Subgroups — C₂³×#124×C₆₁ #29566093

Order 15616.

14334 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×D₄×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×#124×C₆₁	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₂²⋊C₄×C₆₁	C₂	1	4	yes	maximal, complemented
5	C₂³×C₄²×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁶×C₆₁	C₂²	1	2	yes	—
7	C₂³×Q₈×C₆₁	—	2	16	no	—
8	C₂³×Q₈×C₆₁	C₂²	1	28	yes	complemented
9	C₂³×D₄×C₆₁	—	2	16	no	—
10	C₂³×D₄×C₆₁	C₂²	1	28	yes	—
11	C₂⁴×C₄×C₆₁	C₂²	1	2	yes	—
12	C₂⁴×C₄×C₆₁	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#124×C₆₁	C₂²	1	448	yes	direct product factor
15	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	112	yes	complemented
16	C₂²×C₄²×C₆₁	C₂²	1	28	yes	complemented
17	C₂⁵×C₆₁	—	2	16	no	—
18	C₂⁵×C₆₁	—	2	32	no	—
19	C₂⁵×C₆₁	C₂³	1	28	yes	—
20	C₂⁵×C₆₁	C₂³	1	1	yes	characteristic
21	C₂²×Q₈×C₆₁	—	2	224	no	—
22	C₂²×Q₈×C₆₁	C₂³	1	112	yes	complemented
23	C₂²×D₄×C₆₁	—	2	224	no	—
24	C₂²×D₄×C₆₁	C₂³	1	112	yes	—
25	C₂³×C₄×C₆₁	—	2	16	no	—
26	C₂³×C₄×C₆₁	—	2	16	no	—
27	C₂³×C₄×C₆₁	D₄	1	16	yes	—
28	C₂³×C₄×C₆₁	C₂³	1	28	yes	—
29	C₂³×C₄×C₆₁	C₂³	1	28	yes	—
30	C₂³×C₄×C₆₁	C₂³	1	14	yes	—
31	#124×C₆₁	C₂³	1	512	yes	direct product factor
32	C₂×C₂²⋊C₄×C₆₁	C₂³	1	448	yes	complemented
33	C₂×C₄²×C₆₁	C₂³	1	112	yes	complemented
34	C₂⁴×C₆₁	—	2	112	no	—
35	C₂⁴×C₆₁	—	2	224	no	—
36	C₂⁴×C₆₁	—	4	128	no	—
37	C₂⁴×C₆₁	C₂×D₄	1	8	yes	—
38	C₂⁴×C₆₁	#41	1	16	yes	—
39	C₂⁴×C₆₁	C₂⁴	1	56	yes	—
40	C₂⁴×C₆₁	C₂⁴	1	7	yes	—
41	C₂×Q₈×C₆₁	—	2	448	no	—
42	C₂×Q₈×C₆₁	C₂⁴	1	64	yes	complemented
43	C₂×D₄×C₆₁	—	2	448	no	—
44	C₂×D₄×C₆₁	C₂⁴	1	64	yes	—
45	C₂²×C₄×C₆₁	—	2	112	no	—
46	C₂²×C₄×C₆₁	—	2	112	no	—
47	C₂²×C₄×C₆₁	C₂×D₄	1	112	yes	—
48	C₂²×C₄×C₆₁	C₂⁴	1	56	yes	—
49	C₂²×C₄×C₆₁	C₂⁴	1	56	yes	—
50	C₂²×C₄×C₆₁	C₂⁴	1	28	yes	—
51	C₂²⋊C₄×C₆₁	C₂⁴	1	256	yes	complemented
52	C₄²×C₆₁	C₂⁴	1	64	yes	complemented
53	C₂³×C₆₁	—	2	112	no	—
54	C₂³×C₆₁	—	2	224	no	—
55	C₂³×C₆₁	—	4	448	no	—
56	C₂³×C₆₁	#124	1	64	yes	direct product factor
57	C₂³×C₆₁	C₂²×D₄	1	28	yes	—
58	C₂³×C₆₁	C₂×#41	1	56	yes	—
59	C₂³×C₆₁	C₂⁵	1	16	yes	—
60	C₂³×C₆₁	C₂⁵	1	7	yes	—
61	Q₈×C₆₁	—	2	128	no	—
62	D₄×C₆₁	—	2	128	no	—
63	C₂×C₄×C₆₁	—	2	112	no	—
64	C₂×C₄×C₆₁	—	2	112	no	—
65	C₂×C₄×C₆₁	C₂²×D₄	1	112	yes	—
66	C₂×C₄×C₆₁	C₂⁵	1	16	yes	—
67	C₂×C₄×C₆₁	C₂⁵	1	16	yes	—
68	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
69	C₂³×#124	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂²×C₆₁	—	2	16	no	—
71	C₂²×C₆₁	—	2	32	no	—
72	C₂²×C₆₁	—	4	224	no	—
73	C₂²×C₆₁	C₂×#124	1	112	yes	direct product factor
74	C₂²×C₆₁	C₂³×D₄	1	14	yes	—
75	C₂²×C₆₁	C₂²×#41	1	28	yes	—
76	C₂²×C₆₁	C₂⁶	1	1	yes	characteristic
77	C₄×C₆₁	—	2	16	no	—
78	C₄×C₆₁	—	2	16	no	—
79	C₄×C₆₁	C₂³×D₄	1	16	yes	—
80	C₂⁴×Q₈	C₂×C₆₁	1	1	yes	characteristic, complemented
81	C₂⁴×D₄	C₂×C₆₁	1	1	yes	characteristic
82	C₂²×#124	C₂×C₆₁	1	56	yes	direct product factor
83	C₂³×C₂²⋊C₄	C₂×C₆₁	1	4	yes	complemented
84	C₂³×C₄²	C₂×C₆₁	1	1	yes	characteristic, complemented
85	C₂×C₆₁	—	4	16	no	—
86	C₂×C₆₁	C₂²×#124	1	28	yes	direct product factor
87	C₂×C₆₁	C₂⁴×D₄	1	1	yes	characteristic
88	C₂×C₆₁	C₂³×#41	1	2	yes	—
89	C₂⁶	C₂²×C₆₁	1	2	yes	—
90	C₂³×Q₈	—	2	16	no	—
91	C₂³×Q₈	C₂²×C₆₁	1	28	yes	complemented
92	C₂³×D₄	—	2	16	no	—
93	C₂³×D₄	C₂²×C₆₁	1	28	yes	—
94	C₂⁴×C₄	C₂²×C₆₁	1	2	yes	—
95	C₂⁴×C₄	C₂²×C₆₁	1	2	yes	—
96	C₂⁴×C₄	C₂²×C₆₁	1	1	yes	characteristic
97	C₂×#124	C₂²×C₆₁	1	448	yes	direct product factor
98	C₂²×C₂²⋊C₄	C₂²×C₆₁	1	112	yes	complemented
99	C₂²×C₄²	C₂²×C₆₁	1	28	yes	complemented
100	C₆₁	C₂³×#124	1	1	yes	characteristic, direct product factor, Sylow
101	C₂⁵	—	2	16	no	—
102	C₂⁵	—	2	32	no	—
103	C₂⁵	C₂³×C₆₁	1	28	yes	—
104	C₂⁵	C₂³×C₆₁	1	1	yes	characteristic
105	C₂²×Q₈	—	2	224	no	—
106	C₂²×Q₈	C₂³×C₆₁	1	112	yes	complemented
107	C₂²×D₄	—	2	224	no	—
108	C₂²×D₄	C₂³×C₆₁	1	112	yes	—
109	C₂³×C₄	—	2	16	no	—
110	C₂³×C₄	—	2	16	no	—
111	C₂³×C₄	D₄×C₆₁	1	16	yes	—
112	C₂³×C₄	C₂³×C₆₁	1	28	yes	—
113	C₂³×C₄	C₂³×C₆₁	1	28	yes	—
114	C₂³×C₄	C₂³×C₆₁	1	14	yes	—
115	#124	C₂³×C₆₁	1	512	yes	direct product factor
116	C₂×C₂²⋊C₄	C₂³×C₆₁	1	448	yes	complemented
117	C₂×C₄²	C₂³×C₆₁	1	112	yes	complemented
118	C₂⁴	—	2	112	no	—
119	C₂⁴	—	2	224	no	—
120	C₂⁴	—	4	128	no	—
121	C₂⁴	C₂×D₄×C₆₁	1	8	yes	—
122	C₂⁴	#41×C₆₁	1	16	yes	—
123	C₂⁴	C₂⁴×C₆₁	1	56	yes	—
124	C₂⁴	C₂⁴×C₆₁	1	7	yes	—
125	C₂×Q₈	—	2	448	no	—
126	C₂×Q₈	C₂⁴×C₆₁	1	64	yes	complemented
127	C₂×D₄	—	2	448	no	—
128	C₂×D₄	C₂⁴×C₆₁	1	64	yes	—
129	C₂²×C₄	—	2	112	no	—
130	C₂²×C₄	—	2	112	no	—
131	C₂²×C₄	C₂×D₄×C₆₁	1	112	yes	—
132	C₂²×C₄	C₂⁴×C₆₁	1	56	yes	—
133	C₂²×C₄	C₂⁴×C₆₁	1	56	yes	—
134	C₂²×C₄	C₂⁴×C₆₁	1	28	yes	—
135	C₂²⋊C₄	C₂⁴×C₆₁	1	256	yes	complemented
136	C₄²	C₂⁴×C₆₁	1	64	yes	complemented
137	C₂³	—	2	112	no	—
138	C₂³	—	2	224	no	—
139	C₂³	—	4	448	no	—
140	C₂³	#124×C₆₁	1	64	yes	direct product factor
141	C₂³	C₂²×D₄×C₆₁	1	28	yes	—
142	C₂³	C₂×#41×C₆₁	1	56	yes	—
143	C₂³	C₂⁵×C₆₁	1	16	yes	—
144	C₂³	C₂⁵×C₆₁	1	7	yes	—
145	Q₈	—	2	128	no	—
146	D₄	—	2	128	no	—
147	C₂×C₄	—	2	112	no	—
148	C₂×C₄	—	2	112	no	—
149	C₂×C₄	C₂²×D₄×C₆₁	1	112	yes	—
150	C₂×C₄	C₂⁵×C₆₁	1	16	yes	—
151	C₂×C₄	C₂⁵×C₆₁	1	16	yes	—
152	C₂×C₄	C₂⁵×C₆₁	1	8	yes	—
153	C₂²	—	2	16	no	—
154	C₂²	—	2	32	no	—
155	C₂²	—	4	224	no	—
156	C₂²	C₂×#124×C₆₁	1	112	yes	direct product factor
157	C₂²	C₂³×D₄×C₆₁	1	14	yes	—
158	C₂²	C₂²×#41×C₆₁	1	28	yes	—
159	C₂²	C₂⁶×C₆₁	1	1	yes	characteristic
160	C₄	—	2	16	no	—
161	C₄	—	2	16	no	—
162	C₄	C₂³×D₄×C₆₁	1	16	yes	—
163	C₂	—	4	16	no	—
164	C₂	C₂²×#124×C₆₁	1	28	yes	direct product factor
165	C₂	C₂⁴×D₄×C₆₁	1	1	yes	characteristic
166	C₂	C₂³×#41×C₆₁	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂⁴×Q₈×C₆₁	C₂	semidirect
3. C₂²×#124×C₆₁	C₂	direct (N × H)
4. C₂³×C₂²⋊C₄×C₆₁	C₂	semidirect
5. C₂³×C₄²×C₆₁	C₂	semidirect
8. C₂³×Q₈×C₆₁	C₂²	semidirect
14. C₂×#124×C₆₁	C₂²	direct (N × H)
15. C₂²×C₂²⋊C₄×C₆₁	C₂²	semidirect
16. C₂²×C₄²×C₆₁	C₂²	semidirect
22. C₂²×Q₈×C₆₁	C₂³	semidirect
31. #124×C₆₁	C₂³	direct (N × H)
32. C₂×C₂²⋊C₄×C₆₁	C₂³	semidirect
33. C₂×C₄²×C₆₁	C₂³	semidirect
42. C₂×Q₈×C₆₁	C₂⁴	semidirect
51. C₂²⋊C₄×C₆₁	C₂⁴	semidirect
52. C₄²×C₆₁	C₂⁴	semidirect
56. C₂³×C₆₁	#124	direct (N × H)
69. C₂³×#124	C₆₁	direct (N × H)
73. C₂²×C₆₁	C₂×#124	direct (N × H)
80. C₂⁴×Q₈	C₂×C₆₁	semidirect
82. C₂²×#124	C₂×C₆₁	direct (N × H)
83. C₂³×C₂²⋊C₄	C₂×C₆₁	semidirect
84. C₂³×C₄²	C₂×C₆₁	semidirect
86. C₂×C₆₁	C₂²×#124	direct (N × H)
91. C₂³×Q₈	C₂²×C₆₁	semidirect
97. C₂×#124	C₂²×C₆₁	direct (N × H)
98. C₂²×C₂²⋊C₄	C₂²×C₆₁	semidirect
99. C₂²×C₄²	C₂²×C₆₁	semidirect
100. C₆₁	C₂³×#124	direct (N × H)
106. C₂²×Q₈	C₂³×C₆₁	semidirect
115. #124	C₂³×C₆₁	direct (N × H)
116. C₂×C₂²⋊C₄	C₂³×C₆₁	semidirect
117. C₂×C₄²	C₂³×C₆₁	semidirect
126. C₂×Q₈	C₂⁴×C₆₁	semidirect
135. C₂²⋊C₄	C₂⁴×C₆₁	semidirect
136. C₄²	C₂⁴×C₆₁	semidirect
140. C₂³	#124×C₆₁	direct (N × H)
156. C₂²	C₂×#124×C₆₁	direct (N × H)
164. C₂	C₂²×#124×C₆₁	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×Q₈×C₆₁	2
C₂⁴×D₄×C₆₁	2
C₂²×#124×C₆₁	2
C₂³×C₂²⋊C₄×C₆₁	2
C₂³×C₄²×C₆₁	2
C₂³×#124	61

Subgroups — C23×#124×C61 #29566093

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#124×C₆₁ #29566093