Subgroups — C₂³×C₇×C₈² #72183243

Order 3584.

8324 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	C₂³×C₄×C₇×C₈	C₂	1	3	yes	maximal
2	C₂²×C₇×C₈²	C₂	1	28	yes	maximal, direct product factor
3	C₂⁴×C₇×C₈	C₄	1	6	yes	—
4	C₂³×C₄²×C₇	C₂²	1	1	yes	characteristic
5	C₂²×C₄×C₇×C₈	C₄	1	42	yes	—
6	C₂²×C₄×C₇×C₈	C₂²	1	42	yes	—
7	C₂×C₇×C₈²	C₂²	1	112	yes	direct product factor
8	C₂³×C₈²	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂⁴×C₄×C₇	C₂×C₄	1	3	yes	—
10	C₂³×C₇×C₈	C₈	1	96	yes	direct product factor
11	C₂³×C₇×C₈	C₂×C₄	1	84	yes	—
12	C₂²×C₄²×C₇	C₂×C₄	1	21	yes	—
13	C₂²×C₄²×C₇	C₂³	1	7	yes	—
14	C₂×C₄×C₇×C₈	C₂×C₄	1	252	yes	—
15	C₂×C₄×C₇×C₈	C₂³	1	84	yes	—
16	C₇×C₈²	C₂³	1	64	yes	direct product factor
17	C₂³×C₄×C₈	C₂×C₇	1	3	yes	—
18	C₂²×C₈²	C₂×C₇	1	28	yes	direct product factor
19	C₂⁵×C₇	C₄²	1	1	yes	characteristic
20	C₂³×C₄×C₇	C₄²	1	21	yes	—
21	C₂³×C₄×C₇	C₂×C₈	1	48	yes	—
22	C₂³×C₄×C₇	C₂²×C₄	1	21	yes	—
23	C₂²×C₇×C₈	C₂×C₈	1	672	yes	direct product factor
24	C₂²×C₇×C₈	C₂²×C₄	1	168	yes	—
25	C₂×C₄²×C₇	C₄²	1	42	yes	—
26	C₂×C₄²×C₇	C₂²×C₄	1	63	yes	—
27	C₂×C₄²×C₇	C₂⁴	1	7	yes	—
28	C₄×C₇×C₈	C₂²×C₄	1	168	yes	—
29	C₄×C₇×C₈	C₂⁴	1	24	yes	—
30	C₂⁴×C₈	C₄×C₇	1	6	yes	—
31	C₂³×C₄²	C₂²×C₇	1	1	yes	characteristic
32	C₂²×C₄×C₈	C₄×C₇	1	42	yes	—
33	C₂²×C₄×C₈	C₂²×C₇	1	42	yes	—
34	C₂×C₈²	C₂²×C₇	1	112	yes	direct product factor
35	C₂⁴×C₇	C₄×C₈	1	24	yes	—
36	C₂⁴×C₇	C₂×C₄²	1	7	yes	—
37	C₂²×C₄×C₇	C₄×C₈	1	168	yes	—
38	C₂²×C₄×C₇	C₂×C₄²	1	63	yes	—
39	C₂²×C₄×C₇	C₂²×C₈	1	168	yes	—
40	C₂²×C₄×C₇	C₂³×C₄	1	21	yes	—
41	C₂×C₇×C₈	C₂²×C₈	1	672	yes	direct product factor
42	C₂×C₇×C₈	C₂³×C₄	1	48	yes	—
43	C₄²×C₇	C₂×C₄²	1	42	yes	—
44	C₄²×C₇	C₂³×C₄	1	21	yes	—
45	C₄²×C₇	C₂⁵	1	1	yes	characteristic
46	C₂⁴×C₄	C₂×C₄×C₇	1	3	yes	—
47	C₂³×C₈	C₇×C₈	1	96	yes	direct product factor
48	C₂³×C₈	C₂×C₄×C₇	1	84	yes	—
49	C₂²×C₄²	C₂×C₄×C₇	1	21	yes	—
50	C₂²×C₄²	C₂³×C₇	1	7	yes	—
51	C₂×C₄×C₈	C₂×C₄×C₇	1	252	yes	—
52	C₂×C₄×C₈	C₂³×C₇	1	84	yes	—
53	C₈²	C₂³×C₇	1	64	yes	direct product factor
54	C₂³×C₇	C₈²	1	64	yes	direct product factor
55	C₂³×C₇	C₂×C₄×C₈	1	84	yes	—
56	C₂³×C₇	C₂²×C₄²	1	7	yes	—
57	C₂×C₄×C₇	C₂×C₄×C₈	1	252	yes	—
58	C₂×C₄×C₇	C₂²×C₄²	1	21	yes	—
59	C₂×C₄×C₇	C₂³×C₈	1	84	yes	—
60	C₂×C₄×C₇	C₂⁴×C₄	1	3	yes	—
61	C₇×C₈	C₂³×C₈	1	96	yes	direct product factor
62	C₂⁵	C₄²×C₇	1	1	yes	characteristic
63	C₂³×C₄	C₄²×C₇	1	21	yes	—
64	C₂³×C₄	C₂×C₇×C₈	1	48	yes	—
65	C₂³×C₄	C₂²×C₄×C₇	1	21	yes	—
66	C₂²×C₈	C₂×C₇×C₈	1	672	yes	direct product factor
67	C₂²×C₈	C₂²×C₄×C₇	1	168	yes	—
68	C₂×C₄²	C₄²×C₇	1	42	yes	—
69	C₂×C₄²	C₂²×C₄×C₇	1	63	yes	—
70	C₂×C₄²	C₂⁴×C₇	1	7	yes	—
71	C₄×C₈	C₂²×C₄×C₇	1	168	yes	—
72	C₄×C₈	C₂⁴×C₇	1	24	yes	—
73	C₂²×C₇	C₂×C₈²	1	112	yes	direct product factor
74	C₂²×C₇	C₂²×C₄×C₈	1	42	yes	—
75	C₂²×C₇	C₂³×C₄²	1	1	yes	characteristic
76	C₄×C₇	C₂²×C₄×C₈	1	42	yes	—
77	C₄×C₇	C₂⁴×C₈	1	6	yes	—
78	C₂⁴	C₄×C₇×C₈	1	24	yes	—
79	C₂⁴	C₂×C₄²×C₇	1	7	yes	—
80	C₂²×C₄	C₄×C₇×C₈	1	168	yes	—
81	C₂²×C₄	C₂×C₄²×C₇	1	63	yes	—
82	C₂²×C₄	C₂²×C₇×C₈	1	168	yes	—
83	C₂²×C₄	C₂³×C₄×C₇	1	21	yes	—
84	C₂×C₈	C₂²×C₇×C₈	1	672	yes	direct product factor
85	C₂×C₈	C₂³×C₄×C₇	1	48	yes	—
86	C₄²	C₂×C₄²×C₇	1	42	yes	—
87	C₄²	C₂³×C₄×C₇	1	21	yes	—
88	C₄²	C₂⁵×C₇	1	1	yes	characteristic
89	C₂×C₇	C₂²×C₈²	1	28	yes	direct product factor
90	C₂×C₇	C₂³×C₄×C₈	1	3	yes	—
91	C₂³	C₇×C₈²	1	64	yes	direct product factor
92	C₂³	C₂×C₄×C₇×C₈	1	84	yes	—
93	C₂³	C₂²×C₄²×C₇	1	7	yes	—
94	C₂×C₄	C₂×C₄×C₇×C₈	1	252	yes	—
95	C₂×C₄	C₂²×C₄²×C₇	1	21	yes	—
96	C₂×C₄	C₂³×C₇×C₈	1	84	yes	—
97	C₂×C₄	C₂⁴×C₄×C₇	1	3	yes	—
98	C₈	C₂³×C₇×C₈	1	96	yes	direct product factor
99	C₇	C₂³×C₈²	1	1	yes	characteristic, direct product factor, Sylow
100	C₂²	C₂×C₇×C₈²	1	112	yes	direct product factor
101	C₂²	C₂²×C₄×C₇×C₈	1	42	yes	—
102	C₂²	C₂³×C₄²×C₇	1	1	yes	characteristic
103	C₄	C₂²×C₄×C₇×C₈	1	42	yes	—
104	C₄	C₂⁴×C₇×C₈	1	6	yes	—
105	C₂	C₂²×C₇×C₈²	1	28	yes	direct product factor
106	C₂	C₂³×C₄×C₇×C₈	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄²	C₂⁵×C₇
socle, prime radical	C₂⁵×C₇	C₄²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇×C₈²	C₂	direct (N × H)
7. C₂×C₇×C₈²	C₂²	direct (N × H)
8. C₂³×C₈²	C₇	direct (N × H)
10. C₂³×C₇×C₈	C₈	direct (N × H)
16. C₇×C₈²	C₂³	direct (N × H)
18. C₂²×C₈²	C₂×C₇	direct (N × H)
23. C₂²×C₇×C₈	C₂×C₈	direct (N × H)
34. C₂×C₈²	C₂²×C₇	direct (N × H)
41. C₂×C₇×C₈	C₂²×C₈	direct (N × H)
47. C₂³×C₈	C₇×C₈	direct (N × H)
53. C₈²	C₂³×C₇	direct (N × H)
54. C₂³×C₇	C₈²	direct (N × H)
61. C₇×C₈	C₂³×C₈	direct (N × H)
66. C₂²×C₈	C₂×C₇×C₈	direct (N × H)
73. C₂²×C₇	C₂×C₈²	direct (N × H)
84. C₂×C₈	C₂²×C₇×C₈	direct (N × H)
89. C₂×C₇	C₂²×C₈²	direct (N × H)
91. C₂³	C₇×C₈²	direct (N × H)
98. C₈	C₂³×C₇×C₈	direct (N × H)
99. C₇	C₂³×C₈²	direct (N × H)
100. C₂²	C₂×C₇×C₈²	direct (N × H)
105. C₂	C₂²×C₇×C₈²	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×C₄×C₇×C₈	2
C₂²×C₇×C₈²	2
C₂³×C₈²	7

Subgroups — C23×C7×C82 #72183243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₇×C₈² #72183243