Subgroups — C₅₃×#8122 #27215634

Order 13568.

520 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2920	C₂	1	2	yes	maximal
2	C₅₃×#2920	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1307	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
5	C₅₃×#432	—	2	2	no	—
6	C₅₃×#432	—	2	2	no	—
7	C₅₃×#432	C₂²	1	4	yes	—
8	C₅₃×#431	—	2	2	no	—
9	C₅₃×#431	—	2	2	no	—
10	C₅₃×#431	C₄	1	4	yes	—
11	C₅₃×#423	—	2	2	no	—
12	C₅₃×#423	C₂²	1	2	yes	—
13	C₅₃×#407	C₄	1	2	yes	—
14	C₅₃×#407	C₄	1	4	yes	—
15	C₄×M₁₆×C₅₃	—	2	2	no	—
16	C₄×M₁₆×C₅₃	C₄	1	2	yes	—
17	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×M₁₆×C₅₃	—	2	4	no	—
20	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
21	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₅₃	D₄	1	2	yes	—
24	C₂×C₄²×C₅₃	D₄	1	2	yes	—
25	C₂×C₄²×C₅₃	D₄	1	2	yes	—
26	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₅₃	—	2	4	no	—
29	C₄⋊C₈×C₅₃	—	2	4	no	—
30	C₄⋊C₈×C₅₃	—	2	4	no	—
31	C₂²⋊C₈×C₅₃	—	2	4	no	—
32	C₂²⋊C₈×C₅₃	—	2	4	no	—
33	C₂²⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
34	#97×C₅₃	—	2	2	no	—
35	#97×C₅₃	—	2	4	no	—
36	C₄×C₈×C₅₃	—	2	4	no	—
37	C₄×C₈×C₅₃	—	2	2	no	—
38	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	—	4	4	no	—
40	C₂²×C₄×C₅₃	—	4	2	no	—
41	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
42	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
51	M₁₆×C₅₃	—	4	8	no	—
52	M₁₆×C₅₃	—	4	4	no	—
53	C₂×C₈×C₅₃	—	2	4	no	—
54	C₂×C₈×C₅₃	—	2	8	no	—
55	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
59	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
60	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
61	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
62	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₅₃	C₂×D₄	1	2	yes	—
64	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
65	C₂³×C₅₃	—	4	1	no	—
66	C₂³×C₅₃	—	4	2	no	—
67	C₂³×C₅₃	#95	1	2	yes	—
68	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₅₃	—	2	4	no	—
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	—	2	4	no	—
72	C₂×C₄×C₅₃	—	2	4	no	—
73	C₂×C₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	4	1	no	—
75	C₂×C₄×C₅₃	—	4	2	no	—
76	C₂×C₄×C₅₃	—	4	2	no	—
77	C₂×C₄×C₅₃	—	4	2	no	—
78	C₂×C₄×C₅₃	—	4	2	no	—
79	C₂×C₄×C₅₃	#95	1	2	yes	—
80	C₂×C₄×C₅₃	#95	1	2	yes	—
81	C₂×C₄×C₅₃	#95	1	2	yes	—
82	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
83	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
86	C₈×C₅₃	—	4	8	no	—
87	C₈×C₅₃	—	4	4	no	—
88	#8122	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂²×C₅₃	—	2	2	no	—
90	C₂²×C₅₃	—	2	1	no	—
91	C₂²×C₅₃	—	2	2	no	—
92	C₂²×C₅₃	—	2	2	no	—
93	C₂²×C₅₃	—	2	2	no	—
94	C₂²×C₅₃	—	4	4	no	—
95	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
96	C₄×C₅₃	—	2	2	no	—
97	C₄×C₅₃	—	2	1	no	—
98	C₄×C₅₃	—	2	2	no	—
99	C₄×C₅₃	—	2	2	no	—
100	C₄×C₅₃	—	2	2	no	—
101	C₄×C₅₃	#344	1	4	yes	—
102	C₄×C₅₃	#413	1	2	yes	—
103	#2920	C₂×C₅₃	1	2	yes	—
104	#2920	C₂×C₅₃	1	1	yes	characteristic
105	#1307	C₂×C₅₃	1	4	yes	complemented
106	C₂×C₅₃	—	4	2	no	—
107	C₂×C₅₃	—	4	1	no	—
108	C₂×C₅₃	#1742	1	1	yes	characteristic
109	C₂×C₅₃	C₂×#344	1	2	yes	—
110	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
111	#432	—	2	2	no	—
112	#432	—	2	2	no	—
113	#432	C₂²×C₅₃	1	4	yes	—
114	#431	—	2	2	no	—
115	#431	—	2	2	no	—
116	#431	C₄×C₅₃	1	4	yes	—
117	#423	—	2	2	no	—
118	#423	C₂²×C₅₃	1	2	yes	—
119	#407	C₄×C₅₃	1	2	yes	—
120	#407	C₄×C₅₃	1	4	yes	—
121	C₄×M₁₆	—	2	2	no	—
122	C₄×M₁₆	C₄×C₅₃	1	2	yes	—
123	C₅₃	#8122	1	1	yes	characteristic, direct product factor, Sylow
124	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
125	C₂³×C₄	C₂×C₄×C₅₃	1	2	yes	—
126	C₂×M₁₆	—	2	4	no	—
127	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
128	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
129	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
130	C₂×C₄²	D₄×C₅₃	1	2	yes	—
131	C₂×C₄²	D₄×C₅₃	1	2	yes	—
132	C₂×C₄²	D₄×C₅₃	1	2	yes	—
133	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
134	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
135	C₄⋊C₈	—	2	4	no	—
136	C₄⋊C₈	—	2	4	no	—
137	C₄⋊C₈	—	2	4	no	—
138	C₂²⋊C₈	—	2	4	no	—
139	C₂²⋊C₈	—	2	4	no	—
140	C₂²⋊C₈	C₂×C₄×C₅₃	1	8	yes	—
141	#97	—	2	4	no	—
142	#97	—	2	2	no	—
143	C₄×C₈	—	2	4	no	—
144	C₄×C₈	—	2	2	no	—
145	C₂⁴	C₄²×C₅₃	1	1	yes	characteristic
146	C₂²×C₄	—	4	4	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
149	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
150	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
151	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
152	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
153	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
154	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
155	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
156	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
157	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
158	M₁₆	—	4	8	no	—
159	M₁₆	—	4	4	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	2	8	no	—
162	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
163	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
164	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
165	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
166	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
167	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
168	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
169	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
170	C₄²	C₂×D₄×C₅₃	1	2	yes	—
171	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
172	C₂³	—	4	1	no	—
173	C₂³	—	4	2	no	—
174	C₂³	#95×C₅₃	1	2	yes	—
175	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	#95×C₅₃	1	2	yes	—
187	C₂×C₄	#95×C₅₃	1	2	yes	—
188	C₂×C₄	#95×C₅₃	1	2	yes	—
189	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
190	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
191	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
192	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
193	C₈	—	4	8	no	—
194	C₈	—	4	4	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	4	no	—
201	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	2	no	—
206	C₄	—	2	2	no	—
207	C₄	C₅₃×#344	1	4	yes	—
208	C₄	C₅₃×#413	1	2	yes	—
209	C₂	—	4	2	no	—
210	C₂	—	4	1	no	—
211	C₂	C₅₃×#1742	1	1	yes	characteristic
212	C₂	C₂×C₅₃×#344	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#95
commutator	C₂³	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1307	C₂	semidirect
88. #8122	C₅₃	direct (N × H)
105. #1307	C₂×C₅₃	semidirect
123. C₅₃	#8122	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#2920	2
C₅₃×#1307	2
#8122	53

Subgroups — C53×#8122 #27215634

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8122 #27215634