Subgroups — C₂×C₃×#5067 #11873574

Order 1152.

1089 conjugacy classes of subgroups.

The subgroup lattice has 289 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#874	C₂	1	1	yes	maximal, characteristic
3	C₃×#5067	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×#809	C₂	1	4	yes	maximal, complemented
5	C₂×C₃×#803	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×#490	—	3	1	no	maximal
7	C₂×#5067	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃×D₁₂	—	2	2	no	—
9	C₂²×C₃×D₁₂	C₂²	1	2	yes	—
10	C₃²×#125	—	2	2	no	—
11	C₃²×#125	C₂²	1	4	yes	complemented
12	C₂×C₃²×C₄⋊C₄	—	2	2	no	—
13	C₂×C₃²×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
16	C₃×#874	—	2	2	no	—
17	C₃×#874	C₂²	1	4	yes	—
18	C₃×#809	C₂²	1	16	yes	complemented
19	C₃×#803	C₂²	1	4	yes	complemented
20	C₂×C₃×#127	—	3	1	no	—
21	C₂×C₃×#125	—	2	1	no	—
22	C₂×C₃×#125	D₃	1	1	yes	characteristic, complemented
23	C₂×C₃×#125	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×#874	C₂×C₃	1	1	yes	characteristic
25	C₃×#490	—	3	8	no	—
26	C₂×C₃×#102	—	3	4	no	—
27	C₂×C₃×#97	—	3	1	no	—
28	#5067	C₂×C₃	1	8	yes	direct product factor
29	C₂×#809	C₂×C₃	1	4	yes	complemented
30	C₂×#803	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂³×C₃×D₃	—	2	2	no	—
32	C₂²×C₃²×C₄	—	2	2	no	—
33	C₂²×C₃²×C₄	D₄	1	2	yes	—
34	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
35	C₂×C₃×D₁₂	—	2	8	no	—
36	C₂×C₃×D₁₂	—	2	8	no	—
37	C₂×C₃×D₁₂	—	2	2	no	—
38	C₂×C₃×D₁₂	—	4	8	no	—
39	C₂×C₃×D₁₂	C₂³	1	4	yes	—
40	C₃²×C₄⋊C₄	—	2	8	no	—
41	C₃²×C₄⋊C₄	—	2	2	no	—
42	C₃²×C₄⋊C₄	C₂³	1	4	yes	—
43	C₃²×C₄²	D₄	1	2	yes	—
44	C₃²×C₄²	C₂³	1	2	yes	—
45	C₂×C₃×C₃⋊C₈	—	2	4	no	—
46	C₂×C₃×C₃⋊C₈	C₂³	1	4	yes	—
47	C₂×#490	—	3	1	no	Sylow
48	C₂²×C₃×D₄	—	3	2	no	—
49	C₂²×C₃×D₄	—	6	2	no	—
50	C₂²×D₁₂	—	2	2	no	—
51	C₂²×D₁₂	C₂²×C₃	1	2	yes	—
52	C₂²×C₃×C₈	—	3	2	no	—
53	C₃×#127	—	3	4	no	—
54	C₃×#127	—	6	2	no	—
55	C₃×#125	—	2	4	no	—
56	C₃×#125	—	2	2	no	—
57	C₃×#125	—	2	2	no	—
58	C₃×#125	—	2	4	no	—
59	C₃×#125	C₂×D₃	1	4	yes	complemented
60	C₃×#125	C₂²×C₃	1	4	yes	complemented
61	C₂×C₃×C₄⋊C₄	—	2	4	no	—
62	C₂×C₃×C₄⋊C₄	—	2	2	no	—
63	C₂×C₃×C₄⋊C₄	—	2	2	no	—
64	C₂×C₃×C₄⋊C₄	—	2	2	no	—
65	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
66	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
67	C₂×C₃×C₄²	—	2	1	no	—
68	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
69	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
70	C₂²×C₃⋊C₈	C₂²×C₃	1	2	yes	—
71	#874	—	2	2	no	—
72	#874	C₂²×C₃	1	4	yes	—
73	C₃×#102	—	3	16	no	—
74	C₃×#97	—	3	4	no	—
75	#809	C₂²×C₃	1	16	yes	complemented
76	#803	C₂²×C₃	1	4	yes	complemented
77	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₃×D₃	—	2	4	no	—
79	C₂²×C₃×D₃	—	4	4	no	—
80	C₂²×C₃×D₃	—	4	8	no	—
81	C₂×C₃²×C₄	—	2	4	no	—
82	C₂×C₃²×C₄	—	2	4	no	—
83	C₂×C₃²×C₄	—	4	4	no	—
84	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
85	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃²×C₄	#41	1	4	yes	—
87	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
88	C₃×D₁₂	—	2	8	no	—
89	C₃×D₁₂	—	4	16	no	—
90	C₃×D₁₂	—	4	4	no	—
91	C₃×C₃⋊C₈	—	2	4	no	—
92	C₂×#127	—	3	1	no	—
93	C₂×#125	C₃×D₃	1	1	yes	characteristic, complemented
94	#490	—	3	8	no	—
95	C₂×#102	—	3	4	no	—
96	C₂×#97	—	3	1	no	—
97	C₂⁴×C₃	—	6	2	no	—
98	C₂³×D₃	—	2	2	no	—
99	C₂×C₃×D₄	—	3	4	no	—
100	C₂×C₃×D₄	—	6	8	no	—
101	C₂×C₃×D₄	—	6	8	no	—
102	C₂×C₃×D₄	—	6	2	no	—
103	C₂×C₃×D₄	—	12	8	no	—
104	C₂²×C₃×C₄	—	2	4	no	—
105	C₂²×C₃×C₄	—	2	2	no	—
106	C₂²×C₃×C₄	—	2	2	no	—
107	C₂²×C₃×C₄	—	2	2	no	—
108	C₂²×C₃×C₄	—	2	1	no	—
109	C₂²×C₃×C₄	#67	1	2	yes	—
110	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
111	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
112	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
113	C₂×D₁₂	—	2	8	no	—
114	C₂×D₁₂	—	2	2	no	—
115	C₂×D₁₂	—	2	8	no	—
116	C₂×D₁₂	—	4	8	no	—
117	C₂×D₁₂	C₂³×C₃	1	4	yes	—
118	C₂×C₃×C₈	—	3	4	no	—
119	C₂×C₃×C₈	—	6	4	no	—
120	C₃×C₄⋊C₄	—	2	16	no	—
121	C₃×C₄⋊C₄	—	2	4	no	—
122	C₃×C₄⋊C₄	—	2	8	no	—
123	C₃×C₄⋊C₄	—	2	2	no	—
124	C₃×C₄⋊C₄	—	2	8	no	—
125	C₃×C₄⋊C₄	—	2	2	no	—
126	C₃×C₄⋊C₄	—	2	4	no	—
127	C₃×C₄⋊C₄	C₂²×D₃	1	4	yes	—
128	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
129	C₃×C₄²	—	2	2	no	—
130	C₃×C₄²	—	2	2	no	—
131	C₃×C₄²	#67	1	2	yes	—
132	C₃×C₄²	C₃×D₄	1	2	yes	—
133	C₃×C₄²	C₂²×D₃	1	2	yes	—
134	C₃×C₄²	C₂³×C₃	1	2	yes	—
135	C₂×C₃⋊C₈	—	2	4	no	—
136	C₂×C₃⋊C₈	C₂³×C₃	1	4	yes	—
137	C₂²×C₃²	#124	1	2	yes	—
138	C₂²×C₃²	AGL(1,ℤ/8)	1	4	yes	—
139	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₃×D₃	—	4	4	no	—
141	C₂×C₃×D₃	—	4	8	no	—
142	C₂×C₃×D₃	—	8	8	no	—
143	C₃²×C₄	—	2	4	no	—
144	C₃²×C₄	—	4	4	no	—
145	C₃²×C₄	#124	1	2	yes	—
146	C₃²×C₄	C₂×#41	1	2	yes	—
147	C₂²×D₄	—	3	2	no	—
148	C₂²×D₄	—	6	2	no	—
149	C₂²×C₈	—	3	2	no	—
150	#127	—	3	4	no	—
151	#127	—	6	2	no	—
152	#125	—	2	2	no	—
153	#125	C₂×C₃×D₃	1	4	yes	complemented
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
156	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
157	#102	—	3	16	no	—
158	#97	—	3	4	no	—
159	C₂³×C₃	—	2	1	no	—
160	C₂³×C₃	—	6	4	no	—
161	C₂³×C₃	—	12	4	no	—
162	C₂³×C₃	—	12	8	no	—
163	C₂³×C₃	C₂×#67	1	1	yes	characteristic
164	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
165	C₂²×D₃	—	2	4	no	—
166	C₂²×D₃	—	4	4	no	—
167	C₂²×D₃	—	4	8	no	—
168	C₃×D₄	—	6	8	no	—
169	C₃×D₄	—	12	16	no	—
170	C₃×D₄	—	12	4	no	—
171	C₂×C₃×C₄	—	2	4	no	—
172	C₂×C₃×C₄	—	2	4	no	—
173	C₂×C₃×C₄	—	2	8	no	—
174	C₂×C₃×C₄	—	2	4	no	—
175	C₂×C₃×C₄	—	2	4	no	—
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	—	2	4	no	—
178	C₂×C₃×C₄	—	2	1	no	—
179	C₂×C₃×C₄	—	2	4	no	—
180	C₂×C₃×C₄	—	4	4	no	—
181	C₂×C₃×C₄	—	4	8	no	—
182	C₂×C₃×C₄	—	4	4	no	—
183	C₂×C₃×C₄	—	4	4	no	—
184	C₂×C₃×C₄	#239	1	4	yes	—
185	C₂×C₃×C₄	C₂×#67	1	4	yes	—
186	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
188	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
189	C₂×C₃×C₄	C₃×#41	1	4	yes	—
190	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
192	D₁₂	—	2	8	no	—
193	D₁₂	—	4	16	no	—
194	D₁₂	—	4	4	no	—
195	C₃×C₈	—	6	4	no	—
196	C₃⋊C₈	—	2	4	no	—
197	C₂×C₃²	#490	1	4	yes	complemented
198	C₂×C₃²	C₂×#124	1	1	yes	characteristic
199	C₂×C₃²	C₂×AGL(1,ℤ/8)	1	2	yes	—
200	C₃×D₃	—	8	4	no	—
201	C₂⁴	—	6	2	no	—
202	C₂×D₄	—	3	4	no	—
203	C₂×D₄	—	6	8	no	—
204	C₂×D₄	—	6	8	no	—
205	C₂×D₄	—	6	2	no	—
206	C₂×D₄	—	12	8	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	C₃×#67	1	2	yes	—
209	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
210	C₂×C₈	—	3	4	no	—
211	C₂×C₈	—	6	4	no	—
212	C₄⋊C₄	—	2	8	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	C₂²×C₃×D₃	1	4	yes	—
215	C₄²	C₃×#67	1	2	yes	—
216	C₄²	C₂²×C₃×D₃	1	2	yes	—
217	C₂²×C₃	—	2	2	no	—
218	C₂²×C₃	—	2	4	no	—
219	C₂²×C₃	—	2	1	no	—
220	C₂²×C₃	—	12	4	no	—
221	C₂²×C₃	—	12	8	no	—
222	C₂²×C₃	—	24	8	no	—
223	C₂²×C₃	#947	1	2	yes	—
224	C₂²×C₃	#949	1	4	yes	—
225	C₂²×C₃	C₃×#124	1	2	yes	—
226	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	4	yes	—
227	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
228	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
229	C₂×D₃	—	4	8	no	—
230	C₂×D₃	—	4	4	no	—
231	C₂×D₃	—	8	8	no	—
232	C₃×C₄	—	2	4	no	—
233	C₃×C₄	—	2	4	no	—
234	C₃×C₄	—	2	2	no	—
235	C₃×C₄	—	2	2	no	—
236	C₃×C₄	—	4	4	no	—
237	C₃×C₄	—	4	8	no	—
238	C₃×C₄	—	4	4	no	—
239	C₃×C₄	—	4	4	no	—
240	C₃×C₄	#947	1	2	yes	—
241	C₃×C₄	C₃×#124	1	2	yes	—
242	C₃×C₄	C₂×#239	1	2	yes	—
243	C₃×C₄	C₂×C₃×#41	1	2	yes	—
244	C₃²	C₂×#490	1	1	yes	characteristic, complemented, Sylow
245	C₂³	—	6	4	no	—
246	C₂³	—	12	4	no	—
247	C₂³	—	12	8	no	—
248	C₂³	C₂×C₃×#67	1	1	yes	characteristic
249	D₄	—	6	8	no	—
250	D₄	—	12	16	no	—
251	D₄	—	12	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	C₃×#239	1	4	yes	—
256	C₂×C₄	C₂×C₃×#67	1	4	yes	—
257	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
258	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
259	C₈	—	6	4	no	—
260	C₂×C₃	—	2	1	no	—
261	C₂×C₃	—	2	2	no	—
262	C₂×C₃	—	2	4	no	—
263	C₂×C₃	—	24	4	no	—
264	C₂×C₃	#5067	1	4	yes	direct product factor
265	C₂×C₃	C₃×#490	1	4	yes	complemented
266	C₂×C₃	C₂×#947	1	1	yes	characteristic
267	C₂×C₃	C₂×#949	1	2	yes	—
268	C₂×C₃	C₂×C₃×#124	1	1	yes	characteristic
269	C₂×C₃	C₂×C₃×AGL(1,ℤ/8)	1	2	yes	—
270	D₃	—	8	4	no	—
271	C₂²	—	12	4	no	—
272	C₂²	—	12	8	no	—
273	C₂²	—	24	8	no	—
274	C₂²	C₃×#947	1	2	yes	—
275	C₂²	C₃×#949	1	4	yes	—
276	C₂²	C₂²×C₃×#67	1	1	yes	characteristic
277	C₄	—	2	4	no	—
278	C₄	—	4	4	no	—
279	C₄	C₃×#947	1	2	yes	—
280	C₄	C₂×C₃×#239	1	2	yes	—
281	C₃	—	2	1	no	—
282	C₃	C₂×#5067	1	1	yes	characteristic, direct product factor
283	C₃	C₂×C₃×#490	1	1	yes	characteristic, complemented
284	C₂	—	24	4	no	—
285	C₂	C₃×#5067	1	4	yes	direct product factor
286	C₂	C₂×C₃×#947	1	1	yes	characteristic
287	C₂	C₂×C₃×#949	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₂×C₃²×#125	C₂
socle	C₂³×C₃²	C₂×D₄
prime radical	C₂×C₃×#874	C₂
hypercenter	C₂×C₃×#125	D₃
hypocenter	C₃	C₂×C₃×#490
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×#125

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×#125	C₂	semidirect
3. C₃×#5067	C₂	direct (N × H)
4. C₂×C₃×#809	C₂	semidirect
5. C₂×C₃×#803	C₂	semidirect
7. C₂×#5067	C₃	direct (N × H)
11. C₃²×#125	C₂²	semidirect
18. C₃×#809	C₂²	semidirect
19. C₃×#803	C₂²	semidirect
22. C₂×C₃×#125	D₃	semidirect
23. C₂×C₃×#125	C₂×C₃	semidirect
28. #5067	C₂×C₃	direct (N × H)
29. C₂×#809	C₂×C₃	semidirect
30. C₂×#803	C₂×C₃	semidirect
59. C₃×#125	C₂×D₃	semidirect
60. C₃×#125	C₂²×C₃	semidirect
75. #809	C₂²×C₃	semidirect
76. #803	C₂²×C₃	semidirect
93. C₂×#125	C₃×D₃	semidirect
153. #125	C₂×C₃×D₃	semidirect
197. C₂×C₃²	#490	semidirect
244. C₃²	C₂×#490	semidirect
264. C₂×C₃	#5067	direct (N × H)
265. C₂×C₃	C₃×#490	semidirect
282. C₃	C₂×#5067	direct (N × H)
283. C₃	C₂×C₃×#490	semidirect
285. C₂	C₃×#5067	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×C₃²×#125	2
C₂×C₃×#874	2
C₃×#5067	2
C₂×C₃×#809	2
C₂×C₃×#803	2
C₂×C₃×#490	3
C₂×#5067	3

Subgroups — C2×C3×#5067 #11873574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#5067 #11873574