Subgroups — #28290720

Order 13568.

11222 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	2₋¹⁺⁴×Dic₅₃	C₂	1	12	yes	maximal, complemented
2	H(5,𝔽₂)×Dic₅₃	C₂	1	20	yes	maximal, complemented
3	#20287918	C₂	1	15	yes	maximal, complemented
4	C₂×#16811106	C₂	1	15	yes	maximal, complemented
5	C₂×C₅₃×#585	C₂	1	1	yes	maximal, characteristic
6	#16811106	C₂²	1	60	yes	complemented
7	#16811106	C₂²	1	120	yes	complemented
8	#16811106	C₂²	1	20	yes	complemented
9	#16811101	C₂²	1	30	yes	complemented
10	#16811098	C₂²	1	90	yes	complemented
11	#41×Dic₅₃	C₂²	1	120	yes	complemented
12	C₂×Q₈×Dic₅₃	C₂²	1	30	yes	complemented
13	C₂×D₄×Dic₅₃	C₂²	1	90	yes	complemented
14	C₂×#13133176	C₂²	1	45	yes	complemented
15	C₂²×C₄⋊Dic₅₃	C₂²	1	15	yes	complemented
16	C₅₃×#585	C₄	1	32	yes	complemented
17	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	6	yes	—
18	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	10	yes	—
19	C₂²×#41×C₅₃	C₂²	1	15	yes	—
20	C₂³×Dic₅₃	C₂³	1	30	yes	—
21	Q₈×Dic₅₃	C₂³	1	120	yes	—
22	Q₈×Dic₅₃	C₂³	1	40	yes	—
23	D₄×Dic₅₃	C₂³	1	360	yes	—
24	D₄×Dic₅₃	C₂³	1	120	yes	—
25	#13133176	C₂³	1	60	yes	—
26	#13133176	C₂³	1	180	yes	—
27	C₂×C₄⋊Dic₅₃	C₂³	1	45	yes	—
28	C₂×C₄⋊Dic₅₃	C₂³	1	90	yes	—
29	C₂×C₄⋊Dic₅₃	C₂³	1	15	yes	—
30	C₂×C₂²⋊Dic₅₃	C₂³	1	90	yes	—
31	C₂×C₄×Dic₅₃	C₂³	1	90	yes	—
32	2₋¹⁺⁴×C₅₃	C₂×C₄	1	96	yes	complemented
33	H(5,𝔽₂)×C₅₃	C₂×C₄	1	160	yes	complemented
34	C₂×#41×C₅₃	C₂×C₄	1	240	yes	complemented
35	C₂×#41×C₅₃	C₂³	1	60	yes	—
36	C₂×#41×C₅₃	C₂³	1	20	yes	—
37	C₂²×Q₈×C₅₃	C₂³	1	15	yes	—
38	C₂²×D₄×C₅₃	C₂³	1	45	yes	—
39	C₂³×C₄×C₅₃	C₂³	1	15	yes	—
40	C₂⁴×C₅₃	C₂⁴	1	15	yes	—
41	#41×C₅₃	C₂²×C₄	1	160	yes	complemented
42	#41×C₅₃	C₂²×C₄	1	480	yes	complemented
43	C₂×Q₈×C₅₃	C₂²×C₄	1	120	yes	complemented
44	C₂×Q₈×C₅₃	C₂⁴	1	20	yes	—
45	C₂×D₄×C₅₃	C₂²×C₄	1	360	yes	complemented
46	C₂×D₄×C₅₃	C₂⁴	1	60	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	120	yes	complemented
48	C₂²×C₄×C₅₃	C₂⁴	1	15	yes	—
49	C₂²×C₄×C₅₃	C₂⁴	1	45	yes	—
50	C₂²×Dic₅₃	—	8	30	no	—
51	C₂²×Dic₅₃	C₂⁴	1	30	yes	—
52	C₂²×Dic₅₃	C₂⁴	1	90	yes	—
53	C₂²⋊Dic₅₃	C₂⁴	1	120	yes	—
54	C₄⋊Dic₅₃	C₂⁴	1	10	yes	—
55	C₄⋊Dic₅₃	C₂⁴	1	6	yes	—
56	C₄⋊Dic₅₃	C₂⁴	1	90	yes	—
57	C₄⋊Dic₅₃	C₂⁴	1	30	yes	—
58	C₄×Dic₅₃	C₂⁴	1	120	yes	—
59	C₂³×C₅₃	—	4	15	no	—
60	C₂³×C₅₃	—	4	15	no	—
61	C₂³×C₅₃	C₂³×C₄	1	60	yes	—
62	C₂³×C₅₃	C₂⁵	1	15	yes	—
63	Q₈×C₅₃	C₂³×C₄	1	80	yes	—
64	D₄×C₅₃	C₂³×C₄	1	240	yes	—
65	C₂×C₄×C₅₃	C₂³×C₄	1	60	yes	—
66	C₂×C₄×C₅₃	C₂³×C₄	1	180	yes	—
67	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂⁵	1	15	yes	—
69	C₂×Dic₅₃	—	4	90	no	—
70	C₂×Dic₅₃	—	4	30	no	—
71	C₂×Dic₅₃	C₂⁵	1	20	yes	—
72	C₂×Dic₅₃	C₂⁵	1	12	yes	—
73	#62654	—	53	1	no	maximal, Sylow
74	C₂²×C₅₃	—	2	15	no	—
75	C₂²×C₅₃	—	2	15	no	—
76	C₂²×C₅₃	—	4	60	no	—
77	C₂²×C₅₃	C₂⁴×C₄	1	30	yes	—
78	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
79	C₄×C₅₃	C₂⁴×C₄	1	2	yes	—
80	C₄×C₅₃	C₂⁴×C₄	1	30	yes	—
81	Dic₅₃	—	2	20	no	—
82	Dic₅₃	—	2	12	no	—
83	C₂×#585	D₅₃	1	1	yes	characteristic
84	C₄×2₋¹⁺⁴	—	53	12	no	—
85	C₄×H(5,𝔽₂)	—	53	20	no	—
86	#3428	—	53	15	no	—
87	C₂×#520	—	53	15	no	—
88	C₂×C₅₃	—	2	30	no	—
89	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
90	C₂×C₅₃	H(7,𝔽₂)	1	1	yes	characteristic
91	C₂×C₅₃	2₋¹⁺⁶	1	1	yes	characteristic
92	#585	Dic₅₃	1	32	yes	complemented
93	C₂×2₋¹⁺⁴	C₂×D₅₃	1	6	yes	—
94	C₂×H(5,𝔽₂)	C₂×D₅₃	1	10	yes	—
95	C₂²×#41	C₂×D₅₃	1	15	yes	—
96	#520	—	53	60	no	—
97	#520	—	53	120	no	—
98	#520	—	53	20	no	—
99	#519	—	53	30	no	—
100	#518	—	53	90	no	—
101	C₄×#41	—	53	120	no	—
102	C₂×C₄×Q₈	—	53	30	no	—
103	C₂×C₄×D₄	—	53	90	no	—
104	C₂×#117	—	53	45	no	—
105	C₂²×C₄⋊C₄	—	53	15	no	—
106	C₅₃	#62654	1	1	yes	characteristic, complemented, Sylow
107	2₋¹⁺⁴	C₂×Dic₅₃	1	96	yes	complemented
108	H(5,𝔽₂)	C₂×Dic₅₃	1	160	yes	complemented
109	C₂×#41	C₂×Dic₅₃	1	240	yes	complemented
110	C₂×#41	C₂²×D₅₃	1	60	yes	—
111	C₂×#41	C₂²×D₅₃	1	20	yes	—
112	C₂²×Q₈	C₂²×D₅₃	1	15	yes	—
113	C₂²×D₄	C₂²×D₅₃	1	45	yes	—
114	C₂³×C₄	—	53	30	no	—
115	C₂³×C₄	C₂²×D₅₃	1	15	yes	—
116	C₄×Q₈	—	53	120	no	—
117	C₄×Q₈	—	53	40	no	—
118	C₄×D₄	—	53	360	no	—
119	C₄×D₄	—	53	120	no	—
120	#117	—	53	60	no	—
121	#117	—	53	180	no	—
122	C₂×C₄⋊C₄	—	53	45	no	—
123	C₂×C₄⋊C₄	—	53	90	no	—
124	C₂×C₄⋊C₄	—	53	15	no	—
125	C₂×C₂²⋊C₄	—	53	90	no	—
126	C₂×C₄²	—	53	90	no	—
127	C₂⁴	C₂³×D₅₃	1	15	yes	—
128	#41	C₂²×Dic₅₃	1	160	yes	complemented
129	#41	C₂²×Dic₅₃	1	480	yes	complemented
130	C₂×Q₈	C₂²×Dic₅₃	1	120	yes	complemented
131	C₂×Q₈	C₂³×D₅₃	1	20	yes	—
132	C₂×D₄	C₂²×Dic₅₃	1	360	yes	complemented
133	C₂×D₄	C₂³×D₅₃	1	60	yes	—
134	C₂²×C₄	—	53	30	no	—
135	C₂²×C₄	—	53	90	no	—
136	C₂²×C₄	—	424	30	no	—
137	C₂²×C₄	C₂²×Dic₅₃	1	120	yes	complemented
138	C₂²×C₄	C₂³×D₅₃	1	45	yes	—
139	C₂²×C₄	C₂³×D₅₃	1	15	yes	—
140	C₄⋊C₄	—	53	10	no	—
141	C₄⋊C₄	—	53	6	no	—
142	C₄⋊C₄	—	53	90	no	—
143	C₄⋊C₄	—	53	30	no	—
144	C₂²⋊C₄	—	53	120	no	—
145	C₄²	—	53	120	no	—
146	C₂³	—	4	15	no	—
147	C₂³	—	4	15	no	—
148	C₂³	C₂³×Dic₅₃	1	60	yes	—
149	C₂³	C₂⁴×D₅₃	1	15	yes	—
150	Q₈	C₂³×Dic₅₃	1	80	yes	—
151	D₄	C₂³×Dic₅₃	1	240	yes	—
152	C₂×C₄	—	53	12	no	—
153	C₂×C₄	—	53	20	no	—
154	C₂×C₄	—	212	30	no	—
155	C₂×C₄	—	212	90	no	—
156	C₂×C₄	C₂³×Dic₅₃	1	60	yes	—
157	C₂×C₄	C₂³×Dic₅₃	1	180	yes	—
158	C₂×C₄	C₂⁴×D₅₃	1	15	yes	—
159	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
160	C₂²	—	2	15	no	—
161	C₂²	—	2	15	no	—
162	C₂²	—	4	60	no	—
163	C₂²	C₂⁴×Dic₅₃	1	30	yes	—
164	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
165	C₄	—	106	20	no	—
166	C₄	—	106	12	no	—
167	C₄	C₂⁴×Dic₅₃	1	2	yes	—
168	C₄	C₂⁴×Dic₅₃	1	30	yes	—
169	C₂	—	2	30	no	—
170	C₂	C₂⁵×Dic₅₃	1	1	yes	characteristic
171	C₂	#20289526	1	1	yes	characteristic
172	C₂	#20289528	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×D₅₃
commutator	C₂×C₅₃	C₂⁵×C₄
Fitting, prime radical	C₂×C₅₃×#585	C₂
socle, cosocle	C₂²×C₅₃	C₂⁶
hypercenter	C₂×#585	D₅₃
hypocenter	C₅₃	#62654
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×#585

Complements

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

N	H ≅ G/N	type
1. 2₋¹⁺⁴×Dic₅₃	C₂	semidirect
2. H(5,𝔽₂)×Dic₅₃	C₂	semidirect
3. #20287918	C₂	semidirect
4. C₂×#16811106	C₂	semidirect
6. #16811106	C₂²	semidirect
7. #16811106	C₂²	semidirect
8. #16811106	C₂²	semidirect
9. #16811101	C₂²	semidirect
10. #16811098	C₂²	semidirect
11. #41×Dic₅₃	C₂²	semidirect
12. C₂×Q₈×Dic₅₃	C₂²	semidirect
13. C₂×D₄×Dic₅₃	C₂²	semidirect
14. C₂×#13133176	C₂²	semidirect
15. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
16. C₅₃×#585	C₄	semidirect
32. 2₋¹⁺⁴×C₅₃	C₂×C₄	semidirect
33. H(5,𝔽₂)×C₅₃	C₂×C₄	semidirect
34. C₂×#41×C₅₃	C₂×C₄	semidirect
41. #41×C₅₃	C₂²×C₄	semidirect
42. #41×C₅₃	C₂²×C₄	semidirect
43. C₂×Q₈×C₅₃	C₂²×C₄	semidirect
45. C₂×D₄×C₅₃	C₂²×C₄	semidirect
47. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
92. #585	Dic₅₃	semidirect
106. C₅₃	#62654	semidirect
107. 2₋¹⁺⁴	C₂×Dic₅₃	semidirect
108. H(5,𝔽₂)	C₂×Dic₅₃	semidirect
109. C₂×#41	C₂×Dic₅₃	semidirect
128. #41	C₂²×Dic₅₃	semidirect
129. #41	C₂²×Dic₅₃	semidirect
130. C₂×Q₈	C₂²×Dic₅₃	semidirect
132. C₂×D₄	C₂²×Dic₅₃	semidirect
137. C₂²×C₄	C₂²×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
2₋¹⁺⁴×Dic₅₃	2
H(5,𝔽₂)×Dic₅₃	2
#20287918	2
C₂×#16811106	2
C₂×C₅₃×#585	2
#62654	53