Subgroups — C₂³×#13133127 #28247474

Order 13568.

4126 conjugacy classes of subgroups.

The subgroup lattice has 138 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	1	yes	maximal, characteristic
2	C₂³×C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13133127	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×Dic₅₃	C₄	1	2	yes	—
6	C₂³×#88322	C₄	1	16	yes	complemented
7	C₂³×#88322	C₂²	1	14	yes	—
8	C₂²×C₄×Dic₅₃	C₄	1	14	yes	—
9	C₂²×C₄×Dic₅₃	C₂²	1	14	yes	—
10	C₂×#13133127	C₂²	1	112	yes	direct product factor
11	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₅₃	C₄	1	16	yes	complemented
13	C₂³×C₈×C₅₃	C₂²	1	14	yes	—
14	C₂³×Dic₅₃	—	2	16	no	—
15	C₂³×Dic₅₃	C₂×C₄	1	28	yes	—
16	C₂²×#88322	C₂×C₄	1	112	yes	complemented
17	C₂²×#88322	C₂³	1	28	yes	—
18	C₂×C₄×Dic₅₃	C₂×C₄	1	84	yes	—
19	C₂×C₄×Dic₅₃	C₂³	1	28	yes	—
20	#13133127	C₂³	1	64	yes	direct product factor
21	C₂⁵×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₅₃	C₂×C₄	1	7	yes	—
23	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
25	C₂³×C₄×C₅₃	C₂³	1	7	yes	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	112	yes	complemented
27	C₂²×C₈×C₅₃	C₂³	1	28	yes	—
28	C₂⁴×C₅₃	C₄²	1	8	yes	—
29	C₂⁴×C₅₃	M₁₆	1	16	yes	—
30	C₂⁴×C₅₃	C₂²×C₄	1	7	yes	—
31	C₂²×C₄×C₅₃	C₄²	1	56	yes	—
32	C₂²×C₄×C₅₃	C₂²×C₄	1	21	yes	—
33	C₂²×C₄×C₅₃	C₂²×C₄	1	28	yes	—
34	C₂²×C₄×C₅₃	C₂²×C₄	1	28	yes	—
35	C₂²×C₄×C₅₃	C₂⁴	1	7	yes	—
36	C₂²×Dic₅₃	—	2	112	no	—
37	C₂²×Dic₅₃	C₂²×C₄	1	56	yes	—
38	C₂×C₈×C₅₃	C₂²×C₄	1	112	yes	complemented
39	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
40	C₄×Dic₅₃	C₂²×C₄	1	56	yes	—
41	C₄×Dic₅₃	C₂⁴	1	8	yes	—
42	C₂×#88322	C₂²×C₄	1	112	yes	complemented
43	C₂×#88322	C₂⁴	1	8	yes	—
44	C₂³×C₅₃	#97	1	64	yes	complemented
45	C₂³×C₅₃	C₂×C₄²	1	28	yes	—
46	C₂³×C₅₃	C₂×M₁₆	1	56	yes	—
47	C₂³×C₅₃	C₂³×C₄	1	7	yes	—
48	C₂×C₄×C₅₃	C₂×C₄²	1	84	yes	—
49	C₂×C₄×C₅₃	C₂³×C₄	1	14	yes	—
50	C₂×C₄×C₅₃	C₂³×C₄	1	7	yes	—
51	C₂×C₄×C₅₃	C₂³×C₄	1	14	yes	—
52	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
53	C₂×Dic₅₃	—	2	112	no	—
54	C₂×Dic₅₃	C₂³×C₄	1	16	yes	—
55	C₈×C₅₃	C₂³×C₄	1	16	yes	complemented
56	#88322	C₂³×C₄	1	16	yes	complemented
57	C₂³×#97	—	53	1	no	maximal, Sylow
58	C₂²×C₅₃	C₂×#97	1	112	yes	complemented
59	C₂²×C₅₃	C₂²×C₄²	1	14	yes	—
60	C₂²×C₅₃	C₂²×M₁₆	1	28	yes	—
61	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
62	C₄×C₅₃	C₂²×C₄²	1	14	yes	—
63	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
64	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
65	Dic₅₃	—	2	16	no	—
66	C₂⁴×C₈	—	53	1	no	—
67	C₂⁴×C₈	D₅₃	1	1	yes	characteristic
68	C₂³×C₄²	—	53	1	no	—
69	C₂²×#97	—	53	28	no	—
70	C₂×C₅₃	C₂²×#97	1	28	yes	complemented
71	C₂×C₅₃	C₂³×C₄²	1	1	yes	characteristic
72	C₂×C₅₃	C₂³×M₁₆	1	2	yes	—
73	C₂⁴×C₄	—	53	2	no	—
74	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
75	C₂³×C₈	—	53	16	no	—
76	C₂³×C₈	—	53	14	no	—
77	C₂³×C₈	Dic₅₃	1	16	yes	complemented
78	C₂³×C₈	C₂×D₅₃	1	14	yes	—
79	C₂²×C₄²	—	53	14	no	—
80	C₂²×C₄²	—	53	14	no	—
81	C₂×#97	—	53	112	no	—
82	C₅₃	C₂³×#97	1	1	yes	characteristic, complemented, Sylow
83	C₂⁵	C₄×D₅₃	1	1	yes	characteristic
84	C₂³×C₄	—	53	28	no	—
85	C₂³×C₄	—	106	16	no	—
86	C₂³×C₄	C₄×D₅₃	1	8	yes	—
87	C₂³×C₄	C₄×D₅₃	1	7	yes	—
88	C₂³×C₄	C₂×Dic₅₃	1	8	yes	—
89	C₂³×C₄	C₂²×D₅₃	1	7	yes	—
90	C₂²×C₈	—	53	112	no	—
91	C₂²×C₈	—	53	28	no	—
92	C₂²×C₈	C₂×Dic₅₃	1	112	yes	complemented
93	C₂²×C₈	C₂²×D₅₃	1	28	yes	—
94	C₂×C₄²	—	53	84	no	—
95	C₂×C₄²	—	53	28	no	—
96	#97	—	53	64	no	—
97	C₂⁴	#11719912	1	16	yes	—
98	C₂⁴	C₄×Dic₅₃	1	8	yes	—
99	C₂⁴	C₂×C₄×D₅₃	1	7	yes	—
100	C₂²×C₄	—	53	56	no	—
101	C₂²×C₄	—	106	112	no	—
102	C₂²×C₄	C₄×Dic₅₃	1	56	yes	—
103	C₂²×C₄	C₂×C₄×D₅₃	1	21	yes	—
104	C₂²×C₄	C₂×C₄×D₅₃	1	28	yes	—
105	C₂²×C₄	C₂²×Dic₅₃	1	28	yes	—
106	C₂²×C₄	C₂³×D₅₃	1	7	yes	—
107	C₂×C₈	—	53	112	no	—
108	C₂×C₈	—	53	8	no	—
109	C₂×C₈	C₂²×Dic₅₃	1	112	yes	complemented
110	C₂×C₈	C₂³×D₅₃	1	8	yes	—
111	C₄²	—	53	56	no	—
112	C₄²	—	53	8	no	—
113	C₂³	#13133127	1	64	yes	direct product factor
114	C₂³	C₂×C₄×Dic₅₃	1	28	yes	—
115	C₂³	C₂×#11719912	1	56	yes	—
116	C₂³	C₂²×C₄×D₅₃	1	7	yes	—
117	C₂×C₄	—	53	16	no	—
118	C₂×C₄	—	106	112	no	—
119	C₂×C₄	C₂×C₄×Dic₅₃	1	84	yes	—
120	C₂×C₄	C₂²×C₄×D₅₃	1	7	yes	—
121	C₂×C₄	C₂²×C₄×D₅₃	1	14	yes	—
122	C₂×C₄	C₂³×Dic₅₃	1	14	yes	—
123	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
124	C₈	—	53	16	no	—
125	C₈	C₂³×Dic₅₃	1	16	yes	complemented
126	C₂²	C₂×#13133127	1	112	yes	direct product factor
127	C₂²	C₂²×C₄×Dic₅₃	1	14	yes	—
128	C₂²	C₂²×#11719912	1	28	yes	—
129	C₂²	C₂³×C₄×D₅₃	1	1	yes	characteristic
130	C₄	—	106	16	no	—
131	C₄	C₂²×C₄×Dic₅₃	1	14	yes	—
132	C₄	C₂³×C₄×D₅₃	1	1	yes	characteristic
133	C₄	C₂⁴×Dic₅₃	1	1	yes	characteristic
134	C₂	C₂²×#13133127	1	28	yes	direct product factor
135	C₂	C₂³×C₄×Dic₅₃	1	1	yes	characteristic
136	C₂	C₂³×#11719912	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₅₃
commutator	C₂×C₅₃	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂⁴×C₈×C₅₃	C₂
socle, prime radical	C₂⁵×C₅₃	C₂×C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂⁴×C₈	D₅₃
hypocenter	C₅₃	C₂³×#97
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
3. C₂²×#13133127	C₂	direct (N × H)
6. C₂³×#88322	C₄	semidirect
10. C₂×#13133127	C₂²	direct (N × H)
12. C₂³×C₈×C₅₃	C₄	semidirect
16. C₂²×#88322	C₂×C₄	semidirect
20. #13133127	C₂³	direct (N × H)
26. C₂²×C₈×C₅₃	C₂×C₄	semidirect
38. C₂×C₈×C₅₃	C₂²×C₄	semidirect
42. C₂×#88322	C₂²×C₄	semidirect
44. C₂³×C₅₃	#97	semidirect
55. C₈×C₅₃	C₂³×C₄	semidirect
56. #88322	C₂³×C₄	semidirect
58. C₂²×C₅₃	C₂×#97	semidirect
70. C₂×C₅₃	C₂²×#97	semidirect
77. C₂³×C₈	Dic₅₃	semidirect
82. C₅₃	C₂³×#97	semidirect
92. C₂²×C₈	C₂×Dic₅₃	semidirect
109. C₂×C₈	C₂²×Dic₅₃	semidirect
113. C₂³	#13133127	direct (N × H)
125. C₈	C₂³×Dic₅₃	semidirect
126. C₂²	C₂×#13133127	direct (N × H)
134. C₂	C₂²×#13133127	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
C₂³×C₄×Dic₅₃	2
C₂²×#13133127	2
C₂⁴×C₈×C₅₃	2
C₂³×#97	53

Subgroups — C23×#13133127 #28247474

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133127 #28247474