Subgroups — C₂×C₄²×C₈×C₁₀₁ #38802476

Order 25856.

2046 conjugacy classes of subgroups.

The subgroup lattice has 108 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₁₀₁	C₂	1	6	yes	maximal
2	C₂×C₄³×C₁₀₁	C₂	1	1	yes	maximal, characteristic
3	C₄²×C₈×C₁₀₁	C₂	1	8	yes	maximal, direct product factor
4	C₂³×C₈×C₁₀₁	C₂²	1	4	yes	—
5	C₂²×C₄²×C₁₀₁	C₄	1	4	yes	—
6	C₂²×C₄²×C₁₀₁	C₂²	1	3	yes	—
7	C₂×C₄×C₈×C₁₀₁	C₄	1	48	yes	direct product factor
8	C₂×C₄×C₈×C₁₀₁	C₂²	1	24	yes	—
9	C₄³×C₁₀₁	C₄	1	4	yes	—
10	C₄³×C₁₀₁	C₂²	1	4	yes	—
11	C₂³×C₄×C₁₀₁	C₂×C₄	1	6	yes	—
12	C₂³×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
13	C₂²×C₈×C₁₀₁	C₂×C₄	1	48	yes	—
14	C₂²×C₈×C₁₀₁	C₂³	1	8	yes	—
15	C₂×C₄²×C₁₀₁	C₈	1	32	yes	direct product factor
16	C₂×C₄²×C₁₀₁	C₂×C₄	1	6	yes	—
17	C₂×C₄²×C₁₀₁	C₂×C₄	1	16	yes	—
18	C₂×C₄²×C₁₀₁	C₂×C₄	1	24	yes	—
19	C₂×C₄²×C₁₀₁	C₂³	1	6	yes	—
20	C₄×C₈×C₁₀₁	C₂×C₄	1	96	yes	direct product factor
21	C₂⁴×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₁	C₄²	1	24	yes	—
23	C₂²×C₄×C₁₀₁	C₂×C₈	1	48	yes	—
24	C₂²×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₀₁	C₂²×C₄	1	12	yes	—
26	C₂²×C₄×C₁₀₁	C₂²×C₄	1	12	yes	—
27	C₂²×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
28	C₂×C₈×C₁₀₁	C₄²	1	64	yes	direct product factor
29	C₂×C₈×C₁₀₁	C₂²×C₄	1	48	yes	—
30	C₄²×C₁₀₁	C₄²	1	24	yes	—
31	C₄²×C₁₀₁	C₂×C₈	1	64	yes	direct product factor
32	C₄²×C₁₀₁	C₂²×C₄	1	24	yes	—
33	C₂³×C₁₀₁	C₂×C₄²	1	6	yes	—
34	C₂³×C₁₀₁	C₂²×C₈	1	8	yes	—
35	C₂³×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₁	C₄×C₈	1	96	yes	direct product factor
37	C₂×C₄×C₁₀₁	C₂×C₄²	1	6	yes	—
38	C₂×C₄×C₁₀₁	C₂×C₄²	1	16	yes	—
39	C₂×C₄×C₁₀₁	C₂×C₄²	1	24	yes	—
40	C₂×C₄×C₁₀₁	C₂²×C₈	1	48	yes	—
41	C₂×C₄×C₁₀₁	C₂³×C₄	1	6	yes	—
42	C₈×C₁₀₁	C₂×C₄²	1	32	yes	direct product factor
43	C₂²×C₁₀₁	C₄³	1	4	yes	—
44	C₂²×C₁₀₁	C₂×C₄×C₈	1	24	yes	—
45	C₂²×C₁₀₁	C₂²×C₄²	1	3	yes	—
46	C₂²×C₁₀₁	C₂³×C₈	1	4	yes	—
47	C₄×C₁₀₁	C₄³	1	4	yes	—
48	C₄×C₁₀₁	C₂×C₄×C₈	1	48	yes	direct product factor
49	C₄×C₁₀₁	C₂²×C₄²	1	4	yes	—
50	C₂×C₄²×C₈	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂×C₁₀₁	C₄²×C₈	1	8	yes	direct product factor
52	C₂×C₁₀₁	C₂×C₄³	1	1	yes	characteristic
53	C₂×C₁₀₁	C₂²×C₄×C₈	1	6	yes	—
54	C₂²×C₄×C₈	C₂×C₁₀₁	1	6	yes	—
55	C₂×C₄³	C₂×C₁₀₁	1	1	yes	characteristic
56	C₄²×C₈	C₂×C₁₀₁	1	8	yes	direct product factor
57	C₁₀₁	C₂×C₄²×C₈	1	1	yes	characteristic, direct product factor, Sylow
58	C₂³×C₈	C₂²×C₁₀₁	1	4	yes	—
59	C₂²×C₄²	C₄×C₁₀₁	1	4	yes	—
60	C₂²×C₄²	C₂²×C₁₀₁	1	3	yes	—
61	C₂×C₄×C₈	C₄×C₁₀₁	1	48	yes	direct product factor
62	C₂×C₄×C₈	C₂²×C₁₀₁	1	24	yes	—
63	C₄³	C₄×C₁₀₁	1	4	yes	—
64	C₄³	C₂²×C₁₀₁	1	4	yes	—
65	C₂³×C₄	C₂×C₄×C₁₀₁	1	6	yes	—
66	C₂³×C₄	C₂³×C₁₀₁	1	1	yes	characteristic
67	C₂²×C₈	C₂×C₄×C₁₀₁	1	48	yes	—
68	C₂²×C₈	C₂³×C₁₀₁	1	8	yes	—
69	C₂×C₄²	C₈×C₁₀₁	1	32	yes	direct product factor
70	C₂×C₄²	C₂×C₄×C₁₀₁	1	6	yes	—
71	C₂×C₄²	C₂×C₄×C₁₀₁	1	16	yes	—
72	C₂×C₄²	C₂×C₄×C₁₀₁	1	24	yes	—
73	C₂×C₄²	C₂³×C₁₀₁	1	6	yes	—
74	C₄×C₈	C₂×C₄×C₁₀₁	1	96	yes	direct product factor
75	C₂⁴	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
76	C₂²×C₄	C₄²×C₁₀₁	1	24	yes	—
77	C₂²×C₄	C₂×C₈×C₁₀₁	1	48	yes	—
78	C₂²×C₄	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
79	C₂²×C₄	C₂²×C₄×C₁₀₁	1	12	yes	—
80	C₂²×C₄	C₂²×C₄×C₁₀₁	1	12	yes	—
81	C₂²×C₄	C₂⁴×C₁₀₁	1	1	yes	characteristic
82	C₂×C₈	C₄²×C₁₀₁	1	64	yes	direct product factor
83	C₂×C₈	C₂²×C₄×C₁₀₁	1	48	yes	—
84	C₄²	C₄²×C₁₀₁	1	24	yes	—
85	C₄²	C₂×C₈×C₁₀₁	1	64	yes	direct product factor
86	C₄²	C₂²×C₄×C₁₀₁	1	24	yes	—
87	C₂³	C₂×C₄²×C₁₀₁	1	6	yes	—
88	C₂³	C₂²×C₈×C₁₀₁	1	8	yes	—
89	C₂³	C₂³×C₄×C₁₀₁	1	1	yes	characteristic
90	C₂×C₄	C₄×C₈×C₁₀₁	1	96	yes	direct product factor
91	C₂×C₄	C₂×C₄²×C₁₀₁	1	6	yes	—
92	C₂×C₄	C₂×C₄²×C₁₀₁	1	16	yes	—
93	C₂×C₄	C₂×C₄²×C₁₀₁	1	24	yes	—
94	C₂×C₄	C₂²×C₈×C₁₀₁	1	48	yes	—
95	C₂×C₄	C₂³×C₄×C₁₀₁	1	6	yes	—
96	C₈	C₂×C₄²×C₁₀₁	1	32	yes	direct product factor
97	C₂²	C₄³×C₁₀₁	1	4	yes	—
98	C₂²	C₂×C₄×C₈×C₁₀₁	1	24	yes	—
99	C₂²	C₂²×C₄²×C₁₀₁	1	3	yes	—
100	C₂²	C₂³×C₈×C₁₀₁	1	4	yes	—
101	C₄	C₄³×C₁₀₁	1	4	yes	—
102	C₄	C₂×C₄×C₈×C₁₀₁	1	48	yes	direct product factor
103	C₄	C₂²×C₄²×C₁₀₁	1	4	yes	—
104	C₂	C₄²×C₈×C₁₀₁	1	8	yes	direct product factor
105	C₂	C₂×C₄³×C₁₀₁	1	1	yes	characteristic
106	C₂	C₂²×C₄×C₈×C₁₀₁	1	6	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₁
socle, prime radical	C₂⁴×C₁₀₁	C₂²×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄²×C₈×C₁₀₁	C₂	direct (N × H)
7. C₂×C₄×C₈×C₁₀₁	C₄	direct (N × H)
15. C₂×C₄²×C₁₀₁	C₈	direct (N × H)
20. C₄×C₈×C₁₀₁	C₂×C₄	direct (N × H)
28. C₂×C₈×C₁₀₁	C₄²	direct (N × H)
31. C₄²×C₁₀₁	C₂×C₈	direct (N × H)
36. C₂×C₄×C₁₀₁	C₄×C₈	direct (N × H)
42. C₈×C₁₀₁	C₂×C₄²	direct (N × H)
48. C₄×C₁₀₁	C₂×C₄×C₈	direct (N × H)
50. C₂×C₄²×C₈	C₁₀₁	direct (N × H)
51. C₂×C₁₀₁	C₄²×C₈	direct (N × H)
56. C₄²×C₈	C₂×C₁₀₁	direct (N × H)
57. C₁₀₁	C₂×C₄²×C₈	direct (N × H)
61. C₂×C₄×C₈	C₄×C₁₀₁	direct (N × H)
69. C₂×C₄²	C₈×C₁₀₁	direct (N × H)
74. C₄×C₈	C₂×C₄×C₁₀₁	direct (N × H)
82. C₂×C₈	C₄²×C₁₀₁	direct (N × H)
85. C₄²	C₂×C₈×C₁₀₁	direct (N × H)
90. C₂×C₄	C₄×C₈×C₁₀₁	direct (N × H)
96. C₈	C₂×C₄²×C₁₀₁	direct (N × H)
102. C₄	C₂×C₄×C₈×C₁₀₁	direct (N × H)
104. C₂	C₄²×C₈×C₁₀₁	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×C₄×C₈×C₁₀₁	2
C₂×C₄³×C₁₀₁	2
C₄²×C₈×C₁₀₁	2
C₂×C₄²×C₈	101

Subgroups — C2×C42×C8×C101 #38802476

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×C₈×C₁₀₁ #38802476