Subgroups — C₂×C₃³×#137 #13180029

Order 1728.

3612 conjugacy classes of subgroups.

The subgroup lattice has 228 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	1	yes	maximal, characteristic
2	C₂²×C₃³×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃³×#137	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃³×Q₁₆	C₂	1	2	yes	maximal, complemented
5	C₂×C₃³×SD₁₆	C₂	1	2	yes	maximal, complemented
6	C₂×C₃³×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃²×#137	C₃	1	13	yes	maximal, direct product factor
8	C₃³×#41	—	2	2	no	—
9	C₃³×#41	C₂²	1	4	yes	—
10	C₂×C₃³×Q₈	—	2	2	no	—
11	C₂×C₃³×Q₈	—	2	1	no	—
12	C₂×C₃³×Q₈	C₂²	1	2	yes	—
13	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃³×Q₈	C₂²	1	4	yes	complemented
15	C₂×C₃³×D₄	—	2	1	no	—
16	C₂×C₃³×D₄	C₂²	1	1	yes	characteristic
17	C₂²×C₃³×C₄	—	2	1	no	—
18	C₂²×C₃³×C₄	—	2	1	no	—
19	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
20	C₃³×Q₁₆	C₂²	1	8	yes	complemented
21	C₃³×SD₁₆	C₂²	1	8	yes	complemented
22	C₃³×M₁₆	C₂²	1	4	yes	complemented
23	C₂×C₃³×C₈	C₂²	1	2	yes	—
24	C₂×C₃²×#41	C₂×C₃	1	13	yes	—
25	C₂²×C₃²×Q₈	C₂×C₃	1	13	yes	complemented
26	C₃²×#137	C₂×C₃	1	104	yes	direct product factor
27	C₂×C₃²×Q₁₆	C₂×C₃	1	26	yes	complemented
28	C₂×C₃²×SD₁₆	C₂×C₃	1	26	yes	complemented
29	C₂×C₃²×M₁₆	C₂×C₃	1	13	yes	complemented
30	C₂³×C₃³	—	2	1	no	—
31	C₂³×C₃³	D₄	1	1	yes	characteristic
32	C₃³×Q₈	—	2	2	no	—
33	C₃³×Q₈	—	2	4	no	—
34	C₃³×Q₈	—	2	1	no	—
35	C₃³×Q₈	C₂³	1	4	yes	—
36	C₃³×Q₈	C₂³	1	2	yes	—
37	C₃³×D₄	—	2	4	no	—
38	C₃³×D₄	—	2	1	no	—
39	C₃³×D₄	C₂³	1	2	yes	—
40	C₂×C₃³×C₄	—	2	4	no	—
41	C₂×C₃³×C₄	—	2	1	no	—
42	C₂×C₃³×C₄	—	2	2	no	—
43	C₂×C₃³×C₄	—	2	4	no	—
44	C₂×C₃³×C₄	D₄	1	2	yes	—
45	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
46	C₂×C₃³×C₄	C₂³	1	2	yes	—
47	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
48	C₃³×C₈	C₂³	1	4	yes	—
49	C₂×C₃×#137	C₃²	1	13	yes	direct product factor
50	C₃²×#41	—	2	26	no	—
51	C₃²×#41	C₂²×C₃	1	52	yes	—
52	C₂×C₃²×Q₈	—	2	13	no	—
53	C₂×C₃²×Q₈	—	2	26	no	—
54	C₂×C₃²×Q₈	C₂²×C₃	1	26	yes	—
55	C₂×C₃²×Q₈	C₂²×C₃	1	13	yes	—
56	C₂×C₃²×Q₈	C₂²×C₃	1	52	yes	complemented
57	C₂×C₃²×D₄	—	2	13	no	—
58	C₂×C₃²×D₄	C₂²×C₃	1	13	yes	—
59	C₂²×C₃²×C₄	—	2	13	no	—
60	C₂²×C₃²×C₄	—	2	13	no	—
61	C₂²×C₃²×C₄	C₂²×C₃	1	13	yes	—
62	C₃²×Q₁₆	C₂²×C₃	1	104	yes	complemented
63	C₃²×SD₁₆	C₂²×C₃	1	104	yes	complemented
64	C₃²×M₁₆	C₂²×C₃	1	52	yes	complemented
65	C₂×C₃²×C₈	C₂²×C₃	1	26	yes	—
66	C₂²×C₃³	—	2	2	no	—
67	C₂²×C₃³	—	2	2	no	—
68	C₂²×C₃³	—	4	2	no	—
69	C₂²×C₃³	C₂×D₄	1	2	yes	—
70	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
71	C₃³×C₄	—	2	2	no	—
72	C₃³×C₄	—	2	4	no	—
73	C₃³×C₄	C₂×D₄	1	2	yes	—
74	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
75	C₃³×C₄	C₂⁴	1	1	yes	characteristic
76	C₂×C₃×#41	C₂×C₃²	1	13	yes	—
77	C₂²×C₃×Q₈	C₂×C₃²	1	13	yes	complemented
78	C₃×#137	C₂×C₃²	1	104	yes	direct product factor
79	C₂×C₃×Q₁₆	C₂×C₃²	1	26	yes	complemented
80	C₂×C₃×SD₁₆	C₂×C₃²	1	26	yes	complemented
81	C₂×C₃×M₁₆	C₂×C₃²	1	13	yes	complemented
82	C₂³×C₃²	—	2	13	no	—
83	C₂³×C₃²	C₃×D₄	1	13	yes	—
84	C₃²×Q₈	—	2	26	no	—
85	C₃²×Q₈	—	2	52	no	—
86	C₃²×Q₈	—	2	13	no	—
87	C₃²×Q₈	C₂³×C₃	1	52	yes	—
88	C₃²×Q₈	C₂³×C₃	1	26	yes	—
89	C₃²×D₄	—	2	52	no	—
90	C₃²×D₄	—	2	13	no	—
91	C₃²×D₄	C₂³×C₃	1	26	yes	—
92	C₂×C₃²×C₄	—	2	13	no	—
93	C₂×C₃²×C₄	—	2	26	no	—
94	C₂×C₃²×C₄	—	2	52	no	—
95	C₂×C₃²×C₄	—	2	52	no	—
96	C₂×C₃²×C₄	C₃×D₄	1	13	yes	—
97	C₂×C₃²×C₄	C₃×D₄	1	26	yes	—
98	C₂×C₃²×C₄	C₂³×C₃	1	13	yes	—
99	C₂×C₃²×C₄	C₂³×C₃	1	26	yes	—
100	C₃²×C₈	C₂³×C₃	1	52	yes	—
101	C₂×#137	C₃³	1	1	yes	characteristic, direct product factor, Sylow
102	C₂×C₃³	—	2	2	no	—
103	C₂×C₃³	—	4	2	no	—
104	C₂×C₃³	#137	1	2	yes	direct product factor
105	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
106	C₃×#41	—	2	26	no	—
107	C₃×#41	C₂²×C₃²	1	52	yes	—
108	C₂×C₃×Q₈	—	2	13	no	—
109	C₂×C₃×Q₈	—	2	26	no	—
110	C₂×C₃×Q₈	C₂²×C₃²	1	26	yes	—
111	C₂×C₃×Q₈	C₂²×C₃²	1	13	yes	—
112	C₂×C₃×Q₈	C₂²×C₃²	1	52	yes	complemented
113	C₂×C₃×D₄	—	2	13	no	—
114	C₂×C₃×D₄	C₂²×C₃²	1	13	yes	—
115	C₂²×C₃×C₄	—	2	13	no	—
116	C₂²×C₃×C₄	—	2	13	no	—
117	C₂²×C₃×C₄	C₂²×C₃²	1	13	yes	—
118	C₃×Q₁₆	C₂²×C₃²	1	104	yes	complemented
119	C₃×SD₁₆	C₂²×C₃²	1	104	yes	complemented
120	C₃×M₁₆	C₂²×C₃²	1	52	yes	complemented
121	C₂×C₃×C₈	C₂²×C₃²	1	26	yes	—
122	C₂²×C₃²	—	2	26	no	—
123	C₂²×C₃²	—	2	26	no	—
124	C₂²×C₃²	—	4	26	no	—
125	C₂²×C₃²	C₂×C₃×D₄	1	26	yes	—
126	C₂²×C₃²	C₂×C₃×D₄	1	13	yes	—
127	C₃²×C₄	—	2	26	no	—
128	C₃²×C₄	—	2	52	no	—
129	C₃²×C₄	C₂×C₃×D₄	1	26	yes	—
130	C₃²×C₄	C₂×C₃×D₄	1	13	yes	—
131	C₃²×C₄	C₂⁴×C₃	1	13	yes	—
132	C₂×#41	C₂×C₃³	1	1	yes	characteristic
133	C₂²×Q₈	C₂×C₃³	1	1	yes	characteristic, complemented
134	#137	C₂×C₃³	1	8	yes	direct product factor
135	C₂×Q₁₆	C₂×C₃³	1	2	yes	complemented
136	C₂×SD₁₆	C₂×C₃³	1	2	yes	complemented
137	C₂×M₁₆	C₂×C₃³	1	1	yes	characteristic, complemented
138	C₃³	C₂×#137	1	1	yes	characteristic, direct product factor, Sylow
139	C₂³×C₃	—	2	13	no	—
140	C₂³×C₃	C₃²×D₄	1	13	yes	—
141	C₃×Q₈	—	2	26	no	—
142	C₃×Q₈	—	2	52	no	—
143	C₃×Q₈	—	2	13	no	—
144	C₃×Q₈	C₂³×C₃²	1	52	yes	—
145	C₃×Q₈	C₂³×C₃²	1	26	yes	—
146	C₃×D₄	—	2	52	no	—
147	C₃×D₄	—	2	13	no	—
148	C₃×D₄	C₂³×C₃²	1	26	yes	—
149	C₂×C₃×C₄	—	2	13	no	—
150	C₂×C₃×C₄	—	2	26	no	—
151	C₂×C₃×C₄	—	2	52	no	—
152	C₂×C₃×C₄	—	2	52	no	—
153	C₂×C₃×C₄	C₃²×D₄	1	13	yes	—
154	C₂×C₃×C₄	C₃²×D₄	1	26	yes	—
155	C₂×C₃×C₄	C₂³×C₃²	1	13	yes	—
156	C₂×C₃×C₄	C₂³×C₃²	1	26	yes	—
157	C₃×C₈	C₂³×C₃²	1	52	yes	—
158	C₂×C₃²	—	2	26	no	—
159	C₂×C₃²	—	4	26	no	—
160	C₂×C₃²	C₃×#137	1	26	yes	direct product factor
161	C₂×C₃²	C₂²×C₃×D₄	1	13	yes	—
162	#41	—	2	2	no	—
163	#41	C₂²×C₃³	1	4	yes	—
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	C₂²×C₃³	1	2	yes	—
167	C₂×Q₈	C₂²×C₃³	1	1	yes	characteristic
168	C₂×Q₈	C₂²×C₃³	1	4	yes	complemented
169	C₂×D₄	—	2	1	no	—
170	C₂×D₄	C₂²×C₃³	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	C₂²×C₃³	1	1	yes	characteristic
174	Q₁₆	C₂²×C₃³	1	8	yes	complemented
175	SD₁₆	C₂²×C₃³	1	8	yes	complemented
176	M₁₆	C₂²×C₃³	1	4	yes	complemented
177	C₂×C₈	C₂²×C₃³	1	2	yes	—
178	C₂²×C₃	—	2	26	no	—
179	C₂²×C₃	—	2	26	no	—
180	C₂²×C₃	—	4	26	no	—
181	C₂²×C₃	C₂×C₃²×D₄	1	26	yes	—
182	C₂²×C₃	C₂×C₃²×D₄	1	13	yes	—
183	C₃×C₄	—	2	26	no	—
184	C₃×C₄	—	2	52	no	—
185	C₃×C₄	C₂×C₃²×D₄	1	26	yes	—
186	C₃×C₄	C₂×C₃²×D₄	1	13	yes	—
187	C₃×C₄	C₂⁴×C₃²	1	13	yes	—
188	C₃²	C₂×C₃×#137	1	13	yes	direct product factor
189	C₂³	—	2	1	no	—
190	C₂³	C₃³×D₄	1	1	yes	characteristic
191	Q₈	—	2	2	no	—
192	Q₈	—	2	4	no	—
193	Q₈	—	2	1	no	—
194	Q₈	C₂³×C₃³	1	4	yes	—
195	Q₈	C₂³×C₃³	1	2	yes	—
196	D₄	—	2	4	no	—
197	D₄	—	2	1	no	—
198	D₄	C₂³×C₃³	1	2	yes	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	C₃³×D₄	1	1	yes	characteristic
204	C₂×C₄	C₃³×D₄	1	2	yes	—
205	C₂×C₄	C₂³×C₃³	1	1	yes	characteristic
206	C₂×C₄	C₂³×C₃³	1	2	yes	—
207	C₈	C₂³×C₃³	1	4	yes	—
208	C₂×C₃	—	2	26	no	—
209	C₂×C₃	—	4	26	no	—
210	C₂×C₃	C₃²×#137	1	26	yes	direct product factor
211	C₂×C₃	C₂²×C₃²×D₄	1	13	yes	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	C₂×C₃³×D₄	1	2	yes	—
216	C₂²	C₂×C₃³×D₄	1	1	yes	characteristic
217	C₄	—	2	2	no	—
218	C₄	—	2	4	no	—
219	C₄	C₂×C₃³×D₄	1	2	yes	—
220	C₄	C₂×C₃³×D₄	1	1	yes	characteristic
221	C₄	C₂⁴×C₃³	1	1	yes	characteristic
222	C₃	C₂×C₃²×#137	1	13	yes	direct product factor
223	C₂	—	2	2	no	—
224	C₂	—	4	2	no	—
225	C₂	C₃³×#137	1	2	yes	direct product factor
226	C₂	C₂²×C₃³×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃³	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₃³
prime radical	C₂×C₃³×#41	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃³ ◁ C₂²×C₃³×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃³×Q₈	C₂	semidirect
3. C₃³×#137	C₂	direct (N × H)
4. C₂×C₃³×Q₁₆	C₂	semidirect
5. C₂×C₃³×SD₁₆	C₂	semidirect
6. C₂×C₃³×M₁₆	C₂	semidirect
7. C₂×C₃²×#137	C₃	direct (N × H)
14. C₂×C₃³×Q₈	C₂²	semidirect
20. C₃³×Q₁₆	C₂²	semidirect
21. C₃³×SD₁₆	C₂²	semidirect
22. C₃³×M₁₆	C₂²	semidirect
25. C₂²×C₃²×Q₈	C₂×C₃	semidirect
26. C₃²×#137	C₂×C₃	direct (N × H)
27. C₂×C₃²×Q₁₆	C₂×C₃	semidirect
28. C₂×C₃²×SD₁₆	C₂×C₃	semidirect
29. C₂×C₃²×M₁₆	C₂×C₃	semidirect
49. C₂×C₃×#137	C₃²	direct (N × H)
56. C₂×C₃²×Q₈	C₂²×C₃	semidirect
62. C₃²×Q₁₆	C₂²×C₃	semidirect
63. C₃²×SD₁₆	C₂²×C₃	semidirect
64. C₃²×M₁₆	C₂²×C₃	semidirect
77. C₂²×C₃×Q₈	C₂×C₃²	semidirect
78. C₃×#137	C₂×C₃²	direct (N × H)
79. C₂×C₃×Q₁₆	C₂×C₃²	semidirect
80. C₂×C₃×SD₁₆	C₂×C₃²	semidirect
81. C₂×C₃×M₁₆	C₂×C₃²	semidirect
101. C₂×#137	C₃³	direct (N × H)
104. C₂×C₃³	#137	direct (N × H)
112. C₂×C₃×Q₈	C₂²×C₃²	semidirect
118. C₃×Q₁₆	C₂²×C₃²	semidirect
119. C₃×SD₁₆	C₂²×C₃²	semidirect
120. C₃×M₁₆	C₂²×C₃²	semidirect
133. C₂²×Q₈	C₂×C₃³	semidirect
134. #137	C₂×C₃³	direct (N × H)
135. C₂×Q₁₆	C₂×C₃³	semidirect
136. C₂×SD₁₆	C₂×C₃³	semidirect
137. C₂×M₁₆	C₂×C₃³	semidirect
138. C₃³	C₂×#137	direct (N × H)
160. C₂×C₃²	C₃×#137	direct (N × H)
168. C₂×Q₈	C₂²×C₃³	semidirect
174. Q₁₆	C₂²×C₃³	semidirect
175. SD₁₆	C₂²×C₃³	semidirect
176. M₁₆	C₂²×C₃³	semidirect
188. C₃²	C₂×C₃×#137	direct (N × H)
210. C₂×C₃	C₃²×#137	direct (N × H)
222. C₃	C₂×C₃²×#137	direct (N × H)
225. C₂	C₃³×#137	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
C₂²×C₃³×Q₈	2
C₃³×#137	2
C₂×C₃³×Q₁₆	2
C₂×C₃³×SD₁₆	2
C₂×C₃³×M₁₆	2
C₂×C₃²×#137	3

Subgroups — C2×C33×#137 #13180029

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃³×#137 #13180029