Subgroups — C₂×C₆₁×C₂²⋊C₃₂ #29516985

Order 15616.

342 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	C₂³×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₂×C₆₁	C₂	1	2	yes	maximal, complemented
3	C₆₁×C₂²⋊C₃₂	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₈×C₆₁	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₆₁	—	2	4	no	—
6	C₂²×C₁₆×C₆₁	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₆₁	C₄	1	2	yes	—
8	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
9	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×C₃₂×C₆₁	—	2	4	no	—
11	C₂×C₃₂×C₆₁	C₂²	1	4	yes	complemented
12	C₂³×C₄×C₆₁	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₆₁	—	2	2	no	—
14	C₂²×C₈×C₆₁	—	2	2	no	—
15	C₂²×C₈×C₆₁	C₈	1	1	yes	characteristic
16	C₂²×C₈×C₆₁	C₈	1	2	yes	—
17	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₆₁	—	2	2	no	—
20	C₂×C₁₆×C₆₁	—	2	4	no	—
21	C₂×C₁₆×C₆₁	—	2	4	no	—
22	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₆₁	D₄	1	4	yes	—
25	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
26	C₃₂×C₆₁	—	2	4	no	—
27	C₂⁴×C₆₁	C₁₆	1	1	yes	characteristic
28	C₂²×C₄×C₆₁	—	2	2	no	—
29	C₂²×C₄×C₆₁	—	2	2	no	—
30	C₂²×C₄×C₆₁	C₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₆₁	C₁₆	1	2	yes	—
32	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
33	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₆₁	—	2	2	no	—
35	C₂×C₈×C₆₁	—	2	4	no	—
36	C₂×C₈×C₆₁	—	2	2	no	—
37	C₂×C₈×C₆₁	—	2	2	no	—
38	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
39	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
42	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₁₆×C₆₁	—	2	2	no	—
44	C₁₆×C₆₁	C₂²⋊C₄	1	2	yes	—
45	C₁₆×C₆₁	C₂×D₄	1	2	yes	—
46	C₂³×C₆₁	—	2	2	no	—
47	C₂³×C₆₁	—	2	2	no	—
48	C₂³×C₆₁	C₃₂	1	4	yes	complemented
49	C₂³×C₆₁	C₂×C₁₆	1	2	yes	—
50	C₂³×C₆₁	C₂×C₁₆	1	1	yes	characteristic
51	C₂×C₄×C₆₁	—	2	2	no	—
52	C₂×C₄×C₆₁	—	2	4	no	—
53	C₂×C₄×C₆₁	—	2	2	no	—
54	C₂×C₄×C₆₁	—	2	2	no	—
55	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
56	C₂×C₄×C₆₁	C₂×C₁₆	1	2	yes	—
57	C₂×C₄×C₆₁	C₂×C₁₆	1	1	yes	characteristic
58	C₂×C₄×C₆₁	M₃₂	1	2	yes	—
59	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
60	C₈×C₆₁	—	2	2	no	—
61	C₈×C₆₁	C₂²⋊C₈	1	2	yes	—
62	C₈×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₈×C₆₁	C₂×M₁₆	1	1	yes	characteristic
64	C₂×C₂²⋊C₃₂	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂²×C₆₁	—	2	2	no	—
66	C₂²×C₆₁	—	2	8	no	—
67	C₂²×C₆₁	—	2	2	no	—
68	C₂²×C₆₁	C₂²⋊C₁₆	1	2	yes	—
69	C₂²×C₆₁	C₂×C₃₂	1	2	yes	—
70	C₂²×C₆₁	C₂×C₃₂	1	4	yes	complemented
71	C₂²×C₆₁	M₆₄	1	2	yes	—
72	C₂²×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
73	C₄×C₆₁	—	2	2	no	—
74	C₄×C₆₁	C₂²⋊C₁₆	1	2	yes	—
75	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
76	C₄×C₆₁	C₂×M₃₂	1	1	yes	characteristic
77	C₂³×C₁₆	C₂×C₆₁	1	1	yes	characteristic
78	C₂²×C₃₂	C₂×C₆₁	1	2	yes	complemented
79	C₂²⋊C₃₂	C₂×C₆₁	1	4	yes	direct product factor
80	C₂×C₆₁	—	2	4	no	—
81	C₂×C₆₁	C₂²⋊C₃₂	1	4	yes	direct product factor
82	C₂×C₆₁	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
83	C₂×C₆₁	C₂²×C₃₂	1	1	yes	characteristic
84	C₂×C₆₁	C₂×M₆₄	1	1	yes	characteristic
85	C₂³×C₈	C₄×C₆₁	1	1	yes	characteristic
86	C₂²×C₁₆	—	2	4	no	—
87	C₂²×C₁₆	C₄×C₆₁	1	1	yes	characteristic
88	C₂²×C₁₆	C₄×C₆₁	1	2	yes	—
89	C₂²×C₁₆	C₂²×C₆₁	1	2	yes	—
90	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
91	C₂×C₃₂	—	2	4	no	—
92	C₂×C₃₂	C₂²×C₆₁	1	4	yes	complemented
93	C₆₁	C₂×C₂²⋊C₃₂	1	1	yes	characteristic, direct product factor, Sylow
94	C₂³×C₄	C₈×C₆₁	1	1	yes	characteristic
95	C₂²×C₈	—	2	2	no	—
96	C₂²×C₈	—	2	2	no	—
97	C₂²×C₈	C₈×C₆₁	1	1	yes	characteristic
98	C₂²×C₈	C₈×C₆₁	1	2	yes	—
99	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
100	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	2	4	no	—
103	C₂×C₁₆	—	2	4	no	—
104	C₂×C₁₆	C₂×C₄×C₆₁	1	2	yes	—
105	C₂×C₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
106	C₂×C₁₆	D₄×C₆₁	1	4	yes	—
107	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
108	C₃₂	—	2	4	no	—
109	C₂⁴	C₁₆×C₆₁	1	1	yes	characteristic
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	C₁₆×C₆₁	1	1	yes	characteristic
113	C₂²×C₄	C₁₆×C₆₁	1	2	yes	—
114	C₂²×C₄	C₂×C₈×C₆₁	1	2	yes	—
115	C₂²×C₄	C₂×C₈×C₆₁	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	4	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
121	C₂×C₈	C₂×C₈×C₆₁	1	2	yes	—
122	C₂×C₈	C₂×C₈×C₆₁	1	1	yes	characteristic
123	C₂×C₈	M₁₆×C₆₁	1	2	yes	—
124	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
125	C₁₆	—	2	2	no	—
126	C₁₆	C₂²⋊C₄×C₆₁	1	2	yes	—
127	C₁₆	C₂×D₄×C₆₁	1	2	yes	—
128	C₂³	—	2	2	no	—
129	C₂³	—	2	2	no	—
130	C₂³	C₃₂×C₆₁	1	4	yes	complemented
131	C₂³	C₂×C₁₆×C₆₁	1	2	yes	—
132	C₂³	C₂×C₁₆×C₆₁	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	C₂²⋊C₈×C₆₁	1	2	yes	—
138	C₂×C₄	C₂×C₁₆×C₆₁	1	2	yes	—
139	C₂×C₄	C₂×C₁₆×C₆₁	1	1	yes	characteristic
140	C₂×C₄	M₃₂×C₆₁	1	2	yes	—
141	C₂×C₄	C₂²×C₈×C₆₁	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	C₂²⋊C₈×C₆₁	1	2	yes	—
144	C₈	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
145	C₈	C₂×M₁₆×C₆₁	1	1	yes	characteristic
146	C₂²	—	2	2	no	—
147	C₂²	—	2	8	no	—
148	C₂²	—	2	2	no	—
149	C₂²	C₆₁×C₂²⋊C₁₆	1	2	yes	—
150	C₂²	C₂×C₃₂×C₆₁	1	2	yes	—
151	C₂²	C₂×C₃₂×C₆₁	1	4	yes	complemented
152	C₂²	C₆₁×M₆₄	1	2	yes	—
153	C₂²	C₂²×C₁₆×C₆₁	1	1	yes	characteristic
154	C₄	—	2	2	no	—
155	C₄	C₆₁×C₂²⋊C₁₆	1	2	yes	—
156	C₄	C₂×C₂²⋊C₈×C₆₁	1	1	yes	characteristic
157	C₄	C₂×M₃₂×C₆₁	1	1	yes	characteristic
158	C₂	—	2	4	no	—
159	C₂	C₆₁×C₂²⋊C₃₂	1	4	yes	direct product factor
160	C₂	C₂×C₆₁×C₂²⋊C₁₆	1	1	yes	characteristic
161	C₂	C₂²×C₃₂×C₆₁	1	1	yes	characteristic
162	C₂	C₂×C₆₁×M₆₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₁₆×C₆₁	C₂²
commutator	C₂	C₂²×C₃₂×C₆₁
Frattini, cosocle	C₂×C₁₆	C₂³×C₆₁
socle	C₂³×C₆₁	C₂×C₁₆
prime radical	C₂⁴×C₆₁	C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃₂×C₆₁	C₂	semidirect
3. C₆₁×C₂²⋊C₃₂	C₂	direct (N × H)
11. C₂×C₃₂×C₆₁	C₂²	semidirect
48. C₂³×C₆₁	C₃₂	semidirect
64. C₂×C₂²⋊C₃₂	C₆₁	direct (N × H)
70. C₂²×C₆₁	C₂×C₃₂	semidirect
78. C₂²×C₃₂	C₂×C₆₁	semidirect
79. C₂²⋊C₃₂	C₂×C₆₁	direct (N × H)
81. C₂×C₆₁	C₂²⋊C₃₂	direct (N × H)
92. C₂×C₃₂	C₂²×C₆₁	semidirect
93. C₆₁	C₂×C₂²⋊C₃₂	direct (N × H)
130. C₂³	C₃₂×C₆₁	semidirect
151. C₂²	C₂×C₃₂×C₆₁	semidirect
159. C₂	C₆₁×C₂²⋊C₃₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₆₁	2
C₂²×C₃₂×C₆₁	2
C₆₁×C₂²⋊C₃₂	2
C₂×C₂²⋊C₃₂	61

Subgroups — C2×C61×C22⋊C32 #29516985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×C₂²⋊C₃₂ #29516985