Subgroups — C₃×#5833 #10595930

Order 576.

670 conjugacy classes of subgroups.

The subgroup lattice has 305 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#235	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#946	C₂	1	2	yes	maximal, complemented
4	C₃×#939	C₂	1	4	yes	maximal, complemented
5	C₃×#938	C₂	1	2	yes	maximal, complemented
6	C₃×#930	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#924	C₂	1	2	yes	maximal, complemented
9	C₃×#537	—	3	1	no	maximal
10	#5833	C₃	1	1	yes	maximal, characteristic, direct product factor
11	C₃²×#41	—	2	4	no	—
12	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic
14	C₂×C₃²×D₄	C₂²	1	2	yes	—
15	C₂²×C₃²×C₄	C₂²	1	2	yes	—
16	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
17	C₂×C₃×#67	C₂²	1	2	yes	—
18	C₂×C₃×#67	C₂²	1	4	yes	—
19	C₂²×C₃×Dic₃	C₂²	1	2	yes	—
20	C₃×#235	—	2	4	no	—
21	C₂×C₃×D₁₂	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
23	C₂×C₃×Dic₆	C₂²	1	1	yes	characteristic
24	C₃×#217	C₂²	1	4	yes	—
25	C₃×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
26	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
27	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
28	C₃×Dic₃⋊C₄	C₂²	1	4	yes	—
29	C₂×C₃×#41	—	2	1	no	—
30	C₂×C₃×#41	—	3	1	no	—
31	C₂×C₃×#41	D₃	1	1	yes	characteristic, complemented
32	C₂×C₃×#41	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂×#235	C₂×C₃	1	1	yes	characteristic, complemented
34	C₃×C₂²⋊Q₈	—	3	2	no	—
35	C₃×C₂²⋊Q₈	—	3	2	no	—
36	C₃×#121	—	3	2	no	—
37	C₃×#121	—	3	4	no	—
38	C₃×#121	—	3	2	no	—
39	C₂×C₃×C₄⋊C₄	—	3	1	no	—
40	#946	C₂×C₃	1	2	yes	complemented
41	#939	C₂×C₃	1	4	yes	complemented
42	#938	C₂×C₃	1	2	yes	complemented
43	#930	C₂×C₃	1	2	yes	complemented
44	C₂×C₄⋊Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
45	#924	C₂×C₃	1	2	yes	complemented
46	C₂³×C₃²	C₂³	1	2	yes	—
47	C₂³×C₃²	C₂³	1	1	yes	characteristic
48	C₂²×C₃×D₃	C₂³	1	2	yes	—
49	C₃²×Q₈	—	2	2	no	—
50	C₃²×D₄	—	2	4	no	—
51	C₃²×D₄	—	2	2	no	—
52	C₂×C₃²×C₄	—	2	4	no	—
53	C₂×C₃²×C₄	D₄	1	4	yes	complemented
54	C₂×C₃²×C₄	C₂³	1	2	yes	—
55	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
56	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
57	C₃×#67	—	2	4	no	—
58	C₃×#67	—	4	4	no	—
59	C₂×C₃×Dic₃	—	4	2	no	—
60	C₂×C₃×Dic₃	C₂³	1	2	yes	—
61	C₂×C₃×Dic₃	C₂³	1	4	yes	—
62	C₃×D₁₂	—	2	2	no	—
63	C₃×C₄×D₃	—	2	4	no	—
64	C₃×Dic₆	—	2	2	no	—
65	#537	—	3	1	no	Sylow
66	C₃×#41	—	2	8	no	—
67	C₃×#41	—	2	4	no	—
68	C₃×#41	—	2	4	no	—
69	C₃×#41	—	6	4	no	—
70	C₂×C₃×Q₈	—	2	1	no	—
71	C₂×C₃×Q₈	—	3	1	no	—
72	C₂×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
73	C₂×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
74	C₂×C₃×D₄	—	2	1	no	—
75	C₂×C₃×D₄	—	2	2	no	—
76	C₂×C₃×D₄	—	3	2	no	—
77	C₂×C₃×D₄	—	3	4	no	—
78	C₂×C₃×D₄	—	3	1	no	—
79	C₂×C₃×D₄	C₂×D₃	1	2	yes	—
80	C₂×C₃×D₄	C₂×D₃	1	1	yes	characteristic
81	C₂×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₃×D₄	C₂²×C₃	1	2	yes	—
83	C₂²×C₃×C₄	—	2	2	no	—
84	C₂²×C₃×C₄	—	2	1	no	—
85	C₂²×C₃×C₄	—	3	2	no	—
86	C₂²×C₃×C₄	—	3	2	no	—
87	C₂²×C₃×C₄	C₂×D₃	1	2	yes	—
88	C₂²×C₃×C₄	C₂×D₃	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₂²×C₃	1	2	yes	—
90	C₂²×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
91	C₂×#67	C₂²×C₃	1	2	yes	—
92	C₂×#67	C₂²×C₃	1	4	yes	—
93	C₂²×Dic₃	C₂²×C₃	1	2	yes	—
94	#235	—	2	4	no	—
95	C₂×D₁₂	C₂²×C₃	1	1	yes	characteristic
96	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
97	C₂×Dic₆	C₂²×C₃	1	1	yes	characteristic
98	C₃×C₄⋊C₄	—	3	4	no	—
99	C₃×C₄⋊C₄	—	3	2	no	—
100	C₃×C₄⋊C₄	—	3	2	no	—
101	C₃×C₂²⋊C₄	—	3	4	no	—
102	C₃×C₂²⋊C₄	—	3	4	no	—
103	#217	C₂²×C₃	1	4	yes	—
104	(C₂×D₃)⋊C₄	C₂²×C₃	1	4	yes	—
105	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
106	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
107	Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
108	C₂²×C₃²	—	2	2	no	—
109	C₂²×C₃²	—	2	1	no	—
110	C₂²×C₃²	—	2	1	no	—
111	C₂²×C₃²	—	2	2	no	—
112	C₂²×C₃²	—	2	2	no	—
113	C₂²×C₃²	C₂×D₄	1	2	yes	—
114	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
115	C₂×C₃×D₃	—	2	2	no	—
116	C₂×C₃×D₃	—	2	2	no	—
117	C₂×C₃×D₃	—	2	2	no	—
118	C₃²×C₄	—	2	2	no	—
119	C₃²×C₄	C₂×D₄	1	2	yes	—
120	C₃²×C₄	C₂×D₄	1	2	yes	—
121	C₃×Dic₃	—	2	2	no	—
122	C₃×Dic₃	—	2	4	no	—
123	C₂×#41	—	3	1	no	—
124	C₂×#41	C₃×D₃	1	1	yes	characteristic, complemented
125	C₂²⋊Q₈	—	3	2	no	—
126	C₂²⋊Q₈	—	3	2	no	—
127	#121	—	3	2	no	—
128	#121	—	3	4	no	—
129	#121	—	3	2	no	—
130	C₂×C₄⋊C₄	—	3	1	no	—
131	C₂³×C₃	—	2	2	no	—
132	C₂³×C₃	—	2	1	no	—
133	C₂³×C₃	—	3	2	no	—
134	C₂³×C₃	C₂²×D₃	1	2	yes	—
135	C₂³×C₃	C₂²×D₃	1	1	yes	characteristic
136	C₂³×C₃	C₂³×C₃	1	2	yes	—
137	C₂³×C₃	C₂³×C₃	1	1	yes	characteristic
138	C₂²×D₃	C₂³×C₃	1	2	yes	—
139	C₃×Q₈	—	2	4	no	—
140	C₃×Q₈	—	2	2	no	—
141	C₃×Q₈	—	2	2	no	—
142	C₃×Q₈	—	6	2	no	—
143	C₃×D₄	—	2	8	no	—
144	C₃×D₄	—	2	4	no	—
145	C₃×D₄	—	2	4	no	—
146	C₃×D₄	—	2	4	no	—
147	C₃×D₄	—	2	2	no	—
148	C₃×D₄	—	2	2	no	—
149	C₃×D₄	—	6	4	no	—
150	C₃×D₄	—	6	2	no	—
151	C₃×D₄	—	12	4	no	—
152	C₂×C₃×C₄	—	2	8	no	—
153	C₂×C₃×C₄	—	2	4	no	—
154	C₂×C₃×C₄	—	2	4	no	—
155	C₂×C₃×C₄	—	2	4	no	—
156	C₂×C₃×C₄	—	2	1	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	3	4	no	—
160	C₂×C₃×C₄	—	3	2	no	—
161	C₂×C₃×C₄	—	6	4	no	—
162	C₂×C₃×C₄	—	12	2	no	—
163	C₂×C₃×C₄	#67	1	4	yes	complemented
164	C₂×C₃×C₄	C₃×D₄	1	4	yes	complemented
165	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
166	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
169	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
171	#67	—	2	4	no	—
172	#67	—	4	4	no	—
173	C₂×Dic₃	—	4	2	no	—
174	C₂×Dic₃	C₂³×C₃	1	2	yes	—
175	C₂×Dic₃	C₂³×C₃	1	4	yes	—
176	D₁₂	—	2	2	no	—
177	C₄×D₃	—	2	4	no	—
178	Dic₆	—	2	2	no	—
179	C₂×C₃²	—	2	2	no	—
180	C₂×C₃²	—	4	2	no	—
181	C₂×C₃²	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₃²	H(5,𝔽₂)	1	1	yes	characteristic
183	C₂×C₃²	2₋¹⁺⁴	1	1	yes	characteristic
184	C₃×D₃	—	4	2	no	—
185	#41	—	2	4	no	—
186	#41	—	6	4	no	—
187	C₂×Q₈	—	3	1	no	—
188	C₂×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
189	C₂×D₄	—	3	2	no	—
190	C₂×D₄	—	3	4	no	—
191	C₂×D₄	—	3	1	no	—
192	C₂×D₄	C₂×C₃×D₃	1	2	yes	—
193	C₂×D₄	C₂×C₃×D₃	1	1	yes	characteristic
194	C₂²×C₄	—	3	2	no	—
195	C₂²×C₄	—	3	2	no	—
196	C₂²×C₄	C₂×C₃×D₃	1	2	yes	—
197	C₂²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
198	C₄⋊C₄	—	3	2	no	—
199	C₄⋊C₄	—	3	2	no	—
200	C₄⋊C₄	—	3	4	no	—
201	C₂²⋊C₄	—	3	4	no	—
202	C₂²⋊C₄	—	3	4	no	—
203	C₂²×C₃	—	2	1	no	—
204	C₂²×C₃	—	2	1	no	—
205	C₂²×C₃	—	2	1	no	—
206	C₂²×C₃	—	2	1	no	—
207	C₂²×C₃	—	2	4	no	—
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	2	1	no	—
215	C₂²×C₃	—	2	2	no	—
216	C₂²×C₃	—	4	1	no	—
217	C₂²×C₃	—	4	1	no	—
218	C₂²×C₃	—	4	2	no	—
219	C₂²×C₃	—	4	2	no	—
220	C₂²×C₃	—	6	2	no	—
221	C₂²×C₃	—	6	2	no	—
222	C₂²×C₃	—	6	2	no	—
223	C₂²×C₃	C₂×#67	1	2	yes	—
224	C₂²×C₃	C₂×C₃×D₄	1	2	yes	—
225	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
226	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
227	C₂×D₃	—	2	2	no	—
228	C₂×D₃	—	2	2	no	—
229	C₂×D₃	—	2	2	no	—
230	C₃×C₄	—	2	4	no	—
231	C₃×C₄	—	2	2	no	—
232	C₃×C₄	—	2	2	no	—
233	C₃×C₄	—	2	2	no	—
234	C₃×C₄	—	2	2	no	—
235	C₃×C₄	—	6	4	no	—
236	C₃×C₄	—	6	2	no	—
237	C₃×C₄	C₂×#67	1	2	yes	—
238	C₃×C₄	C₂×#67	1	2	yes	—
239	C₃×C₄	C₂×C₃×D₄	1	2	yes	—
240	C₃×C₄	C₂×C₃×D₄	1	2	yes	—
241	Dic₃	—	2	2	no	—
242	Dic₃	—	2	4	no	—
243	C₃²	#537	1	1	yes	characteristic, complemented, Sylow
244	C₂³	—	3	2	no	—
245	C₂³	C₂²×C₃×D₃	1	2	yes	—
246	C₂³	C₂²×C₃×D₃	1	1	yes	characteristic
247	Q₈	—	2	2	no	—
248	Q₈	—	6	2	no	—
249	D₄	—	2	4	no	—
250	D₄	—	2	2	no	—
251	D₄	—	6	4	no	—
252	D₄	—	6	2	no	—
253	D₄	—	12	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	3	4	no	—
256	C₂×C₄	—	3	2	no	—
257	C₂×C₄	—	6	4	no	—
258	C₂×C₄	—	12	2	no	—
259	C₂×C₄	C₃×#67	1	4	yes	complemented
260	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
261	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
263	C₂×C₃	—	2	2	no	—
264	C₂×C₃	—	2	2	no	—
265	C₂×C₃	—	2	1	no	—
266	C₂×C₃	—	2	1	no	—
267	C₂×C₃	—	2	1	no	—
268	C₂×C₃	—	4	2	no	—
269	C₂×C₃	—	4	4	no	—
270	C₂×C₃	—	4	2	no	—
271	C₂×C₃	—	4	2	no	—
272	C₂×C₃	—	12	2	no	—
273	C₂×C₃	#1009	1	1	yes	characteristic
274	C₂×C₃	#1010	1	1	yes	characteristic
275	C₂×C₃	C₂²×#67	1	1	yes	characteristic
276	C₂×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
277	C₂×C₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic
278	C₂×C₃	C₃×2₋¹⁺⁴	1	1	yes	characteristic
279	D₃	—	4	2	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	6	2	no	—
286	C₂²	—	6	2	no	—
287	C₂²	—	6	2	no	—
288	C₂²	C₂×C₃×#67	1	2	yes	—
289	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	6	4	no	—
292	C₄	—	6	2	no	—
293	C₄	C₂×C₃×#67	1	2	yes	—
294	C₄	C₂×C₃×#67	1	2	yes	—
295	C₃	—	2	1	no	—
296	C₃	#5833	1	1	yes	characteristic, direct product factor
297	C₃	C₃×#537	1	1	yes	characteristic, complemented
298	C₂	—	2	2	no	—
299	C₂	—	4	2	no	—
300	C₂	—	12	2	no	—
301	C₂	C₃×#1009	1	1	yes	characteristic
302	C₂	C₃×#1010	1	1	yes	characteristic
303	C₂	C₂²×C₃×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₃
Fitting	C₂×C₃²×#41	C₂
socle	C₂²×C₃²	C₂⁴
hypercenter	C₂×C₃×#41	D₃
hypocenter	C₃	C₃×#537
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×#41

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×#41	C₂	semidirect
2. C₂×C₃×#235	C₂	semidirect
3. C₃×#946	C₂	semidirect
4. C₃×#939	C₂	semidirect
5. C₃×#938	C₂	semidirect
6. C₃×#930	C₂	semidirect
7. C₂×C₃×C₄⋊Dic₃	C₂	semidirect
8. C₃×#924	C₂	semidirect
10. #5833	C₃	direct (N × H)
31. C₂×C₃×#41	D₃	semidirect
32. C₂×C₃×#41	C₂×C₃	semidirect
33. C₂×#235	C₂×C₃	semidirect
40. #946	C₂×C₃	semidirect
41. #939	C₂×C₃	semidirect
42. #938	C₂×C₃	semidirect
43. #930	C₂×C₃	semidirect
44. C₂×C₄⋊Dic₃	C₂×C₃	semidirect
45. #924	C₂×C₃	semidirect
53. C₂×C₃²×C₄	D₄	semidirect
124. C₂×#41	C₃×D₃	semidirect
163. C₂×C₃×C₄	#67	semidirect
164. C₂×C₃×C₄	C₃×D₄	semidirect
243. C₃²	#537	semidirect
259. C₂×C₄	C₃×#67	semidirect
296. C₃	#5833	direct (N × H)
297. C₃	C₃×#537	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃²×#41	2
C₂×C₃×#235	2
C₃×#946	2
C₃×#939	2
C₃×#938	2
C₃×#930	2
C₂×C₃×C₄⋊Dic₃	2
C₃×#924	2
C₃×#537	3
#5833	3

Subgroups — C3×#5833 #10595930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5833 #10595930