Subgroups — C₂³×C₄×#88322 #28247472

Order 13568.

4638 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	C₂⁴×#88322	C₂	1	2	yes	maximal
2	C₂²×C₄×#88322	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#88322	C₄	1	32	yes	direct product factor
5	C₂³×#88322	C₂²	1	28	yes	—
6	C₂×C₄×#88322	C₂²	1	112	yes	direct product factor
7	C₂⁴×C₄×C₅₃	C₄	1	2	yes	—
8	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₅₃	C₄	1	14	yes	—
10	C₂²×C₄²×C₅₃	C₂²	1	14	yes	—
11	C₂²×#88322	C₂×C₄	1	224	yes	direct product factor
12	C₂²×#88322	C₂³	1	56	yes	—
13	C₄×#88322	C₂³	1	64	yes	direct product factor
14	C₂⁵×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₅₃	C₈	1	32	yes	complemented
16	C₂³×C₄×C₅₃	C₂×C₄	1	7	yes	—
17	C₂³×C₄×C₅₃	C₂×C₄	1	16	yes	—
18	C₂³×C₄×C₅₃	C₂×C₄	1	28	yes	—
19	C₂³×C₄×C₅₃	C₂³	1	7	yes	—
20	C₂×C₄²×C₅₃	C₂×C₄	1	84	yes	—
21	C₂×C₄²×C₅₃	C₂³	1	28	yes	—
22	C₂⁴×C₅₃	C₄²	1	8	yes	—
23	C₂⁴×C₅₃	C₂×C₈	1	16	yes	—
24	C₂⁴×C₅₃	C₂²×C₄	1	7	yes	—
25	C₂²×C₄×C₅₃	C₄²	1	56	yes	—
26	C₂²×C₄×C₅₃	C₂×C₈	1	224	yes	complemented
27	C₂²×C₄×C₅₃	C₂²×C₄	1	21	yes	—
28	C₂²×C₄×C₅₃	C₂²×C₄	1	56	yes	—
29	C₂²×C₄×C₅₃	C₂²×C₄	1	56	yes	—
30	C₂²×C₄×C₅₃	C₂⁴	1	7	yes	—
31	C₄²×C₅₃	C₂²×C₄	1	56	yes	—
32	C₄²×C₅₃	C₂⁴	1	8	yes	—
33	C₂×#88322	C₂²×C₄	1	224	yes	direct product factor
34	C₂×#88322	C₂⁴	1	16	yes	—
35	C₂³×C₅₃	C₄×C₈	1	64	yes	complemented
36	C₂³×C₅₃	C₂×C₄²	1	28	yes	—
37	C₂³×C₅₃	C₂²×C₈	1	56	yes	—
38	C₂³×C₅₃	C₂³×C₄	1	7	yes	—
39	C₂×C₄×C₅₃	C₂×C₄²	1	84	yes	—
40	C₂×C₄×C₅₃	C₂²×C₈	1	224	yes	complemented
41	C₂×C₄×C₅₃	C₂³×C₄	1	28	yes	—
42	C₂×C₄×C₅₃	C₂³×C₄	1	16	yes	—
43	C₂×C₄×C₅₃	C₂³×C₄	1	7	yes	—
44	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
45	#88322	C₂³×C₄	1	32	yes	direct product factor
46	C₂³×C₄×C₈	—	53	1	no	maximal, Sylow
47	C₂²×C₅₃	C₂×C₄×C₈	1	112	yes	complemented
48	C₂²×C₅₃	C₂²×C₄²	1	14	yes	—
49	C₂²×C₅₃	C₂³×C₈	1	28	yes	—
50	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
51	C₄×C₅₃	C₂²×C₄²	1	14	yes	—
52	C₄×C₅₃	C₂³×C₈	1	32	yes	complemented
53	C₄×C₅₃	C₂⁴×C₄	1	2	yes	—
54	C₂⁴×C₈	—	53	2	no	—
55	C₂³×C₄²	D₅₃	1	1	yes	characteristic
56	C₂²×C₄×C₈	—	53	28	no	—
57	C₂×C₅₃	C₂²×C₄×C₈	1	28	yes	complemented
58	C₂×C₅₃	C₂³×C₄²	1	1	yes	characteristic
59	C₂×C₅₃	C₂⁴×C₈	1	2	yes	—
60	C₂⁴×C₄	Dic₅₃	1	2	yes	—
61	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
62	C₂³×C₈	—	53	32	no	—
63	C₂³×C₈	—	53	28	no	—
64	C₂²×C₄²	Dic₅₃	1	14	yes	—
65	C₂²×C₄²	C₂×D₅₃	1	14	yes	—
66	C₂×C₄×C₈	—	53	112	no	—
67	C₅₃	C₂³×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
68	C₂⁵	C₂×Dic₅₃	1	1	yes	characteristic
69	C₂³×C₄	#88322	1	32	yes	direct product factor
70	C₂³×C₄	C₄×D₅₃	1	16	yes	—
71	C₂³×C₄	C₂×Dic₅₃	1	28	yes	—
72	C₂³×C₄	C₂×Dic₅₃	1	7	yes	—
73	C₂³×C₄	C₂²×D₅₃	1	7	yes	—
74	C₂²×C₈	—	53	224	no	—
75	C₂²×C₈	—	53	56	no	—
76	C₂×C₄²	C₂×Dic₅₃	1	84	yes	—
77	C₂×C₄²	C₂²×D₅₃	1	28	yes	—
78	C₄×C₈	—	53	64	no	—
79	C₂⁴	C₂×#88322	1	16	yes	—
80	C₂⁴	C₄×Dic₅₃	1	8	yes	—
81	C₂⁴	C₂²×Dic₅₃	1	7	yes	—
82	C₂²×C₄	C₂×#88322	1	224	yes	direct product factor
83	C₂²×C₄	C₄×Dic₅₃	1	56	yes	—
84	C₂²×C₄	C₂×C₄×D₅₃	1	56	yes	—
85	C₂²×C₄	C₂²×Dic₅₃	1	21	yes	—
86	C₂²×C₄	C₂²×Dic₅₃	1	56	yes	—
87	C₂²×C₄	C₂³×D₅₃	1	7	yes	—
88	C₂×C₈	—	53	224	no	—
89	C₂×C₈	—	53	16	no	—
90	C₄²	C₂²×Dic₅₃	1	56	yes	—
91	C₄²	C₂³×D₅₃	1	8	yes	—
92	C₂³	C₄×#88322	1	64	yes	direct product factor
93	C₂³	C₂×C₄×Dic₅₃	1	28	yes	—
94	C₂³	C₂²×#88322	1	56	yes	—
95	C₂³	C₂³×Dic₅₃	1	7	yes	—
96	C₂×C₄	C₂×C₄×Dic₅₃	1	84	yes	—
97	C₂×C₄	C₂²×#88322	1	224	yes	direct product factor
98	C₂×C₄	C₂²×C₄×D₅₃	1	28	yes	—
99	C₂×C₄	C₂³×Dic₅₃	1	7	yes	—
100	C₂×C₄	C₂³×Dic₅₃	1	16	yes	—
101	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
102	C₈	—	53	32	no	—
103	C₂²	C₂×C₄×#88322	1	112	yes	direct product factor
104	C₂²	C₂²×C₄×Dic₅₃	1	14	yes	—
105	C₂²	C₂³×#88322	1	28	yes	—
106	C₂²	C₂⁴×Dic₅₃	1	1	yes	characteristic
107	C₄	C₂²×C₄×Dic₅₃	1	14	yes	—
108	C₄	C₂³×#88322	1	32	yes	direct product factor
109	C₄	C₂³×C₄×D₅₃	1	2	yes	—
110	C₂	C₂²×C₄×#88322	1	28	yes	direct product factor
111	C₂	C₂³×C₄×Dic₅₃	1	1	yes	characteristic
112	C₂	C₂⁴×#88322	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂³×C₄²	D₅₃
commutator, hypocenter	C₅₃	C₂³×C₄×C₈
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂³×C₄²×C₅₃	C₂
socle, prime radical	C₂⁵×C₅₃	C₂×C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×#88322	C₂	direct (N × H)
4. C₂³×#88322	C₄	direct (N × H)
6. C₂×C₄×#88322	C₂²	direct (N × H)
11. C₂²×#88322	C₂×C₄	direct (N × H)
13. C₄×#88322	C₂³	direct (N × H)
15. C₂³×C₄×C₅₃	C₈	semidirect
26. C₂²×C₄×C₅₃	C₂×C₈	semidirect
33. C₂×#88322	C₂²×C₄	direct (N × H)
35. C₂³×C₅₃	C₄×C₈	semidirect
40. C₂×C₄×C₅₃	C₂²×C₈	semidirect
45. #88322	C₂³×C₄	direct (N × H)
47. C₂²×C₅₃	C₂×C₄×C₈	semidirect
52. C₄×C₅₃	C₂³×C₈	semidirect
57. C₂×C₅₃	C₂²×C₄×C₈	semidirect
67. C₅₃	C₂³×C₄×C₈	semidirect
69. C₂³×C₄	#88322	direct (N × H)
82. C₂²×C₄	C₂×#88322	direct (N × H)
92. C₂³	C₄×#88322	direct (N × H)
97. C₂×C₄	C₂²×#88322	direct (N × H)
103. C₂²	C₂×C₄×#88322	direct (N × H)
108. C₄	C₂³×#88322	direct (N × H)
110. C₂	C₂²×C₄×#88322	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×#88322	2
C₂²×C₄×#88322	2
C₂³×C₄²×C₅₃	2
C₂³×C₄×C₈	53

Subgroups — C23×C4×#88322 #28247472

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄×#88322 #28247472