Subgroups — C₅₃×#22243 #27229755

Order 13568.

666 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×#446	C₂	1	2	yes	maximal
2	C₅₃×#2399	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1985	C₂	1	2	yes	maximal
4	C₅₃×#1917	C₂	1	8	yes	maximal
5	C₅₃×#1832	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1746	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
8	C₅₃×#446	—	2	8	no	—
9	C₂×C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
10	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
11	C₂×#97×C₅₃	C₂²	1	4	yes	—
12	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#398	C₄	1	4	yes	—
15	C₅₃×#395	C₄	1	4	yes	—
16	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
17	C₅₃×#376	C₂²	1	1	yes	characteristic
18	C₅₃×#336	C₂²	1	8	yes	—
19	C₅₃×#336	C₂²	1	4	yes	—
20	C₅₃×#336	C₂²	1	4	yes	—
21	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
23	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
24	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
25	C₄×Q₈×C₅₃	—	2	4	no	—
26	C₄×Q₈×C₅₃	—	2	4	no	—
27	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
28	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
30	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
31	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₅₃	—	2	8	no	—
34	C₄⋊C₈×C₅₃	—	2	4	no	—
35	#97×C₅₃	—	2	8	no	—
36	#97×C₅₃	—	2	2	no	—
37	C₄×C₈×C₅₃	D₄	1	4	yes	—
38	#95×C₅₃	C₂×C₄	1	4	yes	—
39	#95×C₅₃	C₂×C₄	1	4	yes	—
40	#95×C₅₃	C₂×C₄	1	4	yes	—
41	C₂×Q₈×C₅₃	—	2	4	no	—
42	C₂×Q₈×C₅₃	—	4	4	no	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
48	C₂×C₈×C₅₃	—	2	4	no	—
49	C₂×C₈×C₅₃	—	2	4	no	—
50	C₂×C₈×C₅₃	—	2	2	no	—
51	C₂×C₈×C₅₃	—	2	8	no	—
52	C₂×C₈×C₅₃	—	2	4	no	—
53	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
54	C₄⋊C₄×C₅₃	—	2	8	no	—
55	C₄⋊C₄×C₅₃	—	2	4	no	—
56	C₄⋊C₄×C₅₃	—	4	8	no	—
57	C₄²×C₅₃	—	2	4	no	—
58	C₄²×C₅₃	—	2	4	no	—
59	C₄²×C₅₃	—	2	4	no	—
60	C₄²×C₅₃	C₂×D₄	1	2	yes	—
61	C₄²×C₅₃	#41	1	2	yes	—
62	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
63	Q₈×C₅₃	—	4	8	no	—
64	C₂×C₄×C₅₃	—	2	4	no	—
65	C₂×C₄×C₅₃	—	2	4	no	—
66	C₂×C₄×C₅₃	—	2	4	no	—
67	C₂×C₄×C₅₃	—	2	4	no	—
68	C₂×C₄×C₅₃	—	2	4	no	—
69	C₂×C₄×C₅₃	—	2	4	no	—
70	C₂×C₄×C₅₃	—	2	8	no	—
71	C₂×C₄×C₅₃	—	2	8	no	—
72	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
73	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
75	C₂×C₄×C₅₃	2₋¹⁺⁴	1	4	yes	—
76	C₈×C₅₃	—	2	4	no	—
77	C₈×C₅₃	—	4	4	no	—
78	C₈×C₅₃	—	4	2	no	—
79	#22243	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
81	C₂²×C₅₃	#519	1	2	yes	—
82	C₂²×C₅₃	#568	1	4	yes	—
83	C₄×C₅₃	—	2	4	no	—
84	C₄×C₅₃	—	4	4	no	—
85	C₄×C₅₃	—	4	4	no	—
86	C₄×C₅₃	—	4	4	no	—
87	C₄×C₅₃	#536	1	2	yes	—
88	C₄×C₅₃	#541	1	2	yes	—
89	C₂×#446	C₂×C₅₃	1	2	yes	—
90	#2399	C₂×C₅₃	1	1	yes	characteristic
91	#1985	C₂×C₅₃	1	2	yes	—
92	#1917	C₂×C₅₃	1	8	yes	—
93	#1832	C₂×C₅₃	1	1	yes	characteristic
94	#1746	C₂×C₅₃	1	1	yes	characteristic
95	C₂×C₅₃	#2327	1	1	yes	characteristic
96	C₂×C₅₃	#2930	1	2	yes	—
97	C₂×C₅₃	#2987	1	4	yes	—
98	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
99	#446	—	2	8	no	—
100	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
101	C₂×C₄⋊C₈	C₂²×C₅₃	1	4	yes	—
102	C₂×#97	C₂²×C₅₃	1	4	yes	—
103	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
105	#398	C₄×C₅₃	1	4	yes	—
106	#395	C₄×C₅₃	1	4	yes	—
107	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
108	#376	C₂²×C₅₃	1	1	yes	characteristic
109	#336	C₂²×C₅₃	1	4	yes	—
110	#336	C₂²×C₅₃	1	8	yes	—
111	#336	C₂²×C₅₃	1	4	yes	—
112	C₅₃	#22243	1	1	yes	characteristic, direct product factor, Sylow
113	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
114	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
115	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
116	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
117	C₄×Q₈	—	2	4	no	—
118	C₄×Q₈	—	2	4	no	—
119	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
120	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
121	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	8	yes	—
122	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
123	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
124	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
125	C₄⋊C₈	—	2	8	no	—
126	C₄⋊C₈	—	2	4	no	—
127	#97	—	2	8	no	—
128	#97	—	2	2	no	—
129	C₄×C₈	D₄×C₅₃	1	4	yes	—
130	#95	C₂×C₄×C₅₃	1	4	yes	—
131	#95	C₂×C₄×C₅₃	1	4	yes	—
132	#95	C₂×C₄×C₅₃	1	4	yes	—
133	C₂×Q₈	—	2	4	no	—
134	C₂×Q₈	—	4	4	no	—
135	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
136	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
137	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
138	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
139	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	8	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	2	8	no	—
148	C₄⋊C₄	—	4	8	no	—
149	C₄²	—	2	4	no	—
150	C₄²	—	2	4	no	—
151	C₄²	—	2	4	no	—
152	C₄²	C₂×D₄×C₅₃	1	2	yes	—
153	C₄²	#41×C₅₃	1	2	yes	—
154	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
155	Q₈	—	4	8	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	—	2	8	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
165	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
166	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
167	C₂×C₄	2₋¹⁺⁴×C₅₃	1	4	yes	—
168	C₈	—	2	4	no	—
169	C₈	—	4	4	no	—
170	C₈	—	4	2	no	—
171	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
172	C₂²	C₅₃×#519	1	2	yes	—
173	C₂²	C₅₃×#568	1	4	yes	—
174	C₄	—	2	4	no	—
175	C₄	—	4	4	no	—
176	C₄	—	4	4	no	—
177	C₄	—	4	4	no	—
178	C₄	C₅₃×#536	1	2	yes	—
179	C₄	C₅₃×#541	1	2	yes	—
180	C₂	C₅₃×#2327	1	1	yes	characteristic
181	C₂	C₅₃×#2930	1	2	yes	—
182	C₂	C₅₃×#2987	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle, prime radical	C₂³×C₅₃	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
79. #22243	C₅₃	direct (N × H)
112. C₅₃	#22243	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×#446	2
C₅₃×#2399	2
C₅₃×#1985	2
C₅₃×#1917	2
C₅₃×#1832	2
C₅₃×#1746	2
#22243	53

Subgroups — C53×#22243 #27229755

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22243 #27229755