Subgroups — C₃×#4727 #10591824

Order 576.

341 conjugacy classes of subgroups.

The subgroup lattice has 287 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	C₂×C₃×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×D₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#830	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×(C₂×D₃)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×(C₂×D₃)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#408	—	3	1	no	maximal
9	#4727	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃×#67	C₄	1	1	yes	characteristic
12	C₂×C₃×#67	C₄	1	1	yes	characteristic
13	C₃×#235	—	2	2	no	—
14	C₃×#235	—	2	2	no	—
15	C₂×C₃×D₁₂	C₄	1	1	yes	characteristic
16	C₂×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
18	C₂×C₃×Dic₆	C₄	1	1	yes	characteristic
19	C₂×C₃²×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃²×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
23	C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
24	C₃×D₃×C₈	—	2	2	no	—
25	C₂×C₃×#41	—	3	1	no	—
26	C₂×#235	C₂×C₃	1	1	yes	characteristic
27	C₂×C₃×M₁₆	—	3	1	no	—
28	C₂²×C₃×C₈	—	3	1	no	—
29	C₂×ΓL(1,𝔽₂₅)	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂×D₃×C₈	C₂×C₃	1	1	yes	characteristic, complemented
31	C₃×C₂²⋊C₈	—	2	1	no	—
32	C₃×C₂²⋊C₈	—	3	1	no	—
33	C₃×C₂²⋊C₈	—	3	1	no	—
34	C₃×C₂²⋊C₈	—	3	1	no	—
35	C₃×C₂²⋊C₈	D₃	1	1	yes	characteristic, complemented
36	C₃×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
37	#830	C₂×C₃	1	1	yes	characteristic, complemented
38	(C₂×D₃)⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
39	(C₂×D₃)⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂³×C₃²	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₃×D₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₃×D₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃²×C₄	—	2	1	no	—
44	C₂×C₃²×C₄	—	2	1	no	—
45	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
47	C₃×#67	—	2	2	no	—
48	C₃×#67	—	2	2	no	—
49	C₂×C₃×Dic₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₃×Dic₃	C₂×C₄	1	1	yes	characteristic
51	C₃×D₁₂	—	2	2	no	—
52	C₃×C₄×D₃	—	2	1	no	—
53	C₃×C₄×D₃	—	2	1	no	—
54	C₃×C₄×D₃	D₄	1	2	yes	—
55	C₃×C₄×D₃	D₄	1	2	yes	—
56	C₃×Dic₆	—	2	2	no	—
57	C₃²×C₈	—	2	1	no	—
58	C₃²×C₈	—	2	1	no	—
59	C₃×C₃⋊C₈	—	2	1	no	—
60	C₃×C₃⋊C₈	—	2	1	no	—
61	#408	—	3	1	no	Sylow
62	C₃×#41	—	6	2	no	—
63	C₃×#41	—	6	2	no	—
64	C₂×C₃×Q₈	—	3	1	no	—
65	C₂×C₃×D₄	—	3	1	no	—
66	C₂×C₃×D₄	—	3	1	no	—
67	C₂×C₃×D₄	—	3	1	no	—
68	C₂²×C₃×C₄	—	2	1	no	—
69	C₂²×C₃×C₄	—	3	1	no	—
70	C₂²×C₃×C₄	—	3	1	no	—
71	C₂²×C₃×C₄	C₂×D₃	1	1	yes	characteristic
72	C₂²×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
73	C₂×#67	C₃×C₄	1	1	yes	characteristic
74	C₂×#67	C₃×C₄	1	1	yes	characteristic
75	#235	—	2	2	no	—
76	#235	—	2	2	no	—
77	C₂×D₁₂	C₃×C₄	1	1	yes	characteristic
78	C₂×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
80	C₂×Dic₆	C₃×C₄	1	1	yes	characteristic
81	C₃×M₁₆	—	6	2	no	—
82	C₂×C₃×C₈	—	2	1	no	—
83	C₂×C₃×C₈	—	2	1	no	—
84	C₂×C₃×C₈	—	3	1	no	—
85	C₂×C₃×C₈	—	3	1	no	—
86	C₂×C₃×C₈	—	6	2	no	—
87	C₂×C₃×C₈	C₂×D₃	1	1	yes	characteristic
88	C₂×C₃×C₈	C₂×D₃	1	1	yes	characteristic
89	C₂×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
90	C₂×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
91	C₂×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
92	C₂×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
93	ΓL(1,𝔽₂₅)	—	2	2	no	—
94	D₃×C₈	—	2	2	no	—
95	C₂²×C₃²	—	2	1	no	—
96	C₂²×C₃²	—	2	1	no	—
97	C₂²×C₃²	—	2	1	no	—
98	C₂²×C₃²	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₃×D₃	—	2	1	no	—
100	C₂×C₃×D₃	—	2	1	no	—
101	C₂×C₃×D₃	—	2	1	no	—
102	C₂×C₃×D₃	—	2	1	no	—
103	C₂×C₃×D₃	—	2	1	no	—
104	C₂×C₃×D₃	C₂²⋊C₄	1	2	yes	—
105	C₃²×C₄	—	2	1	no	—
106	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
107	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
108	C₃×Dic₃	—	2	1	no	—
109	C₃×Dic₃	C₂²⋊C₄	1	2	yes	—
110	C₂×#41	—	3	1	no	—
111	C₂×M₁₆	—	3	1	no	—
112	C₂²×C₈	—	3	1	no	—
113	C₂²⋊C₈	—	3	1	no	—
114	C₂²⋊C₈	—	3	1	no	—
115	C₂²⋊C₈	—	3	1	no	—
116	C₂²⋊C₈	C₃×D₃	1	1	yes	characteristic, complemented
117	C₂³×C₃	—	2	1	no	—
118	C₂³×C₃	—	3	1	no	—
119	C₂³×C₃	—	3	1	no	—
120	C₂³×C₃	C₄×D₃	1	1	yes	characteristic
121	C₂³×C₃	C₂×C₃×C₄	1	1	yes	characteristic
122	C₂²×D₃	C₂×C₃×C₄	1	1	yes	characteristic
123	C₂²×D₃	C₂×C₃×C₄	1	1	yes	characteristic
124	C₃×Q₈	—	6	2	no	—
125	C₃×D₄	—	6	2	no	—
126	C₃×D₄	—	6	2	no	—
127	C₃×D₄	—	6	2	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	3	1	no	—
135	C₂×C₃×C₄	—	3	2	no	—
136	C₂×C₃×C₄	—	3	2	no	—
137	C₂×C₃×C₄	—	3	1	no	—
138	C₂×C₃×C₄	—	4	1	no	—
139	C₂×C₃×C₄	—	4	1	no	—
140	C₂×C₃×C₄	—	6	1	no	—
141	C₂×C₃×C₄	—	6	1	no	—
142	C₂×C₃×C₄	C₄×D₃	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
146	#67	—	2	2	no	—
147	#67	—	2	2	no	—
148	C₂×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
149	C₂×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
150	D₁₂	—	2	2	no	—
151	C₄×D₃	—	2	1	no	—
152	C₄×D₃	—	2	1	no	—
153	C₄×D₃	C₃×D₄	1	2	yes	—
154	C₄×D₃	C₃×D₄	1	2	yes	—
155	Dic₆	—	2	2	no	—
156	C₃×C₈	—	2	1	no	—
157	C₃×C₈	—	2	1	no	—
158	C₃×C₈	—	2	1	no	—
159	C₃×C₈	—	2	1	no	—
160	C₃×C₈	—	4	1	no	—
161	C₃×C₈	—	4	1	no	—
162	C₃×C₈	—	6	1	no	—
163	C₃×C₈	—	6	1	no	—
164	C₃⋊C₈	—	2	1	no	—
165	C₃⋊C₈	—	2	1	no	—
166	C₂×C₃²	—	4	1	no	—
167	C₂×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₃²	#131	1	1	yes	characteristic
169	C₂×C₃²	#131	1	1	yes	characteristic
170	C₃×D₃	—	2	2	no	—
171	C₃×D₃	—	4	1	no	—
172	#41	—	6	2	no	—
173	#41	—	6	2	no	—
174	C₂×Q₈	—	3	1	no	—
175	C₂×D₄	—	3	1	no	—
176	C₂×D₄	—	3	1	no	—
177	C₂×D₄	—	3	1	no	—
178	C₂²×C₄	—	3	1	no	—
179	C₂²×C₄	—	3	1	no	—
180	C₂²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
181	M₁₆	—	6	2	no	—
182	C₂×C₈	—	3	1	no	—
183	C₂×C₈	—	3	1	no	—
184	C₂×C₈	—	6	2	no	—
185	C₂×C₈	C₂×C₃×D₃	1	1	yes	characteristic
186	C₂×C₈	C₂×C₃×D₃	1	1	yes	characteristic
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	—	2	1	no	—
189	C₂²×C₃	—	2	1	no	—
190	C₂²×C₃	—	2	1	no	—
191	C₂²×C₃	—	2	1	no	—
192	C₂²×C₃	—	2	1	no	—
193	C₂²×C₃	—	2	1	no	—
194	C₂²×C₃	—	2	1	no	—
195	C₂²×C₃	—	3	2	no	—
196	C₂²×C₃	—	4	1	no	—
197	C₂²×C₃	—	4	1	no	—
198	C₂²×C₃	—	6	1	no	—
199	C₂²×C₃	—	6	1	no	—
200	C₂²×C₃	—	6	1	no	—
201	C₂²×C₃	—	6	1	no	—
202	C₂²×C₃	—	6	1	no	—
203	C₂²×C₃	C₂×C₄×D₃	1	1	yes	characteristic
204	C₂²×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
205	C₂×D₃	—	2	1	no	—
206	C₂×D₃	—	2	1	no	—
207	C₂×D₃	—	2	1	no	—
208	C₂×D₃	—	2	1	no	—
209	C₂×D₃	—	2	1	no	—
210	C₂×D₃	C₃×C₂²⋊C₄	1	2	yes	—
211	C₃×C₄	—	2	1	no	—
212	C₃×C₄	—	2	1	no	—
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	3	2	no	—
216	C₃×C₄	—	4	1	no	—
217	C₃×C₄	—	6	1	no	—
218	C₃×C₄	D₃×D₄	1	1	yes	characteristic
219	C₃×C₄	D₃×D₄	1	1	yes	characteristic
220	C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
221	C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
222	Dic₃	—	2	1	no	—
223	Dic₃	C₃×C₂²⋊C₄	1	2	yes	—
224	C₃²	#408	1	1	yes	characteristic, complemented, Sylow
225	C₂³	—	3	1	no	—
226	C₂³	—	3	1	no	—
227	C₂³	C₃×C₄×D₃	1	1	yes	characteristic
228	Q₈	—	6	2	no	—
229	D₄	—	6	2	no	—
230	D₄	—	6	2	no	—
231	D₄	—	6	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	3	1	no	—
235	C₂×C₄	—	3	2	no	—
236	C₂×C₄	—	3	2	no	—
237	C₂×C₄	—	3	1	no	—
238	C₂×C₄	—	6	1	no	—
239	C₂×C₄	—	6	1	no	—
240	C₂×C₄	C₃×C₄×D₃	1	1	yes	characteristic
241	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	6	1	no	—
245	C₈	—	6	1	no	—
246	C₂×C₃	—	2	1	no	—
247	C₂×C₃	—	2	1	no	—
248	C₂×C₃	—	2	1	no	—
249	C₂×C₃	—	4	2	no	—
250	C₂×C₃	—	4	1	no	—
251	C₂×C₃	—	4	1	no	—
252	C₂×C₃	—	6	2	no	—
253	C₂×C₃	—	12	1	no	—
254	C₂×C₃	D₃×C₂²⋊C₄	1	1	yes	characteristic
255	C₂×C₃	#901	1	1	yes	characteristic
256	C₂×C₃	#907	1	1	yes	characteristic
257	C₂×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×C₃	C₃×#131	1	1	yes	characteristic
259	C₂×C₃	C₃×#131	1	1	yes	characteristic
260	D₃	—	2	2	no	—
261	D₃	—	4	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	3	2	no	—
266	C₂²	—	6	1	no	—
267	C₂²	—	6	1	no	—
268	C₂²	—	6	1	no	—
269	C₂²	—	6	1	no	—
270	C₂²	—	6	1	no	—
271	C₂²	C₂×C₃×C₄×D₃	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	3	2	no	—
274	C₄	—	6	1	no	—
275	C₄	C₃×D₃×D₄	1	1	yes	characteristic
276	C₄	C₃×D₃×D₄	1	1	yes	characteristic
277	C₃	—	2	1	no	—
278	C₃	#4727	1	1	yes	characteristic, direct product factor
279	C₃	C₃×#408	1	1	yes	characteristic, complemented
280	C₂	—	4	1	no	—
281	C₂	—	6	2	no	—
282	C₂	—	12	1	no	—
283	C₂	C₃×D₃×C₂²⋊C₄	1	1	yes	characteristic
284	C₂	C₃×#901	1	1	yes	characteristic
285	C₂	C₃×#907	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×D₃
Fitting	C₃²×C₂²⋊C₈	C₂
socle	C₂²×C₃²	C₂²×C₄
prime radical	C₂×C₃×#235	C₂
cosocle	C₂×C₃×C₄	C₂³×C₃
hypercenter	C₃×C₂²⋊C₈	D₃
hypocenter	C₃	C₃×#408
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×ΓL(1,𝔽₂₅)	C₂	semidirect
3. C₂×C₃×D₃×C₈	C₂	semidirect
4. C₃²×C₂²⋊C₈	C₂	semidirect
5. C₃×#830	C₂	semidirect
6. C₃×(C₂×D₃)⋊C₈	C₂	semidirect
7. C₃×(C₂×D₃)⋊C₈	C₂	semidirect
9. #4727	C₃	direct (N × H)
29. C₂×ΓL(1,𝔽₂₅)	C₂×C₃	semidirect
30. C₂×D₃×C₈	C₂×C₃	semidirect
35. C₃×C₂²⋊C₈	D₃	semidirect
36. C₃×C₂²⋊C₈	C₂×C₃	semidirect
37. #830	C₂×C₃	semidirect
38. (C₂×D₃)⋊C₈	C₂×C₃	semidirect
39. (C₂×D₃)⋊C₈	C₂×C₃	semidirect
116. C₂²⋊C₈	C₃×D₃	semidirect
224. C₃²	#408	semidirect
278. C₃	#4727	direct (N × H)
279. C₃	C₃×#408	semidirect

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
C₂×C₃×ΓL(1,𝔽₂₅)	2
C₂×C₃×D₃×C₈	2
C₃²×C₂²⋊C₈	2
C₃×#830	2
C₃×(C₂×D₃)⋊C₈	2
C₃×#408	3
#4727	3

Subgroups — C3×#4727 #10591824

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#4727 #10591824