Subgroups — #28310193

Order 13568.

478 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811510	C₂	1	2	yes	maximal
2	C₄×D₅₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×#6157	C₂	1	2	yes	maximal
4	#20290371	C₂	1	1	yes	maximal, characteristic
5	#346×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×D₅₃⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×D₅₃⋊C₁₆	C₂²	1	1	yes	characteristic
8	#16811582	C₄	1	1	yes	characteristic
9	C₂×#13133271	C₄	1	2	yes	—
10	C₂×C₈×#6157	C₂²	1	2	yes	—
11	#16811559	C₄	1	1	yes	characteristic
12	#16811510	C₄	1	8	yes	complemented
13	#16811509	C₄	1	2	yes	complemented
14	C₄×#11719910	C₄	1	2	yes	complemented
15	C₁₆×#6157	C₄	1	8	yes	complemented
16	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
17	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
18	#16810399	C₄	1	1	yes	characteristic, complemented
19	#16810398	C₄	1	2	yes	complemented
20	C₅₃×#346	C₄	1	1	yes	characteristic, complemented
21	C₂×D₅₃⋊C₈	C₂×C₄	1	2	yes	—
22	C₂×C₄×#6157	C₂×C₄	1	2	yes	—
23	D₅₃⋊C₁₆	C₂×C₄	1	4	yes	—
24	D₅₃⋊C₁₆	C₂×C₄	1	4	yes	—
25	C₂×#11719910	C₂×C₄	1	2	yes	—
26	C₂×#11719910	C₂×C₄	1	2	yes	—
27	#13133272	C₈	1	4	yes	—
28	#13133271	C₈	1	8	yes	—
29	C₈×#6157	C₂×C₄	1	4	yes	—
30	C₈×#6157	C₂×C₄	1	4	yes	—
31	(C₂×#6157)⋊C₄	C₈	1	4	yes	—
32	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
34	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×#11719908	C₂×C₄	1	2	yes	—
36	C₁₆×D₅₃	C₂×C₄	1	4	yes	—
37	C₁₆×D₅₃	C₂×C₄	1	4	yes	—
38	C₄×#88322	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×#6157	C₂×C₈	1	4	yes	—
44	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
45	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×#88324	C₂×C₈	1	4	yes	—
47	C₄×#6157	C₄²	1	4	yes	—
48	C₄×#6157	C₂×C₈	1	4	yes	—
49	D₅₃⋊C₈	C₄²	1	4	yes	—
50	D₅₃⋊C₈	C₂×C₈	1	4	yes	—
51	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₅₃	C₄²	1	1	yes	characteristic
54	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
55	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
56	C₂×#88322	C₄²	1	1	yes	characteristic
57	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
58	C₈×D₅₃	C₂×C₈	1	4	yes	—
59	C₈×D₅₃	C₂²×C₄	1	2	yes	—
60	C₈×D₅₃	C₂²×C₄	1	2	yes	—
61	#11719910	C₄²	1	4	yes	—
62	#11719910	C₄²	1	4	yes	—
63	C₁₆×C₅₃	C₄²	1	4	yes	—
64	#11719908	C₄²	1	4	yes	—
65	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×#6157	—	2	8	no	—
67	C₂×#6157	C₄×C₈	1	8	yes	—
68	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
69	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
70	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
71	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
72	C₄×D₅₃	—	2	4	no	—
73	C₄×D₅₃	C₂×C₄²	1	2	yes	—
74	C₄×D₅₃	C₂²×C₈	1	2	yes	—
75	#88324	C₄×C₈	1	8	yes	—
76	C₈×C₅₃	C₄×C₈	1	2	yes	—
77	C₈×C₅₃	C₂×C₄²	1	2	yes	—
78	#88322	C₄×C₈	1	2	yes	—
79	#88322	C₂×C₄²	1	1	yes	characteristic
80	#88322	C₂×C₄²	1	1	yes	characteristic
81	#12539	—	53	1	no	maximal, Sylow
82	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
83	C₂×D₅₃	C₂×C₄×C₈	1	2	yes	—
84	C₂×D₅₃	#504	1	4	yes	—
85	#6157	—	2	8	no	—
86	C₄×C₅₃	—	2	2	no	—
87	C₄×C₅₃	C₄³	1	1	yes	characteristic
88	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
89	Dic₅₃	—	2	2	no	—
90	Dic₅₃	C₂×C₄×C₈	1	2	yes	—
91	C₂×#346	—	53	2	no	—
92	C₂×#346	—	53	1	no	—
93	C₂×C₄×C₁₆	—	53	2	no	—
94	C₂×C₄×C₁₆	—	53	1	no	—
95	#1726	—	53	1	no	—
96	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
97	C₂×C₅₃	#2108	1	2	yes	—
98	D₅₃	#2108	1	4	yes	—
99	C₂²×C₁₆	—	53	1	no	—
100	C₂²×C₁₆	—	53	1	no	—
101	C₂²×C₁₆	—	53	2	no	—
102	C₂×#97	—	53	2	no	—
103	C₂×#97	—	53	1	no	—
104	C₂×C₄×C₈	—	53	2	no	—
105	C₂×C₄×C₈	—	53	1	no	—
106	#376	—	53	1	no	—
107	#346	—	53	8	no	—
108	#346	—	53	2	no	—
109	#346	—	53	1	no	—
110	#346	#6157	1	1	yes	characteristic, complemented
111	C₄×C₁₆	—	53	8	no	—
112	C₄×C₁₆	—	53	2	no	—
113	C₄×C₁₆	—	53	2	no	—
114	C₅₃	#12539	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	—	53	2	no	—
116	C₂²×C₈	—	53	1	no	—
117	C₂²×C₈	—	53	1	no	—
118	C₂×C₄²	—	53	2	no	—
119	C₂×C₄²	—	53	1	no	—
120	C₂×C₁₆	—	53	2	no	—
121	C₂×C₁₆	—	53	2	no	—
122	C₂×C₁₆	—	53	4	no	—
123	C₂×C₁₆	—	53	4	no	—
124	C₂×C₁₆	—	53	2	no	—
125	C₂×C₁₆	—	53	4	no	—
126	C₂×C₁₆	—	53	4	no	—
127	C₂×C₁₆	C₂×#6157	1	2	yes	—
128	#97	—	53	8	no	—
129	#97	—	53	4	no	—
130	C₄×C₈	—	53	4	no	—
131	C₄×C₈	—	53	4	no	—
132	C₄×C₈	—	53	1	no	—
133	C₄×C₈	—	53	1	no	—
134	C₄×C₈	—	53	1	no	—
135	C₄×C₈	C₂×#6157	1	1	yes	characteristic
136	#95	—	53	4	no	—
137	C₂²×C₄	—	53	1	no	—
138	C₂²×C₄	—	53	4	no	—
139	C₂²×C₄	—	53	2	no	—
140	C₂×C₈	—	53	4	no	—
141	C₂×C₈	—	53	4	no	—
142	C₂×C₈	—	53	4	no	—
143	C₂×C₈	—	53	1	no	—
144	C₂×C₈	—	53	1	no	—
145	C₂×C₈	—	53	4	no	—
146	C₂×C₈	—	53	2	no	—
147	C₂×C₈	—	53	2	no	—
148	C₂×C₈	C₄×#6157	1	1	yes	characteristic
149	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
150	C₄²	—	53	4	no	—
151	C₄²	—	53	4	no	—
152	C₄²	—	53	2	no	—
153	C₄²	—	53	1	no	—
154	C₄²	C₄×#6157	1	1	yes	characteristic
155	C₁₆	—	53	4	no	—
156	C₁₆	—	53	4	no	—
157	C₁₆	—	53	4	no	—
158	C₁₆	C₄×#6157	1	4	yes	—
159	C₂³	—	53	1	no	—
160	C₂×C₄	—	53	1	no	—
161	C₂×C₄	—	53	2	no	—
162	C₂×C₄	—	53	2	no	—
163	C₂×C₄	—	53	8	no	—
164	C₂×C₄	—	53	2	no	—
165	C₂×C₄	—	106	4	no	—
166	C₂×C₄	—	106	8	no	—
167	C₂×C₄	C₈×#6157	1	2	yes	—
168	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
169	C₈	—	53	8	no	—
170	C₈	—	53	1	no	—
171	C₈	—	53	1	no	—
172	C₈	—	53	2	no	—
173	C₈	C₈×#6157	1	2	yes	—
174	C₈	C₂×C₄×#6157	1	2	yes	—
175	C₂²	—	53	2	no	—
176	C₂²	—	53	4	no	—
177	C₂²	C₂×C₈×#6157	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	53	2	no	—
180	C₄	—	106	8	no	—
181	C₄	—	106	2	no	—
182	C₄	C₄²×#6157	1	1	yes	characteristic
183	C₄	C₂×C₈×#6157	1	1	yes	characteristic
184	C₂	—	53	4	no	—
185	C₂	C₄×C₈×#6157	1	1	yes	characteristic
186	C₂	#20290927	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#6157
commutator	C₂×C₅₃	C₄²×C₈
Fitting	C₅₃×#346	C₄
socle	C₂²×C₅₃	C₂×C₄×C₈
prime radical	C₂²×D₅₃	C₂²×C₈
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	#346	#6157
hypocenter	C₅₃	#12539
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
12. #16811510	C₄	semidirect
13. #16811509	C₄	semidirect
14. C₄×#11719910	C₄	semidirect
15. C₁₆×#6157	C₄	semidirect
18. #16810399	C₄	semidirect
19. #16810398	C₄	semidirect
20. C₅₃×#346	C₄	semidirect
110. #346	#6157	semidirect
114. C₅₃	#12539	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811510	2
C₄×D₅₃⋊C₁₆	2
C₂×C₁₆×#6157	2
#20290371	2
#346×D₅₃	2
#12539	53