Subgroups — C₂×C₃×#901 #10595835

Order 576.

570 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#235	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#208	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#901	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×ΓL(1,𝔽₂₅)	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×D₃×C₈	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#131	—	3	1	no	maximal
8	C₂×#901	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
10	C₂×C₃×#67	C₄	1	2	yes	—
11	C₃×#235	C₄	1	4	yes	—
12	C₃×#235	C₂²	1	4	yes	—
13	C₂×C₃×D₁₂	C₄	1	1	yes	characteristic
14	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
15	C₂×C₃×Dic₆	C₄	1	1	yes	characteristic
16	C₂×C₃²×C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃²×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃²×C₈	C₂²	1	4	yes	complemented
19	C₃×#208	C₂²	1	4	yes	complemented
20	C₂×C₃×C₃⋊C₈	C₂²	1	2	yes	—
21	C₃×ΓL(1,𝔽₂₅)	C₂²	1	8	yes	complemented
22	C₃×D₃×C₈	C₂²	1	8	yes	complemented
23	C₂×C₃×#41	—	3	1	no	—
24	C₂×#235	C₂×C₃	1	1	yes	characteristic
25	C₃×#131	—	3	8	no	—
26	C₂×C₃×M₁₆	—	3	1	no	—
27	C₂×C₃×M₁₆	—	3	2	no	—
28	C₂²×C₃×C₈	—	2	1	no	—
29	C₂²×C₃×C₈	—	3	2	no	—
30	C₂²×C₃×C₈	D₃	1	1	yes	characteristic, complemented
31	C₂²×C₃×C₈	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂×#208	C₂×C₃	1	1	yes	characteristic, complemented
33	#901	C₂×C₃	1	8	yes	direct product factor
34	C₂×ΓL(1,𝔽₂₅)	C₂×C₃	1	2	yes	complemented
35	C₂×D₃×C₈	C₂×C₃	1	2	yes	complemented
36	C₂³×C₃²	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₃×D₃	C₂×C₄	1	2	yes	—
38	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃²×C₄	C₂×C₄	1	2	yes	—
40	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
41	C₂×C₃²×C₄	C₂³	1	2	yes	—
42	C₃×#67	C₂×C₄	1	8	yes	—
43	C₂×C₃×Dic₃	C₂×C₄	1	2	yes	—
44	C₃×D₁₂	C₂×C₄	1	4	yes	—
45	C₃×C₄×D₃	C₂×C₄	1	4	yes	—
46	C₃×C₄×D₃	C₂³	1	4	yes	—
47	C₃×Dic₆	C₂×C₄	1	4	yes	—
48	C₃²×C₈	C₂³	1	2	yes	—
49	C₃²×C₈	C₂³	1	2	yes	—
50	C₃×C₃⋊C₈	C₂³	1	4	yes	—
51	C₂×#131	—	3	1	no	Sylow
52	C₃×#41	—	3	4	no	—
53	C₃×#41	—	3	4	no	—
54	C₂×C₃×Q₈	—	3	1	no	—
55	C₂×C₃×D₄	—	3	2	no	—
56	C₂×C₃×D₄	—	3	1	no	—
57	C₂²×C₃×C₄	—	2	1	no	—
58	C₂²×C₃×C₄	—	3	2	no	—
59	C₂²×C₃×C₄	C₂×D₃	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
61	C₂×#67	C₃×C₄	1	2	yes	—
62	#235	C₃×C₄	1	4	yes	—
63	#235	C₂²×C₃	1	4	yes	—
64	C₂×D₁₂	C₃×C₄	1	1	yes	characteristic
65	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
66	C₂×Dic₆	C₃×C₄	1	1	yes	characteristic
67	C₃×M₁₆	—	3	4	no	—
68	C₃×M₁₆	—	3	8	no	—
69	C₂×C₃×C₈	—	2	1	no	—
70	C₂×C₃×C₈	—	2	1	no	—
71	C₂×C₃×C₈	—	2	4	no	—
72	C₂×C₃×C₈	—	3	2	no	—
73	C₂×C₃×C₈	—	3	8	no	—
74	C₂×C₃×C₈	C₂×D₃	1	1	yes	characteristic
75	C₂×C₃×C₈	C₂×D₃	1	1	yes	characteristic
76	C₂×C₃×C₈	C₂×D₃	1	4	yes	complemented
77	C₂×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
78	C₂×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₃×C₈	C₂²×C₃	1	4	yes	complemented
80	#208	C₂²×C₃	1	4	yes	complemented
81	C₂×C₃⋊C₈	C₂²×C₃	1	2	yes	—
82	ΓL(1,𝔽₂₅)	C₂²×C₃	1	8	yes	complemented
83	D₃×C₈	C₂²×C₃	1	8	yes	complemented
84	C₂²×C₃²	—	2	2	no	—
85	C₂²×C₃²	C₂²×C₄	1	2	yes	—
86	C₂²×C₃²	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×D₃	—	2	4	no	—
88	C₂×C₃×D₃	C₂²×C₄	1	4	yes	—
89	C₃²×C₄	C₂²×C₄	1	2	yes	—
90	C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
91	C₃²×C₄	C₂⁴	1	1	yes	characteristic
92	C₃×Dic₃	C₂²×C₄	1	4	yes	—
93	C₂×#41	—	3	1	no	—
94	#131	—	3	8	no	—
95	C₂×M₁₆	—	3	1	no	—
96	C₂×M₁₆	—	3	2	no	—
97	C₂²×C₈	—	3	2	no	—
98	C₂²×C₈	C₃×D₃	1	1	yes	characteristic, complemented
99	C₂³×C₃	—	2	1	no	—
100	C₂³×C₃	—	3	2	no	—
101	C₂³×C₃	C₄×D₃	1	1	yes	characteristic
102	C₂³×C₃	C₂×C₃×C₄	1	1	yes	characteristic
103	C₂²×D₃	C₂×C₃×C₄	1	2	yes	—
104	C₃×Q₈	—	3	4	no	—
105	C₃×D₄	—	3	8	no	—
106	C₃×D₄	—	3	4	no	—
107	C₂×C₃×C₄	—	2	1	no	—
108	C₂×C₃×C₄	—	2	2	no	—
109	C₂×C₃×C₄	—	2	1	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	—	3	2	no	—
112	C₂×C₃×C₄	—	3	4	no	—
113	C₂×C₃×C₄	—	3	4	no	—
114	C₂×C₃×C₄	C₄×D₃	1	1	yes	characteristic
115	C₂×C₃×C₄	C₄×D₃	1	2	yes	—
116	C₂×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
117	C₂×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
118	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
120	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
122	#67	C₂×C₃×C₄	1	8	yes	—
123	C₂×Dic₃	C₂×C₃×C₄	1	2	yes	—
124	D₁₂	C₂×C₃×C₄	1	4	yes	—
125	C₄×D₃	C₂×C₃×C₄	1	4	yes	—
126	C₄×D₃	C₂³×C₃	1	4	yes	—
127	Dic₆	C₂×C₃×C₄	1	4	yes	—
128	C₃×C₈	—	2	2	no	—
129	C₃×C₈	—	2	2	no	—
130	C₃×C₈	—	3	4	no	—
131	C₃×C₈	C₂²×D₃	1	2	yes	—
132	C₃×C₈	C₂²×D₃	1	2	yes	—
133	C₃×C₈	C₂³×C₃	1	2	yes	—
134	C₃×C₈	C₂³×C₃	1	2	yes	—
135	C₃⋊C₈	C₂³×C₃	1	4	yes	—
136	C₂×C₃²	—	2	2	no	—
137	C₂×C₃²	#131	1	2	yes	complemented
138	C₂×C₃²	C₂³×C₄	1	1	yes	characteristic
139	C₃×D₃	—	2	4	no	—
140	#41	—	3	4	no	—
141	#41	—	3	4	no	—
142	C₂×Q₈	—	3	1	no	—
143	C₂×D₄	—	3	2	no	—
144	C₂×D₄	—	3	1	no	—
145	C₂²×C₄	—	3	2	no	—
146	C₂²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
147	M₁₆	—	3	4	no	—
148	M₁₆	—	3	8	no	—
149	C₂×C₈	—	3	2	no	—
150	C₂×C₈	—	3	8	no	—
151	C₂×C₈	C₂×C₃×D₃	1	1	yes	characteristic
152	C₂×C₈	C₂×C₃×D₃	1	1	yes	characteristic
153	C₂×C₈	C₂×C₃×D₃	1	4	yes	complemented
154	C₂²×C₃	—	2	4	no	—
155	C₂²×C₃	—	2	2	no	—
156	C₂²×C₃	—	2	2	no	—
157	C₂²×C₃	—	2	2	no	—
158	C₂²×C₃	—	2	1	no	—
159	C₂²×C₃	—	3	4	no	—
160	C₂²×C₃	—	6	4	no	—
161	C₂²×C₃	C₂×C₄×D₃	1	2	yes	—
162	C₂²×C₃	C₂×C₄×D₃	1	1	yes	characteristic
163	C₂²×C₃	C₂²×C₃×C₄	1	2	yes	—
164	C₂²×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
165	C₂×D₃	—	2	4	no	—
166	C₂×D₃	C₂²×C₃×C₄	1	4	yes	—
167	C₃×C₄	—	2	1	no	—
168	C₃×C₄	—	2	1	no	—
169	C₃×C₄	—	2	2	no	—
170	C₃×C₄	—	3	4	no	—
171	C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
172	C₃×C₄	C₂×C₄×D₃	1	2	yes	—
173	C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
174	C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
175	C₃×C₄	C₂³×D₃	1	1	yes	characteristic
176	C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
177	Dic₃	C₂²×C₃×C₄	1	4	yes	—
178	C₃²	C₂×#131	1	1	yes	characteristic, complemented, Sylow
179	C₂³	—	3	2	no	—
180	C₂³	C₃×C₄×D₃	1	1	yes	characteristic
181	Q₈	—	3	4	no	—
182	D₄	—	3	8	no	—
183	D₄	—	3	4	no	—
184	C₂×C₄	—	3	2	no	—
185	C₂×C₄	—	3	4	no	—
186	C₂×C₄	—	3	4	no	—
187	C₂×C₄	C₃×C₄×D₃	1	1	yes	characteristic
188	C₂×C₄	C₃×C₄×D₃	1	2	yes	—
189	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
190	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
191	C₈	—	3	4	no	—
192	C₈	C₂²×C₃×D₃	1	2	yes	—
193	C₈	C₂²×C₃×D₃	1	2	yes	—
194	C₂×C₃	—	2	4	no	—
195	C₂×C₃	—	2	2	no	—
196	C₂×C₃	—	2	2	no	—
197	C₂×C₃	—	2	2	no	—
198	C₂×C₃	—	2	1	no	—
199	C₂×C₃	—	6	4	no	—
200	C₂×C₃	#901	1	2	yes	direct product factor
201	C₂×C₃	C₃×#131	1	2	yes	complemented
202	C₂×C₃	C₂²×C₄×D₃	1	1	yes	characteristic
203	C₂×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
204	D₃	—	2	4	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	3	4	no	—
207	C₂²	—	6	4	no	—
208	C₂²	C₂×C₃×C₄×D₃	1	2	yes	—
209	C₂²	C₂×C₃×C₄×D₃	1	1	yes	characteristic
210	C₄	—	3	4	no	—
211	C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
212	C₄	C₂×C₃×C₄×D₃	1	2	yes	—
213	C₄	C₂³×C₃×D₃	1	1	yes	characteristic
214	C₃	—	2	1	no	—
215	C₃	C₂×#901	1	1	yes	characteristic, direct product factor
216	C₃	C₂×C₃×#131	1	1	yes	characteristic, complemented
217	C₂	—	2	2	no	—
218	C₂	—	6	4	no	—
219	C₂	C₃×#901	1	2	yes	direct product factor
220	C₂	C₂²×C₃×C₄×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂×D₃
commutator	C₂×C₃	C₂³×C₃×C₄
Frattini	C₄	C₂³×C₃×D₃
Fitting	C₂²×C₃²×C₈	C₂
socle	C₂²×C₃²	C₂²×C₄
prime radical	C₂×C₃×#235	C₂
cosocle	C₃×C₄	C₂⁴×C₃
hypercenter	C₂²×C₃×C₈	D₃
hypocenter	C₃	C₂×C₃×#131
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃²×C₈	C₂	semidirect
3. C₂×C₃×#208	C₂	semidirect
4. C₃×#901	C₂	direct (N × H)
5. C₂×C₃×ΓL(1,𝔽₂₅)	C₂	semidirect
6. C₂×C₃×D₃×C₈	C₂	semidirect
8. C₂×#901	C₃	direct (N × H)
18. C₂×C₃²×C₈	C₂²	semidirect
19. C₃×#208	C₂²	semidirect
21. C₃×ΓL(1,𝔽₂₅)	C₂²	semidirect
22. C₃×D₃×C₈	C₂²	semidirect
30. C₂²×C₃×C₈	D₃	semidirect
31. C₂²×C₃×C₈	C₂×C₃	semidirect
32. C₂×#208	C₂×C₃	semidirect
33. #901	C₂×C₃	direct (N × H)
34. C₂×ΓL(1,𝔽₂₅)	C₂×C₃	semidirect
35. C₂×D₃×C₈	C₂×C₃	semidirect
76. C₂×C₃×C₈	C₂×D₃	semidirect
79. C₂×C₃×C₈	C₂²×C₃	semidirect
80. #208	C₂²×C₃	semidirect
82. ΓL(1,𝔽₂₅)	C₂²×C₃	semidirect
83. D₃×C₈	C₂²×C₃	semidirect
98. C₂²×C₈	C₃×D₃	semidirect
137. C₂×C₃²	#131	semidirect
153. C₂×C₈	C₂×C₃×D₃	semidirect
178. C₃²	C₂×#131	semidirect
200. C₂×C₃	#901	direct (N × H)
201. C₂×C₃	C₃×#131	semidirect
215. C₃	C₂×#901	direct (N × H)
216. C₃	C₂×C₃×#131	semidirect
219. C₂	C₃×#901	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×C₃×#235	2
C₂²×C₃²×C₈	2
C₂×C₃×#208	2
C₃×#901	2
C₂×C₃×ΓL(1,𝔽₂₅)	2
C₂×C₃×D₃×C₈	2
C₂×C₃×#131	3
C₂×#901	3

Subgroups — C2×C3×#901 #10595835

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#901 #10595835