Subgroups — C₃³×#535 #13179990

Order 1728.

4676 conjugacy classes of subgroups.

The subgroup lattice has 232 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, complemented
2	C₂²×C₃³×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃³×#127	C₂	1	2	yes	maximal, complemented
4	C₃³×#124	C₂	1	2	yes	maximal, complemented
5	C₃³×#121	C₂	1	4	yes	maximal, complemented
6	C₃³×#120	C₂	1	4	yes	maximal, complemented
7	C₃³×#117	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#535	C₃	1	13	yes	maximal, direct product factor
9	C₂⁴×C₃³	C₂²	1	2	yes	—
10	C₃³×#41	—	2	4	no	—
11	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₃³×D₄	—	2	4	no	—
13	C₂×C₃³×D₄	C₂²	1	2	yes	—
14	C₂×C₃³×D₄	C₂²	1	2	yes	—
15	C₂×C₃³×D₄	C₂²	1	2	yes	—
16	C₂×C₃³×D₄	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₃³×D₄	C₂²	1	8	yes	complemented
18	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
19	C₂²×C₃³×C₄	C₂²	1	2	yes	complemented
20	C₃³×C₄⋊C₄	C₂²	1	2	yes	complemented
21	C₃³×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₃³×C₂²⋊C₄	C₂²	1	8	yes	complemented
23	C₃³×C₄²	C₂²	1	2	yes	complemented
24	C₂×C₃²×#41	C₂×C₃	1	13	yes	complemented
25	C₂²×C₃²×D₄	C₂×C₃	1	13	yes	complemented
26	C₃²×#127	C₂×C₃	1	26	yes	complemented
27	C₃²×#124	C₂×C₃	1	26	yes	complemented
28	C₃²×#121	C₂×C₃	1	52	yes	complemented
29	C₃²×#120	C₂×C₃	1	52	yes	complemented
30	C₃²×#117	C₂×C₃	1	13	yes	complemented
31	C₂³×C₃³	—	2	4	no	—
32	C₂³×C₃³	—	4	4	no	—
33	C₂³×C₃³	C₂³	1	4	yes	—
34	C₂³×C₃³	C₂³	1	2	yes	—
35	C₂³×C₃³	C₂³	1	1	yes	characteristic
36	C₃³×Q₈	—	2	2	no	—
37	C₃³×D₄	—	2	4	no	—
38	C₃³×D₄	—	2	4	no	—
39	C₃³×D₄	—	2	4	no	—
40	C₃³×D₄	—	2	2	no	—
41	C₃³×D₄	—	4	8	no	—
42	C₂×C₃³×C₄	—	2	4	no	—
43	C₂×C₃³×C₄	D₄	1	4	yes	complemented
44	C₂×C₃³×C₄	C₂³	1	4	yes	—
45	C₂×C₃³×C₄	C₂³	1	2	yes	—
46	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
47	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
48	C₃×#535	C₃²	1	13	yes	direct product factor
49	C₂⁴×C₃²	C₂²×C₃	1	26	yes	—
50	C₃²×#41	—	2	52	no	—
51	C₂×C₃²×Q₈	C₂²×C₃	1	13	yes	complemented
52	C₂×C₃²×D₄	—	2	52	no	—
53	C₂×C₃²×D₄	C₂²×C₃	1	26	yes	—
54	C₂×C₃²×D₄	C₂²×C₃	1	26	yes	—
55	C₂×C₃²×D₄	C₂²×C₃	1	26	yes	—
56	C₂×C₃²×D₄	C₂²×C₃	1	13	yes	complemented
57	C₂×C₃²×D₄	C₂²×C₃	1	104	yes	complemented
58	C₂²×C₃²×C₄	C₂²×C₃	1	13	yes	—
59	C₂²×C₃²×C₄	C₂²×C₃	1	26	yes	complemented
60	C₃²×C₄⋊C₄	C₂²×C₃	1	26	yes	complemented
61	C₃²×C₂²⋊C₄	C₂²×C₃	1	26	yes	—
62	C₃²×C₂²⋊C₄	C₂²×C₃	1	104	yes	complemented
63	C₃²×C₄²	C₂²×C₃	1	26	yes	complemented
64	C₂²×C₃³	—	2	2	no	—
65	C₂²×C₃³	—	2	2	no	—
66	C₂²×C₃³	—	2	4	no	—
67	C₂²×C₃³	—	2	4	no	—
68	C₂²×C₃³	—	2	8	no	—
69	C₂²×C₃³	—	4	8	no	—
70	C₂²×C₃³	C₂×D₄	1	2	yes	—
71	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
72	C₃³×C₄	—	2	2	no	—
73	C₃³×C₄	—	2	4	no	—
74	C₃³×C₄	C₂×D₄	1	2	yes	—
75	C₃³×C₄	C₂×D₄	1	2	yes	—
76	C₂×C₃×#41	C₂×C₃²	1	13	yes	complemented
77	C₂²×C₃×D₄	C₂×C₃²	1	13	yes	complemented
78	C₃×#127	C₂×C₃²	1	26	yes	complemented
79	C₃×#124	C₂×C₃²	1	26	yes	complemented
80	C₃×#121	C₂×C₃²	1	52	yes	complemented
81	C₃×#120	C₂×C₃²	1	52	yes	complemented
82	C₃×#117	C₂×C₃²	1	13	yes	complemented
83	C₂³×C₃²	—	2	52	no	—
84	C₂³×C₃²	—	4	52	no	—
85	C₂³×C₃²	C₂³×C₃	1	52	yes	—
86	C₂³×C₃²	C₂³×C₃	1	26	yes	—
87	C₂³×C₃²	C₂³×C₃	1	13	yes	—
88	C₃²×Q₈	—	2	26	no	—
89	C₃²×D₄	—	2	52	no	—
90	C₃²×D₄	—	2	52	no	—
91	C₃²×D₄	—	2	52	no	—
92	C₃²×D₄	—	2	26	no	—
93	C₃²×D₄	—	4	104	no	—
94	C₂×C₃²×C₄	—	2	52	no	—
95	C₂×C₃²×C₄	C₃×D₄	1	52	yes	complemented
96	C₂×C₃²×C₄	C₂³×C₃	1	52	yes	—
97	C₂×C₃²×C₄	C₂³×C₃	1	26	yes	—
98	C₂×C₃²×C₄	C₂³×C₃	1	13	yes	—
99	C₂×C₃²×C₄	C₂³×C₃	1	13	yes	—
100	#535	C₃³	1	1	yes	characteristic, direct product factor, Sylow
101	C₂×C₃³	—	2	2	no	—
102	C₂×C₃³	—	4	2	no	—
103	C₂×C₃³	—	4	4	no	—
104	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₃³	H(5,𝔽₂)	1	2	yes	—
106	C₂⁴×C₃	C₂²×C₃²	1	26	yes	—
107	C₃×#41	—	2	52	no	—
108	C₂×C₃×Q₈	C₂²×C₃²	1	13	yes	complemented
109	C₂×C₃×D₄	—	2	52	no	—
110	C₂×C₃×D₄	C₂²×C₃²	1	26	yes	—
111	C₂×C₃×D₄	C₂²×C₃²	1	26	yes	—
112	C₂×C₃×D₄	C₂²×C₃²	1	26	yes	—
113	C₂×C₃×D₄	C₂²×C₃²	1	13	yes	complemented
114	C₂×C₃×D₄	C₂²×C₃²	1	104	yes	complemented
115	C₂²×C₃×C₄	C₂²×C₃²	1	13	yes	—
116	C₂²×C₃×C₄	C₂²×C₃²	1	26	yes	complemented
117	C₃×C₄⋊C₄	C₂²×C₃²	1	26	yes	complemented
118	C₃×C₂²⋊C₄	C₂²×C₃²	1	26	yes	—
119	C₃×C₂²⋊C₄	C₂²×C₃²	1	104	yes	complemented
120	C₃×C₄²	C₂²×C₃²	1	26	yes	complemented
121	C₂²×C₃²	—	2	26	no	—
122	C₂²×C₃²	—	2	26	no	—
123	C₂²×C₃²	—	2	52	no	—
124	C₂²×C₃²	—	2	52	no	—
125	C₂²×C₃²	—	2	104	no	—
126	C₂²×C₃²	—	4	104	no	—
127	C₂²×C₃²	C₂×C₃×D₄	1	26	yes	—
128	C₂²×C₃²	C₂⁴×C₃	1	13	yes	—
129	C₃²×C₄	—	2	26	no	—
130	C₃²×C₄	—	2	52	no	—
131	C₃²×C₄	C₂×C₃×D₄	1	26	yes	—
132	C₃²×C₄	C₂×C₃×D₄	1	26	yes	—
133	C₂×#41	C₂×C₃³	1	1	yes	characteristic, complemented
134	C₂²×D₄	C₂×C₃³	1	1	yes	characteristic, complemented
135	#127	C₂×C₃³	1	2	yes	complemented
136	#124	C₂×C₃³	1	2	yes	complemented
137	#121	C₂×C₃³	1	4	yes	complemented
138	#120	C₂×C₃³	1	4	yes	complemented
139	#117	C₂×C₃³	1	1	yes	characteristic, complemented
140	C₃³	#535	1	1	yes	characteristic, direct product factor, Sylow
141	C₂³×C₃	—	2	52	no	—
142	C₂³×C₃	—	4	52	no	—
143	C₂³×C₃	C₂³×C₃²	1	52	yes	—
144	C₂³×C₃	C₂³×C₃²	1	26	yes	—
145	C₂³×C₃	C₂³×C₃²	1	13	yes	—
146	C₃×Q₈	—	2	26	no	—
147	C₃×D₄	—	2	52	no	—
148	C₃×D₄	—	2	52	no	—
149	C₃×D₄	—	2	52	no	—
150	C₃×D₄	—	2	26	no	—
151	C₃×D₄	—	4	104	no	—
152	C₂×C₃×C₄	—	2	52	no	—
153	C₂×C₃×C₄	C₃²×D₄	1	52	yes	complemented
154	C₂×C₃×C₄	C₂³×C₃²	1	52	yes	—
155	C₂×C₃×C₄	C₂³×C₃²	1	26	yes	—
156	C₂×C₃×C₄	C₂³×C₃²	1	13	yes	—
157	C₂×C₃×C₄	C₂³×C₃²	1	13	yes	—
158	C₂×C₃²	—	2	26	no	—
159	C₂×C₃²	—	4	26	no	—
160	C₂×C₃²	—	4	52	no	—
161	C₂×C₃²	C₂²×C₃×D₄	1	13	yes	—
162	C₂×C₃²	C₃×H(5,𝔽₂)	1	26	yes	—
163	C₂⁴	C₂²×C₃³	1	2	yes	—
164	#41	—	2	4	no	—
165	C₂×Q₈	C₂²×C₃³	1	1	yes	characteristic, complemented
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	C₂²×C₃³	1	2	yes	—
168	C₂×D₄	C₂²×C₃³	1	2	yes	—
169	C₂×D₄	C₂²×C₃³	1	2	yes	—
170	C₂×D₄	C₂²×C₃³	1	1	yes	characteristic, complemented
171	C₂×D₄	C₂²×C₃³	1	8	yes	complemented
172	C₂²×C₄	C₂²×C₃³	1	1	yes	characteristic
173	C₂²×C₄	C₂²×C₃³	1	2	yes	complemented
174	C₄⋊C₄	C₂²×C₃³	1	2	yes	complemented
175	C₂²⋊C₄	C₂²×C₃³	1	2	yes	—
176	C₂²⋊C₄	C₂²×C₃³	1	8	yes	complemented
177	C₄²	C₂²×C₃³	1	2	yes	complemented
178	C₂²×C₃	—	2	26	no	—
179	C₂²×C₃	—	2	26	no	—
180	C₂²×C₃	—	2	52	no	—
181	C₂²×C₃	—	2	52	no	—
182	C₂²×C₃	—	2	104	no	—
183	C₂²×C₃	—	4	104	no	—
184	C₂²×C₃	C₂×C₃²×D₄	1	26	yes	—
185	C₂²×C₃	C₂⁴×C₃²	1	13	yes	—
186	C₃×C₄	—	2	26	no	—
187	C₃×C₄	—	2	52	no	—
188	C₃×C₄	C₂×C₃²×D₄	1	26	yes	—
189	C₃×C₄	C₂×C₃²×D₄	1	26	yes	—
190	C₃²	C₃×#535	1	13	yes	direct product factor
191	C₂³	—	2	4	no	—
192	C₂³	—	4	4	no	—
193	C₂³	C₂³×C₃³	1	4	yes	—
194	C₂³	C₂³×C₃³	1	2	yes	—
195	C₂³	C₂³×C₃³	1	1	yes	characteristic
196	Q₈	—	2	2	no	—
197	D₄	—	2	4	no	—
198	D₄	—	2	4	no	—
199	D₄	—	2	4	no	—
200	D₄	—	2	2	no	—
201	D₄	—	4	8	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	C₃³×D₄	1	4	yes	complemented
204	C₂×C₄	C₂³×C₃³	1	4	yes	—
205	C₂×C₄	C₂³×C₃³	1	2	yes	—
206	C₂×C₄	C₂³×C₃³	1	1	yes	characteristic
207	C₂×C₄	C₂³×C₃³	1	1	yes	characteristic
208	C₂×C₃	—	2	26	no	—
209	C₂×C₃	—	4	52	no	—
210	C₂×C₃	—	4	26	no	—
211	C₂×C₃	C₂²×C₃²×D₄	1	13	yes	—
212	C₂×C₃	C₃²×H(5,𝔽₂)	1	26	yes	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	4	no	—
217	C₂²	—	2	8	no	—
218	C₂²	—	4	8	no	—
219	C₂²	C₂×C₃³×D₄	1	2	yes	—
220	C₂²	C₂⁴×C₃³	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	4	no	—
223	C₄	C₂×C₃³×D₄	1	2	yes	—
224	C₄	C₂×C₃³×D₄	1	2	yes	—
225	C₃	C₃²×#535	1	13	yes	direct product factor
226	C₂	—	2	2	no	—
227	C₂	—	4	4	no	—
228	C₂	—	4	2	no	—
229	C₂	C₂²×C₃³×D₄	1	1	yes	characteristic
230	C₂	C₃³×H(5,𝔽₂)	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃³×#41	C₂	semidirect
2. C₂²×C₃³×D₄	C₂	semidirect
3. C₃³×#127	C₂	semidirect
4. C₃³×#124	C₂	semidirect
5. C₃³×#121	C₂	semidirect
6. C₃³×#120	C₂	semidirect
7. C₃³×#117	C₂	semidirect
8. C₃²×#535	C₃	direct (N × H)
11. C₂×C₃³×Q₈	C₂²	semidirect
16. C₂×C₃³×D₄	C₂²	semidirect
17. C₂×C₃³×D₄	C₂²	semidirect
19. C₂²×C₃³×C₄	C₂²	semidirect
20. C₃³×C₄⋊C₄	C₂²	semidirect
22. C₃³×C₂²⋊C₄	C₂²	semidirect
23. C₃³×C₄²	C₂²	semidirect
24. C₂×C₃²×#41	C₂×C₃	semidirect
25. C₂²×C₃²×D₄	C₂×C₃	semidirect
26. C₃²×#127	C₂×C₃	semidirect
27. C₃²×#124	C₂×C₃	semidirect
28. C₃²×#121	C₂×C₃	semidirect
29. C₃²×#120	C₂×C₃	semidirect
30. C₃²×#117	C₂×C₃	semidirect
43. C₂×C₃³×C₄	D₄	semidirect
48. C₃×#535	C₃²	direct (N × H)
51. C₂×C₃²×Q₈	C₂²×C₃	semidirect
56. C₂×C₃²×D₄	C₂²×C₃	semidirect
57. C₂×C₃²×D₄	C₂²×C₃	semidirect
59. C₂²×C₃²×C₄	C₂²×C₃	semidirect
60. C₃²×C₄⋊C₄	C₂²×C₃	semidirect
62. C₃²×C₂²⋊C₄	C₂²×C₃	semidirect
63. C₃²×C₄²	C₂²×C₃	semidirect
76. C₂×C₃×#41	C₂×C₃²	semidirect
77. C₂²×C₃×D₄	C₂×C₃²	semidirect
78. C₃×#127	C₂×C₃²	semidirect
79. C₃×#124	C₂×C₃²	semidirect
80. C₃×#121	C₂×C₃²	semidirect
81. C₃×#120	C₂×C₃²	semidirect
82. C₃×#117	C₂×C₃²	semidirect
95. C₂×C₃²×C₄	C₃×D₄	semidirect
100. #535	C₃³	direct (N × H)
108. C₂×C₃×Q₈	C₂²×C₃²	semidirect
113. C₂×C₃×D₄	C₂²×C₃²	semidirect
114. C₂×C₃×D₄	C₂²×C₃²	semidirect
116. C₂²×C₃×C₄	C₂²×C₃²	semidirect
117. C₃×C₄⋊C₄	C₂²×C₃²	semidirect
119. C₃×C₂²⋊C₄	C₂²×C₃²	semidirect
120. C₃×C₄²	C₂²×C₃²	semidirect
133. C₂×#41	C₂×C₃³	semidirect
134. C₂²×D₄	C₂×C₃³	semidirect
135. #127	C₂×C₃³	semidirect
136. #124	C₂×C₃³	semidirect
137. #121	C₂×C₃³	semidirect
138. #120	C₂×C₃³	semidirect
139. #117	C₂×C₃³	semidirect
140. C₃³	#535	direct (N × H)
153. C₂×C₃×C₄	C₃²×D₄	semidirect
165. C₂×Q₈	C₂²×C₃³	semidirect
170. C₂×D₄	C₂²×C₃³	semidirect
171. C₂×D₄	C₂²×C₃³	semidirect
173. C₂²×C₄	C₂²×C₃³	semidirect
174. C₄⋊C₄	C₂²×C₃³	semidirect
176. C₂²⋊C₄	C₂²×C₃³	semidirect
177. C₄²	C₂²×C₃³	semidirect
190. C₃²	C₃×#535	direct (N × H)
203. C₂×C₄	C₃³×D₄	semidirect
225. C₃	C₃²×#535	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×D₄	2
C₃³×#127	2
C₃³×#124	2
C₃³×#121	2
C₃³×#120	2
C₃³×#117	2
C₃²×#535	3

Subgroups — C33×#535 #13179990

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×#535 #13179990