Subgroups — C₃×#67723 #11772486

Order 1152.

337 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#107	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#812	C₂	1	1	yes	maximal, characteristic
4	C₃×#1387	—	3	1	no	maximal
5	#67723	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃²×C₄⋊C₄	—	2	1	no	—
8	C₂×C₃×#208	—	2	1	no	—
9	C₃²×#107	—	2	1	no	—
10	C₃²×#107	C₄	1	2	yes	—
11	C₃×#819	—	2	1	no	—
12	C₃×#819	C₄	1	2	yes	complemented
13	C₃×#812	—	2	1	no	—
14	C₃×#812	C₄	1	2	yes	complemented
15	C₂×C₃×C₃⋊C₁₆	—	2	1	no	—
16	C₂×C₃×M₃₂	—	3	1	no	—
17	C₂×C₃×#108	—	3	1	no	—
18	C₂×C₃×#107	—	2	1	no	—
19	C₂×C₃×#107	D₃	1	1	yes	characteristic
20	C₂×C₃×#107	C₂×C₃	1	1	yes	characteristic
21	C₂×#819	C₂×C₃	1	1	yes	characteristic
22	C₂×#812	C₂×C₃	1	1	yes	characteristic
23	C₂²×C₃²×C₄	—	4	1	no	—
24	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
25	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
29	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
31	C₃²×C₄⋊C₄	—	2	2	no	—
32	C₃²×C₄⋊C₄	—	4	1	no	—
33	C₃×#208	—	2	2	no	—
34	C₃×#208	—	4	1	no	—
35	C₂×C₃×C₃⋊C₈	—	4	1	no	—
36	C₃×C₃⋊C₁₆	—	2	2	no	—
37	#1387	—	3	1	no	Sylow
38	C₂×C₃×M₁₆	—	6	1	no	—
39	C₂²×C₃×C₈	—	2	1	no	—
40	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₃×C₄⋊C₄	—	2	1	no	—
44	C₂×C₃×C₄⋊C₄	—	2	1	no	—
45	C₂×#208	—	2	1	no	—
46	C₃×M₃₂	—	3	2	no	—
47	C₃×M₃₂	—	6	1	no	—
48	C₂×C₃×C₁₆	—	6	1	no	—
49	C₃×#108	—	3	2	no	—
50	C₃×#108	—	6	1	no	—
51	C₃×#107	—	2	2	no	—
52	C₃×#107	—	2	1	no	—
53	C₃×#107	—	2	1	no	—
54	C₃×#107	—	2	2	no	—
55	C₃×#107	Dic₃	1	2	yes	—
56	C₃×#107	C₃×C₄	1	2	yes	—
57	#819	—	2	1	no	—
58	#819	C₃×C₄	1	2	yes	complemented
59	#812	—	2	1	no	—
60	#812	C₃×C₄	1	2	yes	complemented
61	C₂×C₃⋊C₁₆	—	2	1	no	—
62	C₂³×C₃²	SD₁₆	1	1	yes	characteristic
63	C₂×C₃²×C₄	—	4	2	no	—
64	C₂×C₃²×C₄	—	8	2	no	—
65	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
69	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
70	C₂×C₃²×C₄	Q₁₆	1	1	yes	characteristic
71	C₃²×C₈	C₄²	1	1	yes	characteristic
72	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
74	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
75	C₃×C₃⋊C₈	—	4	2	no	—
76	C₂×M₃₂	—	3	1	no	—
77	C₂×#108	—	3	1	no	—
78	C₂×#107	C₃×D₃	1	1	yes	characteristic
79	C₂²×C₃×C₄	—	2	1	no	—
80	C₂²×C₃×C₄	—	4	2	no	—
81	C₂²×C₃×C₄	—	4	1	no	—
82	C₂²×C₃×C₄	—	4	1	no	—
83	C₂²×C₃×C₄	#67	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
85	C₃×M₁₆	—	6	2	no	—
86	C₃×M₁₆	—	12	1	no	—
87	C₂×C₃×C₈	—	2	1	no	—
88	C₂×C₃×C₈	—	2	1	no	—
89	C₂×C₃×C₈	—	2	1	no	—
90	C₂×C₃×C₈	—	2	1	no	—
91	C₂×C₃×C₈	—	2	1	no	—
92	C₂×C₃×C₈	—	2	1	no	—
93	C₂×C₃×C₈	—	12	1	no	—
94	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
95	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
96	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
97	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
98	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
99	C₂×C₃×C₈	#67	1	1	yes	characteristic
100	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
103	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
104	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
105	C₂×C₃×C₈	C₃×Q₈	1	1	yes	characteristic
106	C₃×C₄⋊C₄	—	2	4	no	—
107	C₃×C₄⋊C₄	—	2	2	no	—
108	C₃×C₄⋊C₄	—	2	2	no	—
109	C₃×C₄⋊C₄	—	4	2	no	—
110	C₃×C₄⋊C₄	—	4	1	no	—
111	C₃×C₄⋊C₄	—	4	1	no	—
112	#208	—	2	2	no	—
113	#208	—	4	1	no	—
114	C₂×C₃⋊C₈	—	4	1	no	—
115	C₃×C₁₆	—	6	2	no	—
116	C₃⋊C₁₆	—	2	2	no	—
117	C₂²×C₃²	—	2	1	no	—
118	C₂²×C₃²	—	2	1	no	—
119	C₂²×C₃²	#102	1	1	yes	characteristic
120	C₂²×C₃²	#103	1	1	yes	characteristic
121	C₂²×C₃²	#106	1	1	yes	characteristic
122	C₃²×C₄	—	8	2	no	—
123	C₃²×C₄	#95	1	1	yes	characteristic
124	C₃²×C₄	#102	1	1	yes	characteristic
125	C₃²×C₄	#103	1	1	yes	characteristic
126	C₃²×C₄	#107	1	1	yes	characteristic
127	C₂×M₁₆	—	6	1	no	—
128	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
129	C₂×C₄⋊C₄	—	2	1	no	—
130	M₃₂	—	3	2	no	—
131	M₃₂	—	6	1	no	—
132	C₂×C₁₆	—	6	1	no	—
133	#108	—	3	2	no	—
134	#108	—	6	1	no	—
135	#107	—	2	1	no	—
136	#107	C₃×Dic₃	1	2	yes	—
137	C₂³×C₃	—	2	1	no	—
138	C₂³×C₃	#214	1	1	yes	characteristic
139	C₂³×C₃	C₃×SD₁₆	1	1	yes	characteristic
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	4	4	no	—
147	C₂×C₃×C₄	—	4	2	no	—
148	C₂×C₃×C₄	—	4	2	no	—
149	C₂×C₃×C₄	—	8	4	no	—
150	C₂×C₃×C₄	—	8	2	no	—
151	C₂×C₃×C₄	—	8	2	no	—
152	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
153	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
154	C₂×C₃×C₄	#213	1	1	yes	characteristic
155	C₂×C₃×C₄	#215	1	1	yes	characteristic
156	C₂×C₃×C₄	#216	1	1	yes	characteristic
157	C₂×C₃×C₄	#217	1	1	yes	characteristic
158	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₃×C₄	C₃×D₈	1	1	yes	characteristic
162	C₂×C₃×C₄	C₃×SD₁₆	1	1	yes	characteristic
163	C₂×C₃×C₄	C₃×Q₁₆	1	1	yes	characteristic
164	C₃×C₈	—	2	1	no	—
165	C₃×C₈	—	2	1	no	—
166	C₃×C₈	—	2	1	no	—
167	C₃×C₈	—	2	1	no	—
168	C₃×C₈	—	12	2	no	—
169	C₃×C₈	C₄×Dic₃	1	1	yes	characteristic
170	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
171	C₃×C₈	C₄⋊Dic₃	1	1	yes	characteristic
172	C₃×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
173	C₃×C₈	C₃×C₄²	1	1	yes	characteristic
174	C₃×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
175	C₃×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
176	C₃×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
177	C₃⋊C₈	—	4	2	no	—
178	C₂×C₃²	—	2	1	no	—
179	C₂×C₃²	—	2	1	no	—
180	C₂×C₃²	#340	1	1	yes	characteristic
181	C₂×C₃²	#361	1	1	yes	characteristic
182	C₂×C₃²	#362	1	1	yes	characteristic
183	C₂²×C₄	—	4	1	no	—
184	C₂²×C₄	C₃×#67	1	1	yes	characteristic
185	M₁₆	—	6	2	no	—
186	M₁₆	—	12	1	no	—
187	C₂×C₈	—	12	1	no	—
188	C₂×C₈	C₃×Dic₆	1	1	yes	characteristic
189	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
190	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
191	C₂×C₈	C₃×D₁₂	1	1	yes	characteristic
192	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
193	C₂×C₈	C₃×#67	1	1	yes	characteristic
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	4	1	no	—
196	C₁₆	—	6	2	no	—
197	C₂²×C₃	—	2	2	no	—
198	C₂²×C₃	—	2	2	no	—
199	C₂²×C₃	—	2	1	no	—
200	C₂²×C₃	—	2	1	no	—
201	C₂²×C₃	—	2	1	no	—
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	2	1	no	—
204	C₂²×C₃	—	2	1	no	—
205	C₂²×C₃	—	2	1	no	—
206	C₂²×C₃	#808	1	1	yes	characteristic
207	C₂²×C₃	#810	1	1	yes	characteristic
208	C₂²×C₃	#832	1	1	yes	characteristic
209	C₂²×C₃	C₃×#102	1	1	yes	characteristic
210	C₂²×C₃	C₃×#103	1	1	yes	characteristic
211	C₂²×C₃	C₃×#106	1	1	yes	characteristic
212	C₃×C₄	—	2	1	no	—
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	2	1	no	—
216	C₃×C₄	—	8	4	no	—
217	C₃×C₄	—	8	2	no	—
218	C₃×C₄	—	8	2	no	—
219	C₃×C₄	#807	1	1	yes	characteristic
220	C₃×C₄	#809	1	1	yes	characteristic
221	C₃×C₄	C₃⋊#95	1	1	yes	characteristic
222	C₃×C₄	#835	1	1	yes	characteristic
223	C₃×C₄	C₃×#95	1	1	yes	characteristic
224	C₃×C₄	C₃×#102	1	1	yes	characteristic
225	C₃×C₄	C₃×#103	1	1	yes	characteristic
226	C₃×C₄	C₃×#107	1	1	yes	characteristic
227	C₃²	#1387	1	1	yes	characteristic, complemented, Sylow
228	C₂³	C₃×#214	1	1	yes	characteristic
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	C₃×Dic₃⋊C₄	1	1	yes	characteristic
232	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
233	C₂×C₄	C₃×#213	1	1	yes	characteristic
234	C₂×C₄	C₃×#215	1	1	yes	characteristic
235	C₂×C₄	C₃×#216	1	1	yes	characteristic
236	C₂×C₄	C₃×#217	1	1	yes	characteristic
237	C₈	—	12	2	no	—
238	C₈	C₃×C₄×Dic₃	1	1	yes	characteristic
239	C₈	C₃×Dic₃⋊C₄	1	1	yes	characteristic
240	C₈	C₃×C₄⋊Dic₃	1	1	yes	characteristic
241	C₈	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
242	C₂×C₃	—	2	2	no	—
243	C₂×C₃	—	2	2	no	—
244	C₂×C₃	—	2	1	no	—
245	C₂×C₃	—	2	1	no	—
246	C₂×C₃	—	2	1	no	—
247	C₂×C₃	—	2	1	no	—
248	C₂×C₃	—	2	1	no	—
249	C₂×C₃	—	2	1	no	—
250	C₂×C₃	—	2	1	no	—
251	C₂×C₃	#4529	1	1	yes	characteristic
252	C₂×C₃	#4563	1	1	yes	characteristic
253	C₂×C₃	#4565	1	1	yes	characteristic
254	C₂×C₃	C₃×#340	1	1	yes	characteristic
255	C₂×C₃	C₃×#361	1	1	yes	characteristic
256	C₂×C₃	C₃×#362	1	1	yes	characteristic
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	C₃×#808	1	1	yes	characteristic
260	C₂²	C₃×#810	1	1	yes	characteristic
261	C₂²	C₃×#832	1	1	yes	characteristic
262	C₄	—	8	2	no	—
263	C₄	C₃×#807	1	1	yes	characteristic
264	C₄	C₃×#809	1	1	yes	characteristic
265	C₄	C₃×C₃⋊#95	1	1	yes	characteristic
266	C₄	C₃×#835	1	1	yes	characteristic
267	C₃	—	2	1	no	—
268	C₃	#67723	1	1	yes	characteristic, direct product factor
269	C₃	C₃×#1387	1	1	yes	characteristic, complemented
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	C₃×#4529	1	1	yes	characteristic
273	C₂	C₃×#4563	1	1	yes	characteristic
274	C₂	C₃×#4565	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	#832
commutator	C₃×C₈	C₃×C₄²
Frattini	C₂²×C₈	C₂×C₃×D₃
Fitting	C₂×C₃²×#107	C₂
socle	C₂²×C₃²	#102
prime radical	C₂³×C₃²	SD₁₆
cosocle	C₂²×C₃×C₈	C₂²×C₃
hypercenter	C₂×C₃×#107	D₃
hypocenter	C₃	C₃×#1387
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈ ◁ C₂×C₃×#107

Complements

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

N	H ≅ G/N	type
5. #67723	C₃	direct (N × H)
12. C₃×#819	C₄	semidirect
14. C₃×#812	C₄	semidirect
58. #819	C₃×C₄	semidirect
60. #812	C₃×C₄	semidirect
227. C₃²	#1387	semidirect
268. C₃	#67723	direct (N × H)
269. C₃	C₃×#1387	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#107	2
C₂×C₃×#819	2
C₂×C₃×#812	2
C₃×#1387	3
#67723	3

Subgroups — C3×#67723 #11772486

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#67723 #11772486