C₄⋊C₄	C₂²×C₃²	semidirect
103. C₃×C₂²⋊C₄	C₂²×C₃²	semidirect
104. C₃×C₂²⋊C₄	C₂²×C₃²	semidirect
118. C₂³×C₄	C₂×C₃³	semidirect
119. #123	C₂×C₃³	direct (N × H)
120. C₂×C₄⋊C₄	C₂×C₃³	semidirect
121. C₂×C₂²⋊C₄	C₂×C₃³	semidirect
122. C₂×C₂²⋊C₄	C₂×C₃³	semidirect
123. C₃³	C₂×#123	direct (N × H)
141. C₂×C₃²	C₃×#123	direct (N × H)
152. C₂²×C₄	C₂²×C₃³	semidirect
153. C₄⋊C₄	C₂²×C₃³	semidirect
154. C₂²⋊C₄	C₂²×C₃³	semidirect
155. C₂²⋊C₄	C₂²×C₃³	semidirect
168. C₃²	C₂×C₃×#123	direct (N × H)
186. C₂×C₃	C₃²×#123	direct (N × H)
201. C₃	C₂×C₃²×#123	direct (N × H)
204. C₂	C₃³×#123	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃³×D₄	2
C₂³×C₃³×C₄	2
C₃³×#123	2
C₂×C₃³×C₄⋊C₄	2
C₂×C₃³×C₂²⋊C₄	2
C₂×C₃²×#123	3

Subgroups — C2×C33×#123 #13179979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃³×#123 #13179979