Subgroups — C₂³×#13133176 #28290652

Order 13568.

14334 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133176	C₂	1	56	yes	maximal, direct product factor
2	C₂³×C₄⋊Dic₅₃	C₂	1	2	yes	maximal, complemented
3	C₂³×C₂²⋊Dic₅₃	C₂	1	2	yes	maximal
4	C₂³×C₄×Dic₅₃	C₂	1	2	yes	maximal, complemented
5	C₂⁵×C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×Dic₅₃	C₂²	1	4	yes	—
7	C₂×#13133176	C₂²	1	448	yes	direct product factor
8	C₂²×C₄⋊Dic₅₃	C₂²	1	56	yes	complemented
9	C₂²×C₂²⋊Dic₅₃	C₂²	1	56	yes	—
10	C₂²×C₄×Dic₅₃	C₂²	1	56	yes	complemented
11	C₂⁶×C₅₃	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₅₃	C₄	1	32	yes	complemented
13	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₅₃	C₂²	1	28	yes	—
16	C₂³×Dic₅₃	—	2	32	no	—
17	C₂³×Dic₅₃	C₂³	1	56	yes	—
18	#13133176	C₂³	1	512	yes	direct product factor
19	C₂×C₄⋊Dic₅₃	C₂³	1	224	yes	complemented
20	C₂×C₂²⋊Dic₅₃	C₂³	1	224	yes	—
21	C₂×C₄×Dic₅₃	C₂³	1	224	yes	complemented
22	C₂⁵×C₅₃	—	2	16	no	—
23	C₂⁵×C₅₃	C₂×C₄	1	16	yes	—
24	C₂⁵×C₅₃	C₂³	1	14	yes	—
25	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
26	C₂³×C₄×C₅₃	C₂×C₄	1	16	yes	—
27	C₂³×C₄×C₅₃	C₂×C₄	1	16	yes	—
28	C₂³×C₄×C₅₃	C₂×C₄	1	448	yes	complemented
29	C₂³×C₄×C₅₃	C₂³	1	14	yes	—
30	C₂³×C₄×C₅₃	C₂³	1	14	yes	—
31	C₂³×C₄×C₅₃	C₂³	1	112	yes	—
32	C₂⁴×C₅₃	—	2	128	no	—
33	C₂⁴×C₅₃	—	2	112	no	—
34	C₂⁴×C₅₃	C₂²×C₄	1	112	yes	—
35	C₂⁴×C₅₃	C₂²×C₄	1	8	yes	—
36	C₂⁴×C₅₃	#41	1	16	yes	—
37	C₂⁴×C₅₃	C₂⁴	1	28	yes	—
38	C₂⁴×C₅₃	C₂⁴	1	7	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	112	yes	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	112	yes	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	896	yes	complemented
42	C₂²×C₄×C₅₃	C₂⁴	1	28	yes	—
43	C₂²×C₄×C₅₃	C₂⁴	1	28	yes	—
44	C₂²×C₄×C₅₃	C₂⁴	1	64	yes	—
45	C₂²×Dic₅₃	—	2	224	no	—
46	C₂²×Dic₅₃	C₂⁴	1	112	yes	—
47	C₂²⋊Dic₅₃	C₂⁴	1	128	yes	—
48	C₄⋊Dic₅₃	C₂⁴	1	128	yes	complemented
49	C₄×Dic₅₃	C₂⁴	1	128	yes	complemented
50	C₂³×C₅₃	—	2	448	no	—
51	C₂³×C₅₃	—	2	112	no	—
52	C₂³×C₅₃	#117	1	64	yes	complemented
53	C₂³×C₅₃	C₂³×C₄	1	112	yes	—
54	C₂³×C₅₃	C₂³×C₄	1	28	yes	—
55	C₂³×C₅₃	C₂×#41	1	56	yes	—
56	C₂³×C₅₃	C₂⁵	1	8	yes	—
57	C₂³×C₅₃	C₂⁵	1	7	yes	—
58	C₂×C₄×C₅₃	C₂³×C₄	1	112	yes	—
59	C₂×C₄×C₅₃	C₂³×C₄	1	112	yes	—
60	C₂×C₄×C₅₃	C₂³×C₄	1	256	yes	complemented
61	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
62	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
63	C₂×Dic₅₃	—	2	224	no	—
64	C₂×Dic₅₃	C₂⁵	1	32	yes	—
65	C₂³×#117	—	53	1	no	maximal, Sylow
66	C₂²×C₅₃	—	2	16	no	—
67	C₂²×C₅₃	—	2	224	no	—
68	C₂²×C₅₃	C₂×#117	1	112	yes	complemented
69	C₂²×C₅₃	C₂⁴×C₄	1	16	yes	—
70	C₂²×C₅₃	C₂⁴×C₄	1	14	yes	—
71	C₂²×C₅₃	C₂²×#41	1	28	yes	—
72	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
73	C₄×C₅₃	C₂⁴×C₄	1	16	yes	—
74	C₄×C₅₃	C₂⁴×C₄	1	16	yes	—
75	Dic₅₃	—	2	32	no	—
76	C₂⁵×C₄	D₅₃	1	1	yes	characteristic
77	C₂²×#117	—	53	56	no	—
78	C₂³×C₄⋊C₄	—	53	2	no	—
79	C₂³×C₂²⋊C₄	—	53	2	no	—
80	C₂³×C₄²	—	53	2	no	—
81	C₂×C₅₃	—	2	16	no	—
82	C₂×C₅₃	C₂²×#117	1	28	yes	complemented
83	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
84	C₂×C₅₃	C₂³×#41	1	2	yes	—
85	C₂⁶	C₂×D₅₃	1	1	yes	characteristic
86	C₂⁴×C₄	—	53	4	no	—
87	C₂⁴×C₄	Dic₅₃	1	32	yes	complemented
88	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
89	C₂⁴×C₄	C₂×D₅₃	1	28	yes	—
90	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
91	C₂×#117	—	53	448	no	—
92	C₂²×C₄⋊C₄	—	53	56	no	—
93	C₂²×C₂²⋊C₄	—	53	56	no	—
94	C₂²×C₄²	—	53	56	no	—
95	C₅₃	C₂³×#117	1	1	yes	characteristic, complemented, Sylow
96	C₂⁵	—	2	16	no	—
97	C₂⁵	C₂×Dic₅₃	1	16	yes	—
98	C₂⁵	C₂²×D₅₃	1	14	yes	—
99	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
100	C₂³×C₄	—	53	56	no	—
101	C₂³×C₄	—	106	32	no	—
102	C₂³×C₄	C₂×Dic₅₃	1	16	yes	—
103	C₂³×C₄	C₂×Dic₅₃	1	16	yes	—
104	C₂³×C₄	C₂×Dic₅₃	1	448	yes	complemented
105	C₂³×C₄	C₂²×D₅₃	1	14	yes	—
106	C₂³×C₄	C₂²×D₅₃	1	14	yes	—
107	C₂³×C₄	C₂²×D₅₃	1	112	yes	—
108	#117	—	53	512	no	—
109	C₂×C₄⋊C₄	—	53	224	no	—
110	C₂×C₂²⋊C₄	—	53	224	no	—
111	C₂×C₄²	—	53	224	no	—
112	C₂⁴	—	2	128	no	—
113	C₂⁴	—	2	112	no	—
114	C₂⁴	#11719945	1	16	yes	—
115	C₂⁴	C₂²×Dic₅₃	1	112	yes	—
116	C₂⁴	C₂²×Dic₅₃	1	8	yes	—
117	C₂⁴	C₂³×D₅₃	1	28	yes	—
118	C₂⁴	C₂³×D₅₃	1	7	yes	—
119	C₂²×C₄	—	53	112	no	—
120	C₂²×C₄	—	106	224	no	—
121	C₂²×C₄	C₂²×Dic₅₃	1	112	yes	—
122	C₂²×C₄	C₂²×Dic₅₃	1	112	yes	—
123	C₂²×C₄	C₂²×Dic₅₃	1	896	yes	complemented
124	C₂²×C₄	C₂³×D₅₃	1	28	yes	—
125	C₂²×C₄	C₂³×D₅₃	1	28	yes	—
126	C₂²×C₄	C₂³×D₅₃	1	64	yes	—
127	C₄⋊C₄	—	53	128	no	—
128	C₂²⋊C₄	—	53	128	no	—
129	C₄²	—	53	128	no	—
130	C₂³	—	2	448	no	—
131	C₂³	—	2	112	no	—
132	C₂³	#13133176	1	64	yes	direct product factor
133	C₂³	C₂³×Dic₅₃	1	112	yes	—
134	C₂³	C₂³×Dic₅₃	1	28	yes	—
135	C₂³	C₂×#11719945	1	56	yes	—
136	C₂³	C₂⁴×D₅₃	1	8	yes	—
137	C₂³	C₂⁴×D₅₃	1	7	yes	—
138	C₂×C₄	—	53	32	no	—
139	C₂×C₄	—	106	224	no	—
140	C₂×C₄	C₂³×Dic₅₃	1	112	yes	—
141	C₂×C₄	C₂³×Dic₅₃	1	112	yes	—
142	C₂×C₄	C₂³×Dic₅₃	1	256	yes	complemented
143	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
144	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
145	C₂²	—	2	224	no	—
146	C₂²	—	2	16	no	—
147	C₂²	C₂×#13133176	1	112	yes	direct product factor
148	C₂²	C₂⁴×Dic₅₃	1	16	yes	—
149	C₂²	C₂⁴×Dic₅₃	1	14	yes	—
150	C₂²	C₂²×#11719945	1	28	yes	—
151	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
152	C₄	—	106	32	no	—
153	C₄	C₂⁴×Dic₅₃	1	16	yes	—
154	C₄	C₂⁴×Dic₅₃	1	16	yes	—
155	C₂	—	2	16	no	—
156	C₂	C₂²×#13133176	1	28	yes	direct product factor
157	C₂	C₂⁵×Dic₅₃	1	1	yes	characteristic
158	C₂	C₂³×#11719945	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₅₃
commutator	C₂×C₅₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂⁵×C₄×C₅₃	C₂
socle	C₂⁵×C₅₃	C₂³
prime radical	C₂⁶×C₅₃	C₂²
cosocle	C₂²×C₅₃	C₂⁶
hypercenter	C₂⁵×C₄	D₅₃
hypocenter	C₅₃	C₂³×#117
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133176	C₂	direct (N × H)
2. C₂³×C₄⋊Dic₅₃	C₂	semidirect
4. C₂³×C₄×Dic₅₃	C₂	semidirect
7. C₂×#13133176	C₂²	direct (N × H)
8. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
10. C₂²×C₄×Dic₅₃	C₂²	semidirect
12. C₂⁴×C₄×C₅₃	C₄	semidirect
18. #13133176	C₂³	direct (N × H)
19. C₂×C₄⋊Dic₅₃	C₂³	semidirect
21. C₂×C₄×Dic₅₃	C₂³	semidirect
28. C₂³×C₄×C₅₃	C₂×C₄	semidirect
41. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
48. C₄⋊Dic₅₃	C₂⁴	semidirect
49. C₄×Dic₅₃	C₂⁴	semidirect
52. C₂³×C₅₃	#117	semidirect
60. C₂×C₄×C₅₃	C₂³×C₄	semidirect
68. C₂²×C₅₃	C₂×#117	semidirect
82. C₂×C₅₃	C₂²×#117	semidirect
87. C₂⁴×C₄	Dic₅₃	semidirect
95. C₅₃	C₂³×#117	semidirect
104. C₂³×C₄	C₂×Dic₅₃	semidirect
123. C₂²×C₄	C₂²×Dic₅₃	semidirect
132. C₂³	#13133176	direct (N × H)
142. C₂×C₄	C₂³×Dic₅₃	semidirect
147. C₂²	C₂×#13133176	direct (N × H)
156. C₂	C₂²×#13133176	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133176	2
C₂³×C₄⋊Dic₅₃	2
C₂³×C₂²⋊Dic₅₃	2
C₂³×C₄×Dic₅₃	2
C₂⁵×C₄×C₅₃	2
C₂³×#117	53

Subgroups — C23×#13133176 #28290652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133176 #28290652