Subgroups — C₅₃×#12660 #27220172

Order 13568.

330 conjugacy classes of subgroups.

The subgroup lattice has 182 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	C₅₃×#1726	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	C₂²	1	1	yes	characteristic
8	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₃	C₄	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	C₅₃×#376	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	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	M₃₂×C₅₃	—	2	4	no	—
23	M₃₂×C₅₃	C₂×C₄	1	8	yes	—
24	C₂×C₁₆×C₅₃	—	2	4	no	—
25	#97×C₅₃	—	2	2	no	—
26	#97×C₅₃	—	2	2	no	—
27	#97×C₅₃	D₄	1	2	yes	—
28	#97×C₅₃	Q₈	1	2	yes	—
29	C₄×C₈×C₅₃	—	2	2	no	—
30	C₄×C₈×C₅₃	—	2	2	no	—
31	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
32	#95×C₅₃	—	2	2	no	—
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	C₄²	1	2	yes	—
55	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
56	C₁₆×C₅₃	—	2	8	no	—
57	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
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	—	4	2	no	—
62	C₂×C₄×C₅₃	—	4	2	no	—
63	C₂×C₄×C₅₃	—	4	2	no	—
64	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×M₁₆	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	C₈×C₅₃	—	2	2	no	—
71	C₈×C₅₃	—	2	2	no	—
72	C₈×C₅₃	#97	1	4	yes	—
73	C₈×C₅₃	C₄×D₄	1	2	yes	—
74	C₈×C₅₃	C₄×Q₈	1	2	yes	—
75	#12660	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂²×C₅₃	—	2	2	no	—
77	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
78	C₂²×C₅₃	#423	1	1	yes	characteristic
79	C₂²×C₅₃	#432	1	1	yes	characteristic
80	C₄×C₅₃	—	4	2	no	—
81	C₄×C₅₃	—	4	2	no	—
82	C₄×C₅₃	—	4	2	no	—
83	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
84	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
85	C₄×C₅₃	#423	1	1	yes	characteristic
86	C₄×C₅₃	#432	1	1	yes	characteristic
87	C₂×#364	C₂×C₅₃	1	2	yes	—
88	C₂×#347	C₂×C₅₃	1	2	yes	—
89	#1726	C₂×C₅₃	1	1	yes	characteristic
90	#1377	C₂×C₅₃	1	2	yes	—
91	C₂×C₅₃	—	2	1	no	—
92	C₂×C₅₃	—	2	1	no	—
93	C₂×C₅₃	#1767	1	1	yes	characteristic
94	C₂×C₅₃	#2166	1	2	yes	—
95	C₂×M₃₂	C₂²×C₅₃	1	4	yes	—
96	C₂×#97	C₄×C₅₃	1	1	yes	characteristic
97	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
98	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
99	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
100	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
101	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
102	#376	C₄×C₅₃	1	1	yes	characteristic
103	#364	C₄×C₅₃	1	8	yes	complemented
104	#347	—	2	4	no	—
105	C₅₃	#12660	1	1	yes	characteristic, direct product factor, Sylow
106	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
107	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
108	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
109	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
110	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
111	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
112	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
113	M₃₂	—	2	4	no	—
114	M₃₂	C₂×C₄×C₅₃	1	8	yes	—
115	C₂×C₁₆	—	2	4	no	—
116	#97	—	2	2	no	—
117	#97	—	2	2	no	—
118	#97	D₄×C₅₃	1	2	yes	—
119	#97	Q₈×C₅₃	1	2	yes	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
123	#95	—	2	2	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	M₁₆×C₅₃	1	2	yes	—
127	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	C₄²×C₅₃	1	2	yes	—
133	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
134	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
135	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
136	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
137	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
138	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
139	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
140	C₂×C₈	#41×C₅₃	1	2	yes	—
141	C₄²	—	2	1	no	—
142	C₄²	—	2	1	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	2	1	no	—
145	C₄²	C₄²×C₅₃	1	2	yes	—
146	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
147	C₁₆	—	2	8	no	—
148	C₂³	C₂×M₁₆×C₅₃	1	1	yes	characteristic
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
157	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
158	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
159	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
160	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
161	C₈	—	2	2	no	—
162	C₈	—	2	2	no	—
163	C₈	#97×C₅₃	1	4	yes	—
164	C₈	C₄×D₄×C₅₃	1	2	yes	—
165	C₈	C₄×Q₈×C₅₃	1	2	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	C₄×M₁₆×C₅₃	1	1	yes	characteristic
168	C₂²	C₅₃×#423	1	1	yes	characteristic
169	C₂²	C₅₃×#432	1	1	yes	characteristic
170	C₄	—	4	2	no	—
171	C₄	—	4	2	no	—
172	C₄	—	4	2	no	—
173	C₄	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
174	C₄	C₂×#97×C₅₃	1	1	yes	characteristic
175	C₄	C₅₃×#423	1	1	yes	characteristic
176	C₄	C₅₃×#432	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	C₅₃×#1767	1	1	yes	characteristic
180	C₂	C₅₃×#2166	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
75. #12660	C₅₃	direct (N × H)
103. #364	C₄×C₅₃	semidirect
105. C₅₃	#12660	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
C₅₃×#1726	2
C₅₃×#1377	2
#12660	53

Subgroups — C53×#12660 #27220172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12660 #27220172