Subgroups — C₂×C₄×M₃₂ #33322

Order 256.

459 conjugacy classes of subgroups.

The subgroup lattice has 114 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×M₃₂	C₂	1	8	yes	maximal, direct product factor
4	C₂×#346	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₁₆	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈	C₄	1	1	yes	characteristic
7	C₂³×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄²	C₄	1	1	yes	characteristic
9	C₂×M₃₂	C₄	1	16	yes	direct product factor
10	C₂×M₃₂	C₂²	1	8	yes	—
11	C₂²×C₁₆	C₂²	1	4	yes	—
12	C₂×C₄×C₈	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈	C₄	1	4	yes	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈	C₂²	1	4	yes	—
17	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
18	#346	C₂²	1	8	yes	complemented
19	C₄×C₁₆	C₂²	1	8	yes	complemented
20	C₂³×C₄	C₈	1	2	yes	—
21	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈	C₈	1	8	yes	—
23	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₈	C₂×C₄	1	8	yes	—
26	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈	C₂×C₄	1	4	yes	—
28	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈	C₂³	1	2	yes	—
30	C₂²×C₈	C₂³	1	1	yes	characteristic
31	C₂×C₄²	C₈	1	4	yes	—
32	C₂×C₄²	C₈	1	2	yes	—
33	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²	C₂×C₄	1	4	yes	—
35	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
36	M₃₂	C₂×C₄	1	32	yes	direct product factor
37	C₂×C₁₆	C₂×C₄	1	16	yes	complemented
38	C₂×C₁₆	C₂³	1	8	yes	—
39	C₄×C₈	C₂×C₄	1	2	yes	—
40	C₄×C₈	C₂×C₄	1	4	yes	—
41	C₄×C₈	C₂×C₄	1	4	yes	—
42	C₄×C₈	C₂×C₄	1	2	yes	—
43	C₄×C₈	C₂³	1	2	yes	—
44	C₄×C₈	C₂³	1	2	yes	—
45	C₂⁴	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄	—	2	8	no	—
47	C₂²×C₄	C₄²	1	4	yes	—
48	C₂²×C₄	C₂×C₈	1	2	yes	—
49	C₂²×C₄	C₂×C₈	1	8	yes	—
50	C₂²×C₄	C₂×C₈	1	2	yes	—
51	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄	C₂×C₈	1	4	yes	—
53	C₂²×C₄	C₂×C₈	1	2	yes	—
54	C₂²×C₄	C₂²×C₄	1	2	yes	—
55	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈	C₄²	1	8	yes	—
57	C₂×C₈	C₄²	1	4	yes	—
58	C₂×C₈	C₂×C₈	1	16	yes	—
59	C₂×C₈	C₂×C₈	1	4	yes	—
60	C₂×C₈	C₂×C₈	1	4	yes	—
61	C₂×C₈	C₂²×C₄	1	2	yes	—
62	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈	C₂²×C₄	1	4	yes	—
64	C₂×C₈	C₂²×C₄	1	8	yes	—
65	C₂×C₈	C₂²×C₄	1	2	yes	—
66	C₂×C₈	C₂²×C₄	1	2	yes	—
67	C₂×C₈	C₂⁴	1	1	yes	characteristic
68	C₄²	C₂×C₈	1	2	yes	—
69	C₄²	C₂×C₈	1	8	yes	—
70	C₄²	C₂×C₈	1	2	yes	—
71	C₄²	C₂²×C₄	1	2	yes	—
72	C₄²	C₂²×C₄	1	2	yes	—
73	C₁₆	C₂²×C₄	1	16	yes	complemented
74	C₂³	—	2	4	no	—
75	C₂³	C₄×C₈	1	4	yes	—
76	C₂³	C₂²×C₈	1	2	yes	—
77	C₂³	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄	—	2	4	no	—
79	C₂×C₄	—	2	16	no	—
80	C₂×C₄	C₄×C₈	1	8	yes	—
81	C₂×C₄	C₄×C₈	1	4	yes	—
82	C₂×C₄	M₃₂	1	8	yes	direct product factor
83	C₂×C₄	C₂×C₄²	1	4	yes	—
84	C₂×C₄	C₂×C₄²	1	2	yes	—
85	C₂×C₄	C₂²×C₈	1	4	yes	—
86	C₂×C₄	C₂²×C₈	1	4	yes	—
87	C₂×C₄	C₂²×C₈	1	2	yes	—
88	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄	C₂²×C₈	1	2	yes	—
90	C₂×C₄	C₂²×C₈	1	4	yes	—
91	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
92	C₈	C₂×C₄²	1	2	yes	—
93	C₈	C₂×C₄²	1	4	yes	—
94	C₈	C₂²×C₈	1	4	yes	—
95	C₈	C₂²×C₈	1	4	yes	—
96	C₈	C₂³×C₄	1	2	yes	—
97	C₂²	—	2	8	no	—
98	C₂²	—	2	4	no	—
99	C₂²	C₂×C₄×C₈	1	4	yes	—
100	C₂²	C₂×C₄×C₈	1	2	yes	—
101	C₂²	C₂×M₃₂	1	4	yes	—
102	C₂²	C₂³×C₈	1	1	yes	characteristic
103	C₄	—	2	4	no	—
104	C₄	C₂×C₄×C₈	1	2	yes	—
105	C₄	C₂×C₄×C₈	1	4	yes	—
106	C₄	C₂×M₃₂	1	8	yes	direct product factor
107	C₄	C₂²×C₄²	1	1	yes	characteristic
108	C₄	C₂³×C₈	1	1	yes	characteristic
109	C₂	—	2	4	no	—
110	C₂	C₄×M₃₂	1	4	yes	direct product factor
111	C₂	C₂²×C₄×C₈	1	1	yes	characteristic
112	C₂	C₂²×M₃₂	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂²
commutator	C₂	C₂²×C₄×C₈
Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂³	C₂²×C₈
prime radical	C₂⁴	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. C₄×M₃₂	C₂	direct (N × H)
4. C₂×#346	C₂	semidirect
5. C₂×C₄×C₁₆	C₂	semidirect
9. C₂×M₃₂	C₄	direct (N × H)
18. #346	C₂²	semidirect
19. C₄×C₁₆	C₂²	semidirect
36. M₃₂	C₂×C₄	direct (N × H)
37. C₂×C₁₆	C₂×C₄	semidirect
73. C₁₆	C₂²×C₄	semidirect
82. C₂×C₄	M₃₂	direct (N × H)
106. C₄	C₂×M₃₂	direct (N × H)
110. C₂	C₄×M₃₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×M₃₂	2
C₂²×C₄×C₈	2
C₄×M₃₂	2
C₂×#346	2
C₂×C₄×C₁₆	2

Subgroups — C2×C4×M32 #33322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×M₃₂ #33322