Subgroups — #27349842

Order 13568.

712 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283348	C₂	1	2	yes	maximal
2	#20273018	C₂	1	4	yes	maximal, complemented
3	#20273015	C₂	1	8	yes	maximal, complemented
4	C₅₃×#2920	C₂	1	1	yes	maximal, characteristic
5	#16810680	C₄	1	8	yes	—
6	#16810676	C₂²	1	8	yes	—
7	#16810647	C₄	1	4	yes	—
8	#16810587	C₄	1	4	yes	—
9	C₄×#11719917	C₂²	1	4	yes	—
10	#16810345	C₂²	1	8	yes	complemented
11	#16810344	C₂²	1	4	yes	complemented
12	C₈×#88322	C₂²	1	4	yes	complemented
13	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#432	C₄	1	4	yes	—
15	C₅₃×#431	C₂²	1	4	yes	—
16	C₅₃×#423	C₄	1	2	yes	—
17	C₅₃×#407	C₄	1	2	yes	—
18	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
19	C₂×#11719917	C₂×C₄	1	8	yes	—
20	C₄⋊#88322	C₂×C₄	1	16	yes	—
21	C₂²⋊#88322	C₂×C₄	1	16	yes	—
22	#13133128	C₂×C₄	1	8	yes	—
23	C₄×#88322	C₂³	1	8	yes	—
24	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
33	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
34	C₂²⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
35	#97×C₅₃	C₂×C₄	1	4	yes	—
36	C₄×C₈×C₅₃	C₂³	1	4	yes	—
37	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	—	4	4	no	—
39	C₂²×C₄×C₅₃	—	4	2	no	—
40	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
42	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
43	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	M₁₆×C₅₃	—	4	4	no	—
49	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
50	C₄²×C₅₃	C₄²	1	4	yes	—
51	C₄²×C₅₃	C₄²	1	2	yes	—
52	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
53	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
54	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
56	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
57	#11719917	—	4	8	no	—
58	C₂×#88322	C₂²×C₄	1	16	yes	—
59	C₂³×C₅₃	—	4	2	no	—
60	C₂³×C₅₃	—	4	1	no	—
61	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
62	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₅₃	—	2	8	no	—
64	C₂×C₄×C₅₃	—	2	4	no	—
65	C₂×C₄×C₅₃	—	2	4	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	4	2	no	—
68	C₂×C₄×C₅₃	—	4	4	no	—
69	C₂×C₄×C₅₃	—	4	1	no	—
70	C₂×C₄×C₅₃	—	4	2	no	—
71	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
72	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
74	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
75	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
76	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
77	C₈×C₅₃	—	2	8	no	—
78	#88322	—	2	16	no	—
79	#17339	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	—	2	4	no	—
81	C₂²×C₅₃	—	2	2	no	—
82	C₂²×C₅₃	—	2	2	no	—
83	C₂²×C₅₃	—	2	1	no	—
84	C₂²×C₅₃	—	4	4	no	—
85	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
86	C₄×C₅₃	—	2	4	no	—
87	C₄×C₅₃	—	2	2	no	—
88	C₄×C₅₃	—	2	2	no	—
89	C₄×C₅₃	—	2	1	no	—
90	C₄×C₅₃	#568	1	4	yes	—
91	C₄×C₅₃	#568	1	2	yes	—
92	#2920	—	53	2	no	—
93	#2920	D₅₃	1	1	yes	characteristic
94	#1455	—	53	4	no	—
95	#1454	—	53	8	no	—
96	C₂×C₅₃	—	4	2	no	—
97	C₂×C₅₃	—	4	1	no	—
98	C₂×C₅₃	#2873	1	2	yes	—
99	C₂×C₅₃	#2873	1	1	yes	characteristic
100	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
101	#432	—	53	8	no	—
102	#432	Dic₅₃	1	4	yes	—
103	#431	—	53	8	no	—
104	#431	C₂×D₅₃	1	4	yes	—
105	#423	—	53	4	no	—
106	#423	Dic₅₃	1	2	yes	—
107	#407	—	53	4	no	—
108	#407	Dic₅₃	1	2	yes	—
109	C₄×M₁₆	—	53	4	no	—
110	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
111	#322	—	53	4	no	—
112	#322	—	53	8	no	—
113	C₈²	—	53	4	no	—
114	C₅₃	#17339	1	1	yes	characteristic, complemented, Sylow
115	C₂³×C₄	C₄×D₅₃	1	2	yes	—
116	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
117	C₂×M₁₆	—	53	8	no	—
118	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
119	C₂×C₄²	C₄×D₅₃	1	2	yes	—
120	C₂×C₄²	C₄×D₅₃	1	4	yes	—
121	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
122	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
123	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
124	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
125	C₄⋊C₈	—	53	16	no	—
126	C₄⋊C₈	C₂×Dic₅₃	1	8	yes	—
127	C₂²⋊C₈	—	53	16	no	—
128	C₂²⋊C₈	C₂×Dic₅₃	1	8	yes	—
129	#97	—	53	8	no	—
130	#97	C₂×Dic₅₃	1	4	yes	—
131	C₄×C₈	—	53	8	no	—
132	C₄×C₈	C₂²×D₅₃	1	4	yes	—
133	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
134	C₂²×C₄	—	4	4	no	—
135	C₂²×C₄	—	4	2	no	—
136	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
137	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
138	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
139	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
140	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
141	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
142	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
143	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
144	M₁₆	—	4	4	no	—
145	M₁₆	—	212	8	no	—
146	C₂×C₈	—	53	16	no	—
147	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
148	C₄²	C₄×Dic₅₃	1	4	yes	—
149	C₄²	C₄×Dic₅₃	1	2	yes	—
150	C₄²	C₂×C₄×D₅₃	1	2	yes	—
151	C₄²	C₂×C₄×D₅₃	1	4	yes	—
152	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
153	C₄²	C₂²×Dic₅₃	1	2	yes	—
154	C₄²	C₂³×D₅₃	1	1	yes	characteristic
155	C₂³	—	4	2	no	—
156	C₂³	—	4	1	no	—
157	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
158	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
168	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
169	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
170	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
171	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
172	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
173	C₈	—	2	8	no	—
174	C₈	—	106	16	no	—
175	C₂²	—	2	2	no	—
176	C₂²	—	2	4	no	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	4	4	no	—
180	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
181	C₄	—	2	2	no	—
182	C₄	—	2	4	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	2	no	—
185	C₄	#16811389	1	4	yes	—
186	C₄	#16811390	1	2	yes	—
187	C₂	—	4	1	no	—
188	C₂	—	4	2	no	—
189	C₂	#20283133	1	2	yes	—
190	C₂	#20283134	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Fitting	C₅₃×#2920	C₂
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#2920	D₅₃
hypocenter	C₅₃	#17339
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
2. #20273018	C₂	semidirect
3. #20273015	C₂	semidirect
10. #16810345	C₂²	semidirect
11. #16810344	C₂²	semidirect
12. C₈×#88322	C₂²	semidirect
114. C₅₃	#17339	semidirect

Maximal subgroups

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

subgroup	index
#20283348	2
#20273018	2
#20273015	2
C₅₃×#2920	2
#17339	53