Subgroups — C₂³×#88128 #16769606

Order 3328.

24030 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₅₂	C₂	1	6	yes	maximal, complemented
2	C₂²×#88128	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄²×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁵×D₁₃	C₂²	1	4	yes	—
5	C₂⁴×C₄×C₁₃	C₂²	1	3	yes	—
6	C₂³×D₅₂	—	2	96	no	—
7	C₂³×D₅₂	C₂²	1	168	yes	complemented
8	C₂²×C₄²×C₁₃	C₂²	1	28	yes	complemented
9	C₂×#88128	C₂²	1	448	yes	direct product factor
10	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
11	C₂⁴×D₁₃	—	2	96	no	—
12	C₂⁴×D₁₃	C₂³	1	56	yes	—
13	C₂³×C₄×C₁₃	D₄	1	48	yes	complemented
14	C₂³×C₄×C₁₃	C₂³	1	42	yes	—
15	C₂²×D₅₂	—	2	1344	no	—
16	C₂²×D₅₂	C₂³	1	672	yes	complemented
17	C₂×C₄²×C₁₃	C₂³	1	112	yes	complemented
18	#88128	C₂³	1	512	yes	direct product factor
19	C₂³×#127	—	13	1	no	maximal, Sylow
20	C₂⁴×C₁₃	C₂×D₄	1	24	yes	—
21	C₂⁴×C₁₃	C₂⁴	1	7	yes	—
22	C₂³×D₁₃	—	2	672	no	—
23	C₂³×D₁₃	—	4	256	no	—
24	C₂³×D₁₃	C₂⁴	1	112	yes	—
25	C₂²×C₄×C₁₃	C₂×D₄	1	336	yes	complemented
26	C₂²×C₄×C₁₃	C₂⁴	1	84	yes	—
27	C₂×D₅₂	—	2	2688	no	—
28	C₂×D₅₂	C₂⁴	1	384	yes	complemented
29	C₄²×C₁₃	C₂⁴	1	64	yes	complemented
30	C₂⁴×D₄	—	13	6	no	—
31	C₂²×#127	—	13	56	no	—
32	C₂³×C₄²	D₁₃	1	1	yes	characteristic, complemented
33	C₂³×C₁₃	#127	1	64	yes	complemented
34	C₂³×C₁₃	C₂²×D₄	1	84	yes	—
35	C₂³×C₁₃	C₂⁵	1	7	yes	—
36	C₂²×D₁₃	—	2	672	no	—
37	C₂²×D₁₃	—	4	896	no	—
38	C₂²×D₁₃	C₂⁵	1	32	yes	—
39	C₂×C₄×C₁₃	C₂²×D₄	1	336	yes	complemented
40	C₂×C₄×C₁₃	C₂⁵	1	24	yes	—
41	D₅₂	—	2	768	no	—
42	C₂⁶	—	13	4	no	—
43	C₂³×D₄	—	13	168	no	—
44	C₂³×D₄	—	26	96	no	—
45	C₂⁴×C₄	C₂×D₁₃	1	3	yes	—
46	C₂×#127	—	13	448	no	—
47	C₂²×C₄²	C₂×D₁₃	1	28	yes	complemented
48	C₂²×C₁₃	C₂×#127	1	112	yes	complemented
49	C₂²×C₁₃	C₂³×D₄	1	42	yes	—
50	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
51	C₂×D₁₃	—	2	96	no	—
52	C₂×D₁₃	—	4	448	no	—
53	C₄×C₁₃	C₂³×D₄	1	48	yes	complemented
54	C₂⁵	—	13	56	no	—
55	C₂⁵	—	26	96	no	—
56	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
57	C₂²×D₄	—	13	672	no	—
58	C₂²×D₄	—	26	1344	no	—
59	C₂³×C₄	D₅₂	1	48	yes	complemented
60	C₂³×C₄	C₂²×D₁₃	1	42	yes	—
61	#127	—	13	512	no	—
62	C₂×C₄²	C₂²×D₁₃	1	112	yes	complemented
63	C₂×C₁₃	C₂²×#127	1	28	yes	complemented
64	C₂×C₁₃	C₂⁴×D₄	1	3	yes	—
65	D₁₃	—	4	32	no	—
66	C₂⁴	—	13	112	no	—
67	C₂⁴	—	26	672	no	—
68	C₂⁴	—	52	256	no	—
69	C₂⁴	C₂×D₅₂	1	24	yes	—
70	C₂⁴	C₂³×D₁₃	1	7	yes	—
71	C₂×D₄	—	13	384	no	—
72	C₂×D₄	—	26	2688	no	—
73	C₂²×C₄	C₂×D₅₂	1	336	yes	complemented
74	C₂²×C₄	C₂³×D₁₃	1	84	yes	—
75	C₄²	C₂³×D₁₃	1	64	yes	complemented
76	C₁₃	C₂³×#127	1	1	yes	characteristic, complemented, Sylow
77	C₂³	—	13	32	no	—
78	C₂³	—	26	672	no	—
79	C₂³	—	52	896	no	—
80	C₂³	#88128	1	64	yes	direct product factor
81	C₂³	C₂²×D₅₂	1	84	yes	—
82	C₂³	C₂⁴×D₁₃	1	7	yes	—
83	D₄	—	26	768	no	—
84	C₂×C₄	C₂²×D₅₂	1	336	yes	complemented
85	C₂×C₄	C₂⁴×D₁₃	1	24	yes	—
86	C₂²	—	26	96	no	—
87	C₂²	—	52	448	no	—
88	C₂²	C₂×#88128	1	112	yes	direct product factor
89	C₂²	C₂³×D₅₂	1	42	yes	—
90	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
91	C₄	C₂³×D₅₂	1	48	yes	complemented
92	C₂	—	52	32	no	—
93	C₂	C₂²×#88128	1	28	yes	direct product factor
94	C₂	C₂⁴×D₅₂	1	3	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂³×C₄²×C₁₃	C₂
socle	C₂⁵×C₁₃	C₂³
hypercenter	C₂³×C₄²	D₁₃
hypocenter	C₁₃	C₂³×#127
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₅₂	C₂	semidirect
2. C₂²×#88128	C₂	direct (N × H)
3. C₂³×C₄²×C₁₃	C₂	semidirect
7. C₂³×D₅₂	C₂²	semidirect
8. C₂²×C₄²×C₁₃	C₂²	semidirect
9. C₂×#88128	C₂²	direct (N × H)
13. C₂³×C₄×C₁₃	D₄	semidirect
16. C₂²×D₅₂	C₂³	semidirect
17. C₂×C₄²×C₁₃	C₂³	semidirect
18. #88128	C₂³	direct (N × H)
25. C₂²×C₄×C₁₃	C₂×D₄	semidirect
28. C₂×D₅₂	C₂⁴	semidirect
29. C₄²×C₁₃	C₂⁴	semidirect
32. C₂³×C₄²	D₁₃	semidirect
33. C₂³×C₁₃	#127	semidirect
39. C₂×C₄×C₁₃	C₂²×D₄	semidirect
47. C₂²×C₄²	C₂×D₁₃	semidirect
48. C₂²×C₁₃	C₂×#127	semidirect
53. C₄×C₁₃	C₂³×D₄	semidirect
59. C₂³×C₄	D₅₂	semidirect
62. C₂×C₄²	C₂²×D₁₃	semidirect
63. C₂×C₁₃	C₂²×#127	semidirect
73. C₂²×C₄	C₂×D₅₂	semidirect
75. C₄²	C₂³×D₁₃	semidirect
76. C₁₃	C₂³×#127	semidirect
80. C₂³	#88128	direct (N × H)
84. C₂×C₄	C₂²×D₅₂	semidirect
88. C₂²	C₂×#88128	direct (N × H)
91. C₄	C₂³×D₅₂	semidirect
93. C₂	C₂²×#88128	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₅₂	2
C₂²×#88128	2
C₂³×C₄²×C₁₃	2
C₂³×#127	13

Subgroups — C23×#88128 #16769606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88128 #16769606