Subgroups — D₅₃×#1312 #27293077

Order 13568.

350 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₆×D₅₃	C₂	1	2	yes	maximal
2	C₈²×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20272634	C₂	1	2	yes	maximal, complemented
4	#20272633	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₆×D₅₃	C₄	1	4	yes	—
7	C₄×C₈×D₅₃	C₄	1	1	yes	characteristic
8	C₄×C₈×D₅₃	C₄	1	1	yes	characteristic
9	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	C₄×#11719908	C₂²	1	2	yes	—
11	C₁₆×Dic₅₃	C₄	1	4	yes	—
12	C₈×#88322	C₄	1	1	yes	characteristic
13	C₈×#88322	C₄	1	1	yes	characteristic
14	C₈×#88322	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
16	C₈²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₅₃	C₈	1	2	yes	—
18	C₂×C₈×D₅₃	C₈	1	2	yes	—
19	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
20	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×#11719908	C₂×C₄	1	4	yes	—
22	C₁₆×D₅₃	—	2	8	no	—
23	C₄×#88322	C₂×C₄	1	1	yes	characteristic
24	C₄×#88322	C₂×C₄	1	1	yes	characteristic
25	C₄×#88322	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₅₃	C₈	1	2	yes	—
27	C₈×Dic₅₃	C₈	1	2	yes	—
28	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
34	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
39	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
41	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
42	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
43	C₂×#88322	C₄²	1	2	yes	—
44	C₂×#88322	C₂×C₈	1	2	yes	—
45	C₂×#88322	C₂×C₈	1	2	yes	—
46	C₈×D₅₃	—	2	4	no	—
47	C₈×D₅₃	C₁₆	1	8	yes	complemented
48	C₈×D₅₃	M₁₆	1	8	yes	—
49	C₁₆×C₅₃	—	2	4	no	—
50	#11719908	—	2	4	no	—
51	C₂²×D₅₃	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
55	C₂×Dic₅₃	C₄×C₈	1	1	yes	characteristic
56	C₂×Dic₅₃	C₄×C₈	1	1	yes	characteristic
57	C₂×Dic₅₃	C₄×C₈	1	1	yes	characteristic
58	C₄×D₅₃	#97	1	4	yes	—
59	C₄×D₅₃	C₂×C₁₆	1	4	yes	—
60	C₄×D₅₃	M₃₂	1	4	yes	—
61	C₈×C₅₃	—	2	2	no	—
62	C₈×C₅₃	C₂×C₁₆	1	4	yes	complemented
63	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
64	#88322	—	2	2	no	—
65	#88322	#97	1	4	yes	—
66	#88322	C₂×C₁₆	1	4	yes	complemented
67	C₂×#1312	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
69	C₂×D₅₃	#322	1	2	yes	—
70	C₂×D₅₃	C₄×C₁₆	1	2	yes	—
71	C₂×D₅₃	#346	1	2	yes	—
72	C₄×C₅₃	C₂×#97	1	2	yes	—
73	C₄×C₅₃	C₂²×C₁₆	1	2	yes	—
74	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
75	Dic₅₃	#322	1	2	yes	—
76	Dic₅₃	C₄×C₁₆	1	2	yes	—
77	Dic₅₃	#346	1	2	yes	—
78	C₂×C₄×C₁₆	—	53	2	no	—
79	C₂×C₈²	—	53	1	no	—
80	#1312	—	53	2	no	—
81	#1312	—	53	1	no	—
82	#1312	D₅₃	1	1	yes	characteristic, direct product factor
83	C₂×C₅₃	C₂×#322	1	1	yes	characteristic
84	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
85	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
86	D₅₃	#1312	1	4	yes	direct product factor
87	C₂²×C₁₆	—	53	4	no	—
88	C₂×C₄×C₈	—	53	1	no	—
89	C₂×C₄×C₈	—	53	1	no	—
90	C₂×C₄×C₈	—	53	1	no	—
91	C₄×C₁₆	—	53	4	no	—
92	C₄×C₁₆	—	53	2	no	—
93	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
94	C₈²	—	53	1	no	—
95	C₈²	—	53	1	no	—
96	C₈²	—	53	1	no	—
97	C₈²	C₂×D₅₃	1	1	yes	characteristic
98	C₅₃	C₂×#1312	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₈	—	53	2	no	—
100	C₂²×C₈	—	53	2	no	—
101	C₂²×C₈	—	53	2	no	—
102	C₂×C₄²	—	53	1	no	—
103	C₂×C₁₆	—	53	4	no	—
104	C₂×C₁₆	—	106	8	no	—
105	C₂×C₁₆	C₄×D₅₃	1	4	yes	—
106	C₄×C₈	—	53	2	no	—
107	C₄×C₈	—	53	2	no	—
108	C₄×C₈	—	53	2	no	—
109	C₄×C₈	—	53	1	no	—
110	C₄×C₈	—	53	1	no	—
111	C₄×C₈	—	53	1	no	—
112	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
113	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
114	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
115	C₂²×C₄	—	53	1	no	—
116	C₂²×C₄	—	53	1	no	—
117	C₂²×C₄	—	53	1	no	—
118	C₂×C₈	—	53	2	no	—
119	C₂×C₈	—	53	2	no	—
120	C₂×C₈	—	53	8	no	—
121	C₂×C₈	—	53	8	no	—
122	C₂×C₈	—	53	2	no	—
123	C₂×C₈	—	106	4	no	—
124	C₂×C₈	C₈×D₅₃	1	2	yes	—
125	C₂×C₈	C₈×D₅₃	1	2	yes	—
126	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
127	C₄²	—	53	1	no	—
128	C₄²	—	53	1	no	—
129	C₄²	—	53	1	no	—
130	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
131	C₁₆	—	2	4	no	—
132	C₁₆	—	106	4	no	—
133	C₂³	—	53	1	no	—
134	C₂×C₄	—	53	1	no	—
135	C₂×C₄	—	53	1	no	—
136	C₂×C₄	—	53	1	no	—
137	C₂×C₄	—	53	4	no	—
138	C₂×C₄	—	53	4	no	—
139	C₂×C₄	—	53	4	no	—
140	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
141	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
142	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	—	53	4	no	—
145	C₈	—	53	4	no	—
146	C₈	—	106	2	no	—
147	C₈	C₁₆×D₅₃	1	4	yes	complemented
148	C₈	M₁₆×D₅₃	1	4	yes	—
149	C₂²	—	53	2	no	—
150	C₂²	—	53	2	no	—
151	C₂²	—	53	2	no	—
152	C₂²	C₄×C₈×D₅₃	1	1	yes	characteristic
153	C₄	—	53	2	no	—
154	C₄	—	53	2	no	—
155	C₄	—	53	2	no	—
156	C₄	#97×D₅₃	1	2	yes	—
157	C₄	C₂×C₁₆×D₅₃	1	2	yes	—
158	C₄	M₃₂×D₅₃	1	2	yes	—
159	C₂	—	53	4	no	—
160	C₂	#322×D₅₃	1	1	yes	characteristic
161	C₂	C₄×C₁₆×D₅₃	1	1	yes	characteristic
162	C₂	#346×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₅₃
commutator	C₂×C₅₃	C₂×C₄×C₁₆
Fitting	C₅₃×#1312	C₂
socle	C₂²×C₅₃	C₂×C₄×C₈
prime radical	C₂²×D₅₃	C₄×C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1312	D₅₃
hypocenter	C₅₃	C₂×#1312
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ #1312

Complements

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

N	H ≅ G/N	type
3. #20272634	C₂	semidirect
4. #20272633	C₂	semidirect
5. C₅₃×#1312	C₂	semidirect
47. C₈×D₅₃	C₁₆	semidirect
62. C₈×C₅₃	C₂×C₁₆	semidirect
66. #88322	C₂×C₁₆	semidirect
82. #1312	D₅₃	direct (N × H)
86. D₅₃	#1312	direct (N × H)
98. C₅₃	C₂×#1312	semidirect
147. C₈	C₁₆×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₁₆×D₅₃	2
C₈²×D₅₃	2
#20272634	2
#20272633	2
C₅₃×#1312	2
C₂×#1312	53

Subgroups — D53×#1312 #27293077

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#1312 #27293077