Subgroups — C₂³×#14519735 #30586572

Order 15616.

14334 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×D₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14519735	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×#11758198	C₂	1	2	yes	maximal
5	C₂³×C₄×Dic₆₁	C₂	1	2	yes	maximal, complemented
6	C₂³×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₆₁	C₂²	1	1	yes	characteristic
8	C₂⁴×Dic₆₁	C₂²	1	2	yes	—
9	C₂⁴×Dic₆₁	C₂²	1	1	yes	characteristic
10	C₂³×C₄×D₆₁	C₄	1	32	yes	complemented
11	C₂³×C₄×D₆₁	C₂²	1	28	yes	—
12	C₂×#14519735	C₂²	1	448	yes	direct product factor
13	C₂²×C₄⋊Dic₆₁	C₂²	1	28	yes	complemented
14	C₂²×#11758198	C₂²	1	56	yes	—
15	C₂²×C₄×Dic₆₁	C₂²	1	56	yes	complemented
16	C₂⁴×C₄×C₆₁	C₂²	1	2	yes	—
17	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₆₁	C₂²	1	28	yes	complemented
19	C₂⁴×D₆₁	—	2	8	no	—
20	C₂⁴×D₆₁	—	2	8	no	—
21	C₂⁴×D₆₁	C₂×C₄	1	16	yes	—
22	C₂⁴×D₆₁	C₂³	1	14	yes	—
23	C₂³×Dic₆₁	—	2	16	no	—
24	C₂³×Dic₆₁	C₂×C₄	1	16	yes	—
25	C₂³×Dic₆₁	C₂³	1	28	yes	—
26	C₂³×Dic₆₁	C₂³	1	14	yes	—
27	C₂²×C₄×D₆₁	C₂×C₄	1	448	yes	complemented
28	C₂²×C₄×D₆₁	C₂³	1	112	yes	—
29	#14519735	C₂³	1	512	yes	direct product factor
30	C₂×C₄⋊Dic₆₁	C₂³	1	112	yes	complemented
31	C₂×#11758198	C₂³	1	224	yes	—
32	C₂×C₄×Dic₆₁	C₂³	1	224	yes	complemented
33	C₂⁵×C₆₁	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₆₁	—	2	16	no	—
35	C₂³×C₄×C₆₁	C₂×C₄	1	16	yes	—
36	C₂³×C₄×C₆₁	C₂³	1	28	yes	—
37	C₂³×C₄×C₆₁	C₂³	1	14	yes	—
38	C₂×C₄⋊C₄×C₆₁	C₂³	1	112	yes	complemented
39	C₂⁴×C₆₁	C₂²×C₄	1	8	yes	—
40	C₂⁴×C₆₁	#41	1	8	yes	—
41	C₂⁴×C₆₁	#41	1	8	yes	—
42	C₂⁴×C₆₁	C₂⁴	1	7	yes	—
43	C₂³×D₆₁	—	2	128	no	—
44	C₂³×D₆₁	—	2	56	no	—
45	C₂³×D₆₁	—	2	56	no	—
46	C₂³×D₆₁	C₂²×C₄	1	112	yes	—
47	C₂³×D₆₁	C₂⁴	1	28	yes	—
48	C₂²×C₄×C₆₁	—	2	112	no	—
49	C₂²×C₄×C₆₁	C₂²×C₄	1	112	yes	—
50	C₂²×C₄×C₆₁	C₂⁴	1	56	yes	—
51	C₂²×C₄×C₆₁	C₂⁴	1	28	yes	—
52	C₂²×Dic₆₁	—	2	112	no	—
53	C₂²×Dic₆₁	C₂²×C₄	1	112	yes	—
54	C₂²×Dic₆₁	C₂⁴	1	56	yes	—
55	C₂²×Dic₆₁	C₂⁴	1	28	yes	—
56	C₂×C₄×D₆₁	C₂²×C₄	1	896	yes	complemented
57	C₂×C₄×D₆₁	C₂⁴	1	64	yes	—
58	C₄⋊C₄×C₆₁	C₂⁴	1	64	yes	complemented
59	#11758198	C₂⁴	1	128	yes	—
60	C₄⋊Dic₆₁	C₂⁴	1	64	yes	complemented
61	C₄×Dic₆₁	C₂⁴	1	128	yes	complemented
62	C₂³×C₆₁	#117	1	64	yes	complemented
63	C₂³×C₆₁	C₂³×C₄	1	28	yes	—
64	C₂³×C₆₁	C₂×#41	1	28	yes	—
65	C₂³×C₆₁	C₂×#41	1	28	yes	—
66	C₂³×C₆₁	C₂⁵	1	7	yes	—
67	C₂²×D₆₁	—	2	448	no	—
68	C₂²×D₆₁	—	2	56	no	—
69	C₂²×D₆₁	—	2	56	no	—
70	C₂²×D₆₁	C₂³×C₄	1	112	yes	—
71	C₂²×D₆₁	C₂⁵	1	8	yes	—
72	C₂×C₄×C₆₁	—	2	112	no	—
73	C₂×C₄×C₆₁	C₂³×C₄	1	112	yes	—
74	C₂×C₄×C₆₁	C₂⁵	1	16	yes	—
75	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
76	C₂×Dic₆₁	—	2	112	no	—
77	C₂×Dic₆₁	C₂³×C₄	1	112	yes	—
78	C₂×Dic₆₁	C₂⁵	1	16	yes	—
79	C₂×Dic₆₁	C₂⁵	1	8	yes	—
80	C₄×D₆₁	C₂³×C₄	1	256	yes	complemented
81	C₂³×#117	—	61	1	no	maximal, Sylow
82	C₂²×C₆₁	C₂×#117	1	112	yes	complemented
83	C₂²×C₆₁	C₂⁴×C₄	1	14	yes	—
84	C₂²×C₆₁	C₂²×#41	1	14	yes	—
85	C₂²×C₆₁	C₂²×#41	1	14	yes	—
86	C₂²×C₆₁	C₂⁶	1	1	yes	characteristic
87	C₂×D₆₁	—	2	8	no	—
88	C₂×D₆₁	—	2	8	no	—
89	C₂×D₆₁	—	2	224	no	—
90	C₂×D₆₁	C₂⁴×C₄	1	16	yes	—
91	C₄×C₆₁	—	2	16	no	—
92	C₄×C₆₁	C₂⁴×C₄	1	16	yes	—
93	Dic₆₁	—	2	16	no	—
94	Dic₆₁	C₂⁴×C₄	1	16	yes	—
95	C₂⁵×C₄	—	61	1	no	—
96	C₂²×#117	—	61	56	no	—
97	C₂³×C₄⋊C₄	—	61	1	no	—
98	C₂³×C₄⋊C₄	D₆₁	1	1	yes	characteristic, complemented
99	C₂³×C₂²⋊C₄	—	61	2	no	—
100	C₂³×C₄²	—	61	2	no	—
101	C₂×C₆₁	C₂²×#117	1	28	yes	complemented
102	C₂×C₆₁	C₂⁵×C₄	1	1	yes	characteristic
103	C₂×C₆₁	C₂³×#41	1	1	yes	characteristic
104	C₂×C₆₁	C₂³×#41	1	1	yes	characteristic
105	D₆₁	—	2	16	no	—
106	C₂⁶	—	61	1	no	—
107	C₂⁴×C₄	—	61	1	no	—
108	C₂⁴×C₄	—	61	32	no	—
109	C₂⁴×C₄	—	61	28	no	—
110	C₂⁴×C₄	—	61	2	no	—
111	C₂⁴×C₄	C₂×D₆₁	1	2	yes	—
112	C₂⁴×C₄	C₂×D₆₁	1	1	yes	characteristic
113	C₂×#117	—	61	448	no	—
114	C₂²×C₄⋊C₄	—	61	28	no	—
115	C₂²×C₄⋊C₄	C₂×D₆₁	1	28	yes	complemented
116	C₂²×C₂²⋊C₄	—	61	56	no	—
117	C₂²×C₄²	—	61	56	no	—
118	C₆₁	C₂³×#117	1	1	yes	characteristic, complemented, Sylow
119	C₂⁵	—	61	16	no	—
120	C₂⁵	—	61	14	no	—
121	C₂⁵	—	122	8	no	—
122	C₂⁵	—	122	8	no	—
123	C₂⁵	C₂²×D₆₁	1	1	yes	characteristic
124	C₂³×C₄	—	2	16	no	—
125	C₂³×C₄	—	61	16	no	—
126	C₂³×C₄	—	61	14	no	—
127	C₂³×C₄	—	61	448	no	—
128	C₂³×C₄	—	61	112	no	—
129	C₂³×C₄	—	61	28	no	—
130	C₂³×C₄	—	122	16	no	—
131	C₂³×C₄	C₄×D₆₁	1	16	yes	—
132	C₂³×C₄	C₂²×D₆₁	1	28	yes	—
133	C₂³×C₄	C₂²×D₆₁	1	14	yes	—
134	#117	—	61	512	no	—
135	C₂×C₄⋊C₄	—	61	112	no	—
136	C₂×C₄⋊C₄	C₂²×D₆₁	1	112	yes	complemented
137	C₂×C₂²⋊C₄	—	61	224	no	—
138	C₂×C₄²	—	61	224	no	—
139	C₂⁴	—	61	112	no	—
140	C₂⁴	—	61	28	no	—
141	C₂⁴	—	122	56	no	—
142	C₂⁴	—	122	56	no	—
143	C₂⁴	—	122	128	no	—
144	C₂⁴	C₂×C₄×D₆₁	1	8	yes	—
145	C₂⁴	#11758224	1	8	yes	—
146	C₂⁴	#11758226	1	8	yes	—
147	C₂⁴	C₂³×D₆₁	1	7	yes	—
148	C₂²×C₄	—	2	112	no	—
149	C₂²×C₄	—	61	112	no	—
150	C₂²×C₄	—	61	28	no	—
151	C₂²×C₄	—	61	896	no	—
152	C₂²×C₄	—	61	64	no	—
153	C₂²×C₄	—	61	56	no	—
154	C₂²×C₄	—	122	112	no	—
155	C₂²×C₄	C₂×C₄×D₆₁	1	112	yes	—
156	C₂²×C₄	C₂³×D₆₁	1	56	yes	—
157	C₂²×C₄	C₂³×D₆₁	1	28	yes	—
158	C₄⋊C₄	—	61	64	no	—
159	C₄⋊C₄	C₂³×D₆₁	1	64	yes	complemented
160	C₂²⋊C₄	—	61	128	no	—
161	C₄²	—	61	128	no	—
162	C₂³	—	61	112	no	—
163	C₂³	—	61	8	no	—
164	C₂³	—	122	448	no	—
165	C₂³	—	122	56	no	—
166	C₂³	—	122	56	no	—
167	C₂³	#14519735	1	64	yes	direct product factor
168	C₂³	C₂²×C₄×D₆₁	1	28	yes	—
169	C₂³	C₂×#11758224	1	28	yes	—
170	C₂³	C₂×#11758226	1	28	yes	—
171	C₂³	C₂⁴×D₆₁	1	7	yes	—
172	C₂×C₄	—	2	112	no	—
173	C₂×C₄	—	61	112	no	—
174	C₂×C₄	—	61	8	no	—
175	C₂×C₄	—	61	256	no	—
176	C₂×C₄	—	61	16	no	—
177	C₂×C₄	—	122	112	no	—
178	C₂×C₄	C₂²×C₄×D₆₁	1	112	yes	—
179	C₂×C₄	C₂⁴×D₆₁	1	16	yes	—
180	C₂×C₄	C₂⁴×D₆₁	1	8	yes	—
181	C₂²	—	61	16	no	—
182	C₂²	—	122	224	no	—
183	C₂²	—	122	8	no	—
184	C₂²	—	122	8	no	—
185	C₂²	C₂×#14519735	1	112	yes	direct product factor
186	C₂²	C₂³×C₄×D₆₁	1	14	yes	—
187	C₂²	C₂²×#11758224	1	14	yes	—
188	C₂²	C₂²×#11758226	1	14	yes	—
189	C₂²	C₂⁵×D₆₁	1	1	yes	characteristic
190	C₄	—	2	16	no	—
191	C₄	—	61	16	no	—
192	C₄	—	122	16	no	—
193	C₄	C₂³×C₄×D₆₁	1	16	yes	—
194	C₂	—	122	16	no	—
195	C₂	C₂²×#14519735	1	28	yes	direct product factor
196	C₂	C₂⁴×C₄×D₆₁	1	1	yes	characteristic
197	C₂	C₂³×#11758224	1	1	yes	characteristic
198	C₂	C₂³×#11758226	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₆₁
commutator	C₂×C₆₁	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₆₁
Fitting	C₂³×C₄⋊C₄×C₆₁	C₂
socle	C₂⁵×C₆₁	C₂³
prime radical	C₂⁵×D₆₁	C₂²
cosocle	C₂²×C₆₁	C₂⁶
hypercenter	C₂³×C₄⋊C₄	D₆₁
hypocenter	C₆₁	C₂³×#117
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×#14519735	C₂	direct (N × H)
3. C₂³×C₄⋊Dic₆₁	C₂	semidirect
5. C₂³×C₄×Dic₆₁	C₂	semidirect
6. C₂³×C₄⋊C₄×C₆₁	C₂	semidirect
10. C₂³×C₄×D₆₁	C₄	semidirect
12. C₂×#14519735	C₂²	direct (N × H)
13. C₂²×C₄⋊Dic₆₁	C₂²	semidirect
15. C₂²×C₄×Dic₆₁	C₂²	semidirect
18. C₂²×C₄⋊C₄×C₆₁	C₂²	semidirect
27. C₂²×C₄×D₆₁	C₂×C₄	semidirect
29. #14519735	C₂³	direct (N × H)
30. C₂×C₄⋊Dic₆₁	C₂³	semidirect
32. C₂×C₄×Dic₆₁	C₂³	semidirect
38. C₂×C₄⋊C₄×C₆₁	C₂³	semidirect
56. C₂×C₄×D₆₁	C₂²×C₄	semidirect
58. C₄⋊C₄×C₆₁	C₂⁴	semidirect
60. C₄⋊Dic₆₁	C₂⁴	semidirect
61. C₄×Dic₆₁	C₂⁴	semidirect
62. C₂³×C₆₁	#117	semidirect
80. C₄×D₆₁	C₂³×C₄	semidirect
82. C₂²×C₆₁	C₂×#117	semidirect
98. C₂³×C₄⋊C₄	D₆₁	semidirect
101. C₂×C₆₁	C₂²×#117	semidirect
115. C₂²×C₄⋊C₄	C₂×D₆₁	semidirect
118. C₆₁	C₂³×#117	semidirect
136. C₂×C₄⋊C₄	C₂²×D₆₁	semidirect
159. C₄⋊C₄	C₂³×D₆₁	semidirect
167. C₂³	#14519735	direct (N × H)
185. C₂²	C₂×#14519735	direct (N × H)
195. C₂	C₂²×#14519735	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₄×D₆₁	2
C₂²×#14519735	2
C₂³×C₄⋊Dic₆₁	2
C₂³×#11758198	2
C₂³×C₄×Dic₆₁	2
C₂³×C₄⋊C₄×C₆₁	2
C₂³×#117	61

Subgroups — C23×#14519735 #30586572

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14519735 #30586572