Subgroups — C₂²×C₃×#953 #11900246

Order 1152.

11913 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×C₃²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₃×#67	C₂	1	3	yes	maximal, complemented
3	C₂×C₃×#953	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₃×#217	C₂	1	3	yes	maximal, complemented
5	C₂²×C₃×#120	—	3	1	no	maximal
6	C₂²×#953	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂⁵×C₃²	—	2	24	no	—
8	C₂⁵×C₃²	C₂²	1	3	yes	—
9	C₂⁵×C₃²	C₂²	1	12	yes	complemented
10	C₂⁴×C₃×D₃	C₂²	1	1	yes	characteristic, complemented
11	C₂²×C₃×#67	—	2	24	no	—
12	C₂²×C₃×#67	C₂²	1	36	yes	complemented
13	C₂³×C₃×Dic₃	C₂²	1	3	yes	complemented
14	C₃×#953	C₂²	1	64	yes	direct product factor
15	C₂×C₃×#217	C₂²	1	36	yes	complemented
16	C₂⁶×C₃	—	2	1	no	—
17	C₂⁶×C₃	D₃	1	1	yes	characteristic, complemented
18	C₂⁶×C₃	C₂×C₃	1	1	yes	characteristic, complemented
19	C₂³×C₃×D₄	—	3	3	no	—
20	C₂³×#67	C₂×C₃	1	3	yes	complemented
21	C₂×C₃×#120	—	3	24	no	—
22	C₂²×C₃×C₂²⋊C₄	—	3	3	no	—
23	C₂×#953	C₂×C₃	1	24	yes	direct product factor
24	C₂²×#217	C₂×C₃	1	3	yes	complemented
25	C₂⁴×C₃²	—	2	128	no	—
26	C₂⁴×C₃²	—	2	24	no	—
27	C₂⁴×C₃²	—	2	144	no	—
28	C₂⁴×C₃²	D₄	1	24	yes	complemented
29	C₂⁴×C₃²	C₂³	1	18	yes	—
30	C₂⁴×C₃²	C₂³	1	1	yes	characteristic
31	C₂⁴×C₃²	C₂³	1	16	yes	complemented
32	C₂³×C₃×D₃	—	2	12	no	—
33	C₂³×C₃×D₃	C₂³	1	6	yes	complemented
34	C₂×C₃×#67	—	2	144	no	—
35	C₂×C₃×#67	C₂³	1	48	yes	complemented
36	C₂²×C₃×Dic₃	—	2	12	no	—
37	C₂²×C₃×Dic₃	C₂³	1	18	yes	complemented
38	C₃×#217	C₂³	1	48	yes	complemented
39	C₂²×#120	—	3	1	no	Sylow
40	C₂⁵×C₃	—	2	48	no	—
41	C₂⁵×C₃	—	2	24	no	—
42	C₂⁵×C₃	—	2	24	no	—
43	C₂⁵×C₃	—	2	3	no	—
44	C₂⁵×C₃	—	2	12	no	—
45	C₂⁵×C₃	—	3	1	no	—
46	C₂⁵×C₃	C₂×D₃	1	3	yes	—
47	C₂⁵×C₃	C₂×D₃	1	12	yes	complemented
48	C₂⁵×C₃	C₂²×C₃	1	3	yes	—
49	C₂⁵×C₃	C₂²×C₃	1	12	yes	complemented
50	C₂⁴×D₃	C₂²×C₃	1	1	yes	characteristic, complemented
51	C₂²×C₃×D₄	—	3	36	no	—
52	C₂²×C₃×D₄	—	6	24	no	—
53	C₂³×C₃×C₄	—	3	3	no	—
54	C₂²×#67	—	2	24	no	—
55	C₂²×#67	C₂²×C₃	1	36	yes	complemented
56	C₂³×Dic₃	C₂²×C₃	1	3	yes	complemented
57	C₃×#120	—	3	64	no	—
58	C₂×C₃×C₂²⋊C₄	—	3	36	no	—
59	#953	C₂²×C₃	1	64	yes	direct product factor
60	C₂×#217	C₂²×C₃	1	36	yes	complemented
61	C₂³×C₃²	—	2	384	no	—
62	C₂³×C₃²	—	2	96	no	—
63	C₂³×C₃²	—	2	72	no	—
64	C₂³×C₃²	—	2	96	no	—
65	C₂³×C₃²	C₂×D₄	1	12	yes	—
66	C₂³×C₃²	C₂×D₄	1	72	yes	complemented
67	C₂³×C₃²	C₂⁴	1	12	yes	—
68	C₂³×C₃²	C₂⁴	1	3	yes	—
69	C₂²×C₃×D₃	—	2	36	no	—
70	C₂²×C₃×D₃	—	4	16	no	—
71	C₂²×C₃×D₃	C₂⁴	1	4	yes	complemented
72	C₃×#67	—	2	96	no	—
73	C₂×C₃×Dic₃	—	2	36	no	—
74	C₂×C₃×Dic₃	C₂⁴	1	12	yes	complemented
75	C₂⁶	C₃×D₃	1	1	yes	characteristic, complemented
76	C₂³×D₄	—	3	3	no	—
77	C₂×#120	—	3	24	no	—
78	C₂²×C₂²⋊C₄	—	3	3	no	—
79	C₂⁴×C₃	—	2	256	no	—
80	C₂⁴×C₃	—	2	128	no	—
81	C₂⁴×C₃	—	2	128	no	—
82	C₂⁴×C₃	—	2	48	no	—
83	C₂⁴×C₃	—	2	288	no	—
84	C₂⁴×C₃	—	2	24	no	—
85	C₂⁴×C₃	—	2	144	no	—
86	C₂⁴×C₃	—	2	24	no	—
87	C₂⁴×C₃	—	2	144	no	—
88	C₂⁴×C₃	—	2	24	no	—
89	C₂⁴×C₃	—	2	18	no	—
90	C₂⁴×C₃	—	2	1	no	—
91	C₂⁴×C₃	—	2	16	no	—
92	C₂⁴×C₃	—	3	6	no	—
93	C₂⁴×C₃	—	6	12	no	—
94	C₂⁴×C₃	#67	1	24	yes	complemented
95	C₂⁴×C₃	C₃×D₄	1	24	yes	complemented
96	C₂⁴×C₃	C₂²×D₃	1	18	yes	—
97	C₂⁴×C₃	C₂²×D₃	1	1	yes	characteristic
98	C₂⁴×C₃	C₂²×D₃	1	16	yes	complemented
99	C₂⁴×C₃	C₂³×C₃	1	18	yes	—
100	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
101	C₂⁴×C₃	C₂³×C₃	1	16	yes	complemented
102	C₂³×D₃	—	2	12	no	—
103	C₂³×D₃	C₂³×C₃	1	6	yes	complemented
104	C₂×C₃×D₄	—	3	48	no	—
105	C₂×C₃×D₄	—	6	144	no	—
106	C₂²×C₃×C₄	—	3	18	no	—
107	C₂²×C₃×C₄	—	6	12	no	—
108	C₂×#67	—	2	144	no	—
109	C₂×#67	C₂³×C₃	1	48	yes	complemented
110	C₂²×Dic₃	—	2	12	no	—
111	C₂²×Dic₃	C₂³×C₃	1	18	yes	complemented
112	C₃×C₂²⋊C₄	—	3	48	no	—
113	#217	C₂³×C₃	1	48	yes	complemented
114	C₂²×C₃²	—	2	128	no	—
115	C₂²×C₃²	—	2	144	no	—
116	C₂²×C₃²	—	2	24	no	—
117	C₂²×C₃²	#120	1	16	yes	complemented
118	C₂²×C₃²	C₂²×D₄	1	18	yes	—
119	C₂²×C₃²	C₂²×D₄	1	24	yes	complemented
120	C₂²×C₃²	C₂⁵	1	1	yes	characteristic
121	C₂×C₃×D₃	—	2	12	no	—
122	C₂×C₃×D₃	—	4	24	no	—
123	C₃×Dic₃	—	2	12	no	—
124	C₂⁵	—	2	24	no	—
125	C₂⁵	—	3	1	no	—
126	C₂⁵	C₂×C₃×D₃	1	3	yes	—
127	C₂⁵	C₂×C₃×D₃	1	12	yes	complemented
128	C₂²×D₄	—	3	36	no	—
129	C₂²×D₄	—	6	24	no	—
130	C₂³×C₄	—	3	3	no	—
131	#120	—	3	64	no	—
132	C₂×C₂²⋊C₄	—	3	36	no	—
133	C₂³×C₃	—	2	768	no	—
134	C₂³×C₃	—	2	384	no	—
135	C₂³×C₃	—	2	384	no	—
136	C₂³×C₃	—	2	192	no	—
137	C₂³×C₃	—	2	144	no	—
138	C₂³×C₃	—	2	192	no	—
139	C₂³×C₃	—	2	96	no	—
140	C₂³×C₃	—	2	96	no	—
141	C₂³×C₃	—	2	72	no	—
142	C₂³×C₃	—	2	96	no	—
143	C₂³×C₃	—	2	72	no	—
144	C₂³×C₃	—	2	96	no	—
145	C₂³×C₃	—	2	12	no	—
146	C₂³×C₃	—	2	3	no	—
147	C₂³×C₃	—	2	12	no	—
148	C₂³×C₃	—	2	72	no	—
149	C₂³×C₃	—	3	4	no	—
150	C₂³×C₃	—	6	36	no	—
151	C₂³×C₃	—	12	16	no	—
152	C₂³×C₃	C₂×#67	1	12	yes	—
153	C₂³×C₃	C₂×#67	1	72	yes	complemented
154	C₂³×C₃	C₂×C₃×D₄	1	12	yes	—
155	C₂³×C₃	C₂×C₃×D₄	1	72	yes	complemented
156	C₂³×C₃	C₂³×D₃	1	12	yes	—
157	C₂³×C₃	C₂³×D₃	1	3	yes	—
158	C₂³×C₃	C₂⁴×C₃	1	12	yes	—
159	C₂³×C₃	C₂⁴×C₃	1	3	yes	—
160	C₂²×D₃	—	2	36	no	—
161	C₂²×D₃	—	4	16	no	—
162	C₂²×D₃	C₂⁴×C₃	1	4	yes	complemented
163	C₃×D₄	—	6	96	no	—
164	C₂×C₃×C₄	—	3	12	no	—
165	C₂×C₃×C₄	—	6	36	no	—
166	#67	—	2	96	no	—
167	C₂×Dic₃	—	2	36	no	—
168	C₂×Dic₃	C₂⁴×C₃	1	12	yes	complemented
169	C₂×C₃²	—	2	24	no	—
170	C₂×C₃²	C₂×#120	1	12	yes	complemented
171	C₂×C₃²	C₂³×D₄	1	3	yes	—
172	C₃×D₃	—	4	4	no	—
173	C₂⁴	—	2	128	no	—
174	C₂⁴	—	2	24	no	—
175	C₂⁴	—	2	144	no	—
176	C₂⁴	—	3	6	no	—
177	C₂⁴	—	6	12	no	—
178	C₂⁴	C₃×#67	1	24	yes	complemented
179	C₂⁴	C₂²×C₃×D₃	1	18	yes	—
180	C₂⁴	C₂²×C₃×D₃	1	1	yes	characteristic
181	C₂⁴	C₂²×C₃×D₃	1	16	yes	complemented
182	C₂×D₄	—	3	48	no	—
183	C₂×D₄	—	6	144	no	—
184	C₂²×C₄	—	3	18	no	—
185	C₂²×C₄	—	6	12	no	—
186	C₂²⋊C₄	—	3	48	no	—
187	C₂²×C₃	—	2	256	no	—
188	C₂²×C₃	—	2	128	no	—
189	C₂²×C₃	—	2	128	no	—
190	C₂²×C₃	—	2	288	no	—
191	C₂²×C₃	—	2	48	no	—
192	C₂²×C₃	—	2	24	no	—
193	C₂²×C₃	—	2	24	no	—
194	C₂²×C₃	—	2	144	no	—
195	C₂²×C₃	—	2	144	no	—
196	C₂²×C₃	—	2	16	no	—
197	C₂²×C₃	—	2	18	no	—
198	C₂²×C₃	—	2	24	no	—
199	C₂²×C₃	—	2	1	no	—
200	C₂²×C₃	—	6	12	no	—
201	C₂²×C₃	—	12	24	no	—
202	C₂²×C₃	#953	1	16	yes	direct product factor
203	C₂²×C₃	C₃×#120	1	16	yes	complemented
204	C₂²×C₃	C₂²×#67	1	18	yes	—
205	C₂²×C₃	C₂²×#67	1	24	yes	complemented
206	C₂²×C₃	C₂²×C₃×D₄	1	18	yes	—
207	C₂²×C₃	C₂²×C₃×D₄	1	24	yes	complemented
208	C₂²×C₃	C₂⁴×D₃	1	1	yes	characteristic
209	C₂²×C₃	C₂⁵×C₃	1	1	yes	characteristic
210	C₂×D₃	—	2	12	no	—
211	C₂×D₃	—	4	24	no	—
212	C₃×C₄	—	6	12	no	—
213	Dic₃	—	2	12	no	—
214	C₃²	C₂²×#120	1	1	yes	characteristic, complemented, Sylow
215	C₂³	—	2	384	no	—
216	C₂³	—	2	96	no	—
217	C₂³	—	2	72	no	—
218	C₂³	—	2	96	no	—
219	C₂³	—	3	4	no	—
220	C₂³	—	6	36	no	—
221	C₂³	—	12	16	no	—
222	C₂³	C₂×C₃×#67	1	12	yes	—
223	C₂³	C₂×C₃×#67	1	72	yes	complemented
224	C₂³	C₂³×C₃×D₃	1	12	yes	—
225	C₂³	C₂³×C₃×D₃	1	3	yes	—
226	D₄	—	6	96	no	—
227	C₂×C₄	—	3	12	no	—
228	C₂×C₄	—	6	36	no	—
229	C₂×C₃	—	2	48	no	—
230	C₂×C₃	—	2	24	no	—
231	C₂×C₃	—	2	24	no	—
232	C₂×C₃	—	2	12	no	—
233	C₂×C₃	—	2	3	no	—
234	C₂×C₃	—	12	4	no	—
235	C₂×C₃	C₂×#953	1	12	yes	direct product factor
236	C₂×C₃	C₂×C₃×#120	1	12	yes	complemented
237	C₂×C₃	C₂³×#67	1	3	yes	—
238	C₂×C₃	C₂³×C₃×D₄	1	3	yes	—
239	D₃	—	4	4	no	—
240	C₂²	—	2	128	no	—
241	C₂²	—	2	144	no	—
242	C₂²	—	2	24	no	—
243	C₂²	—	6	12	no	—
244	C₂²	—	12	24	no	—
245	C₂²	C₃×#953	1	16	yes	direct product factor
246	C₂²	C₂²×C₃×#67	1	18	yes	—
247	C₂²	C₂²×C₃×#67	1	24	yes	complemented
248	C₂²	C₂⁴×C₃×D₃	1	1	yes	characteristic
249	C₄	—	6	12	no	—
250	C₃	—	2	1	no	—
251	C₃	C₂²×#953	1	1	yes	characteristic, direct product factor
252	C₃	C₂²×C₃×#120	1	1	yes	characteristic, complemented
253	C₂	—	2	24	no	—
254	C₂	—	12	4	no	—
255	C₂	C₂×C₃×#953	1	12	yes	direct product factor
256	C₂	C₂³×C₃×#67	1	3	yes	—

	subgroup	G/H
center	C₂⁴×C₃	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁵×C₃
Frattini	C₂²	C₂⁴×C₃×D₃
Fitting	C₂⁶×C₃²	C₂
socle	C₂⁴×C₃²	C₂³
hypercenter	C₂⁶×C₃	D₃
hypocenter	C₃	C₂²×C₃×#120
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂⁶×C₃

Complements

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

N	H ≅ G/N	type
1. C₂⁶×C₃²	C₂	semidirect
2. C₂³×C₃×#67	C₂	semidirect
3. C₂×C₃×#953	C₂	direct (N × H)
4. C₂²×C₃×#217	C₂	semidirect
6. C₂²×#953	C₃	direct (N × H)
9. C₂⁵×C₃²	C₂²	semidirect
10. C₂⁴×C₃×D₃	C₂²	semidirect
12. C₂²×C₃×#67	C₂²	semidirect
13. C₂³×C₃×Dic₃	C₂²	semidirect
14. C₃×#953	C₂²	direct (N × H)
15. C₂×C₃×#217	C₂²	semidirect
17. C₂⁶×C₃	D₃	semidirect
18. C₂⁶×C₃	C₂×C₃	semidirect
20. C₂³×#67	C₂×C₃	semidirect
23. C₂×#953	C₂×C₃	direct (N × H)
24. C₂²×#217	C₂×C₃	semidirect
28. C₂⁴×C₃²	D₄	semidirect
31. C₂⁴×C₃²	C₂³	semidirect
33. C₂³×C₃×D₃	C₂³	semidirect
35. C₂×C₃×#67	C₂³	semidirect
37. C₂²×C₃×Dic₃	C₂³	semidirect
38. C₃×#217	C₂³	semidirect
47. C₂⁵×C₃	C₂×D₃	semidirect
49. C₂⁵×C₃	C₂²×C₃	semidirect
50. C₂⁴×D₃	C₂²×C₃	semidirect
55. C₂²×#67	C₂²×C₃	semidirect
56. C₂³×Dic₃	C₂²×C₃	semidirect
59. #953	C₂²×C₃	direct (N × H)
60. C₂×#217	C₂²×C₃	semidirect
66. C₂³×C₃²	C₂×D₄	semidirect
71. C₂²×C₃×D₃	C₂⁴	semidirect
74. C₂×C₃×Dic₃	C₂⁴	semidirect
75. C₂⁶	C₃×D₃	semidirect
94. C₂⁴×C₃	#67	semidirect
95. C₂⁴×C₃	C₃×D₄	semidirect
98. C₂⁴×C₃	C₂²×D₃	semidirect
101. C₂⁴×C₃	C₂³×C₃	semidirect
103. C₂³×D₃	C₂³×C₃	semidirect
109. C₂×#67	C₂³×C₃	semidirect
111. C₂²×Dic₃	C₂³×C₃	semidirect
113. #217	C₂³×C₃	semidirect
117. C₂²×C₃²	#120	semidirect
119. C₂²×C₃²	C₂²×D₄	semidirect
127. C₂⁵	C₂×C₃×D₃	semidirect
153. C₂³×C₃	C₂×#67	semidirect
155. C₂³×C₃	C₂×C₃×D₄	semidirect
162. C₂²×D₃	C₂⁴×C₃	semidirect
168. C₂×Dic₃	C₂⁴×C₃	semidirect
170. C₂×C₃²	C₂×#120	semidirect
178. C₂⁴	C₃×#67	semidirect
181. C₂⁴	C₂²×C₃×D₃	semidirect
202. C₂²×C₃	#953	direct (N × H)
203. C₂²×C₃	C₃×#120	semidirect
205. C₂²×C₃	C₂²×#67	semidirect
207. C₂²×C₃	C₂²×C₃×D₄	semidirect
214. C₃²	C₂²×#120	semidirect
223. C₂³	C₂×C₃×#67	semidirect
235. C₂×C₃	C₂×#953	direct (N × H)
236. C₂×C₃	C₂×C₃×#120	semidirect
245. C₂²	C₃×#953	direct (N × H)
247. C₂²	C₂²×C₃×#67	semidirect
251. C₃	C₂²×#953	direct (N × H)
252. C₃	C₂²×C₃×#120	semidirect
255. C₂	C₂×C₃×#953	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁶×C₃²	2
C₂³×C₃×#67	2
C₂×C₃×#953	2
C₂²×C₃×#217	2
C₂²×C₃×#120	3
C₂²×#953	3

Subgroups — C22×C3×#953 #11900246

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#953 #11900246