Subgroups — C₅₃×#12659 #27220171

Order 13568.

346 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#364	C₂	1	2	yes	maximal
2	C₂×C₅₃×#347	C₂	1	2	yes	maximal
3	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1377	C₂	1	2	yes	maximal
5	C₂×M₃₂×C₅₃	C₂²	1	4	yes	—
6	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
7	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
8	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₄³×C₅₃	C₄	1	1	yes	characteristic
13	C₅₃×#364	C₄	1	8	yes	complemented
14	C₅₃×#347	—	2	4	no	—
15	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₅₃	—	2	2	no	—
20	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	M₃₂×C₅₃	—	2	4	no	—
24	M₃₂×C₅₃	C₂×C₄	1	8	yes	—
25	C₂×C₁₆×C₅₃	—	2	4	no	—
26	#97×C₅₃	—	2	2	no	—
27	#97×C₅₃	—	2	2	no	—
28	#97×C₅₃	—	2	2	no	—
29	#97×C₅₃	D₄	1	2	yes	—
30	#97×C₅₃	Q₈	1	2	yes	—
31	C₄×C₈×C₅₃	—	2	2	no	—
32	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₅₃	—	2	1	no	—
34	C₂²×C₄×C₅₃	—	2	1	no	—
35	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	—	2	1	no	—
38	C₂×C₈×C₅₃	—	2	1	no	—
39	C₂×C₈×C₅₃	—	2	1	no	—
40	C₂×C₈×C₅₃	—	2	1	no	—
41	C₂×C₈×C₅₃	C₄²	1	2	yes	—
42	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
44	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
45	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
46	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
47	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₅₃	#41	1	2	yes	—
50	C₄²×C₅₃	—	2	1	no	—
51	C₄²×C₅₃	—	2	1	no	—
52	C₄²×C₅₃	—	2	1	no	—
53	C₄²×C₅₃	—	2	1	no	—
54	C₄²×C₅₃	—	4	4	no	—
55	C₄²×C₅₃	C₄²	1	2	yes	—
56	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
57	C₁₆×C₅₃	—	2	8	no	—
58	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
59	C₂×C₄×C₅₃	—	2	4	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	4	2	no	—
64	C₂×C₄×C₅₃	—	4	2	no	—
65	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
71	C₈×C₅₃	—	2	2	no	—
72	C₈×C₅₃	—	2	2	no	—
73	C₈×C₅₃	#97	1	4	yes	—
74	C₈×C₅₃	C₄×D₄	1	2	yes	—
75	C₈×C₅₃	C₄×Q₈	1	2	yes	—
76	#12659	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂²×C₅₃	—	2	2	no	—
78	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
79	C₂²×C₅₃	#423	1	1	yes	characteristic
80	C₂²×C₅₃	#432	1	1	yes	characteristic
81	C₄×C₅₃	—	2	4	no	—
82	C₄×C₅₃	—	4	2	no	—
83	C₄×C₅₃	—	4	2	no	—
84	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
85	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
86	C₄×C₅₃	#423	1	1	yes	characteristic
87	C₄×C₅₃	#432	1	1	yes	characteristic
88	C₂×#364	C₂×C₅₃	1	2	yes	—
89	C₂×#347	C₂×C₅₃	1	2	yes	—
90	C₄×#97	C₂×C₅₃	1	1	yes	characteristic
91	#1377	C₂×C₅₃	1	2	yes	—
92	C₂×C₅₃	—	2	1	no	—
93	C₂×C₅₃	—	2	1	no	—
94	C₂×C₅₃	#1767	1	1	yes	characteristic
95	C₂×C₅₃	#2165	1	2	yes	—
96	C₂×M₃₂	C₂²×C₅₃	1	4	yes	—
97	C₂×#97	C₄×C₅₃	1	1	yes	characteristic
98	C₂×#97	C₄×C₅₃	1	1	yes	characteristic
99	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
100	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
101	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
103	C₄³	C₄×C₅₃	1	1	yes	characteristic
104	#364	C₄×C₅₃	1	8	yes	complemented
105	#347	—	2	4	no	—
106	C₅₃	#12659	1	1	yes	characteristic, direct product factor, Sylow
107	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
108	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
109	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
110	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
111	C₂×C₄²	—	2	2	no	—
112	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
113	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
114	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
115	M₃₂	—	2	4	no	—
116	M₃₂	C₂×C₄×C₅₃	1	8	yes	—
117	C₂×C₁₆	—	2	4	no	—
118	#97	—	2	2	no	—
119	#97	—	2	2	no	—
120	#97	—	2	2	no	—
121	#97	D₄×C₅₃	1	2	yes	—
122	#97	Q₈×C₅₃	1	2	yes	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	M₁₆×C₅₃	1	2	yes	—
128	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	C₄²×C₅₃	1	2	yes	—
134	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
135	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
136	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
137	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
138	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
139	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
140	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
141	C₂×C₈	#41×C₅₃	1	2	yes	—
142	C₄²	—	2	1	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	2	1	no	—
145	C₄²	—	2	1	no	—
146	C₄²	—	4	4	no	—
147	C₄²	C₄²×C₅₃	1	2	yes	—
148	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
149	C₁₆	—	2	8	no	—
150	C₂³	C₂×M₁₆×C₅₃	1	1	yes	characteristic
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
159	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
160	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
161	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
162	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
163	C₈	—	2	2	no	—
164	C₈	—	2	2	no	—
165	C₈	#97×C₅₃	1	4	yes	—
166	C₈	C₄×D₄×C₅₃	1	2	yes	—
167	C₈	C₄×Q₈×C₅₃	1	2	yes	—
168	C₂²	—	2	2	no	—
169	C₂²	C₄×M₁₆×C₅₃	1	1	yes	characteristic
170	C₂²	C₅₃×#423	1	1	yes	characteristic
171	C₂²	C₅₃×#432	1	1	yes	characteristic
172	C₄	—	2	4	no	—
173	C₄	—	4	2	no	—
174	C₄	—	4	2	no	—
175	C₄	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
176	C₄	C₂×#97×C₅₃	1	1	yes	characteristic
177	C₄	C₅₃×#423	1	1	yes	characteristic
178	C₄	C₅₃×#432	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	C₅₃×#1767	1	1	yes	characteristic
182	C₂	C₅₃×#2165	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	#117
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	#432
prime radical	C₂³×C₅₃	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×#97×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₅₃×#364	C₄	semidirect
76. #12659	C₅₃	direct (N × H)
104. #364	C₄×C₅₃	semidirect
106. C₅₃	#12659	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#364	2
C₂×C₅₃×#347	2
C₄×#97×C₅₃	2
C₅₃×#1377	2
#12659	53

Subgroups — C53×#12659 #27220171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12659 #27220171