Subgroups — C₂×C₃×#4713 #11873571

Order 1152.

1049 conjugacy classes of subgroups.

The subgroup lattice has 283 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#874	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#870	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃²×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#821	C₂	1	4	yes	maximal, complemented
5	C₃×#4713	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃×#490	—	3	1	no	maximal
7	C₂×#4713	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	C₂²×C₃²×C₈	C₂²	1	2	yes	—
11	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₂×C₃×Dic₃⋊C₄	—	2	2	no	—
14	C₃×#874	—	2	2	no	—
15	C₃×#874	C₂²	1	4	yes	—
16	C₃×#870	—	2	2	no	—
17	C₃×#870	C₂²	1	4	yes	complemented
18	C₃²×#97	C₂²	1	4	yes	complemented
19	C₃×#821	C₂²	1	16	yes	complemented
20	C₂×C₃×#127	—	3	1	no	—
21	C₂×C₃×#125	—	3	1	no	—
22	C₂×#874	C₂×C₃	1	1	yes	characteristic
23	C₂×#870	C₂×C₃	1	1	yes	characteristic, complemented
24	C₃×#490	—	3	8	no	—
25	C₂×C₃×#102	—	3	4	no	—
26	C₂×C₃×#97	—	2	1	no	—
27	C₂×C₃×#97	D₃	1	1	yes	characteristic, complemented
28	C₂×C₃×#97	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂×#821	C₂×C₃	1	4	yes	complemented
30	#4713	C₂×C₃	1	8	yes	direct product factor
31	C₂³×C₃×D₃	—	2	2	no	—
32	C₂²×C₃²×C₄	D₄	1	2	yes	—
33	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
34	C₂²×C₃×Dic₃	—	2	2	no	—
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	C₂×C₃²×C₈	—	2	4	no	—
41	C₂×C₃²×C₈	C₂³	1	4	yes	—
42	C₃²×C₄²	D₄	1	2	yes	—
43	C₃²×C₄²	C₂³	1	2	yes	—
44	C₃×C₄⋊Dic₃	—	2	2	no	—
45	C₃×C₄⋊Dic₃	C₂³	1	4	yes	—
46	C₃×Dic₃⋊C₄	—	2	8	no	—
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	—	2	2	no	—
53	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
54	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
55	C₃×#127	—	3	4	no	—
56	C₃×#127	—	6	2	no	—
57	C₃×#125	—	3	4	no	—
58	C₃×#125	—	6	2	no	—
59	C₂×C₃×C₄⋊C₄	—	3	2	no	—
60	C₂×C₃×C₄⋊C₄	—	6	2	no	—
61	C₂×C₃×C₄²	—	2	1	no	—
62	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
63	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
65	C₂×Dic₃⋊C₄	—	2	2	no	—
66	#874	—	2	2	no	—
67	#874	C₂²×C₃	1	4	yes	—
68	#870	—	2	2	no	—
69	#870	C₂²×C₃	1	4	yes	complemented
70	C₃×#102	—	3	16	no	—
71	C₃×#97	—	2	4	no	—
72	C₃×#97	C₂×D₃	1	4	yes	complemented
73	C₃×#97	C₂²×C₃	1	4	yes	complemented
74	#821	C₂²×C₃	1	16	yes	complemented
75	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₃×D₃	—	2	4	no	—
77	C₂²×C₃×D₃	—	4	4	no	—
78	C₂²×C₃×D₃	—	4	8	no	—
79	C₂×C₃²×C₄	—	2	4	no	—
80	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
81	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃²×C₄	#41	1	4	yes	—
83	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
84	C₂×C₃×Dic₃	—	2	4	no	—
85	C₂×C₃×Dic₃	—	4	4	no	—
86	C₃×D₁₂	—	2	8	no	—
87	C₃×D₁₂	—	4	16	no	—
88	C₃×D₁₂	—	4	4	no	—
89	C₃²×C₈	—	2	4	no	—
90	C₂×#127	—	3	1	no	—
91	C₂×#125	—	3	1	no	—
92	#490	—	3	8	no	—
93	C₂×#102	—	3	4	no	—
94	C₂×#97	C₃×D₃	1	1	yes	characteristic, complemented
95	C₂⁴×C₃	—	6	2	no	—
96	C₂³×D₃	—	2	2	no	—
97	C₂×C₃×D₄	—	3	4	no	—
98	C₂×C₃×D₄	—	6	8	no	—
99	C₂×C₃×D₄	—	6	2	no	—
100	C₂×C₃×D₄	—	6	8	no	—
101	C₂×C₃×D₄	—	12	8	no	—
102	C₂²×C₃×C₄	—	2	2	no	—
103	C₂²×C₃×C₄	—	2	1	no	—
104	C₂²×C₃×C₄	—	6	2	no	—
105	C₂²×C₃×C₄	D₁₂	1	2	yes	—
106	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
107	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
109	C₂²×Dic₃	—	2	2	no	—
110	C₂×D₁₂	—	2	8	no	—
111	C₂×D₁₂	—	2	2	no	—
112	C₂×D₁₂	—	2	8	no	—
113	C₂×D₁₂	—	4	8	no	—
114	C₂×D₁₂	C₂³×C₃	1	4	yes	—
115	C₂×C₃×C₈	—	2	8	no	—
116	C₂×C₃×C₈	—	2	4	no	—
117	C₂×C₃×C₈	—	2	4	no	—
118	C₂×C₃×C₈	—	2	4	no	—
119	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
120	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
121	C₃×C₄⋊C₄	—	3	4	no	—
122	C₃×C₄⋊C₄	—	6	8	no	—
123	C₃×C₄⋊C₄	—	6	2	no	—
124	C₃×C₄²	—	2	2	no	—
125	C₃×C₄²	—	2	2	no	—
126	C₃×C₄²	D₁₂	1	2	yes	—
127	C₃×C₄²	C₃×D₄	1	2	yes	—
128	C₃×C₄²	C₂²×D₃	1	2	yes	—
129	C₃×C₄²	C₂³×C₃	1	2	yes	—
130	C₄⋊Dic₃	—	2	2	no	—
131	C₄⋊Dic₃	C₂³×C₃	1	4	yes	—
132	Dic₃⋊C₄	—	2	8	no	—
133	C₂²×C₃²	#124	1	2	yes	—
134	C₂²×C₃²	AGL(1,ℤ/8)	1	4	yes	—
135	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₃×D₃	—	4	4	no	—
137	C₂×C₃×D₃	—	4	8	no	—
138	C₂×C₃×D₃	—	8	8	no	—
139	C₃²×C₄	—	2	4	no	—
140	C₃²×C₄	#124	1	2	yes	—
141	C₃²×C₄	C₂×#41	1	2	yes	—
142	C₃×Dic₃	—	4	4	no	—
143	C₂²×D₄	—	3	2	no	—
144	C₂²×D₄	—	6	2	no	—
145	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
146	#127	—	3	4	no	—
147	#127	—	6	2	no	—
148	#125	—	3	4	no	—
149	#125	—	6	2	no	—
150	C₂×C₄⋊C₄	—	3	2	no	—
151	C₂×C₄⋊C₄	—	6	2	no	—
152	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
153	#102	—	3	16	no	—
154	#97	C₂×C₃×D₃	1	4	yes	complemented
155	C₂³×C₃	—	2	1	no	—
156	C₂³×C₃	—	6	4	no	—
157	C₂³×C₃	—	12	4	no	—
158	C₂³×C₃	—	12	8	no	—
159	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
160	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
161	C₂²×D₃	—	2	4	no	—
162	C₂²×D₃	—	4	4	no	—
163	C₂²×D₃	—	4	8	no	—
164	C₃×D₄	—	6	8	no	—
165	C₃×D₄	—	12	16	no	—
166	C₃×D₄	—	12	4	no	—
167	C₂×C₃×C₄	—	2	4	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	4	no	—
172	C₂×C₃×C₄	—	2	4	no	—
173	C₂×C₃×C₄	—	4	4	no	—
174	C₂×C₃×C₄	—	6	4	no	—
175	C₂×C₃×C₄	—	12	4	no	—
176	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
177	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
178	C₂×C₃×C₄	#235	1	4	yes	—
179	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
180	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
181	C₂×C₃×C₄	C₃×#41	1	4	yes	—
182	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
184	C₂×Dic₃	—	2	4	no	—
185	C₂×Dic₃	—	4	4	no	—
186	D₁₂	—	2	8	no	—
187	D₁₂	—	4	16	no	—
188	D₁₂	—	4	4	no	—
189	C₃×C₈	—	2	8	no	—
190	C₃×C₈	—	2	4	no	—
191	C₃×C₈	—	2	4	no	—
192	C₂×C₃²	#490	1	4	yes	complemented
193	C₂×C₃²	C₂×#124	1	1	yes	characteristic
194	C₂×C₃²	C₂×AGL(1,ℤ/8)	1	2	yes	—
195	C₃×D₃	—	8	4	no	—
196	C₂⁴	—	6	2	no	—
197	C₂×D₄	—	3	4	no	—
198	C₂×D₄	—	6	8	no	—
199	C₂×D₄	—	6	2	no	—
200	C₂×D₄	—	6	8	no	—
201	C₂×D₄	—	12	8	no	—
202	C₂²×C₄	—	6	2	no	—
203	C₂²×C₄	C₃×D₁₂	1	2	yes	—
204	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	C₂²×C₃×D₃	1	4	yes	—
207	C₄⋊C₄	—	3	4	no	—
208	C₄⋊C₄	—	6	2	no	—
209	C₄⋊C₄	—	6	8	no	—
210	C₄²	C₃×D₁₂	1	2	yes	—
211	C₄²	C₂²×C₃×D₃	1	2	yes	—
212	C₂²×C₃	—	2	4	no	—
213	C₂²×C₃	—	2	1	no	—
214	C₂²×C₃	—	2	2	no	—
215	C₂²×C₃	—	12	4	no	—
216	C₂²×C₃	—	12	8	no	—
217	C₂²×C₃	—	24	8	no	—
218	C₂²×C₃	#875	1	2	yes	—
219	C₂²×C₃	#908	1	4	yes	—
220	C₂²×C₃	C₃×#124	1	2	yes	—
221	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	4	yes	—
222	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
223	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
224	C₂×D₃	—	4	8	no	—
225	C₂×D₃	—	4	4	no	—
226	C₂×D₃	—	8	8	no	—
227	C₃×C₄	—	2	4	no	—
228	C₃×C₄	—	2	4	no	—
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	2	2	no	—
231	C₃×C₄	—	4	4	no	—
232	C₃×C₄	—	12	4	no	—
233	C₃×C₄	#875	1	2	yes	—
234	C₃×C₄	C₃×#124	1	2	yes	—
235	C₃×C₄	C₂×#235	1	2	yes	—
236	C₃×C₄	C₂×C₃×#41	1	2	yes	—
237	Dic₃	—	4	4	no	—
238	C₃²	C₂×#490	1	1	yes	characteristic, complemented, Sylow
239	C₂³	—	6	4	no	—
240	C₂³	—	12	8	no	—
241	C₂³	—	12	4	no	—
242	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
243	D₄	—	6	8	no	—
244	D₄	—	12	16	no	—
245	D₄	—	12	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	6	4	no	—
248	C₂×C₄	—	12	4	no	—
249	C₂×C₄	C₂×C₃×D₁₂	1	4	yes	—
250	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
251	C₂×C₄	C₃×#235	1	4	yes	—
252	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
253	C₈	—	2	4	no	—
254	C₂×C₃	—	2	2	no	—
255	C₂×C₃	—	2	4	no	—
256	C₂×C₃	—	2	1	no	—
257	C₂×C₃	—	24	4	no	—
258	C₂×C₃	#4713	1	4	yes	direct product factor
259	C₂×C₃	C₃×#490	1	4	yes	complemented
260	C₂×C₃	C₂×#875	1	1	yes	characteristic
261	C₂×C₃	C₂×#908	1	2	yes	—
262	C₂×C₃	C₂×C₃×#124	1	1	yes	characteristic
263	C₂×C₃	C₂×C₃×AGL(1,ℤ/8)	1	2	yes	—
264	D₃	—	8	4	no	—
265	C₂²	—	12	8	no	—
266	C₂²	—	12	4	no	—
267	C₂²	—	24	8	no	—
268	C₂²	C₃×#875	1	2	yes	—
269	C₂²	C₃×#908	1	4	yes	—
270	C₂²	C₂²×C₃×D₁₂	1	1	yes	characteristic
271	C₄	—	2	4	no	—
272	C₄	—	12	4	no	—
273	C₄	C₃×#875	1	2	yes	—
274	C₄	C₂×C₃×#235	1	2	yes	—
275	C₃	—	2	1	no	—
276	C₃	C₂×#4713	1	1	yes	characteristic, direct product factor
277	C₃	C₂×C₃×#490	1	1	yes	characteristic, complemented
278	C₂	—	24	4	no	—
279	C₂	C₃×#4713	1	4	yes	direct product factor
280	C₂	C₂×C₃×#875	1	1	yes	characteristic
281	C₂	C₂×C₃×#908	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₂×C₃²×#97	C₂
socle	C₂³×C₃²	C₂×D₄
prime radical	C₂×C₃×#874	C₂
hypercenter	C₂×C₃×#97	D₃
hypocenter	C₃	C₂×C₃×#490
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×#97

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#870	C₂	semidirect
3. C₂×C₃²×#97	C₂	semidirect
4. C₂×C₃×#821	C₂	semidirect
5. C₃×#4713	C₂	direct (N × H)
7. C₂×#4713	C₃	direct (N × H)
17. C₃×#870	C₂²	semidirect
18. C₃²×#97	C₂²	semidirect
19. C₃×#821	C₂²	semidirect
23. C₂×#870	C₂×C₃	semidirect
27. C₂×C₃×#97	D₃	semidirect
28. C₂×C₃×#97	C₂×C₃	semidirect
29. C₂×#821	C₂×C₃	semidirect
30. #4713	C₂×C₃	direct (N × H)
69. #870	C₂²×C₃	semidirect
72. C₃×#97	C₂×D₃	semidirect
73. C₃×#97	C₂²×C₃	semidirect
74. #821	C₂²×C₃	semidirect
94. C₂×#97	C₃×D₃	semidirect
154. #97	C₂×C₃×D₃	semidirect
192. C₂×C₃²	#490	semidirect
238. C₃²	C₂×#490	semidirect
258. C₂×C₃	#4713	direct (N × H)
259. C₂×C₃	C₃×#490	semidirect
276. C₃	C₂×#4713	direct (N × H)
277. C₃	C₂×C₃×#490	semidirect
279. C₂	C₃×#4713	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×#874	2
C₂×C₃×#870	2
C₂×C₃²×#97	2
C₂×C₃×#821	2
C₃×#4713	2
C₂×C₃×#490	3
C₂×#4713	3

Subgroups — C2×C3×#4713 #11873571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#4713 #11873571