Subgroups — C₂³×#14519747 #30586645

Order 15616.

42862 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×D₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×D₂₄₄	C₂	1	2	yes	maximal, complemented
3	C₂⁴×C₂²⋊D₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#14519747	C₂	1	56	yes	maximal, direct product factor
5	C₂³×#11758198	C₂	1	2	yes	maximal, complemented
6	C₂³×C₂²⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₆₁	—	2	16	no	—
8	C₂⁵×D₆₁	—	2	32	no	—
9	C₂⁵×D₆₁	C₂²	1	2	yes	—
10	C₂⁵×D₆₁	C₂²	1	28	yes	complemented
11	C₂⁵×D₆₁	C₂²	1	1	yes	characteristic, complemented
12	C₂³×D₂₄₄	—	2	32	no	—
13	C₂³×D₂₄₄	C₂²	1	56	yes	complemented
14	C₂³×C₂²⋊D₆₁	—	2	16	no	—
15	C₂³×C₂²⋊D₆₁	C₂²	1	28	yes	complemented
16	C₂⁴×Dic₆₁	C₂²	1	1	yes	characteristic, complemented
17	C₂×#14519747	C₂²	1	448	yes	direct product factor
18	C₂²×#11758198	C₂²	1	56	yes	complemented
19	C₂⁶×C₆₁	C₂²	1	1	yes	characteristic
20	C₂⁴×C₄×C₆₁	C₂²	1	2	yes	complemented
21	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	28	yes	complemented
22	C₂⁴×D₆₁	—	2	512	no	—
23	C₂⁴×D₆₁	—	2	16	no	—
24	C₂⁴×D₆₁	—	2	16	no	—
25	C₂⁴×D₆₁	—	2	224	no	—
26	C₂⁴×D₆₁	—	2	448	no	—
27	C₂⁴×D₆₁	—	2	8	no	—
28	C₂⁴×D₆₁	—	2	16	no	—
29	C₂⁴×D₆₁	D₄	1	32	yes	complemented
30	C₂⁴×D₆₁	C₂³	1	28	yes	—
31	C₂⁴×D₆₁	C₂³	1	112	yes	complemented
32	C₂⁴×D₆₁	C₂³	1	14	yes	complemented
33	C₂²×D₂₄₄	—	2	448	no	—
34	C₂²×D₂₄₄	C₂³	1	224	yes	complemented
35	C₂²×C₂²⋊D₆₁	—	2	224	no	—
36	C₂²×C₂²⋊D₆₁	C₂³	1	112	yes	complemented
37	C₂³×Dic₆₁	—	2	8	no	—
38	C₂³×Dic₆₁	C₂³	1	14	yes	complemented
39	#14519747	C₂³	1	512	yes	direct product factor
40	C₂×#11758198	C₂³	1	224	yes	complemented
41	C₂⁵×C₆₁	—	2	16	no	—
42	C₂⁵×C₆₁	D₄	1	16	yes	complemented
43	C₂⁵×C₆₁	C₂³	1	14	yes	—
44	C₂⁵×C₆₁	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₆₁	—	2	16	no	—
46	C₂³×C₄×C₆₁	C₂³	1	28	yes	complemented
47	C₂×C₂²⋊C₄×C₆₁	C₂³	1	112	yes	complemented
48	C₂⁴×C₆₁	—	2	128	no	—
49	C₂⁴×C₆₁	—	2	112	no	—
50	C₂⁴×C₆₁	C₂×D₄	1	8	yes	—
51	C₂⁴×C₆₁	C₂×D₄	1	16	yes	—
52	C₂⁴×C₆₁	C₂×D₄	1	112	yes	complemented
53	C₂⁴×C₆₁	C₂⁴	1	28	yes	—
54	C₂⁴×C₆₁	C₂⁴	1	7	yes	—
55	C₂³×D₆₁	—	2	3584	no	—
56	C₂³×D₆₁	—	2	256	no	—
57	C₂³×D₆₁	—	2	112	no	—
58	C₂³×D₆₁	—	2	112	no	—
59	C₂³×D₆₁	—	2	448	no	—
60	C₂³×D₆₁	—	2	896	no	—
61	C₂³×D₆₁	—	2	56	no	—
62	C₂³×D₆₁	—	2	112	no	—
63	C₂³×D₆₁	—	4	64	no	—
64	C₂³×D₆₁	C₂×D₄	1	224	yes	complemented
65	C₂³×D₆₁	C₂⁴	1	56	yes	—
66	C₂³×D₆₁	C₂⁴	1	64	yes	complemented
67	C₂³×D₆₁	C₂⁴	1	28	yes	complemented
68	C₂²×C₄×C₆₁	—	2	112	no	—
69	C₂²×C₄×C₆₁	C₂⁴	1	56	yes	complemented
70	C₂×C₂²⋊D₆₁	—	2	448	no	—
71	C₂×C₂²⋊D₆₁	C₂⁴	1	64	yes	complemented
72	C₂²×Dic₆₁	—	2	56	no	—
73	C₂²×Dic₆₁	C₂⁴	1	28	yes	complemented
74	C₂×D₂₄₄	—	2	896	no	—
75	C₂×D₂₄₄	C₂⁴	1	128	yes	complemented
76	C₂²⋊C₄×C₆₁	C₂⁴	1	64	yes	complemented
77	#11758198	C₂⁴	1	128	yes	complemented
78	C₂³×C₆₁	—	2	448	no	—
79	C₂³×C₆₁	—	2	112	no	—
80	C₂³×C₆₁	#120	1	64	yes	complemented
81	C₂³×C₆₁	C₂²×D₄	1	56	yes	—
82	C₂³×C₆₁	C₂²×D₄	1	28	yes	—
83	C₂³×C₆₁	C₂²×D₄	1	112	yes	complemented
84	C₂³×C₆₁	C₂⁵	1	8	yes	—
85	C₂³×C₆₁	C₂⁵	1	7	yes	—
86	C₂²×D₆₁	—	2	3584	no	—
87	C₂²×D₆₁	—	2	896	no	—
88	C₂²×D₆₁	—	2	112	no	—
89	C₂²×D₆₁	—	2	112	no	—
90	C₂²×D₆₁	—	2	128	no	—
91	C₂²×D₆₁	—	2	256	no	—
92	C₂²×D₆₁	—	2	112	no	—
93	C₂²×D₆₁	—	2	56	no	—
94	C₂²×D₆₁	—	4	224	no	—
95	C₂²×D₆₁	C₂²×D₄	1	224	yes	complemented
96	C₂²×D₆₁	C₂⁵	1	16	yes	—
97	C₂²×D₆₁	C₂⁵	1	8	yes	complemented
98	C₂×C₄×C₆₁	—	2	112	no	—
99	C₂×C₄×C₆₁	C₂⁵	1	16	yes	complemented
100	C₂²⋊D₆₁	—	2	128	no	—
101	C₂×Dic₆₁	—	2	56	no	—
102	C₂×Dic₆₁	C₂⁵	1	8	yes	complemented
103	D₂₄₄	—	2	256	no	—
104	C₂³×#120	—	61	1	no	maximal, Sylow
105	C₂²×C₆₁	—	2	16	no	—
106	C₂²×C₆₁	—	2	224	no	—
107	C₂²×C₆₁	C₂×#120	1	112	yes	complemented
108	C₂²×C₆₁	C₂³×D₄	1	28	yes	—
109	C₂²×C₆₁	C₂³×D₄	1	14	yes	—
110	C₂²×C₆₁	C₂³×D₄	1	16	yes	complemented
111	C₂²×C₆₁	C₂⁶	1	1	yes	characteristic
112	C₂×D₆₁	—	2	512	no	—
113	C₂×D₆₁	—	2	16	no	—
114	C₂×D₆₁	—	2	16	no	—
115	C₂×D₆₁	—	2	448	no	—
116	C₂×D₆₁	—	2	16	no	—
117	C₂×D₆₁	—	2	8	no	—
118	C₂×D₆₁	—	4	112	no	—
119	C₂×D₆₁	C₂³×D₄	1	32	yes	complemented
120	C₄×C₆₁	—	2	16	no	—
121	Dic₆₁	—	2	8	no	—
122	C₂⁷	—	61	1	no	—
123	C₂⁴×D₄	—	61	1	no	—
124	C₂⁴×D₄	—	61	2	no	—
125	C₂²×#120	—	61	56	no	—
126	C₂³×C₂²⋊C₄	—	61	2	no	—
127	C₂³×C₂²⋊C₄	D₆₁	1	1	yes	characteristic, complemented
128	C₂×C₆₁	—	2	16	no	—
129	C₂×C₆₁	C₂²×#120	1	28	yes	complemented
130	C₂×C₆₁	C₂⁴×D₄	1	2	yes	—
131	C₂×C₆₁	C₂⁴×D₄	1	1	yes	characteristic
132	D₆₁	—	2	32	no	—
133	D₆₁	—	4	8	no	—
134	C₂⁶	—	61	2	no	—
135	C₂⁶	—	61	28	no	—
136	C₂⁶	—	61	1	no	—
137	C₂⁶	—	122	32	no	—
138	C₂⁶	—	122	16	no	—
139	C₂⁶	C₂×D₆₁	1	1	yes	characteristic
140	C₂³×D₄	—	61	28	no	—
141	C₂³×D₄	—	61	56	no	—
142	C₂³×D₄	—	122	32	no	—
143	C₂³×D₄	—	122	16	no	—
144	C₂⁴×C₄	—	61	1	no	—
145	C₂⁴×C₄	C₂×D₆₁	1	2	yes	complemented
146	C₂×#120	—	61	448	no	—
147	C₂²×C₂²⋊C₄	—	61	56	no	—
148	C₂²×C₂²⋊C₄	C₂×D₆₁	1	28	yes	complemented
149	C₆₁	C₂³×#120	1	1	yes	characteristic, complemented, Sylow
150	C₂⁵	—	2	16	no	—
151	C₂⁵	—	61	32	no	—
152	C₂⁵	—	61	28	no	—
153	C₂⁵	—	61	112	no	—
154	C₂⁵	—	61	14	no	—
155	C₂⁵	—	122	448	no	—
156	C₂⁵	—	122	512	no	—
157	C₂⁵	—	122	16	no	—
158	C₂⁵	—	122	224	no	—
159	C₂⁵	—	122	16	no	—
160	C₂⁵	—	122	8	no	—
161	C₂⁵	—	122	16	no	—
162	C₂⁵	D₂₄₄	1	16	yes	complemented
163	C₂⁵	C₂²×D₆₁	1	14	yes	—
164	C₂⁵	C₂²×D₆₁	1	1	yes	characteristic
165	C₂²×D₄	—	61	112	no	—
166	C₂²×D₄	—	61	224	no	—
167	C₂²×D₄	—	122	448	no	—
168	C₂²×D₄	—	122	224	no	—
169	C₂³×C₄	—	2	16	no	—
170	C₂³×C₄	—	61	14	no	—
171	C₂³×C₄	—	122	8	no	—
172	C₂³×C₄	C₂²×D₆₁	1	28	yes	complemented
173	#120	—	61	512	no	—
174	C₂×C₂²⋊C₄	—	61	224	no	—
175	C₂×C₂²⋊C₄	C₂²×D₆₁	1	112	yes	complemented
176	C₂⁴	—	2	112	no	—
177	C₂⁴	—	2	128	no	—
178	C₂⁴	—	61	224	no	—
179	C₂⁴	—	61	56	no	—
180	C₂⁴	—	61	64	no	—
181	C₂⁴	—	61	28	no	—
182	C₂⁴	—	122	896	no	—
183	C₂⁴	—	122	3584	no	—
184	C₂⁴	—	122	112	no	—
185	C₂⁴	—	122	112	no	—
186	C₂⁴	—	122	448	no	—
187	C₂⁴	—	122	256	no	—
188	C₂⁴	—	122	112	no	—
189	C₂⁴	—	122	56	no	—
190	C₂⁴	—	244	64	no	—
191	C₂⁴	C₂×D₂₄₄	1	8	yes	—
192	C₂⁴	C₂×D₂₄₄	1	112	yes	complemented
193	C₂⁴	D₄×D₆₁	1	16	yes	—
194	C₂⁴	C₂³×D₆₁	1	28	yes	—
195	C₂⁴	C₂³×D₆₁	1	7	yes	—
196	C₂×D₄	—	61	64	no	—
197	C₂×D₄	—	61	128	no	—
198	C₂×D₄	—	122	896	no	—
199	C₂×D₄	—	122	448	no	—
200	C₂²×C₄	—	2	112	no	—
201	C₂²×C₄	—	61	28	no	—
202	C₂²×C₄	—	122	56	no	—
203	C₂²×C₄	C₂³×D₆₁	1	56	yes	complemented
204	C₂²⋊C₄	—	61	128	no	—
205	C₂²⋊C₄	C₂³×D₆₁	1	64	yes	complemented
206	C₂³	—	2	448	no	—
207	C₂³	—	2	112	no	—
208	C₂³	—	61	224	no	—
209	C₂³	—	61	16	no	—
210	C₂³	—	61	8	no	—
211	C₂³	—	122	256	no	—
212	C₂³	—	122	3584	no	—
213	C₂³	—	122	896	no	—
214	C₂³	—	122	112	no	—
215	C₂³	—	122	112	no	—
216	C₂³	—	122	128	no	—
217	C₂³	—	122	112	no	—
218	C₂³	—	122	56	no	—
219	C₂³	—	244	224	no	—
220	C₂³	#14519747	1	64	yes	direct product factor
221	C₂³	C₂×D₄×D₆₁	1	56	yes	—
222	C₂³	C₂²×D₂₄₄	1	28	yes	—
223	C₂³	C₂²×D₂₄₄	1	112	yes	complemented
224	C₂³	C₂⁴×D₆₁	1	8	yes	—
225	C₂³	C₂⁴×D₆₁	1	7	yes	—
226	D₄	—	122	256	no	—
227	D₄	—	122	128	no	—
228	C₂×C₄	—	2	112	no	—
229	C₂×C₄	—	61	8	no	—
230	C₂×C₄	—	122	56	no	—
231	C₂×C₄	C₂⁴×D₆₁	1	16	yes	complemented
232	C₂²	—	2	224	no	—
233	C₂²	—	2	16	no	—
234	C₂²	—	61	32	no	—
235	C₂²	—	122	512	no	—
236	C₂²	—	122	448	no	—
237	C₂²	—	122	16	no	—
238	C₂²	—	122	16	no	—
239	C₂²	—	122	16	no	—
240	C₂²	—	122	8	no	—
241	C₂²	—	244	112	no	—
242	C₂²	C₂×#14519747	1	112	yes	direct product factor
243	C₂²	C₂²×D₄×D₆₁	1	28	yes	—
244	C₂²	C₂³×D₂₄₄	1	14	yes	—
245	C₂²	C₂³×D₂₄₄	1	16	yes	complemented
246	C₂²	C₂⁵×D₆₁	1	1	yes	characteristic
247	C₄	—	2	16	no	—
248	C₄	—	122	8	no	—
249	C₂	—	2	16	no	—
250	C₂	—	122	32	no	—
251	C₂	—	244	8	no	—
252	C₂	C₂²×#14519747	1	28	yes	direct product factor
253	C₂	C₂³×D₄×D₆₁	1	2	yes	—
254	C₂	C₂⁴×D₂₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₆₁
commutator, cosocle	C₂²×C₆₁	C₂⁶
Frattini	C₂²	C₂⁵×D₆₁
Fitting	C₂³×C₂²⋊C₄×C₆₁	C₂
socle	C₂⁵×C₆₁	C₂³
hypercenter	C₂³×C₂²⋊C₄	D₆₁
hypocenter	C₆₁	C₂³×#120
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁶×D₆₁	C₂	semidirect
2. C₂⁴×D₂₄₄	C₂	semidirect
3. C₂⁴×C₂²⋊D₆₁	C₂	semidirect
4. C₂²×#14519747	C₂	direct (N × H)
5. C₂³×#11758198	C₂	semidirect
6. C₂³×C₂²⋊C₄×C₆₁	C₂	semidirect
10. C₂⁵×D₆₁	C₂²	semidirect
11. C₂⁵×D₆₁	C₂²	semidirect
13. C₂³×D₂₄₄	C₂²	semidirect
15. C₂³×C₂²⋊D₆₁	C₂²	semidirect
16. C₂⁴×Dic₆₁	C₂²	semidirect
17. C₂×#14519747	C₂²	direct (N × H)
18. C₂²×#11758198	C₂²	semidirect
20. C₂⁴×C₄×C₆₁	C₂²	semidirect
21. C₂²×C₂²⋊C₄×C₆₁	C₂²	semidirect
29. C₂⁴×D₆₁	D₄	semidirect
31. C₂⁴×D₆₁	C₂³	semidirect
32. C₂⁴×D₆₁	C₂³	semidirect
34. C₂²×D₂₄₄	C₂³	semidirect
36. C₂²×C₂²⋊D₆₁	C₂³	semidirect
38. C₂³×Dic₆₁	C₂³	semidirect
39. #14519747	C₂³	direct (N × H)
40. C₂×#11758198	C₂³	semidirect
42. C₂⁵×C₆₁	D₄	semidirect
46. C₂³×C₄×C₆₁	C₂³	semidirect
47. C₂×C₂²⋊C₄×C₆₁	C₂³	semidirect
52. C₂⁴×C₆₁	C₂×D₄	semidirect
64. C₂³×D₆₁	C₂×D₄	semidirect
66. C₂³×D₆₁	C₂⁴	semidirect
67. C₂³×D₆₁	C₂⁴	semidirect
69. C₂²×C₄×C₆₁	C₂⁴	semidirect
71. C₂×C₂²⋊D₆₁	C₂⁴	semidirect
73. C₂²×Dic₆₁	C₂⁴	semidirect
75. C₂×D₂₄₄	C₂⁴	semidirect
76. C₂²⋊C₄×C₆₁	C₂⁴	semidirect
77. #11758198	C₂⁴	semidirect
80. C₂³×C₆₁	#120	semidirect
83. C₂³×C₆₁	C₂²×D₄	semidirect
95. C₂²×D₆₁	C₂²×D₄	semidirect
97. C₂²×D₆₁	C₂⁵	semidirect
99. C₂×C₄×C₆₁	C₂⁵	semidirect
102. C₂×Dic₆₁	C₂⁵	semidirect
107. C₂²×C₆₁	C₂×#120	semidirect
110. C₂²×C₆₁	C₂³×D₄	semidirect
119. C₂×D₆₁	C₂³×D₄	semidirect
127. C₂³×C₂²⋊C₄	D₆₁	semidirect
129. C₂×C₆₁	C₂²×#120	semidirect
145. C₂⁴×C₄	C₂×D₆₁	semidirect
148. C₂²×C₂²⋊C₄	C₂×D₆₁	semidirect
149. C₆₁	C₂³×#120	semidirect
162. C₂⁵	D₂₄₄	semidirect
172. C₂³×C₄	C₂²×D₆₁	semidirect
175. C₂×C₂²⋊C₄	C₂²×D₆₁	semidirect
192. C₂⁴	C₂×D₂₄₄	semidirect
203. C₂²×C₄	C₂³×D₆₁	semidirect
205. C₂²⋊C₄	C₂³×D₆₁	semidirect
220. C₂³	#14519747	direct (N × H)
223. C₂³	C₂²×D₂₄₄	semidirect
231. C₂×C₄	C₂⁴×D₆₁	semidirect
242. C₂²	C₂×#14519747	direct (N × H)
245. C₂²	C₂³×D₂₄₄	semidirect
252. C₂	C₂²×#14519747	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×D₆₁	2
C₂⁴×D₂₄₄	2
C₂⁴×C₂²⋊D₆₁	2
C₂²×#14519747	2
C₂³×#11758198	2
C₂³×C₂²⋊C₄×C₆₁	2
C₂³×#120	61

Subgroups — C23×#14519747 #30586645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14519747 #30586645