Subgroups — C₅₃×#8786 #27216298

Order 13568.

378 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#1842	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
3	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1277	C₂	1	4	yes	maximal
5	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
6	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
7	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
8	C₂×#97×C₅₃	C₄	1	2	yes	—
9	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
10	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
12	C₄³×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#336	—	2	2	no	—
14	C₂²×C₈×C₅₃	—	2	2	no	—
15	C₂²×C₈×C₅₃	—	2	2	no	—
16	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₅₃	D₄	1	2	yes	—
21	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
23	C₄⋊C₈×C₅₃	—	2	4	no	—
24	C₄⋊C₈×C₅₃	—	2	4	no	—
25	C₄⋊C₈×C₅₃	—	4	4	no	—
26	#97×C₅₃	—	2	4	no	—
27	#97×C₅₃	—	2	2	no	—
28	#97×C₅₃	C₈	1	4	yes	complemented
29	C₄×C₈×C₅₃	—	2	2	no	—
30	C₄×C₈×C₅₃	—	2	2	no	—
31	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
33	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈×C₅₃	—	2	2	no	—
37	C₂×C₈×C₅₃	—	2	2	no	—
38	C₂×C₈×C₅₃	—	2	2	no	—
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	—	4	2	no	—
41	C₂×C₈×C₅₃	—	4	2	no	—
42	C₂×C₈×C₅₃	—	4	4	no	—
43	C₂×C₈×C₅₃	—	4	2	no	—
44	C₂×C₈×C₅₃	—	4	2	no	—
45	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
46	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
47	C₄²×C₅₃	—	2	2	no	—
48	C₄²×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	—	2	2	no	—
50	C₄²×C₅₃	—	4	2	no	—
51	C₄²×C₅₃	C₂×C₈	1	2	yes	—
52	C₄²×C₅₃	M₁₆	1	2	yes	—
53	C₄²×C₅₃	#41	1	2	yes	—
54	C₄²×C₅₃	#41	1	2	yes	—
55	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
63	C₂×C₄×C₅₃	#117	1	2	yes	—
64	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#125	1	2	yes	—
67	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×C₄×C₅₃	#137	1	2	yes	—
72	C₈×C₅₃	—	2	4	no	—
73	C₈×C₅₃	—	2	4	no	—
74	C₈×C₅₃	—	8	2	no	—
75	C₈×C₅₃	—	8	2	no	—
76	#8786	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂²×C₅₃	#382	1	1	yes	characteristic
78	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
79	C₂²×C₅₃	#423	1	1	yes	characteristic
80	C₂²×C₅₃	#428	1	2	yes	—
81	C₂²×C₅₃	#430	1	2	yes	—
82	C₄×C₅₃	—	2	4	no	—
83	C₄×C₅₃	—	2	2	no	—
84	C₄×C₅₃	—	4	2	no	—
85	C₄×C₅₃	#431	1	2	yes	—
86	C₄×C₅₃	#432	1	2	yes	—
87	C₄×C₅₃	#489	1	2	yes	—
88	C₄×C₅₃	#490	1	1	yes	characteristic
89	C₄×C₅₃	#491	1	1	yes	characteristic
90	#1842	C₂×C₅₃	1	1	yes	characteristic
91	#1831	C₂×C₅₃	1	1	yes	characteristic
92	C₄×#97	C₂×C₅₃	1	1	yes	characteristic
93	#1277	C₂×C₅₃	1	4	yes	—
94	C₂×C₅₃	#1565	1	2	yes	—
95	C₂×C₅₃	#1569	1	2	yes	—
96	C₂×C₅₃	#1831	1	1	yes	characteristic
97	C₂×C₅₃	#1836	1	1	yes	characteristic
98	C₂×C₅₃	#1838	1	1	yes	characteristic
99	C₂×C₄⋊C₈	—	2	2	no	—
100	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
101	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
102	C₂×#97	C₄×C₅₃	1	2	yes	—
103	C₂×#97	C₄×C₅₃	1	1	yes	characteristic
104	C₂×#97	C₄×C₅₃	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
106	C₄³	C₂²×C₅₃	1	1	yes	characteristic
107	#336	—	2	2	no	—
108	C₅₃	#8786	1	1	yes	characteristic, direct product factor, Sylow
109	C₂²×C₈	—	2	2	no	—
110	C₂²×C₈	—	2	2	no	—
111	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
112	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
113	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
114	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
115	C₂×C₄²	D₄×C₅₃	1	2	yes	—
116	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
117	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
118	C₄⋊C₈	—	2	4	no	—
119	C₄⋊C₈	—	2	4	no	—
120	C₄⋊C₈	—	4	4	no	—
121	#97	—	2	4	no	—
122	#97	—	2	2	no	—
123	#97	C₈×C₅₃	1	4	yes	complemented
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	2	2	no	—
126	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
127	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
128	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
129	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
130	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	4	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
141	C₂×C₈	M₁₆×C₅₃	1	4	yes	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	C₂×C₈×C₅₃	1	2	yes	—
147	C₄²	M₁₆×C₅₃	1	2	yes	—
148	C₄²	#41×C₅₃	1	2	yes	—
149	C₄²	#41×C₅₃	1	2	yes	—
150	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	C₄⋊C₈×C₅₃	1	4	yes	—
158	C₂×C₄	#117×C₅₃	1	2	yes	—
159	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
160	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
161	C₂×C₄	#125×C₅₃	1	2	yes	—
162	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
163	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
164	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
165	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
166	C₂×C₄	#137×C₅₃	1	2	yes	—
167	C₈	—	2	4	no	—
168	C₈	—	2	4	no	—
169	C₈	—	8	2	no	—
170	C₈	—	8	2	no	—
171	C₂²	C₅₃×#382	1	1	yes	characteristic
172	C₂²	C₂×C₄⋊C₈×C₅₃	1	1	yes	characteristic
173	C₂²	C₅₃×#423	1	1	yes	characteristic
174	C₂²	C₅₃×#428	1	2	yes	—
175	C₂²	C₅₃×#430	1	2	yes	—
176	C₄	—	2	4	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	2	no	—
179	C₄	C₅₃×#431	1	2	yes	—
180	C₄	C₅₃×#432	1	2	yes	—
181	C₄	C₅₃×#489	1	2	yes	—
182	C₄	C₅₃×#490	1	1	yes	characteristic
183	C₄	C₅₃×#491	1	1	yes	characteristic
184	C₂	C₅₃×#1565	1	2	yes	—
185	C₂	C₅₃×#1569	1	2	yes	—
186	C₂	C₅₃×#1831	1	1	yes	characteristic
187	C₂	C₅₃×#1836	1	1	yes	characteristic
188	C₂	C₅₃×#1838	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
28. #97×C₅₃	C₈	semidirect
76. #8786	C₅₃	direct (N × H)
108. C₅₃	#8786	direct (N × H)
123. #97	C₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#1842	2
C₅₃×#1831	2
C₄×#97×C₅₃	2
C₅₃×#1277	2
#8786	53

Subgroups — C53×#8786 #27216298

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8786 #27216298