Subgroups — C₂³×C₃²×#67 #13181729

Order 1728.

11164 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₃³	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₃²×D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃²×#67	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₃²×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₃²×D₄	—	3	1	no	maximal
6	C₂³×C₃×#67	C₃	1	4	yes	maximal, direct product factor
7	C₂⁴×C₃³	—	2	8	no	—
8	C₂⁴×C₃³	C₂²	1	1	yes	characteristic
9	C₂⁴×C₃³	C₂²	1	14	yes	complemented
10	C₂³×C₃²×D₃	—	2	8	no	—
11	C₂³×C₃²×D₃	C₂²	1	14	yes	complemented
12	C₂×C₃²×#67	C₂²	1	112	yes	direct product factor
13	C₂²×C₃²×Dic₃	C₂²	1	14	yes	complemented
14	C₂⁵×C₃²	—	2	4	no	—
15	C₂⁵×C₃²	—	3	1	no	—
16	C₂⁵×C₃²	D₃	1	1	yes	characteristic, complemented
17	C₂⁵×C₃²	C₂×C₃	1	4	yes	complemented
18	C₂⁴×C₃×D₃	C₂×C₃	1	4	yes	complemented
19	C₂²×C₃²×D₄	—	3	28	no	—
20	C₂³×C₃²×C₄	—	3	1	no	—
21	C₂²×C₃×#67	C₂×C₃	1	112	yes	direct product factor
22	C₂³×C₃×Dic₃	C₂×C₃	1	4	yes	complemented
23	C₂³×C₃³	—	2	56	no	—
24	C₂³×C₃³	D₄	1	8	yes	complemented
25	C₂³×C₃³	C₂³	1	7	yes	—
26	C₂³×C₃³	C₂³	1	28	yes	complemented
27	C₂²×C₃²×D₃	—	2	56	no	—
28	C₂²×C₃²×D₃	C₂³	1	28	yes	complemented
29	C₃²×#67	C₂³	1	64	yes	direct product factor
30	C₂×C₃²×Dic₃	C₂³	1	28	yes	complemented
31	C₂³×C₃×D₄	—	3	4	no	—
32	C₂³×#67	C₃²	1	1	yes	characteristic, direct product factor
33	C₂⁴×C₃²	—	2	64	no	—
34	C₂⁴×C₃²	—	2	4	no	—
35	C₂⁴×C₃²	—	2	56	no	—
36	C₂⁴×C₃²	—	2	8	no	—
37	C₂⁴×C₃²	—	2	32	no	—
38	C₂⁴×C₃²	—	3	14	no	—
39	C₂⁴×C₃²	—	6	8	no	—
40	C₂⁴×C₃²	C₂×D₃	1	1	yes	characteristic
41	C₂⁴×C₃²	C₂×D₃	1	14	yes	complemented
42	C₂⁴×C₃²	C₂²×C₃	1	4	yes	—
43	C₂⁴×C₃²	C₂²×C₃	1	56	yes	complemented
44	C₂³×C₃×D₃	—	2	32	no	—
45	C₂³×C₃×D₃	C₂²×C₃	1	56	yes	complemented
46	C₂×C₃²×D₄	—	3	112	no	—
47	C₂²×C₃²×C₄	—	3	14	no	—
48	C₂×C₃×#67	C₂²×C₃	1	448	yes	direct product factor
49	C₂²×C₃×Dic₃	C₂²×C₃	1	56	yes	complemented
50	C₂²×C₃³	—	2	56	no	—
51	C₂²×C₃³	C₂×D₄	1	28	yes	complemented
52	C₂²×C₃³	C₂⁴	1	7	yes	—
53	C₂²×C₃³	C₂⁴	1	8	yes	complemented
54	C₂×C₃²×D₃	—	2	56	no	—
55	C₂×C₃²×D₃	C₂⁴	1	8	yes	complemented
56	C₃²×Dic₃	C₂⁴	1	8	yes	complemented
57	C₂⁵×C₃	—	2	4	no	—
58	C₂⁵×C₃	—	3	4	no	—
59	C₂⁵×C₃	C₃×D₃	1	4	yes	complemented
60	C₂⁵×C₃	C₂×C₃²	1	1	yes	characteristic, complemented
61	C₂⁴×D₃	C₂×C₃²	1	1	yes	characteristic, complemented
62	C₂²×C₃×D₄	—	3	112	no	—
63	C₂³×C₃×C₄	—	3	4	no	—
64	C₂²×#67	C₂×C₃²	1	28	yes	direct product factor
65	C₂³×Dic₃	C₂×C₃²	1	1	yes	characteristic, complemented
66	C₂³×C₃²	—	2	448	no	—
67	C₂³×C₃²	—	2	56	no	—
68	C₂³×C₃²	—	2	224	no	—
69	C₂³×C₃²	—	2	32	no	—
70	C₂³×C₃²	—	2	28	no	—
71	C₂³×C₃²	—	2	112	no	—
72	C₂³×C₃²	—	3	28	no	—
73	C₂³×C₃²	—	6	56	no	—
74	C₂³×C₃²	#67	1	8	yes	direct product factor
75	C₂³×C₃²	C₃×D₄	1	32	yes	complemented
76	C₂³×C₃²	C₂²×D₃	1	7	yes	—
77	C₂³×C₃²	C₂²×D₃	1	28	yes	complemented
78	C₂³×C₃²	C₂³×C₃	1	28	yes	—
79	C₂³×C₃²	C₂³×C₃	1	112	yes	complemented
80	C₂²×C₃×D₃	—	2	224	no	—
81	C₂²×C₃×D₃	C₂³×C₃	1	112	yes	complemented
82	C₃²×D₄	—	3	64	no	—
83	C₂×C₃²×C₄	—	3	28	no	—
84	C₃×#67	C₂³×C₃	1	256	yes	direct product factor
85	C₂×C₃×Dic₃	C₂³×C₃	1	112	yes	complemented
86	C₂³×D₄	—	3	1	no	Sylow
87	C₂×C₃³	—	2	8	no	—
88	C₂×C₃³	C₂²×D₄	1	14	yes	complemented
89	C₂×C₃³	C₂⁵	1	1	yes	characteristic
90	C₃²×D₃	—	2	8	no	—
91	C₂⁴×C₃	—	2	64	no	—
92	C₂⁴×C₃	—	2	32	no	—
93	C₂⁴×C₃	—	2	8	no	—
94	C₂⁴×C₃	—	2	4	no	—
95	C₂⁴×C₃	—	2	56	no	—
96	C₂⁴×C₃	—	3	56	no	—
97	C₂⁴×C₃	—	6	32	no	—
98	C₂⁴×C₃	C₂×C₃×D₃	1	4	yes	—
99	C₂⁴×C₃	C₂×C₃×D₃	1	56	yes	complemented
100	C₂⁴×C₃	C₂²×C₃²	1	1	yes	characteristic
101	C₂⁴×C₃	C₂²×C₃²	1	14	yes	complemented
102	C₂³×D₃	—	2	8	no	—
103	C₂³×D₃	C₂²×C₃²	1	14	yes	complemented
104	C₂×C₃×D₄	—	3	448	no	—
105	C₂²×C₃×C₄	—	3	56	no	—
106	C₂×#67	C₂²×C₃²	1	112	yes	direct product factor
107	C₂²×Dic₃	C₂²×C₃²	1	14	yes	complemented
108	C₂²×C₃²	—	2	448	no	—
109	C₂²×C₃²	—	2	56	no	—
110	C₂²×C₃²	—	2	224	no	—
111	C₂²×C₃²	—	2	28	no	—
112	C₂²×C₃²	—	2	32	no	—
113	C₂²×C₃²	—	2	112	no	—
114	C₂²×C₃²	—	3	8	no	—
115	C₂²×C₃²	—	6	56	no	—
116	C₂²×C₃²	C₂×#67	1	28	yes	direct product factor
117	C₂²×C₃²	C₂×C₃×D₄	1	112	yes	complemented
118	C₂²×C₃²	C₂³×D₃	1	7	yes	—
119	C₂²×C₃²	C₂³×D₃	1	8	yes	complemented
120	C₂²×C₃²	C₂⁴×C₃	1	28	yes	—
121	C₂²×C₃²	C₂⁴×C₃	1	32	yes	complemented
122	C₂×C₃×D₃	—	2	224	no	—
123	C₂×C₃×D₃	C₂⁴×C₃	1	32	yes	complemented
124	C₃²×C₄	—	3	8	no	—
125	C₃×Dic₃	C₂⁴×C₃	1	32	yes	complemented
126	C₂⁵	—	3	1	no	—
127	C₂⁵	C₃²×D₃	1	1	yes	characteristic, complemented
128	C₂²×D₄	—	3	28	no	—
129	C₂³×C₄	—	3	1	no	—
130	C₃³	C₂³×D₄	1	1	yes	characteristic, complemented, Sylow
131	C₂³×C₃	—	2	448	no	—
132	C₂³×C₃	—	2	224	no	—
133	C₂³×C₃	—	2	56	no	—
134	C₂³×C₃	—	2	32	no	—
135	C₂³×C₃	—	2	28	no	—
136	C₂³×C₃	—	2	112	no	—
137	C₂³×C₃	—	3	112	no	—
138	C₂³×C₃	—	6	224	no	—
139	C₂³×C₃	C₃×#67	1	32	yes	direct product factor
140	C₂³×C₃	C₃²×D₄	1	8	yes	complemented
141	C₂³×C₃	C₂²×C₃×D₃	1	28	yes	—
142	C₂³×C₃	C₂²×C₃×D₃	1	112	yes	complemented
143	C₂³×C₃	C₂³×C₃²	1	7	yes	—
144	C₂³×C₃	C₂³×C₃²	1	28	yes	complemented
145	C₂²×D₃	—	2	56	no	—
146	C₂²×D₃	C₂³×C₃²	1	28	yes	complemented
147	C₃×D₄	—	3	256	no	—
148	C₂×C₃×C₄	—	3	112	no	—
149	#67	C₂³×C₃²	1	64	yes	direct product factor
150	C₂×Dic₃	C₂³×C₃²	1	28	yes	complemented
151	C₂×C₃²	—	2	64	no	—
152	C₂×C₃²	—	2	8	no	—
153	C₂×C₃²	—	2	32	no	—
154	C₂×C₃²	—	2	56	no	—
155	C₂×C₃²	—	2	4	no	—
156	C₂×C₃²	—	6	8	no	—
157	C₂×C₃²	C₂²×#67	1	14	yes	direct product factor
158	C₂×C₃²	C₂²×C₃×D₄	1	56	yes	complemented
159	C₂×C₃²	C₂⁴×D₃	1	1	yes	characteristic
160	C₂×C₃²	C₂⁵×C₃	1	4	yes	—
161	C₃×D₃	—	2	32	no	—
162	C₂⁴	—	2	8	no	—
163	C₂⁴	—	3	14	no	—
164	C₂⁴	—	6	8	no	—
165	C₂⁴	C₂×C₃²×D₃	1	1	yes	characteristic
166	C₂⁴	C₂×C₃²×D₃	1	14	yes	complemented
167	C₂×D₄	—	3	112	no	—
168	C₂²×C₄	—	3	14	no	—
169	C₂²×C₃	—	2	448	no	—
170	C₂²×C₃	—	2	224	no	—
171	C₂²×C₃	—	2	56	no	—
172	C₂²×C₃	—	2	28	no	—
173	C₂²×C₃	—	2	32	no	—
174	C₂²×C₃	—	2	112	no	—
175	C₂²×C₃	—	3	32	no	—
176	C₂²×C₃	—	6	224	no	—
177	C₂²×C₃	C₂×C₃×#67	1	112	yes	direct product factor
178	C₂²×C₃	C₂×C₃²×D₄	1	28	yes	complemented
179	C₂²×C₃	C₂³×C₃×D₃	1	28	yes	—
180	C₂²×C₃	C₂³×C₃×D₃	1	32	yes	complemented
181	C₂²×C₃	C₂⁴×C₃²	1	7	yes	—
182	C₂²×C₃	C₂⁴×C₃²	1	8	yes	complemented
183	C₂×D₃	—	2	56	no	—
184	C₂×D₃	C₂⁴×C₃²	1	8	yes	complemented
185	C₃×C₄	—	3	32	no	—
186	Dic₃	C₂⁴×C₃²	1	8	yes	complemented
187	C₃²	—	2	4	no	—
188	C₃²	C₂³×#67	1	1	yes	characteristic, direct product factor
189	C₃²	C₂³×C₃×D₄	1	4	yes	complemented
190	C₂³	—	2	56	no	—
191	C₂³	—	3	28	no	—
192	C₂³	—	6	56	no	—
193	C₂³	C₃²×#67	1	8	yes	direct product factor
194	C₂³	C₂²×C₃²×D₃	1	7	yes	—
195	C₂³	C₂²×C₃²×D₃	1	28	yes	complemented
196	D₄	—	3	64	no	—
197	C₂×C₄	—	3	28	no	—
198	C₂×C₃	—	2	64	no	—
199	C₂×C₃	—	2	32	no	—
200	C₂×C₃	—	2	8	no	—
201	C₂×C₃	—	2	56	no	—
202	C₂×C₃	—	2	4	no	—
203	C₂×C₃	—	6	32	no	—
204	C₂×C₃	C₂²×C₃×#67	1	56	yes	direct product factor
205	C₂×C₃	C₂²×C₃²×D₄	1	14	yes	complemented
206	C₂×C₃	C₂⁴×C₃×D₃	1	4	yes	—
207	C₂×C₃	C₂⁵×C₃²	1	1	yes	characteristic
208	D₃	—	2	8	no	—
209	C₂²	—	2	56	no	—
210	C₂²	—	3	8	no	—
211	C₂²	—	6	56	no	—
212	C₂²	C₂×C₃²×#67	1	28	yes	direct product factor
213	C₂²	C₂³×C₃²×D₃	1	7	yes	—
214	C₂²	C₂³×C₃²×D₃	1	8	yes	complemented
215	C₄	—	3	8	no	—
216	C₃	—	2	4	no	—
217	C₃	C₂³×C₃×#67	1	4	yes	direct product factor
218	C₃	C₂³×C₃²×D₄	1	1	yes	characteristic, complemented
219	C₂	—	2	8	no	—
220	C₂	—	6	8	no	—
221	C₂	C₂²×C₃²×#67	1	14	yes	direct product factor
222	C₂	C₂⁴×C₃²×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₃²	C₂×D₃
commutator, cosocle	C₂×C₃	C₂⁵×C₃²
Frattini	C₂	C₂⁴×C₃²×D₃
Fitting	C₂⁵×C₃³	C₂
socle	C₂⁴×C₃³	C₂²
hypercenter	C₂⁵×C₃²	D₃
hypocenter	C₃	C₂³×C₃²×D₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴×C₃² ◁ C₂⁵×C₃²

Complements

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

N	H ≅ G/N	type
1. C₂⁵×C₃³	C₂	semidirect
2. C₂⁴×C₃²×D₃	C₂	semidirect
3. C₂²×C₃²×#67	C₂	direct (N × H)
4. C₂³×C₃²×Dic₃	C₂	semidirect
6. C₂³×C₃×#67	C₃	direct (N × H)
9. C₂⁴×C₃³	C₂²	semidirect
11. C₂³×C₃²×D₃	C₂²	semidirect
12. C₂×C₃²×#67	C₂²	direct (N × H)
13. C₂²×C₃²×Dic₃	C₂²	semidirect
16. C₂⁵×C₃²	D₃	semidirect
17. C₂⁵×C₃²	C₂×C₃	semidirect
18. C₂⁴×C₃×D₃	C₂×C₃	semidirect
21. C₂²×C₃×#67	C₂×C₃	direct (N × H)
22. C₂³×C₃×Dic₃	C₂×C₃	semidirect
24. C₂³×C₃³	D₄	semidirect
26. C₂³×C₃³	C₂³	semidirect
28. C₂²×C₃²×D₃	C₂³	semidirect
29. C₃²×#67	C₂³	direct (N × H)
30. C₂×C₃²×Dic₃	C₂³	semidirect
32. C₂³×#67	C₃²	direct (N × H)
41. C₂⁴×C₃²	C₂×D₃	semidirect
43. C₂⁴×C₃²	C₂²×C₃	semidirect
45. C₂³×C₃×D₃	C₂²×C₃	semidirect
48. C₂×C₃×#67	C₂²×C₃	direct (N × H)
49. C₂²×C₃×Dic₃	C₂²×C₃	semidirect
51. C₂²×C₃³	C₂×D₄	semidirect
53. C₂²×C₃³	C₂⁴	semidirect
55. C₂×C₃²×D₃	C₂⁴	semidirect
56. C₃²×Dic₃	C₂⁴	semidirect
59. C₂⁵×C₃	C₃×D₃	semidirect
60. C₂⁵×C₃	C₂×C₃²	semidirect
61. C₂⁴×D₃	C₂×C₃²	semidirect
64. C₂²×#67	C₂×C₃²	direct (N × H)
65. C₂³×Dic₃	C₂×C₃²	semidirect
74. C₂³×C₃²	#67	direct (N × H)
75. C₂³×C₃²	C₃×D₄	semidirect
77. C₂³×C₃²	C₂²×D₃	semidirect
79. C₂³×C₃²	C₂³×C₃	semidirect
81. C₂²×C₃×D₃	C₂³×C₃	semidirect
84. C₃×#67	C₂³×C₃	direct (N × H)
85. C₂×C₃×Dic₃	C₂³×C₃	semidirect
88. C₂×C₃³	C₂²×D₄	semidirect
99. C₂⁴×C₃	C₂×C₃×D₃	semidirect
101. C₂⁴×C₃	C₂²×C₃²	semidirect
103. C₂³×D₃	C₂²×C₃²	semidirect
106. C₂×#67	C₂²×C₃²	direct (N × H)
107. C₂²×Dic₃	C₂²×C₃²	semidirect
116. C₂²×C₃²	C₂×#67	direct (N × H)
117. C₂²×C₃²	C₂×C₃×D₄	semidirect
119. C₂²×C₃²	C₂³×D₃	semidirect
121. C₂²×C₃²	C₂⁴×C₃	semidirect
123. C₂×C₃×D₃	C₂⁴×C₃	semidirect
125. C₃×Dic₃	C₂⁴×C₃	semidirect
127. C₂⁵	C₃²×D₃	semidirect
130. C₃³	C₂³×D₄	semidirect
139. C₂³×C₃	C₃×#67	direct (N × H)
140. C₂³×C₃	C₃²×D₄	semidirect
142. C₂³×C₃	C₂²×C₃×D₃	semidirect
144. C₂³×C₃	C₂³×C₃²	semidirect
146. C₂²×D₃	C₂³×C₃²	semidirect
149. #67	C₂³×C₃²	direct (N × H)
150. C₂×Dic₃	C₂³×C₃²	semidirect
157. C₂×C₃²	C₂²×#67	direct (N × H)
158. C₂×C₃²	C₂²×C₃×D₄	semidirect
166. C₂⁴	C₂×C₃²×D₃	semidirect
177. C₂²×C₃	C₂×C₃×#67	direct (N × H)
178. C₂²×C₃	C₂×C₃²×D₄	semidirect
180. C₂²×C₃	C₂³×C₃×D₃	semidirect
182. C₂²×C₃	C₂⁴×C₃²	semidirect
184. C₂×D₃	C₂⁴×C₃²	semidirect
186. Dic₃	C₂⁴×C₃²	semidirect
188. C₃²	C₂³×#67	direct (N × H)
189. C₃²	C₂³×C₃×D₄	semidirect
193. C₂³	C₃²×#67	direct (N × H)
195. C₂³	C₂²×C₃²×D₃	semidirect
204. C₂×C₃	C₂²×C₃×#67	direct (N × H)
205. C₂×C₃	C₂²×C₃²×D₄	semidirect
212. C₂²	C₂×C₃²×#67	direct (N × H)
214. C₂²	C₂³×C₃²×D₃	semidirect
217. C₃	C₂³×C₃×#67	direct (N × H)
218. C₃	C₂³×C₃²×D₄	semidirect
221. C₂	C₂²×C₃²×#67	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₃³	2
C₂⁴×C₃²×D₃	2
C₂²×C₃²×#67	2
C₂³×C₃²×Dic₃	2
C₂³×C₃²×D₄	3
C₂³×C₃×#67	3

Subgroups — C23×C32×#67 #13181729

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃²×#67 #13181729