Subgroups — C₂²×C₄³×C₇×C₁₇ #72207266

Order 30464.

16188 conjugacy classes of subgroups.

The subgroup lattice has 120 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄²×C₇×C₁₇	C₂	1	7	yes	maximal
2	C₂×C₄³×C₇×C₁₇	C₂	1	24	yes	maximal, direct product factor
3	C₂⁴×C₄×C₇×C₁₇	C₂²	1	7	yes	—
4	C₂²×C₄²×C₇×C₁₇	C₄	1	112	yes	direct product factor
5	C₂²×C₄²×C₇×C₁₇	C₂²	1	84	yes	—
6	C₄³×C₇×C₁₇	C₂²	1	64	yes	direct product factor
7	C₂³×C₄×C₇×C₁₇	C₂×C₄	1	168	yes	—
8	C₂³×C₄×C₇×C₁₇	C₂³	1	42	yes	—
9	C₂⁵×C₇×C₁₇	C₂³	1	1	yes	characteristic
10	C₂×C₄²×C₇×C₁₇	C₂×C₄	1	672	yes	direct product factor
11	C₂×C₄²×C₇×C₁₇	C₂³	1	112	yes	—
12	C₂²×C₄³×C₁₇	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂³×C₄²×C₁₇	C₂×C₇	1	7	yes	—
14	C₂×C₄³×C₁₇	C₂×C₇	1	24	yes	direct product factor
15	C₂⁴×C₇×C₁₇	C₂²×C₄	1	28	yes	—
16	C₂⁴×C₇×C₁₇	C₂⁴	1	3	yes	—
17	C₂²×C₄×C₇×C₁₇	C₄²	1	448	yes	direct product factor
18	C₂²×C₄×C₇×C₁₇	C₂²×C₄	1	504	yes	—
19	C₂²×C₄×C₇×C₁₇	C₂⁴	1	28	yes	—
20	C₄²×C₇×C₁₇	C₂²×C₄	1	448	yes	direct product factor
21	C₂²×C₄³×C₇	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂⁴×C₄×C₁₇	C₂²×C₇	1	7	yes	—
23	C₂²×C₄²×C₁₇	C₄×C₇	1	112	yes	direct product factor
24	C₂²×C₄²×C₁₇	C₂²×C₇	1	84	yes	—
25	C₄³×C₁₇	C₂²×C₇	1	64	yes	direct product factor
26	C₂³×C₇×C₁₇	C₂×C₄²	1	112	yes	—
27	C₂³×C₇×C₁₇	C₂³×C₄	1	42	yes	—
28	C₂³×C₇×C₁₇	C₂⁵	1	1	yes	characteristic
29	C₂×C₄×C₇×C₁₇	C₂×C₄²	1	672	yes	direct product factor
30	C₂×C₄×C₇×C₁₇	C₂³×C₄	1	168	yes	—
31	C₂³×C₄²×C₇	C₂×C₁₇	1	7	yes	—
32	C₂×C₄³×C₇	C₂×C₁₇	1	24	yes	direct product factor
33	C₂⁵×C₁₇	C₂³×C₇	1	1	yes	characteristic
34	C₂³×C₄×C₁₇	C₂×C₄×C₇	1	168	yes	—
35	C₂³×C₄×C₁₇	C₂³×C₇	1	42	yes	—
36	C₂×C₄²×C₁₇	C₂×C₄×C₇	1	672	yes	direct product factor
37	C₂×C₄²×C₁₇	C₂³×C₇	1	112	yes	—
38	C₂²×C₇×C₁₇	C₄³	1	64	yes	direct product factor
39	C₂²×C₇×C₁₇	C₂²×C₄²	1	84	yes	—
40	C₂²×C₇×C₁₇	C₂⁴×C₄	1	7	yes	—
41	C₄×C₇×C₁₇	C₂²×C₄²	1	112	yes	direct product factor
42	C₂⁴×C₄×C₇	C₂²×C₁₇	1	7	yes	—
43	C₂²×C₄²×C₇	C₄×C₁₇	1	112	yes	direct product factor
44	C₂²×C₄²×C₇	C₂²×C₁₇	1	84	yes	—
45	C₄³×C₇	C₂²×C₁₇	1	64	yes	direct product factor
46	C₂⁴×C₁₇	C₂²×C₄×C₇	1	28	yes	—
47	C₂⁴×C₁₇	C₂⁴×C₇	1	3	yes	—
48	C₂²×C₄×C₁₇	C₄²×C₇	1	448	yes	direct product factor
49	C₂²×C₄×C₁₇	C₂²×C₄×C₇	1	504	yes	—
50	C₂²×C₄×C₁₇	C₂⁴×C₇	1	28	yes	—
51	C₄²×C₁₇	C₂²×C₄×C₇	1	448	yes	direct product factor
52	C₂²×C₄³	C₇×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
53	C₂×C₇×C₁₇	C₂×C₄³	1	24	yes	direct product factor
54	C₂×C₇×C₁₇	C₂³×C₄²	1	7	yes	—
55	C₂⁵×C₇	C₂³×C₁₇	1	1	yes	characteristic
56	C₂³×C₄×C₇	C₂×C₄×C₁₇	1	168	yes	—
57	C₂³×C₄×C₇	C₂³×C₁₇	1	42	yes	—
58	C₂×C₄²×C₇	C₂×C₄×C₁₇	1	672	yes	direct product factor
59	C₂×C₄²×C₇	C₂³×C₁₇	1	112	yes	—
60	C₂³×C₁₇	C₂×C₄²×C₇	1	112	yes	—
61	C₂³×C₁₇	C₂³×C₄×C₇	1	42	yes	—
62	C₂³×C₁₇	C₂⁵×C₇	1	1	yes	characteristic
63	C₂×C₄×C₁₇	C₂×C₄²×C₇	1	672	yes	direct product factor
64	C₂×C₄×C₁₇	C₂³×C₄×C₇	1	168	yes	—
65	C₂³×C₄²	C₂×C₇×C₁₇	1	7	yes	—
66	C₂×C₄³	C₂×C₇×C₁₇	1	24	yes	direct product factor
67	C₇×C₁₇	C₂²×C₄³	1	1	yes	characteristic, direct product factor, Hall
68	C₂⁴×C₇	C₂²×C₄×C₁₇	1	28	yes	—
69	C₂⁴×C₇	C₂⁴×C₁₇	1	3	yes	—
70	C₂²×C₄×C₇	C₄²×C₁₇	1	448	yes	direct product factor
71	C₂²×C₄×C₇	C₂²×C₄×C₁₇	1	504	yes	—
72	C₂²×C₄×C₇	C₂⁴×C₁₇	1	28	yes	—
73	C₄²×C₇	C₂²×C₄×C₁₇	1	448	yes	direct product factor
74	C₂²×C₁₇	C₄³×C₇	1	64	yes	direct product factor
75	C₂²×C₁₇	C₂²×C₄²×C₇	1	84	yes	—
76	C₂²×C₁₇	C₂⁴×C₄×C₇	1	7	yes	—
77	C₄×C₁₇	C₂²×C₄²×C₇	1	112	yes	direct product factor
78	C₂⁴×C₄	C₂²×C₇×C₁₇	1	7	yes	—
79	C₂²×C₄²	C₄×C₇×C₁₇	1	112	yes	direct product factor
80	C₂²×C₄²	C₂²×C₇×C₁₇	1	84	yes	—
81	C₄³	C₂²×C₇×C₁₇	1	64	yes	direct product factor
82	C₂³×C₇	C₂×C₄²×C₁₇	1	112	yes	—
83	C₂³×C₇	C₂³×C₄×C₁₇	1	42	yes	—
84	C₂³×C₇	C₂⁵×C₁₇	1	1	yes	characteristic
85	C₂×C₄×C₇	C₂×C₄²×C₁₇	1	672	yes	direct product factor
86	C₂×C₄×C₇	C₂³×C₄×C₁₇	1	168	yes	—
87	C₂×C₁₇	C₂×C₄³×C₇	1	24	yes	direct product factor
88	C₂×C₁₇	C₂³×C₄²×C₇	1	7	yes	—
89	C₂⁵	C₂³×C₇×C₁₇	1	1	yes	characteristic
90	C₂³×C₄	C₂×C₄×C₇×C₁₇	1	168	yes	—
91	C₂³×C₄	C₂³×C₇×C₁₇	1	42	yes	—
92	C₂×C₄²	C₂×C₄×C₇×C₁₇	1	672	yes	direct product factor
93	C₂×C₄²	C₂³×C₇×C₁₇	1	112	yes	—
94	C₂²×C₇	C₄³×C₁₇	1	64	yes	direct product factor
95	C₂²×C₇	C₂²×C₄²×C₁₇	1	84	yes	—
96	C₂²×C₇	C₂⁴×C₄×C₁₇	1	7	yes	—
97	C₄×C₇	C₂²×C₄²×C₁₇	1	112	yes	direct product factor
98	C₁₇	C₂²×C₄³×C₇	1	1	yes	characteristic, direct product factor, Sylow
99	C₂⁴	C₂²×C₄×C₇×C₁₇	1	28	yes	—
100	C₂⁴	C₂⁴×C₇×C₁₇	1	3	yes	—
101	C₂²×C₄	C₄²×C₇×C₁₇	1	448	yes	direct product factor
102	C₂²×C₄	C₂²×C₄×C₇×C₁₇	1	504	yes	—
103	C₂²×C₄	C₂⁴×C₇×C₁₇	1	28	yes	—
104	C₄²	C₂²×C₄×C₇×C₁₇	1	448	yes	direct product factor
105	C₂×C₇	C₂×C₄³×C₁₇	1	24	yes	direct product factor
106	C₂×C₇	C₂³×C₄²×C₁₇	1	7	yes	—
107	C₂³	C₂×C₄²×C₇×C₁₇	1	112	yes	—
108	C₂³	C₂⁵×C₇×C₁₇	1	1	yes	characteristic
109	C₂³	C₂³×C₄×C₇×C₁₇	1	42	yes	—
110	C₂×C₄	C₂×C₄²×C₇×C₁₇	1	672	yes	direct product factor
111	C₂×C₄	C₂³×C₄×C₇×C₁₇	1	168	yes	—
112	C₇	C₂²×C₄³×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
113	C₂²	C₄³×C₇×C₁₇	1	64	yes	direct product factor
114	C₂²	C₂²×C₄²×C₇×C₁₇	1	84	yes	—
115	C₂²	C₂⁴×C₄×C₇×C₁₇	1	7	yes	—
116	C₄	C₂²×C₄²×C₇×C₁₇	1	112	yes	direct product factor
117	C₂	C₂×C₄³×C₇×C₁₇	1	24	yes	direct product factor
118	C₂	C₂³×C₄²×C₇×C₁₇	1	7	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂³	C₂⁵×C₇×C₁₇
socle, prime radical	C₂⁵×C₇×C₁₇	C₂³
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄³×C₇×C₁₇	C₂	direct (N × H)
4. C₂²×C₄²×C₇×C₁₇	C₄	direct (N × H)
6. C₄³×C₇×C₁₇	C₂²	direct (N × H)
10. C₂×C₄²×C₇×C₁₇	C₂×C₄	direct (N × H)
12. C₂²×C₄³×C₁₇	C₇	direct (N × H)
14. C₂×C₄³×C₁₇	C₂×C₇	direct (N × H)
17. C₂²×C₄×C₇×C₁₇	C₄²	direct (N × H)
20. C₄²×C₇×C₁₇	C₂²×C₄	direct (N × H)
21. C₂²×C₄³×C₇	C₁₇	direct (N × H)
23. C₂²×C₄²×C₁₇	C₄×C₇	direct (N × H)
25. C₄³×C₁₇	C₂²×C₇	direct (N × H)
29. C₂×C₄×C₇×C₁₇	C₂×C₄²	direct (N × H)
32. C₂×C₄³×C₇	C₂×C₁₇	direct (N × H)
36. C₂×C₄²×C₁₇	C₂×C₄×C₇	direct (N × H)
38. C₂²×C₇×C₁₇	C₄³	direct (N × H)
41. C₄×C₇×C₁₇	C₂²×C₄²	direct (N × H)
43. C₂²×C₄²×C₇	C₄×C₁₇	direct (N × H)
45. C₄³×C₇	C₂²×C₁₇	direct (N × H)
48. C₂²×C₄×C₁₇	C₄²×C₇	direct (N × H)
51. C₄²×C₁₇	C₂²×C₄×C₇	direct (N × H)
52. C₂²×C₄³	C₇×C₁₇	direct (N × H)
53. C₂×C₇×C₁₇	C₂×C₄³	direct (N × H)
58. C₂×C₄²×C₇	C₂×C₄×C₁₇	direct (N × H)
63. C₂×C₄×C₁₇	C₂×C₄²×C₇	direct (N × H)
66. C₂×C₄³	C₂×C₇×C₁₇	direct (N × H)
67. C₇×C₁₇	C₂²×C₄³	direct (N × H)
70. C₂²×C₄×C₇	C₄²×C₁₇	direct (N × H)
73. C₄²×C₇	C₂²×C₄×C₁₇	direct (N × H)
74. C₂²×C₁₇	C₄³×C₇	direct (N × H)
77. C₄×C₁₇	C₂²×C₄²×C₇	direct (N × H)
79. C₂²×C₄²	C₄×C₇×C₁₇	direct (N × H)
81. C₄³	C₂²×C₇×C₁₇	direct (N × H)
85. C₂×C₄×C₇	C₂×C₄²×C₁₇	direct (N × H)
87. C₂×C₁₇	C₂×C₄³×C₇	direct (N × H)
92. C₂×C₄²	C₂×C₄×C₇×C₁₇	direct (N × H)
94. C₂²×C₇	C₄³×C₁₇	direct (N × H)
97. C₄×C₇	C₂²×C₄²×C₁₇	direct (N × H)
98. C₁₇	C₂²×C₄³×C₇	direct (N × H)
101. C₂²×C₄	C₄²×C₇×C₁₇	direct (N × H)
104. C₄²	C₂²×C₄×C₇×C₁₇	direct (N × H)
105. C₂×C₇	C₂×C₄³×C₁₇	direct (N × H)
110. C₂×C₄	C₂×C₄²×C₇×C₁₇	direct (N × H)
112. C₇	C₂²×C₄³×C₁₇	direct (N × H)
113. C₂²	C₄³×C₇×C₁₇	direct (N × H)
116. C₄	C₂²×C₄²×C₇×C₁₇	direct (N × H)
117. C₂	C₂×C₄³×C₇×C₁₇	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×C₄²×C₇×C₁₇	2
C₂×C₄³×C₇×C₁₇	2
C₂²×C₄³×C₁₇	7
C₂²×C₄³×C₇	17

Subgroups — C22×C43×C7×C17 #72207266

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄³×C₇×C₁₇ #72207266