Subgroups — C₂×C₃³×#135 #13180027

Order 1728.

3724 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×#41	C₂	1	2	yes	maximal, complemented
2	C₃³×#135	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₃³×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃³×SD₁₆	C₂	1	2	yes	maximal, complemented
5	C₂×C₃³×D₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₃³×C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃²×#135	C₃	1	13	yes	maximal, direct product factor
8	C₃³×#41	—	2	4	no	—
9	C₃³×#41	C₂²	1	8	yes	complemented
10	C₂×C₃³×Q₈	C₂²	1	2	yes	—
11	C₂×C₃³×D₄	—	2	2	no	—
12	C₂×C₃³×D₄	C₂²	1	2	yes	—
13	C₂²×C₃³×C₄	—	2	2	no	—
14	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
15	C₃³×Q₁₆	C₂²	1	4	yes	complemented
16	C₃³×SD₁₆	C₂²	1	8	yes	complemented
17	C₃³×D₈	C₂²	1	4	yes	complemented
18	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃³×C₈	C₂²	1	4	yes	complemented
21	C₂×C₃²×#41	C₂×C₃	1	26	yes	complemented
22	C₃²×#135	C₂×C₃	1	104	yes	direct product factor
23	C₂×C₃²×Q₁₆	C₂×C₃	1	13	yes	complemented
24	C₂×C₃²×SD₁₆	C₂×C₃	1	26	yes	complemented
25	C₂×C₃²×D₈	C₂×C₃	1	13	yes	complemented
26	C₂²×C₃²×C₈	C₂×C₃	1	13	yes	complemented
27	C₂³×C₃³	—	2	2	no	—
28	C₂³×C₃³	D₄	1	1	yes	characteristic
29	C₃³×Q₈	—	2	2	no	—
30	C₃³×Q₈	C₂³	1	4	yes	—
31	C₃³×D₄	—	2	8	no	—
32	C₃³×D₄	—	2	2	no	—
33	C₃³×D₄	C₂³	1	4	yes	—
34	C₂×C₃³×C₄	—	2	2	no	—
35	C₂×C₃³×C₄	—	2	8	no	—
36	C₂×C₃³×C₄	D₄	1	2	yes	—
37	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
38	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
39	C₂×C₃³×C₄	C₂³	1	2	yes	—
40	C₃³×C₈	C₂³	1	2	yes	—
41	C₃³×C₈	C₂³	1	2	yes	—
42	C₂×C₃×#135	C₃²	1	13	yes	direct product factor
43	C₃²×#41	—	2	52	no	—
44	C₃²×#41	C₂²×C₃	1	104	yes	complemented
45	C₂×C₃²×Q₈	C₂²×C₃	1	26	yes	—
46	C₂×C₃²×D₄	—	2	26	no	—
47	C₂×C₃²×D₄	C₂²×C₃	1	26	yes	—
48	C₂²×C₃²×C₄	—	2	26	no	—
49	C₂²×C₃²×C₄	C₂²×C₃	1	13	yes	—
50	C₃²×Q₁₆	C₂²×C₃	1	52	yes	complemented
51	C₃²×SD₁₆	C₂²×C₃	1	104	yes	complemented
52	C₃²×D₈	C₂²×C₃	1	52	yes	complemented
53	C₂×C₃²×C₈	C₂²×C₃	1	13	yes	—
54	C₂×C₃²×C₈	C₂²×C₃	1	13	yes	—
55	C₂×C₃²×C₈	C₂²×C₃	1	52	yes	complemented
56	C₂²×C₃³	—	2	2	no	—
57	C₂²×C₃³	—	2	4	no	—
58	C₂²×C₃³	—	4	4	no	—
59	C₂²×C₃³	C₂×D₄	1	2	yes	—
60	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
61	C₃³×C₄	—	2	4	no	—
62	C₃³×C₄	C₂×D₄	1	2	yes	—
63	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
64	C₃³×C₄	C₂⁴	1	1	yes	characteristic
65	C₂×C₃×#41	C₂×C₃²	1	26	yes	complemented
66	C₃×#135	C₂×C₃²	1	104	yes	direct product factor
67	C₂×C₃×Q₁₆	C₂×C₃²	1	13	yes	complemented
68	C₂×C₃×SD₁₆	C₂×C₃²	1	26	yes	complemented
69	C₂×C₃×D₈	C₂×C₃²	1	13	yes	complemented
70	C₂²×C₃×C₈	C₂×C₃²	1	13	yes	complemented
71	C₂³×C₃²	—	2	26	no	—
72	C₂³×C₃²	C₃×D₄	1	13	yes	—
73	C₃²×Q₈	—	2	26	no	—
74	C₃²×Q₈	C₂³×C₃	1	52	yes	—
75	C₃²×D₄	—	2	104	no	—
76	C₃²×D₄	—	2	26	no	—
77	C₃²×D₄	C₂³×C₃	1	52	yes	—
78	C₂×C₃²×C₄	—	2	26	no	—
79	C₂×C₃²×C₄	—	2	104	no	—
80	C₂×C₃²×C₄	C₃×D₄	1	26	yes	—
81	C₂×C₃²×C₄	C₃×D₄	1	13	yes	—
82	C₂×C₃²×C₄	C₂³×C₃	1	13	yes	—
83	C₂×C₃²×C₄	C₂³×C₃	1	26	yes	—
84	C₃²×C₈	C₂³×C₃	1	26	yes	—
85	C₃²×C₈	C₂³×C₃	1	26	yes	—
86	C₂×#135	C₃³	1	1	yes	characteristic, direct product factor, Sylow
87	C₂×C₃³	—	2	2	no	—
88	C₂×C₃³	—	4	4	no	—
89	C₂×C₃³	#135	1	2	yes	direct product factor
90	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
91	C₃×#41	—	2	52	no	—
92	C₃×#41	C₂²×C₃²	1	104	yes	complemented
93	C₂×C₃×Q₈	C₂²×C₃²	1	26	yes	—
94	C₂×C₃×D₄	—	2	26	no	—
95	C₂×C₃×D₄	C₂²×C₃²	1	26	yes	—
96	C₂²×C₃×C₄	—	2	26	no	—
97	C₂²×C₃×C₄	C₂²×C₃²	1	13	yes	—
98	C₃×Q₁₆	C₂²×C₃²	1	52	yes	complemented
99	C₃×SD₁₆	C₂²×C₃²	1	104	yes	complemented
100	C₃×D₈	C₂²×C₃²	1	52	yes	complemented
101	C₂×C₃×C₈	C₂²×C₃²	1	13	yes	—
102	C₂×C₃×C₈	C₂²×C₃²	1	13	yes	—
103	C₂×C₃×C₈	C₂²×C₃²	1	52	yes	complemented
104	C₂²×C₃²	—	2	26	no	—
105	C₂²×C₃²	—	2	52	no	—
106	C₂²×C₃²	—	4	52	no	—
107	C₂²×C₃²	C₂×C₃×D₄	1	26	yes	—
108	C₂²×C₃²	C₂×C₃×D₄	1	13	yes	—
109	C₃²×C₄	—	2	52	no	—
110	C₃²×C₄	C₂×C₃×D₄	1	26	yes	—
111	C₃²×C₄	C₂×C₃×D₄	1	13	yes	—
112	C₃²×C₄	C₂⁴×C₃	1	13	yes	—
113	C₂×#41	C₂×C₃³	1	2	yes	complemented
114	#135	C₂×C₃³	1	8	yes	direct product factor
115	C₂×Q₁₆	C₂×C₃³	1	1	yes	characteristic, complemented
116	C₂×SD₁₆	C₂×C₃³	1	2	yes	complemented
117	C₂×D₈	C₂×C₃³	1	1	yes	characteristic, complemented
118	C₂²×C₈	C₂×C₃³	1	1	yes	characteristic, complemented
119	C₃³	C₂×#135	1	1	yes	characteristic, direct product factor, Sylow
120	C₂³×C₃	—	2	26	no	—
121	C₂³×C₃	C₃²×D₄	1	13	yes	—
122	C₃×Q₈	—	2	26	no	—
123	C₃×Q₈	C₂³×C₃²	1	52	yes	—
124	C₃×D₄	—	2	104	no	—
125	C₃×D₄	—	2	26	no	—
126	C₃×D₄	C₂³×C₃²	1	52	yes	—
127	C₂×C₃×C₄	—	2	26	no	—
128	C₂×C₃×C₄	—	2	104	no	—
129	C₂×C₃×C₄	C₃²×D₄	1	13	yes	—
130	C₂×C₃×C₄	C₃²×D₄	1	26	yes	—
131	C₂×C₃×C₄	C₂³×C₃²	1	13	yes	—
132	C₂×C₃×C₄	C₂³×C₃²	1	26	yes	—
133	C₃×C₈	C₂³×C₃²	1	26	yes	—
134	C₃×C₈	C₂³×C₃²	1	26	yes	—
135	C₂×C₃²	—	2	26	no	—
136	C₂×C₃²	—	4	52	no	—
137	C₂×C₃²	C₃×#135	1	26	yes	direct product factor
138	C₂×C₃²	C₂²×C₃×D₄	1	13	yes	—
139	#41	—	2	4	no	—
140	#41	C₂²×C₃³	1	8	yes	complemented
141	C₂×Q₈	C₂²×C₃³	1	2	yes	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	C₂²×C₃³	1	2	yes	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	C₂²×C₃³	1	1	yes	characteristic
146	Q₁₆	C₂²×C₃³	1	4	yes	complemented
147	SD₁₆	C₂²×C₃³	1	8	yes	complemented
148	D₈	C₂²×C₃³	1	4	yes	complemented
149	C₂×C₈	C₂²×C₃³	1	1	yes	characteristic
150	C₂×C₈	C₂²×C₃³	1	1	yes	characteristic
151	C₂×C₈	C₂²×C₃³	1	4	yes	complemented
152	C₂²×C₃	—	2	26	no	—
153	C₂²×C₃	—	2	52	no	—
154	C₂²×C₃	—	4	52	no	—
155	C₂²×C₃	C₂×C₃²×D₄	1	26	yes	—
156	C₂²×C₃	C₂×C₃²×D₄	1	13	yes	—
157	C₃×C₄	—	2	52	no	—
158	C₃×C₄	C₂×C₃²×D₄	1	26	yes	—
159	C₃×C₄	C₂×C₃²×D₄	1	13	yes	—
160	C₃×C₄	C₂⁴×C₃²	1	13	yes	—
161	C₃²	C₂×C₃×#135	1	13	yes	direct product factor
162	C₂³	—	2	2	no	—
163	C₂³	C₃³×D₄	1	1	yes	characteristic
164	Q₈	—	2	2	no	—
165	Q₈	C₂³×C₃³	1	4	yes	—
166	D₄	—	2	8	no	—
167	D₄	—	2	2	no	—
168	D₄	C₂³×C₃³	1	4	yes	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	8	no	—
171	C₂×C₄	C₃³×D₄	1	1	yes	characteristic
172	C₂×C₄	C₃³×D₄	1	2	yes	—
173	C₂×C₄	C₂³×C₃³	1	1	yes	characteristic
174	C₂×C₄	C₂³×C₃³	1	2	yes	—
175	C₈	C₂³×C₃³	1	2	yes	—
176	C₈	C₂³×C₃³	1	2	yes	—
177	C₂×C₃	—	2	26	no	—
178	C₂×C₃	—	4	52	no	—
179	C₂×C₃	C₃²×#135	1	26	yes	direct product factor
180	C₂×C₃	C₂²×C₃²×D₄	1	13	yes	—
181	C₂²	—	2	2	no	—
182	C₂²	—	2	4	no	—
183	C₂²	—	4	4	no	—
184	C₂²	C₂×C₃³×D₄	1	2	yes	—
185	C₂²	C₂×C₃³×D₄	1	1	yes	characteristic
186	C₄	—	2	4	no	—
187	C₄	C₂×C₃³×D₄	1	2	yes	—
188	C₄	C₂×C₃³×D₄	1	1	yes	characteristic
189	C₄	C₂⁴×C₃³	1	1	yes	characteristic
190	C₃	C₂×C₃²×#135	1	13	yes	direct product factor
191	C₂	—	2	2	no	—
192	C₂	—	4	4	no	—
193	C₂	C₃³×#135	1	2	yes	direct product factor
194	C₂	C₂²×C₃³×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃³×C₄	D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₃³
socle	C₂²×C₃³	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃³×C₄ ◁ C₂²×C₃³×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃³×#41	C₂	semidirect
2. C₃³×#135	C₂	direct (N × H)
3. C₂×C₃³×Q₁₆	C₂	semidirect
4. C₂×C₃³×SD₁₆	C₂	semidirect
5. C₂×C₃³×D₈	C₂	semidirect
6. C₂²×C₃³×C₈	C₂	semidirect
7. C₂×C₃²×#135	C₃	direct (N × H)
9. C₃³×#41	C₂²	semidirect
15. C₃³×Q₁₆	C₂²	semidirect
16. C₃³×SD₁₆	C₂²	semidirect
17. C₃³×D₈	C₂²	semidirect
20. C₂×C₃³×C₈	C₂²	semidirect
21. C₂×C₃²×#41	C₂×C₃	semidirect
22. C₃²×#135	C₂×C₃	direct (N × H)
23. C₂×C₃²×Q₁₆	C₂×C₃	semidirect
24. C₂×C₃²×SD₁₆	C₂×C₃	semidirect
25. C₂×C₃²×D₈	C₂×C₃	semidirect
26. C₂²×C₃²×C₈	C₂×C₃	semidirect
42. C₂×C₃×#135	C₃²	direct (N × H)
44. C₃²×#41	C₂²×C₃	semidirect
50. C₃²×Q₁₆	C₂²×C₃	semidirect
51. C₃²×SD₁₆	C₂²×C₃	semidirect
52. C₃²×D₈	C₂²×C₃	semidirect
55. C₂×C₃²×C₈	C₂²×C₃	semidirect
65. C₂×C₃×#41	C₂×C₃²	semidirect
66. C₃×#135	C₂×C₃²	direct (N × H)
67. C₂×C₃×Q₁₆	C₂×C₃²	semidirect
68. C₂×C₃×SD₁₆	C₂×C₃²	semidirect
69. C₂×C₃×D₈	C₂×C₃²	semidirect
70. C₂²×C₃×C₈	C₂×C₃²	semidirect
86. C₂×#135	C₃³	direct (N × H)
89. C₂×C₃³	#135	direct (N × H)
92. C₃×#41	C₂²×C₃²	semidirect
98. C₃×Q₁₆	C₂²×C₃²	semidirect
99. C₃×SD₁₆	C₂²×C₃²	semidirect
100. C₃×D₈	C₂²×C₃²	semidirect
103. C₂×C₃×C₈	C₂²×C₃²	semidirect
113. C₂×#41	C₂×C₃³	semidirect
114. #135	C₂×C₃³	direct (N × H)
115. C₂×Q₁₆	C₂×C₃³	semidirect
116. C₂×SD₁₆	C₂×C₃³	semidirect
117. C₂×D₈	C₂×C₃³	semidirect
118. C₂²×C₈	C₂×C₃³	semidirect
119. C₃³	C₂×#135	direct (N × H)
137. C₂×C₃²	C₃×#135	direct (N × H)
140. #41	C₂²×C₃³	semidirect
146. Q₁₆	C₂²×C₃³	semidirect
147. SD₁₆	C₂²×C₃³	semidirect
148. D₈	C₂²×C₃³	semidirect
151. C₂×C₈	C₂²×C₃³	semidirect
161. C₃²	C₂×C₃×#135	direct (N × H)
179. C₂×C₃	C₃²×#135	direct (N × H)
190. C₃	C₂×C₃²×#135	direct (N × H)
193. C₂	C₃³×#135	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×C₃³×#41	2
C₃³×#135	2
C₂×C₃³×Q₁₆	2
C₂×C₃³×SD₁₆	2
C₂×C₃³×D₈	2
C₂²×C₃³×C₈	2
C₂×C₃²×#135	3

Subgroups — C2×C33×#135 #13180027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃³×#135 #13180027