Subgroups — C₅₃×#12665 #27220177

Order 13568.

326 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×C₅₃×#364	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2108	C₂	1	2	yes	maximal
3	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1772	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1379	C₂	1	2	yes	maximal
6	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
7	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
8	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#108×C₅₃	C₄	1	1	yes	characteristic
10	C₅₃×#427	C₄	1	1	yes	characteristic
11	C₅₃×#405	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#364	C₄	1	4	yes	complemented
14	C₅₃×#346	—	2	2	no	—
15	C₅₃×#346	C₄	1	4	yes	complemented
16	C₄×C₁₆×C₅₃	—	2	2	no	—
17	C₅₃×#339	C₄	1	2	yes	—
18	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
21	#117×C₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	M₃₂×C₅₃	—	2	2	no	—
24	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
25	C₂×C₁₆×C₅₃	—	2	2	no	—
26	C₂×C₁₆×C₅₃	—	2	1	no	—
27	C₂×C₁₆×C₅₃	—	2	1	no	—
28	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
31	C₂×C₁₆×C₅₃	Q₈	1	2	yes	—
32	#108×C₅₃	C₈	1	4	yes	—
33	#107×C₅₃	C₈	1	2	yes	—
34	#106×C₅₃	C₈	1	2	yes	—
35	#97×C₅₃	—	2	2	no	—
36	C₄×C₈×C₅₃	—	2	2	no	—
37	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	—	2	1	no	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	M₁₆×C₅₃	—	2	2	no	—
42	M₁₆×C₅₃	C₂×C₈	1	4	yes	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	C₄²	1	2	yes	—
45	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
53	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
54	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
55	C₂²⋊C₄×C₅₃	—	2	2	no	—
56	C₄²×C₅₃	—	2	2	no	—
57	C₄²×C₅₃	—	2	1	no	—
58	C₄²×C₅₃	C₄²	1	2	yes	—
59	C₁₆×C₅₃	—	2	2	no	—
60	C₁₆×C₅₃	—	2	4	no	—
61	C₁₆×C₅₃	C₄⋊C₄	1	4	yes	—
62	C₂³×C₅₃	#131	1	1	yes	characteristic
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	4	no	—
65	C₂×C₄×C₅₃	—	4	2	no	—
66	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
72	C₈×C₅₃	—	2	4	no	—
73	C₈×C₅₃	C₄×C₈	1	2	yes	—
74	C₈×C₅₃	C₄×C₈	1	2	yes	—
75	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₈×C₅₃	#117	1	1	yes	characteristic
77	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
78	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
79	#12665	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂²×C₅₃	—	2	2	no	—
81	C₂²×C₅₃	#405	1	1	yes	characteristic
82	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
83	C₂²×C₅₃	C₈×Q₈	1	1	yes	characteristic
84	C₄×C₅₃	—	4	2	no	—
85	C₄×C₅₃	—	4	4	no	—
86	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
87	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
88	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
89	C₄×C₅₃	C₈×Q₈	1	1	yes	characteristic
90	C₂×#364	C₂×C₅₃	1	1	yes	characteristic
91	#2108	C₂×C₅₃	1	2	yes	—
92	C₂×#346	C₂×C₅₃	1	1	yes	characteristic
93	#1772	C₂×C₅₃	1	1	yes	characteristic
94	#1379	C₂×C₅₃	1	2	yes	—
95	C₂×C₅₃	—	2	1	no	—
96	C₂×C₅₃	—	2	1	no	—
97	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₅₃	#2171	1	2	yes	—
99	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
100	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
101	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
102	C₂×#108	C₄×C₅₃	1	1	yes	characteristic
103	#427	C₄×C₅₃	1	1	yes	characteristic
104	#405	C₂²×C₅₃	1	2	yes	—
105	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
106	#364	C₄×C₅₃	1	4	yes	complemented
107	#346	—	2	2	no	—
108	#346	C₄×C₅₃	1	4	yes	complemented
109	C₄×C₁₆	—	2	2	no	—
110	#339	C₄×C₅₃	1	2	yes	—
111	C₅₃	#12665	1	1	yes	characteristic, direct product factor, Sylow
112	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
113	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
114	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
115	#117	C₂×C₄×C₅₃	1	2	yes	—
116	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
117	M₃₂	—	2	2	no	—
118	M₃₂	C₂×C₄×C₅₃	1	4	yes	—
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	2	1	no	—
121	C₂×C₁₆	—	2	1	no	—
122	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
123	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
124	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
125	C₂×C₁₆	Q₈×C₅₃	1	2	yes	—
126	#108	C₈×C₅₃	1	4	yes	—
127	#107	C₈×C₅₃	1	2	yes	—
128	#106	C₈×C₅₃	1	2	yes	—
129	#97	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
132	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
135	M₁₆	—	2	2	no	—
136	M₁₆	C₂×C₈×C₅₃	1	4	yes	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	C₄²×C₅₃	1	2	yes	—
139	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
140	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
141	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
142	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
143	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
144	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
145	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
146	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
147	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
148	C₄⋊C₄	C₂×C₈×C₅₃	1	4	yes	—
149	C₂²⋊C₄	—	2	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	1	no	—
152	C₄²	C₄²×C₅₃	1	2	yes	—
153	C₁₆	—	2	2	no	—
154	C₁₆	—	2	4	no	—
155	C₁₆	C₄⋊C₄×C₅₃	1	4	yes	—
156	C₂³	#131×C₅₃	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
161	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
162	C₂×C₄	C₄×Q₈×C₅₃	1	1	yes	characteristic
163	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
164	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
165	C₂×C₄	#131×C₅₃	1	1	yes	characteristic
166	C₈	—	2	4	no	—
167	C₈	C₄×C₈×C₅₃	1	2	yes	—
168	C₈	C₄×C₈×C₅₃	1	2	yes	—
169	C₈	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
170	C₈	#117×C₅₃	1	1	yes	characteristic
171	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
172	C₈	C₄×Q₈×C₅₃	1	1	yes	characteristic
173	C₂²	—	2	2	no	—
174	C₂²	C₅₃×#405	1	1	yes	characteristic
175	C₂²	C₈×D₄×C₅₃	1	1	yes	characteristic
176	C₂²	C₈×Q₈×C₅₃	1	1	yes	characteristic
177	C₄	—	4	2	no	—
178	C₄	—	4	4	no	—
179	C₄	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
180	C₄	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
181	C₄	C₈×D₄×C₅₃	1	1	yes	characteristic
182	C₄	C₈×Q₈×C₅₃	1	1	yes	characteristic
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	C₈×C₄⋊C₄×C₅₃	1	1	yes	characteristic
186	C₂	C₅₃×#2171	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₅₃	C₂×D₄
commutator	C₄	C₂×C₄×C₈×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₈×D₄
prime radical	C₂³×C₅₃	#131
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₅₃ ◁ C₂²×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₅₃×#364	C₄	semidirect
15. C₅₃×#346	C₄	semidirect
79. #12665	C₅₃	direct (N × H)
106. #364	C₄×C₅₃	semidirect
108. #346	C₄×C₅₃	semidirect
111. C₅₃	#12665	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
C₅₃×#2108	2
C₂×C₅₃×#346	2
C₅₃×#1772	2
C₅₃×#1379	2
#12665	53

Subgroups — C53×#12665 #27220177

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12665 #27220177