Subgroups — C₂²×C₅₃×#493 #27268141

Order 13568.

4638 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₈×C₅₃	C₂	1	4	yes	maximal, complemented
2	C₂²×#127×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₂×C₅₃×#493	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₄×C₅₃	—	2	4	no	—
6	C₂³×D₄×C₅₃	C₂²	1	4	yes	—
7	C₂²×D₈×C₅₃	—	2	32	no	—
8	C₂²×D₈×C₅₃	C₂²	1	48	yes	complemented
9	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
10	C₂×#127×C₅₃	—	2	12	no	—
11	C₂×#127×C₅₃	C₂²	1	24	yes	complemented
12	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#493	C₂²	1	64	yes	direct product factor
14	C₂×C₄×C₈×C₅₃	C₂²	1	12	yes	complemented
15	C₂⁵×C₅₃	—	2	4	no	—
16	C₂²×D₄×C₅₃	—	2	32	no	—
17	C₂²×D₄×C₅₃	—	2	12	no	—
18	C₂²×D₄×C₅₃	—	2	48	no	—
19	C₂²×D₄×C₅₃	—	4	32	no	—
20	C₂²×D₄×C₅₃	C₂³	1	24	yes	—
21	C₂³×C₄×C₅₃	D₄	1	2	yes	—
22	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂×D₈×C₅₃	—	2	192	no	—
24	C₂×D₈×C₅₃	C₂³	1	64	yes	complemented
25	C₂²×C₈×C₅₃	D₄	1	16	yes	complemented
26	C₂²×C₈×C₅₃	C₂³	1	12	yes	—
27	#127×C₅₃	—	2	48	no	—
28	#127×C₅₃	C₂³	1	32	yes	complemented
29	C₂×C₄²×C₅₃	D₄	1	6	yes	—
30	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
31	C₄×C₈×C₅₃	C₂³	1	16	yes	complemented
32	C₂⁴×C₅₃	—	2	24	no	—
33	C₂⁴×C₅₃	—	4	16	no	—
34	C₂⁴×C₅₃	—	4	32	no	—
35	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×D₄×C₅₃	—	2	96	no	—
37	C₂×D₄×C₅₃	—	2	24	no	—
38	C₂×D₄×C₅₃	—	2	64	no	—
39	C₂×D₄×C₅₃	—	4	48	no	—
40	C₂×D₄×C₅₃	—	4	192	no	—
41	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
42	C₂²×C₄×C₅₃	D₈	1	16	yes	complemented
43	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
44	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
45	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
46	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
47	D₈×C₅₃	—	2	128	no	—
48	C₂×C₈×C₅₃	C₂×D₄	1	48	yes	complemented
49	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
50	C₄²×C₅₃	C₂×D₄	1	12	yes	—
51	C₄²×C₅₃	C₂⁴	1	4	yes	—
52	C₂³×C₅₃	—	2	16	no	—
53	C₂³×C₅₃	—	4	48	no	—
54	C₂³×C₅₃	—	4	96	no	—
55	C₂³×C₅₃	—	8	64	no	—
56	C₂³×C₅₃	#127	1	4	yes	—
57	C₂³×C₅₃	C₂×D₈	1	8	yes	—
58	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
59	D₄×C₅₃	—	2	32	no	—
60	D₄×C₅₃	—	4	48	no	—
61	D₄×C₅₃	—	4	128	no	—
62	C₂×C₄×C₅₃	#127	1	12	yes	—
63	C₂×C₄×C₅₃	C₂×D₈	1	48	yes	complemented
64	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
65	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
66	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
67	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
68	C₈×C₅₃	C₂²×D₄	1	16	yes	complemented
69	C₂²×#493	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂²×C₅₃	—	4	16	no	—
71	C₂²×C₅₃	—	4	32	no	—
72	C₂²×C₅₃	—	8	96	no	—
73	C₂²×C₅₃	#493	1	16	yes	direct product factor
74	C₂²×C₅₃	C₂×#127	1	6	yes	—
75	C₂²×C₅₃	C₂²×D₈	1	12	yes	—
76	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
77	C₄×C₅₃	C₂×#127	1	6	yes	—
78	C₄×C₅₃	C₂²×D₈	1	16	yes	complemented
79	C₄×C₅₃	C₂³×D₄	1	2	yes	—
80	C₂³×D₈	C₂×C₅₃	1	4	yes	complemented
81	C₂²×#127	C₂×C₅₃	1	2	yes	complemented
82	C₂×#493	C₂×C₅₃	1	24	yes	direct product factor
83	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
84	C₂×C₅₃	—	8	16	no	—
85	C₂×C₅₃	C₂×#493	1	12	yes	direct product factor
86	C₂×C₅₃	C₂²×#127	1	1	yes	characteristic
87	C₂×C₅₃	C₂³×D₈	1	2	yes	—
88	C₂³×D₄	—	2	4	no	—
89	C₂³×D₄	C₂²×C₅₃	1	4	yes	—
90	C₂²×D₈	—	2	32	no	—
91	C₂²×D₈	C₂²×C₅₃	1	48	yes	complemented
92	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
93	C₂×#127	—	2	12	no	—
94	C₂×#127	C₂²×C₅₃	1	24	yes	complemented
95	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
96	#493	C₂²×C₅₃	1	64	yes	direct product factor
97	C₂×C₄×C₈	C₂²×C₅₃	1	12	yes	complemented
98	C₅₃	C₂²×#493	1	1	yes	characteristic, direct product factor, Sylow
99	C₂⁵	—	2	4	no	—
100	C₂²×D₄	—	2	32	no	—
101	C₂²×D₄	—	2	12	no	—
102	C₂²×D₄	—	2	48	no	—
103	C₂²×D₄	—	4	32	no	—
104	C₂²×D₄	C₂³×C₅₃	1	24	yes	—
105	C₂³×C₄	D₄×C₅₃	1	2	yes	—
106	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
107	C₂×D₈	—	2	192	no	—
108	C₂×D₈	C₂³×C₅₃	1	64	yes	complemented
109	C₂²×C₈	D₄×C₅₃	1	16	yes	complemented
110	C₂²×C₈	C₂³×C₅₃	1	12	yes	—
111	#127	—	2	48	no	—
112	#127	C₂³×C₅₃	1	32	yes	complemented
113	C₂×C₄²	D₄×C₅₃	1	6	yes	—
114	C₂×C₄²	C₂³×C₅₃	1	6	yes	—
115	C₄×C₈	C₂³×C₅₃	1	16	yes	complemented
116	C₂⁴	—	2	24	no	—
117	C₂⁴	—	4	16	no	—
118	C₂⁴	—	4	32	no	—
119	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
120	C₂×D₄	—	2	96	no	—
121	C₂×D₄	—	2	24	no	—
122	C₂×D₄	—	2	64	no	—
123	C₂×D₄	—	4	48	no	—
124	C₂×D₄	—	4	192	no	—
125	C₂×D₄	C₂⁴×C₅₃	1	16	yes	—
126	C₂²×C₄	D₈×C₅₃	1	16	yes	complemented
127	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
128	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
129	C₂²×C₄	C₂×D₄×C₅₃	1	12	yes	—
130	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
131	D₈	—	2	128	no	—
132	C₂×C₈	C₂×D₄×C₅₃	1	48	yes	complemented
133	C₂×C₈	C₂⁴×C₅₃	1	8	yes	—
134	C₄²	C₂×D₄×C₅₃	1	12	yes	—
135	C₄²	C₂⁴×C₅₃	1	4	yes	—
136	C₂³	—	2	16	no	—
137	C₂³	—	4	48	no	—
138	C₂³	—	4	96	no	—
139	C₂³	—	8	64	no	—
140	C₂³	#127×C₅₃	1	4	yes	—
141	C₂³	C₂×D₈×C₅₃	1	8	yes	—
142	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
143	D₄	—	2	32	no	—
144	D₄	—	4	48	no	—
145	D₄	—	4	128	no	—
146	C₂×C₄	#127×C₅₃	1	12	yes	—
147	C₂×C₄	C₂×D₈×C₅₃	1	48	yes	complemented
148	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
149	C₂×C₄	C₂²×D₄×C₅₃	1	12	yes	—
150	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
151	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
152	C₈	C₂²×D₄×C₅₃	1	16	yes	complemented
153	C₂²	—	4	32	no	—
154	C₂²	—	4	16	no	—
155	C₂²	—	8	96	no	—
156	C₂²	C₅₃×#493	1	16	yes	direct product factor
157	C₂²	C₂×#127×C₅₃	1	6	yes	—
158	C₂²	C₂²×D₈×C₅₃	1	12	yes	—
159	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
160	C₄	C₂×#127×C₅₃	1	6	yes	—
161	C₄	C₂²×D₈×C₅₃	1	16	yes	complemented
162	C₄	C₂³×D₄×C₅₃	1	2	yes	—
163	C₂	—	8	16	no	—
164	C₂	C₂×C₅₃×#493	1	12	yes	direct product factor
165	C₂	C₂²×#127×C₅₃	1	1	yes	characteristic
166	C₂	C₂³×D₈×C₅₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×D₈×C₅₃	C₂	semidirect
2. C₂²×#127×C₅₃	C₂	semidirect
3. C₂×C₅₃×#493	C₂	direct (N × H)
4. C₂²×C₄×C₈×C₅₃	C₂	semidirect
8. C₂²×D₈×C₅₃	C₂²	semidirect
11. C₂×#127×C₅₃	C₂²	semidirect
13. C₅₃×#493	C₂²	direct (N × H)
14. C₂×C₄×C₈×C₅₃	C₂²	semidirect
24. C₂×D₈×C₅₃	C₂³	semidirect
25. C₂²×C₈×C₅₃	D₄	semidirect
28. #127×C₅₃	C₂³	semidirect
31. C₄×C₈×C₅₃	C₂³	semidirect
42. C₂²×C₄×C₅₃	D₈	semidirect
48. C₂×C₈×C₅₃	C₂×D₄	semidirect
63. C₂×C₄×C₅₃	C₂×D₈	semidirect
68. C₈×C₅₃	C₂²×D₄	semidirect
69. C₂²×#493	C₅₃	direct (N × H)
73. C₂²×C₅₃	#493	direct (N × H)
78. C₄×C₅₃	C₂²×D₈	semidirect
80. C₂³×D₈	C₂×C₅₃	semidirect
81. C₂²×#127	C₂×C₅₃	semidirect
82. C₂×#493	C₂×C₅₃	direct (N × H)
83. C₂²×C₄×C₈	C₂×C₅₃	semidirect
85. C₂×C₅₃	C₂×#493	direct (N × H)
91. C₂²×D₈	C₂²×C₅₃	semidirect
94. C₂×#127	C₂²×C₅₃	semidirect
96. #493	C₂²×C₅₃	direct (N × H)
97. C₂×C₄×C₈	C₂²×C₅₃	semidirect
98. C₅₃	C₂²×#493	direct (N × H)
108. C₂×D₈	C₂³×C₅₃	semidirect
109. C₂²×C₈	D₄×C₅₃	semidirect
112. #127	C₂³×C₅₃	semidirect
115. C₄×C₈	C₂³×C₅₃	semidirect
126. C₂²×C₄	D₈×C₅₃	semidirect
132. C₂×C₈	C₂×D₄×C₅₃	semidirect
147. C₂×C₄	C₂×D₈×C₅₃	semidirect
152. C₈	C₂²×D₄×C₅₃	semidirect
156. C₂²	C₅₃×#493	direct (N × H)
161. C₄	C₂²×D₈×C₅₃	semidirect
164. C₂	C₂×C₅₃×#493	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×D₈×C₅₃	2
C₂²×#127×C₅₃	2
C₂×C₅₃×#493	2
C₂²×C₄×C₈×C₅₃	2
C₂²×#493	53

Subgroups — C22×C53×#493 #27268141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₃×#493 #27268141