Subgroups — C₂³×C₈×C₁₇×C₂₅ #72204263

Order 27200.

1014 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	C₂³×C₄×C₁₇×C₂₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×C₁₇×C₂₅	C₂	1	14	yes	maximal, direct product factor
3	C₂⁴×C₁₇×C₂₅	C₄	1	1	yes	characteristic
4	C₂²×C₄×C₁₇×C₂₅	C₄	1	7	yes	—
5	C₂²×C₄×C₁₇×C₂₅	C₂²	1	7	yes	—
6	C₂×C₈×C₁₇×C₂₅	C₂²	1	28	yes	direct product factor
7	C₂³×C₅×C₈×C₁₇	C₅	1	1	yes	maximal, characteristic
8	C₂³×C₁₇×C₂₅	C₈	1	8	yes	direct product factor
9	C₂³×C₁₇×C₂₅	C₂×C₄	1	7	yes	—
10	C₂×C₄×C₁₇×C₂₅	C₂×C₄	1	21	yes	—
11	C₂×C₄×C₁₇×C₂₅	C₂³	1	7	yes	—
12	C₈×C₁₇×C₂₅	C₂³	1	8	yes	direct product factor
13	C₂³×C₄×C₅×C₁₇	C₂×C₅	1	1	yes	characteristic
14	C₂²×C₅×C₈×C₁₇	C₂×C₅	1	14	yes	—
15	C₂²×C₁₇×C₂₅	C₂×C₈	1	28	yes	direct product factor
16	C₂²×C₁₇×C₂₅	C₂²×C₄	1	7	yes	—
17	C₄×C₁₇×C₂₅	C₂²×C₄	1	7	yes	—
18	C₄×C₁₇×C₂₅	C₂⁴	1	1	yes	characteristic
19	C₂³×C₈×C₂₅	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂⁴×C₅×C₁₇	C₄×C₅	1	1	yes	characteristic
21	C₂²×C₄×C₅×C₁₇	C₄×C₅	1	7	yes	—
22	C₂²×C₄×C₅×C₁₇	C₂²×C₅	1	7	yes	—
23	C₂×C₅×C₈×C₁₇	C₂²×C₅	1	28	yes	—
24	C₂³×C₈×C₁₇	C₂₅	1	1	yes	characteristic, direct product factor, Hall
25	C₂×C₁₇×C₂₅	C₂²×C₈	1	14	yes	direct product factor
26	C₂×C₁₇×C₂₅	C₂³×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₂₅	C₂×C₁₇	1	1	yes	characteristic
28	C₂²×C₈×C₂₅	C₂×C₁₇	1	14	yes	direct product factor
29	C₂³×C₅×C₁₇	C₅×C₈	1	8	yes	—
30	C₂³×C₅×C₁₇	C₂×C₄×C₅	1	7	yes	—
31	C₂×C₄×C₅×C₁₇	C₂×C₄×C₅	1	21	yes	—
32	C₂×C₄×C₅×C₁₇	C₂³×C₅	1	7	yes	—
33	C₅×C₈×C₁₇	C₂³×C₅	1	8	yes	—
34	C₂³×C₄×C₁₇	C₂×C₂₅	1	1	yes	characteristic
35	C₂²×C₈×C₁₇	C₂×C₂₅	1	14	yes	direct product factor
36	C₁₇×C₂₅	C₂³×C₈	1	1	yes	characteristic, direct product factor, Hall
37	C₂⁴×C₂₅	C₄×C₁₇	1	1	yes	characteristic
38	C₂²×C₄×C₂₅	C₄×C₁₇	1	7	yes	—
39	C₂²×C₄×C₂₅	C₂²×C₁₇	1	7	yes	—
40	C₂×C₈×C₂₅	C₂²×C₁₇	1	28	yes	direct product factor
41	C₂²×C₅×C₁₇	C₂×C₅×C₈	1	28	yes	—
42	C₂²×C₅×C₁₇	C₂²×C₄×C₅	1	7	yes	—
43	C₄×C₅×C₁₇	C₂²×C₄×C₅	1	7	yes	—
44	C₄×C₅×C₁₇	C₂⁴×C₅	1	1	yes	characteristic
45	C₂³×C₅×C₈	C₅×C₁₇	1	1	yes	characteristic
46	C₂⁴×C₁₇	C₄×C₂₅	1	1	yes	characteristic
47	C₂²×C₄×C₁₇	C₄×C₂₅	1	7	yes	—
48	C₂²×C₄×C₁₇	C₂²×C₂₅	1	7	yes	—
49	C₂×C₈×C₁₇	C₂²×C₂₅	1	28	yes	direct product factor
50	C₂³×C₂₅	C₈×C₁₇	1	8	yes	direct product factor
51	C₂³×C₂₅	C₂×C₄×C₁₇	1	7	yes	—
52	C₂×C₄×C₂₅	C₂×C₄×C₁₇	1	21	yes	—
53	C₂×C₄×C₂₅	C₂³×C₁₇	1	7	yes	—
54	C₈×C₂₅	C₂³×C₁₇	1	8	yes	direct product factor
55	C₂×C₅×C₁₇	C₂²×C₅×C₈	1	14	yes	—
56	C₂×C₅×C₁₇	C₂³×C₄×C₅	1	1	yes	characteristic
57	C₂³×C₄×C₅	C₂×C₅×C₁₇	1	1	yes	characteristic
58	C₂²×C₅×C₈	C₂×C₅×C₁₇	1	14	yes	—
59	C₂³×C₁₇	C₈×C₂₅	1	8	yes	direct product factor
60	C₂³×C₁₇	C₂×C₄×C₂₅	1	7	yes	—
61	C₂×C₄×C₁₇	C₂×C₄×C₂₅	1	21	yes	—
62	C₂×C₄×C₁₇	C₂³×C₂₅	1	7	yes	—
63	C₈×C₁₇	C₂³×C₂₅	1	8	yes	direct product factor
64	C₂²×C₂₅	C₂×C₈×C₁₇	1	28	yes	direct product factor
65	C₂²×C₂₅	C₂²×C₄×C₁₇	1	7	yes	—
66	C₄×C₂₅	C₂²×C₄×C₁₇	1	7	yes	—
67	C₄×C₂₅	C₂⁴×C₁₇	1	1	yes	characteristic
68	C₅×C₁₇	C₂³×C₅×C₈	1	1	yes	characteristic
69	C₂⁴×C₅	C₄×C₅×C₁₇	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₄×C₅×C₁₇	1	7	yes	—
71	C₂²×C₄×C₅	C₂²×C₅×C₁₇	1	7	yes	—
72	C₂×C₅×C₈	C₂²×C₅×C₁₇	1	28	yes	—
73	C₂²×C₁₇	C₂×C₈×C₂₅	1	28	yes	direct product factor
74	C₂²×C₁₇	C₂²×C₄×C₂₅	1	7	yes	—
75	C₄×C₁₇	C₂²×C₄×C₂₅	1	7	yes	—
76	C₄×C₁₇	C₂⁴×C₂₅	1	1	yes	characteristic
77	C₂³×C₈	C₁₇×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
78	C₂×C₂₅	C₂²×C₈×C₁₇	1	14	yes	direct product factor
79	C₂×C₂₅	C₂³×C₄×C₁₇	1	1	yes	characteristic
80	C₂³×C₅	C₅×C₈×C₁₇	1	8	yes	—
81	C₂³×C₅	C₂×C₄×C₅×C₁₇	1	7	yes	—
82	C₂×C₄×C₅	C₂×C₄×C₅×C₁₇	1	21	yes	—
83	C₂×C₄×C₅	C₂³×C₅×C₁₇	1	7	yes	—
84	C₅×C₈	C₂³×C₅×C₁₇	1	8	yes	—
85	C₂×C₁₇	C₂²×C₈×C₂₅	1	14	yes	direct product factor
86	C₂×C₁₇	C₂³×C₄×C₂₅	1	1	yes	characteristic
87	C₂³×C₄	C₂×C₁₇×C₂₅	1	1	yes	characteristic
88	C₂²×C₈	C₂×C₁₇×C₂₅	1	14	yes	direct product factor
89	C₂₅	C₂³×C₈×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
90	C₂²×C₅	C₂×C₅×C₈×C₁₇	1	28	yes	—
91	C₂²×C₅	C₂²×C₄×C₅×C₁₇	1	7	yes	—
92	C₄×C₅	C₂²×C₄×C₅×C₁₇	1	7	yes	—
93	C₄×C₅	C₂⁴×C₅×C₁₇	1	1	yes	characteristic
94	C₁₇	C₂³×C₈×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
95	C₂⁴	C₄×C₁₇×C₂₅	1	1	yes	characteristic
96	C₂²×C₄	C₄×C₁₇×C₂₅	1	7	yes	—
97	C₂²×C₄	C₂²×C₁₇×C₂₅	1	7	yes	—
98	C₂×C₈	C₂²×C₁₇×C₂₅	1	28	yes	direct product factor
99	C₂×C₅	C₂²×C₅×C₈×C₁₇	1	14	yes	—
100	C₂×C₅	C₂³×C₄×C₅×C₁₇	1	1	yes	characteristic
101	C₂³	C₈×C₁₇×C₂₅	1	8	yes	direct product factor
102	C₂³	C₂×C₄×C₁₇×C₂₅	1	7	yes	—
103	C₂×C₄	C₂×C₄×C₁₇×C₂₅	1	21	yes	—
104	C₂×C₄	C₂³×C₁₇×C₂₅	1	7	yes	—
105	C₈	C₂³×C₁₇×C₂₅	1	8	yes	direct product factor
106	C₅	C₂³×C₅×C₈×C₁₇	1	1	yes	characteristic
107	C₂²	C₂×C₈×C₁₇×C₂₅	1	28	yes	direct product factor
108	C₂²	C₂²×C₄×C₁₇×C₂₅	1	7	yes	—
109	C₄	C₂²×C₄×C₁₇×C₂₅	1	7	yes	—
110	C₄	C₂⁴×C₁₇×C₂₅	1	1	yes	characteristic
111	C₂	C₂²×C₈×C₁₇×C₂₅	1	14	yes	direct product factor
112	C₂	C₂³×C₄×C₁₇×C₂₅	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄×C₅	C₂⁴×C₅×C₁₇
socle, prime radical	C₂⁴×C₅×C₁₇	C₄×C₅
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₈×C₁₇×C₂₅	C₂	direct (N × H)
6. C₂×C₈×C₁₇×C₂₅	C₂²	direct (N × H)
8. C₂³×C₁₇×C₂₅	C₈	direct (N × H)
12. C₈×C₁₇×C₂₅	C₂³	direct (N × H)
15. C₂²×C₁₇×C₂₅	C₂×C₈	direct (N × H)
19. C₂³×C₈×C₂₅	C₁₇	direct (N × H)
24. C₂³×C₈×C₁₇	C₂₅	direct (N × H)
25. C₂×C₁₇×C₂₅	C₂²×C₈	direct (N × H)
28. C₂²×C₈×C₂₅	C₂×C₁₇	direct (N × H)
35. C₂²×C₈×C₁₇	C₂×C₂₅	direct (N × H)
36. C₁₇×C₂₅	C₂³×C₈	direct (N × H)
40. C₂×C₈×C₂₅	C₂²×C₁₇	direct (N × H)
49. C₂×C₈×C₁₇	C₂²×C₂₅	direct (N × H)
50. C₂³×C₂₅	C₈×C₁₇	direct (N × H)
54. C₈×C₂₅	C₂³×C₁₇	direct (N × H)
59. C₂³×C₁₇	C₈×C₂₅	direct (N × H)
63. C₈×C₁₇	C₂³×C₂₅	direct (N × H)
64. C₂²×C₂₅	C₂×C₈×C₁₇	direct (N × H)
73. C₂²×C₁₇	C₂×C₈×C₂₅	direct (N × H)
77. C₂³×C₈	C₁₇×C₂₅	direct (N × H)
78. C₂×C₂₅	C₂²×C₈×C₁₇	direct (N × H)
85. C₂×C₁₇	C₂²×C₈×C₂₅	direct (N × H)
88. C₂²×C₈	C₂×C₁₇×C₂₅	direct (N × H)
89. C₂₅	C₂³×C₈×C₁₇	direct (N × H)
94. C₁₇	C₂³×C₈×C₂₅	direct (N × H)
98. C₂×C₈	C₂²×C₁₇×C₂₅	direct (N × H)
101. C₂³	C₈×C₁₇×C₂₅	direct (N × H)
105. C₈	C₂³×C₁₇×C₂₅	direct (N × H)
107. C₂²	C₂×C₈×C₁₇×C₂₅	direct (N × H)
111. C₂	C₂²×C₈×C₁₇×C₂₅	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×C₁₇×C₂₅	2
C₂²×C₈×C₁₇×C₂₅	2
C₂³×C₅×C₈×C₁₇	5
C₂³×C₈×C₂₅	17

Subgroups — C23×C8×C17×C25 #72204263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₈×C₁₇×C₂₅ #72204263