Subgroups — C₂×C₈×C₃₂×C₆₁ #72208024

Order 31232.

516 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₃₂×C₆₁	C₂	1	2	yes	maximal
2	C₂×C₈×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₈×C₃₂×C₆₁	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₃₂×C₆₁	C₄	1	4	yes	—
5	C₂×C₄×C₁₆×C₆₁	C₄	1	1	yes	characteristic
6	C₂×C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
7	C₂×C₈²×C₆₁	C₄	1	1	yes	characteristic
8	C₄×C₃₂×C₆₁	C₄	1	4	yes	—
9	C₄×C₃₂×C₆₁	C₂²	1	4	yes	—
10	C₈×C₁₆×C₆₁	C₄	1	2	yes	—
11	C₈×C₁₆×C₆₁	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₆₁	C₈	1	2	yes	—
13	C₂²×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₈×C₆₁	C₈	1	2	yes	—
15	C₂×C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃₂×C₆₁	C₈	1	16	yes	direct product factor
17	C₂×C₃₂×C₆₁	C₂×C₄	1	8	yes	—
18	C₄×C₁₆×C₆₁	C₈	1	2	yes	—
19	C₄×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₄×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
21	C₄×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
22	C₄×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
23	C₈²×C₆₁	C₈	1	2	yes	—
24	C₈²×C₆₁	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₆₁	C₁₆	1	4	yes	—
26	C₂²×C₈×C₆₁	C₄²	1	1	yes	characteristic
27	C₂²×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
29	C₂×C₁₆×C₆₁	C₁₆	1	8	yes	—
30	C₂×C₁₆×C₆₁	C₄²	1	2	yes	—
31	C₂×C₁₆×C₆₁	C₂×C₈	1	4	yes	—
32	C₂×C₁₆×C₆₁	C₂×C₈	1	8	yes	—
33	C₂×C₁₆×C₆₁	C₂²×C₄	1	2	yes	—
34	C₄×C₈×C₆₁	C₁₆	1	4	yes	—
35	C₄×C₈×C₆₁	C₄²	1	1	yes	characteristic
36	C₄×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
37	C₄×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
38	C₄×C₈×C₆₁	C₂×C₈	1	4	yes	—
39	C₄×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₃₂×C₆₁	C₂×C₈	1	16	yes	direct product factor
41	C₂²×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	C₂×C₁₆	1	2	yes	—
43	C₂×C₈×C₆₁	C₃₂	1	16	yes	direct product factor
44	C₂×C₈×C₆₁	C₄×C₈	1	1	yes	characteristic
45	C₂×C₈×C₆₁	C₄×C₈	1	1	yes	characteristic
46	C₂×C₈×C₆₁	C₄×C₈	1	4	yes	—
47	C₂×C₈×C₆₁	C₂×C₁₆	1	4	yes	—
48	C₂×C₈×C₆₁	C₂×C₁₆	1	8	yes	—
49	C₂×C₈×C₆₁	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
51	C₄²×C₆₁	C₄×C₈	1	1	yes	characteristic
52	C₄²×C₆₁	C₂×C₁₆	1	2	yes	—
53	C₄²×C₆₁	C₂²×C₈	1	1	yes	characteristic
54	C₁₆×C₆₁	C₄×C₈	1	4	yes	—
55	C₁₆×C₆₁	C₂×C₁₆	1	8	yes	—
56	C₁₆×C₆₁	C₂²×C₈	1	4	yes	—
57	C₂×C₈×C₃₂	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂³×C₆₁	C₄×C₁₆	1	1	yes	characteristic
59	C₂×C₄×C₆₁	C₈²	1	2	yes	—
60	C₂×C₄×C₆₁	C₄×C₁₆	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₄×C₁₆	1	2	yes	—
62	C₂×C₄×C₆₁	C₄×C₁₆	1	2	yes	—
63	C₂×C₄×C₆₁	C₂×C₃₂	1	8	yes	—
64	C₂×C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
65	C₂×C₄×C₆₁	C₂²×C₁₆	1	2	yes	—
66	C₈×C₆₁	C₈²	1	2	yes	—
67	C₈×C₆₁	C₄×C₁₆	1	2	yes	—
68	C₈×C₆₁	C₂×C₃₂	1	16	yes	direct product factor
69	C₈×C₆₁	C₂×C₄×C₈	1	2	yes	—
70	C₈×C₆₁	C₂²×C₁₆	1	2	yes	—
71	C₂×C₄×C₃₂	C₂×C₆₁	1	2	yes	—
72	C₂×C₈×C₁₆	C₂×C₆₁	1	1	yes	characteristic
73	C₈×C₃₂	C₂×C₆₁	1	4	yes	direct product factor
74	C₂²×C₆₁	C₈×C₁₆	1	2	yes	—
75	C₂²×C₆₁	C₄×C₃₂	1	4	yes	—
76	C₂²×C₆₁	C₂×C₄×C₁₆	1	1	yes	characteristic
77	C₄×C₆₁	C₈×C₁₆	1	2	yes	—
78	C₄×C₆₁	C₄×C₃₂	1	4	yes	—
79	C₄×C₆₁	C₂×C₈²	1	1	yes	characteristic
80	C₄×C₆₁	C₂×C₄×C₁₆	1	1	yes	characteristic
81	C₄×C₆₁	C₂²×C₃₂	1	4	yes	—
82	C₂²×C₃₂	C₄×C₆₁	1	4	yes	—
83	C₂×C₄×C₁₆	C₄×C₆₁	1	1	yes	characteristic
84	C₂×C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
85	C₂×C₈²	C₄×C₆₁	1	1	yes	characteristic
86	C₄×C₃₂	C₄×C₆₁	1	4	yes	—
87	C₄×C₃₂	C₂²×C₆₁	1	4	yes	—
88	C₈×C₁₆	C₄×C₆₁	1	2	yes	—
89	C₈×C₁₆	C₂²×C₆₁	1	2	yes	—
90	C₂×C₆₁	C₈×C₃₂	1	4	yes	direct product factor
91	C₂×C₆₁	C₂×C₈×C₁₆	1	1	yes	characteristic
92	C₂×C₆₁	C₂×C₄×C₃₂	1	2	yes	—
93	C₂²×C₁₆	C₈×C₆₁	1	2	yes	—
94	C₂²×C₁₆	C₂×C₄×C₆₁	1	2	yes	—
95	C₂×C₄×C₈	C₈×C₆₁	1	2	yes	—
96	C₂×C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
97	C₂×C₃₂	C₈×C₆₁	1	16	yes	direct product factor
98	C₂×C₃₂	C₂×C₄×C₆₁	1	8	yes	—
99	C₄×C₁₆	C₈×C₆₁	1	2	yes	—
100	C₄×C₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
101	C₄×C₁₆	C₂×C₄×C₆₁	1	2	yes	—
102	C₄×C₁₆	C₂×C₄×C₆₁	1	2	yes	—
103	C₄×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
104	C₈²	C₈×C₆₁	1	2	yes	—
105	C₈²	C₂×C₄×C₆₁	1	2	yes	—
106	C₆₁	C₂×C₈×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
107	C₂²×C₈	C₁₆×C₆₁	1	4	yes	—
108	C₂²×C₈	C₄²×C₆₁	1	1	yes	characteristic
109	C₂²×C₈	C₂×C₈×C₆₁	1	1	yes	characteristic
110	C₂×C₄²	C₂×C₈×C₆₁	1	1	yes	characteristic
111	C₂×C₁₆	C₁₆×C₆₁	1	8	yes	—
112	C₂×C₁₆	C₄²×C₆₁	1	2	yes	—
113	C₂×C₁₆	C₂×C₈×C₆₁	1	4	yes	—
114	C₂×C₁₆	C₂×C₈×C₆₁	1	8	yes	—
115	C₂×C₁₆	C₂²×C₄×C₆₁	1	2	yes	—
116	C₄×C₈	C₁₆×C₆₁	1	4	yes	—
117	C₄×C₈	C₄²×C₆₁	1	1	yes	characteristic
118	C₄×C₈	C₂×C₈×C₆₁	1	1	yes	characteristic
119	C₄×C₈	C₂×C₈×C₆₁	1	1	yes	characteristic
120	C₄×C₈	C₂×C₈×C₆₁	1	4	yes	—
121	C₄×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
122	C₃₂	C₂×C₈×C₆₁	1	16	yes	direct product factor
123	C₂²×C₄	C₄×C₈×C₆₁	1	1	yes	characteristic
124	C₂²×C₄	C₂×C₁₆×C₆₁	1	2	yes	—
125	C₂×C₈	C₃₂×C₆₁	1	16	yes	direct product factor
126	C₂×C₈	C₄×C₈×C₆₁	1	1	yes	characteristic
127	C₂×C₈	C₄×C₈×C₆₁	1	1	yes	characteristic
128	C₂×C₈	C₄×C₈×C₆₁	1	4	yes	—
129	C₂×C₈	C₂×C₁₆×C₆₁	1	4	yes	—
130	C₂×C₈	C₂×C₁₆×C₆₁	1	8	yes	—
131	C₂×C₈	C₂×C₄²×C₆₁	1	1	yes	characteristic
132	C₂×C₈	C₂²×C₈×C₆₁	1	1	yes	characteristic
133	C₄²	C₄×C₈×C₆₁	1	1	yes	characteristic
134	C₄²	C₂×C₁₆×C₆₁	1	2	yes	—
135	C₄²	C₂²×C₈×C₆₁	1	1	yes	characteristic
136	C₁₆	C₄×C₈×C₆₁	1	4	yes	—
137	C₁₆	C₂×C₁₆×C₆₁	1	8	yes	—
138	C₁₆	C₂²×C₈×C₆₁	1	4	yes	—
139	C₂³	C₄×C₁₆×C₆₁	1	1	yes	characteristic
140	C₂×C₄	C₈²×C₆₁	1	2	yes	—
141	C₂×C₄	C₄×C₁₆×C₆₁	1	1	yes	characteristic
142	C₂×C₄	C₄×C₁₆×C₆₁	1	2	yes	—
143	C₂×C₄	C₄×C₁₆×C₆₁	1	2	yes	—
144	C₂×C₄	C₂×C₃₂×C₆₁	1	8	yes	—
145	C₂×C₄	C₂×C₄×C₈×C₆₁	1	1	yes	characteristic
146	C₂×C₄	C₂²×C₁₆×C₆₁	1	2	yes	—
147	C₈	C₈²×C₆₁	1	2	yes	—
148	C₈	C₄×C₁₆×C₆₁	1	2	yes	—
149	C₈	C₂×C₃₂×C₆₁	1	16	yes	direct product factor
150	C₈	C₂×C₄×C₈×C₆₁	1	2	yes	—
151	C₈	C₂²×C₁₆×C₆₁	1	2	yes	—
152	C₂²	C₈×C₁₆×C₆₁	1	2	yes	—
153	C₂²	C₄×C₃₂×C₆₁	1	4	yes	—
154	C₂²	C₂×C₄×C₁₆×C₆₁	1	1	yes	characteristic
155	C₄	C₈×C₁₆×C₆₁	1	2	yes	—
156	C₄	C₄×C₃₂×C₆₁	1	4	yes	—
157	C₄	C₂×C₈²×C₆₁	1	1	yes	characteristic
158	C₄	C₂×C₄×C₁₆×C₆₁	1	1	yes	characteristic
159	C₄	C₂²×C₃₂×C₆₁	1	4	yes	—
160	C₂	C₈×C₃₂×C₆₁	1	4	yes	direct product factor
161	C₂	C₂×C₈×C₁₆×C₆₁	1	1	yes	characteristic
162	C₂	C₂×C₄×C₃₂×C₆₁	1	2	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄×C₁₆	C₂³×C₆₁
socle, prime radical	C₂³×C₆₁	C₄×C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₈×C₃₂×C₆₁	C₂	direct (N × H)
16. C₂×C₃₂×C₆₁	C₈	direct (N × H)
40. C₃₂×C₆₁	C₂×C₈	direct (N × H)
43. C₂×C₈×C₆₁	C₃₂	direct (N × H)
57. C₂×C₈×C₃₂	C₆₁	direct (N × H)
68. C₈×C₆₁	C₂×C₃₂	direct (N × H)
73. C₈×C₃₂	C₂×C₆₁	direct (N × H)
90. C₂×C₆₁	C₈×C₃₂	direct (N × H)
97. C₂×C₃₂	C₈×C₆₁	direct (N × H)
106. C₆₁	C₂×C₈×C₃₂	direct (N × H)
122. C₃₂	C₂×C₈×C₆₁	direct (N × H)
125. C₂×C₈	C₃₂×C₆₁	direct (N × H)
149. C₈	C₂×C₃₂×C₆₁	direct (N × H)
160. C₂	C₈×C₃₂×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
C₂×C₄×C₃₂×C₆₁	2
C₂×C₈×C₁₆×C₆₁	2
C₈×C₃₂×C₆₁	2
C₂×C₈×C₃₂	61

Subgroups — C2×C8×C32×C61 #72208024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×C₃₂×C₆₁ #72208024