Subgroups — C₂³×D₄×Dic₆₁ #30586578

Order 15616.

19998 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×Dic₆₁	C₂	1	2	yes	maximal, complemented
2	C₂²×D₄×Dic₆₁	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×C₂²⋊Dic₆₁	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄×Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×D₄×C₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₆₁	—	2	32	no	—
8	C₂⁴×Dic₆₁	C₂²	1	1	yes	characteristic
9	C₂⁴×Dic₆₁	C₂²	1	2	yes	—
10	C₂⁴×Dic₆₁	C₂²	1	1	yes	characteristic
11	C₂⁴×Dic₆₁	C₂²	1	56	yes	complemented
12	C₂×D₄×Dic₆₁	C₂²	1	448	yes	direct product factor
13	C₂²×C₄⋊Dic₆₁	C₂²	1	28	yes	complemented
14	C₂²×C₂²⋊Dic₆₁	C₂²	1	56	yes	complemented
15	C₂²×C₄×Dic₆₁	C₂²	1	28	yes	complemented
16	C₂⁶×C₆₁	C₂²	1	2	yes	—
17	C₂³×D₄×C₆₁	C₄	1	32	yes	complemented
18	C₂³×D₄×C₆₁	C₂²	1	28	yes	—
19	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
20	C₂³×Dic₆₁	—	2	16	no	—
21	C₂³×Dic₆₁	—	2	8	no	—
22	C₂³×Dic₆₁	—	2	448	no	—
23	C₂³×Dic₆₁	D₄	1	16	yes	direct product factor
24	C₂³×Dic₆₁	C₂³	1	14	yes	—
25	C₂³×Dic₆₁	C₂³	1	28	yes	—
26	C₂³×Dic₆₁	C₂³	1	14	yes	—
27	C₂³×Dic₆₁	C₂³	1	224	yes	complemented
28	D₄×Dic₆₁	C₂³	1	512	yes	direct product factor
29	C₂×C₄⋊Dic₆₁	C₂³	1	112	yes	complemented
30	C₂×C₂²⋊Dic₆₁	C₂³	1	224	yes	complemented
31	C₂×C₄×Dic₆₁	C₂³	1	112	yes	complemented
32	C₂⁵×C₆₁	—	2	16	no	—
33	C₂⁵×C₆₁	—	2	16	no	—
34	C₂⁵×C₆₁	C₂×C₄	1	32	yes	complemented
35	C₂⁵×C₆₁	C₂³	1	28	yes	—
36	C₂⁵×C₆₁	C₂³	1	1	yes	characteristic
37	C₂²×D₄×C₆₁	C₂×C₄	1	448	yes	complemented
38	C₂²×D₄×C₆₁	C₂³	1	112	yes	—
39	C₂³×C₄×C₆₁	C₂×C₄	1	16	yes	complemented
40	C₂³×C₄×C₆₁	C₂³	1	14	yes	—
41	C₂⁴×C₆₁	—	2	256	no	—
42	C₂⁴×C₆₁	—	2	112	no	—
43	C₂⁴×C₆₁	—	2	112	no	—
44	C₂⁴×C₆₁	C₂²×C₄	1	8	yes	—
45	C₂⁴×C₆₁	C₂²×C₄	1	224	yes	complemented
46	C₂⁴×C₆₁	C₂×D₄	1	8	yes	—
47	C₂⁴×C₆₁	#41	1	8	yes	—
48	C₂⁴×C₆₁	C₂⁴	1	56	yes	—
49	C₂⁴×C₆₁	C₂⁴	1	7	yes	—
50	C₂×D₄×C₆₁	C₂²×C₄	1	896	yes	complemented
51	C₂×D₄×C₆₁	C₂⁴	1	64	yes	—
52	C₂²×C₄×C₆₁	C₂²×C₄	1	112	yes	complemented
53	C₂²×C₄×C₆₁	C₂⁴	1	28	yes	—
54	C₂²×Dic₆₁	—	2	112	no	—
55	C₂²×Dic₆₁	—	2	56	no	—
56	C₂²×Dic₆₁	—	2	896	no	—
57	C₂²×Dic₆₁	C₂×D₄	1	112	yes	direct product factor
58	C₂²×Dic₆₁	C₂⁴	1	56	yes	—
59	C₂²×Dic₆₁	C₂⁴	1	28	yes	—
60	C₂²×Dic₆₁	C₂⁴	1	28	yes	—
61	C₂²×Dic₆₁	C₂⁴	1	128	yes	complemented
62	C₂²⋊Dic₆₁	C₂⁴	1	128	yes	complemented
63	C₄⋊Dic₆₁	C₂⁴	1	64	yes	complemented
64	C₄×Dic₆₁	C₂⁴	1	64	yes	complemented
65	C₂³×C₆₁	—	2	896	no	—
66	C₂³×C₆₁	—	2	112	no	—
67	C₂³×C₆₁	—	2	112	no	—
68	C₂³×C₆₁	C₄×D₄	1	64	yes	complemented
69	C₂³×C₆₁	C₂³×C₄	1	28	yes	—
70	C₂³×C₆₁	C₂³×C₄	1	224	yes	complemented
71	C₂³×C₆₁	C₂²×D₄	1	28	yes	—
72	C₂³×C₆₁	C₂×#41	1	28	yes	—
73	C₂³×C₆₁	C₂⁵	1	16	yes	—
74	C₂³×C₆₁	C₂⁵	1	7	yes	—
75	D₄×C₆₁	C₂³×C₄	1	256	yes	complemented
76	C₂×C₄×C₆₁	C₂³×C₄	1	112	yes	complemented
77	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
78	C₂×Dic₆₁	—	2	112	no	—
79	C₂×Dic₆₁	—	2	56	no	—
80	C₂×Dic₆₁	—	2	256	no	—
81	C₂×Dic₆₁	C₂²×D₄	1	112	yes	direct product factor
82	C₂×Dic₆₁	C₂⁵	1	8	yes	—
83	C₂×Dic₆₁	C₂⁵	1	16	yes	—
84	C₂×Dic₆₁	C₂⁵	1	8	yes	—
85	C₂³×C₄×D₄	—	61	1	no	maximal, Sylow
86	C₂²×C₆₁	—	2	16	no	—
87	C₂²×C₆₁	—	2	16	no	—
88	C₂²×C₆₁	—	2	448	no	—
89	C₂²×C₆₁	C₂×C₄×D₄	1	112	yes	complemented
90	C₂²×C₆₁	C₂⁴×C₄	1	14	yes	—
91	C₂²×C₆₁	C₂⁴×C₄	1	32	yes	complemented
92	C₂²×C₆₁	C₂³×D₄	1	14	yes	—
93	C₂²×C₆₁	C₂²×#41	1	14	yes	—
94	C₂²×C₆₁	C₂⁶	1	1	yes	characteristic
95	C₄×C₆₁	C₂⁴×C₄	1	16	yes	complemented
96	Dic₆₁	—	2	16	no	—
97	Dic₆₁	—	2	8	no	—
98	Dic₆₁	C₂³×D₄	1	16	yes	direct product factor
99	C₂⁴×D₄	D₆₁	1	1	yes	characteristic
100	C₂⁵×C₄	—	61	2	no	—
101	C₂²×C₄×D₄	—	61	56	no	—
102	C₂³×C₄⋊C₄	—	61	1	no	—
103	C₂³×C₂²⋊C₄	—	61	2	no	—
104	C₂³×C₄²	—	61	1	no	—
105	C₂×C₆₁	—	2	32	no	—
106	C₂×C₆₁	C₂²×C₄×D₄	1	28	yes	complemented
107	C₂×C₆₁	C₂⁵×C₄	1	1	yes	characteristic
108	C₂×C₆₁	C₂⁴×D₄	1	1	yes	characteristic
109	C₂×C₆₁	C₂³×#41	1	1	yes	characteristic
110	C₂⁶	C₂×D₆₁	1	2	yes	—
111	C₂³×D₄	Dic₆₁	1	32	yes	direct product factor
112	C₂³×D₄	C₂×D₆₁	1	28	yes	—
113	C₂⁴×C₄	—	61	1	no	—
114	C₂⁴×C₄	—	61	56	no	—
115	C₂⁴×C₄	—	61	2	no	—
116	C₂⁴×C₄	—	61	1	no	—
117	C₂⁴×C₄	—	122	32	no	—
118	C₂⁴×C₄	C₂×D₆₁	1	1	yes	characteristic
119	C₂×C₄×D₄	—	61	448	no	—
120	C₂²×C₄⋊C₄	—	61	28	no	—
121	C₂²×C₂²⋊C₄	—	61	56	no	—
122	C₂²×C₄²	—	61	28	no	—
123	C₆₁	C₂³×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
124	C₂⁵	—	2	16	no	—
125	C₂⁵	—	2	16	no	—
126	C₂⁵	C₂×Dic₆₁	1	32	yes	complemented
127	C₂⁵	C₂²×D₆₁	1	28	yes	—
128	C₂⁵	C₂²×D₆₁	1	1	yes	characteristic
129	C₂²×D₄	C₂×Dic₆₁	1	448	yes	direct product factor
130	C₂²×D₄	C₂²×D₆₁	1	112	yes	—
131	C₂³×C₄	—	61	16	no	—
132	C₂³×C₄	—	61	14	no	—
133	C₂³×C₄	—	61	14	no	—
134	C₂³×C₄	—	61	28	no	—
135	C₂³×C₄	—	61	224	no	—
136	C₂³×C₄	—	122	448	no	—
137	C₂³×C₄	—	122	16	no	—
138	C₂³×C₄	—	122	8	no	—
139	C₂³×C₄	C₂×Dic₆₁	1	16	yes	complemented
140	C₂³×C₄	C₂²×D₆₁	1	14	yes	—
141	C₄×D₄	—	61	512	no	—
142	C₂×C₄⋊C₄	—	61	112	no	—
143	C₂×C₂²⋊C₄	—	61	224	no	—
144	C₂×C₄²	—	61	112	no	—
145	C₂⁴	—	2	112	no	—
146	C₂⁴	—	2	112	no	—
147	C₂⁴	—	2	256	no	—
148	C₂⁴	D₄×D₆₁	1	8	yes	—
149	C₂⁴	#11758224	1	8	yes	—
150	C₂⁴	C₂²×Dic₆₁	1	8	yes	—
151	C₂⁴	C₂²×Dic₆₁	1	224	yes	complemented
152	C₂⁴	C₂³×D₆₁	1	56	yes	—
153	C₂⁴	C₂³×D₆₁	1	7	yes	—
154	C₂×D₄	C₂²×Dic₆₁	1	896	yes	direct product factor
155	C₂×D₄	C₂³×D₆₁	1	64	yes	—
156	C₂²×C₄	—	61	112	no	—
157	C₂²×C₄	—	61	28	no	—
158	C₂²×C₄	—	61	28	no	—
159	C₂²×C₄	—	61	56	no	—
160	C₂²×C₄	—	61	128	no	—
161	C₂²×C₄	—	122	896	no	—
162	C₂²×C₄	—	122	112	no	—
163	C₂²×C₄	—	122	56	no	—
164	C₂²×C₄	C₂²×Dic₆₁	1	112	yes	complemented
165	C₂²×C₄	C₂³×D₆₁	1	28	yes	—
166	C₄⋊C₄	—	61	64	no	—
167	C₂²⋊C₄	—	61	128	no	—
168	C₄²	—	61	64	no	—
169	C₂³	—	2	896	no	—
170	C₂³	—	2	112	no	—
171	C₂³	—	2	112	no	—
172	C₂³	D₄×Dic₆₁	1	64	yes	direct product factor
173	C₂³	C₂³×Dic₆₁	1	28	yes	—
174	C₂³	C₂³×Dic₆₁	1	224	yes	complemented
175	C₂³	C₂×D₄×D₆₁	1	28	yes	—
176	C₂³	C₂×#11758224	1	28	yes	—
177	C₂³	C₂⁴×D₆₁	1	16	yes	—
178	C₂³	C₂⁴×D₆₁	1	7	yes	—
179	D₄	C₂³×Dic₆₁	1	256	yes	direct product factor
180	C₂×C₄	—	61	112	no	—
181	C₂×C₄	—	61	8	no	—
182	C₂×C₄	—	61	8	no	—
183	C₂×C₄	—	61	16	no	—
184	C₂×C₄	—	122	256	no	—
185	C₂×C₄	—	122	112	no	—
186	C₂×C₄	—	122	56	no	—
187	C₂×C₄	C₂³×Dic₆₁	1	112	yes	complemented
188	C₂×C₄	C₂⁴×D₆₁	1	8	yes	—
189	C₂²	—	2	448	no	—
190	C₂²	—	2	16	no	—
191	C₂²	—	2	16	no	—
192	C₂²	C₂×D₄×Dic₆₁	1	112	yes	direct product factor
193	C₂²	C₂⁴×Dic₆₁	1	14	yes	—
194	C₂²	C₂⁴×Dic₆₁	1	32	yes	complemented
195	C₂²	C₂²×D₄×D₆₁	1	14	yes	—
196	C₂²	C₂²×#11758224	1	14	yes	—
197	C₂²	C₂⁵×D₆₁	1	1	yes	characteristic
198	C₄	—	61	16	no	—
199	C₄	—	122	16	no	—
200	C₄	—	122	8	no	—
201	C₄	C₂⁴×Dic₆₁	1	16	yes	complemented
202	C₂	—	2	32	no	—
203	C₂	C₂²×D₄×Dic₆₁	1	28	yes	direct product factor
204	C₂	C₂⁵×Dic₆₁	1	1	yes	characteristic
205	C₂	C₂³×D₄×D₆₁	1	1	yes	characteristic
206	C₂	C₂³×#11758224	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₆₁
commutator	C₂×C₆₁	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₆₁
Fitting, prime radical	C₂⁴×D₄×C₆₁	C₂
socle	C₂⁵×C₆₁	C₂³
cosocle	C₂²×C₆₁	C₂⁶
hypercenter	C₂⁴×D₄	D₆₁
hypocenter	C₆₁	C₂³×C₄×D₄
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×D₄

Complements

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

N	H ≅ G/N	type
1. C₂⁵×Dic₆₁	C₂	semidirect
2. C₂²×D₄×Dic₆₁	C₂	direct (N × H)
3. C₂³×C₄⋊Dic₆₁	C₂	semidirect
4. C₂³×C₂²⋊Dic₆₁	C₂	semidirect
5. C₂³×C₄×Dic₆₁	C₂	semidirect
11. C₂⁴×Dic₆₁	C₂²	semidirect
12. C₂×D₄×Dic₆₁	C₂²	direct (N × H)
13. C₂²×C₄⋊Dic₆₁	C₂²	semidirect
14. C₂²×C₂²⋊Dic₆₁	C₂²	semidirect
15. C₂²×C₄×Dic₆₁	C₂²	semidirect
17. C₂³×D₄×C₆₁	C₄	semidirect
23. C₂³×Dic₆₁	D₄	direct (N × H)
27. C₂³×Dic₆₁	C₂³	semidirect
28. D₄×Dic₆₁	C₂³	direct (N × H)
29. C₂×C₄⋊Dic₆₁	C₂³	semidirect
30. C₂×C₂²⋊Dic₆₁	C₂³	semidirect
31. C₂×C₄×Dic₆₁	C₂³	semidirect
34. C₂⁵×C₆₁	C₂×C₄	semidirect
37. C₂²×D₄×C₆₁	C₂×C₄	semidirect
39. C₂³×C₄×C₆₁	C₂×C₄	semidirect
45. C₂⁴×C₆₁	C₂²×C₄	semidirect
50. C₂×D₄×C₆₁	C₂²×C₄	semidirect
52. C₂²×C₄×C₆₁	C₂²×C₄	semidirect
57. C₂²×Dic₆₁	C₂×D₄	direct (N × H)
61. C₂²×Dic₆₁	C₂⁴	semidirect
62. C₂²⋊Dic₆₁	C₂⁴	semidirect
63. C₄⋊Dic₆₁	C₂⁴	semidirect
64. C₄×Dic₆₁	C₂⁴	semidirect
68. C₂³×C₆₁	C₄×D₄	semidirect
70. C₂³×C₆₁	C₂³×C₄	semidirect
75. D₄×C₆₁	C₂³×C₄	semidirect
76. C₂×C₄×C₆₁	C₂³×C₄	semidirect
81. C₂×Dic₆₁	C₂²×D₄	direct (N × H)
89. C₂²×C₆₁	C₂×C₄×D₄	semidirect
91. C₂²×C₆₁	C₂⁴×C₄	semidirect
95. C₄×C₆₁	C₂⁴×C₄	semidirect
98. Dic₆₁	C₂³×D₄	direct (N × H)
106. C₂×C₆₁	C₂²×C₄×D₄	semidirect
111. C₂³×D₄	Dic₆₁	direct (N × H)
123. C₆₁	C₂³×C₄×D₄	semidirect
126. C₂⁵	C₂×Dic₆₁	semidirect
129. C₂²×D₄	C₂×Dic₆₁	direct (N × H)
139. C₂³×C₄	C₂×Dic₆₁	semidirect
151. C₂⁴	C₂²×Dic₆₁	semidirect
154. C₂×D₄	C₂²×Dic₆₁	direct (N × H)
164. C₂²×C₄	C₂²×Dic₆₁	semidirect
172. C₂³	D₄×Dic₆₁	direct (N × H)
174. C₂³	C₂³×Dic₆₁	semidirect
179. D₄	C₂³×Dic₆₁	direct (N × H)
187. C₂×C₄	C₂³×Dic₆₁	semidirect
192. C₂²	C₂×D₄×Dic₆₁	direct (N × H)
194. C₂²	C₂⁴×Dic₆₁	semidirect
201. C₄	C₂⁴×Dic₆₁	semidirect
203. C₂	C₂²×D₄×Dic₆₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×Dic₆₁	2
C₂²×D₄×Dic₆₁	2
C₂³×C₄⋊Dic₆₁	2
C₂³×C₂²⋊Dic₆₁	2
C₂³×C₄×Dic₆₁	2
C₂⁴×D₄×C₆₁	2
C₂³×C₄×D₄	61

Subgroups — C23×D4×Dic61 #30586578

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₄×Dic₆₁ #30586578