Subgroups — C₅₃×#8028 #27215540

Order 13568.

778 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	C₂×C₄×M₁₆×C₅₃	C₂	1	2	yes	maximal
2	C₅₃×#2285	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1295	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₅₃	—	2	2	no	—
5	C₂×#117×C₅₃	C₄	1	2	yes	—
6	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
7	C₄×M₁₆×C₅₃	—	2	4	no	—
8	C₄×M₁₆×C₅₃	C₄	1	8	yes	complemented
9	C₂×#97×C₅₃	C₂²	1	4	yes	—
10	C₂×C₄×C₈×C₅₃	—	2	2	no	—
11	C₅₃×#384	C₄	1	2	yes	—
12	C₅₃×#382	C₂²	1	2	yes	—
13	C₅₃×#380	—	2	2	no	—
14	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
15	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₅₃	D₄	1	2	yes	—
17	C₂×M₁₆×C₅₃	—	2	16	no	—
18	C₂×M₁₆×C₅₃	—	2	8	no	—
19	C₂²×C₈×C₅₃	—	2	4	no	—
20	#117×C₅₃	—	4	4	no	—
21	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
22	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
23	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
24	C₂×C₄²×C₅₃	—	2	2	no	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	D₄	1	2	yes	—
28	C₂×C₄²×C₅₃	D₄	1	2	yes	—
29	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	#97×C₅₃	—	2	4	no	—
32	#97×C₅₃	C₂×C₄	1	8	yes	—
33	C₄×C₈×C₅₃	—	2	8	no	—
34	#95×C₅₃	—	2	4	no	—
35	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	—	2	2	no	—
38	C₂²×C₄×C₅₃	—	2	8	no	—
39	C₂²×C₄×C₅₃	—	2	4	no	—
40	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
42	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
48	M₁₆×C₅₃	—	2	32	no	—
49	C₂×C₈×C₅₃	—	2	16	no	—
50	C₂×C₈×C₅₃	—	2	8	no	—
51	C₄⋊C₄×C₅₃	—	4	4	no	—
52	C₄⋊C₄×C₅₃	—	4	8	no	—
53	C₂²⋊C₄×C₅₃	—	4	4	no	—
54	C₄²×C₅₃	—	4	4	no	—
55	C₄²×C₅₃	C₄²	1	2	yes	—
56	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
57	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
58	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
59	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
60	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
62	C₂³×C₅₃	—	2	1	no	—
63	C₂³×C₅₃	—	2	1	no	—
64	C₂³×C₅₃	—	2	4	no	—
65	C₂³×C₅₃	#95	1	2	yes	—
66	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	—	2	1	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	—	2	4	no	—
73	C₂×C₄×C₅₃	—	2	4	no	—
74	C₂×C₄×C₅₃	—	4	4	no	—
75	C₂×C₄×C₅₃	—	4	8	no	—
76	C₂×C₄×C₅₃	#95	1	4	yes	—
77	C₂×C₄×C₅₃	#95	1	2	yes	—
78	C₂×C₄×C₅₃	#100	1	2	yes	—
79	C₂×C₄×C₅₃	#101	1	2	yes	—
80	C₂×C₄×C₅₃	#104	1	8	yes	—
81	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
83	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
85	C₈×C₅₃	—	2	16	no	—
86	#8028	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂²×C₅₃	—	2	4	no	—
88	C₂²×C₅₃	—	2	2	no	—
89	C₂²×C₅₃	—	4	2	no	—
90	C₂²×C₅₃	—	4	2	no	—
91	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
92	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
93	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
94	C₂²×C₅₃	C₂×#104	1	4	yes	—
95	C₄×C₅₃	—	2	2	no	—
96	C₄×C₅₃	—	2	2	no	—
97	C₄×C₅₃	—	2	1	no	—
98	C₄×C₅₃	—	2	1	no	—
99	C₄×C₅₃	—	8	4	no	—
100	C₄×C₅₃	#339	1	8	yes	—
101	C₄×C₅₃	#343	1	4	yes	—
102	C₂×C₄×M₁₆	C₂×C₅₃	1	2	yes	—
103	#2285	C₂×C₅₃	1	1	yes	characteristic
104	#1295	C₂×C₅₃	1	4	yes	complemented
105	C₂×C₅₃	—	4	2	no	—
106	C₂×C₅₃	#1525	1	2	yes	—
107	C₂×C₅₃	#1526	1	2	yes	—
108	C₂×C₅₃	C₂×#339	1	2	yes	—
109	C₂×C₅₃	C₂×#343	1	1	yes	characteristic
110	C₂²×M₁₆	—	2	2	no	—
111	C₂×#117	C₄×C₅₃	1	2	yes	—
112	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
113	C₄×M₁₆	—	2	4	no	—
114	C₄×M₁₆	C₄×C₅₃	1	8	yes	complemented
115	C₂×#97	C₂²×C₅₃	1	4	yes	—
116	C₂×C₄×C₈	—	2	2	no	—
117	#384	C₄×C₅₃	1	2	yes	—
118	#382	C₂²×C₅₃	1	2	yes	—
119	#380	—	2	2	no	—
120	C₄×C₄⋊C₄	—	2	2	no	—
121	C₅₃	#8028	1	1	yes	characteristic, direct product factor, Sylow
122	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
123	C₂³×C₄	D₄×C₅₃	1	2	yes	—
124	C₂×M₁₆	—	2	16	no	—
125	C₂×M₁₆	—	2	8	no	—
126	C₂²×C₈	—	2	4	no	—
127	#117	—	4	4	no	—
128	C₂×C₄⋊C₄	—	2	4	no	—
129	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
130	C₂×C₂²⋊C₄	—	2	2	no	—
131	C₂×C₄²	—	2	2	no	—
132	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
133	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
134	C₂×C₄²	D₄×C₅₃	1	2	yes	—
135	C₂×C₄²	D₄×C₅₃	1	2	yes	—
136	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
137	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
138	#97	—	2	4	no	—
139	#97	C₂×C₄×C₅₃	1	8	yes	—
140	C₄×C₈	—	2	8	no	—
141	#95	—	2	4	no	—
142	C₂⁴	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	—	2	8	no	—
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
148	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
149	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
150	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
151	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
152	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
153	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
155	M₁₆	—	2	32	no	—
156	C₂×C₈	—	2	16	no	—
157	C₂×C₈	—	2	8	no	—
158	C₄⋊C₄	—	4	4	no	—
159	C₄⋊C₄	—	4	8	no	—
160	C₂²⋊C₄	—	4	4	no	—
161	C₄²	—	4	4	no	—
162	C₄²	C₄²×C₅₃	1	2	yes	—
163	C₄²	C₂²⋊C₄×C₅₃	1	4	yes	—
164	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
165	C₄²	C₄⋊C₄×C₅₃	1	4	yes	—
166	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
167	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
168	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
169	C₂³	—	2	1	no	—
170	C₂³	—	2	1	no	—
171	C₂³	—	2	4	no	—
172	C₂³	#95×C₅₃	1	2	yes	—
173	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	8	no	—
183	C₂×C₄	#95×C₅₃	1	4	yes	—
184	C₂×C₄	#95×C₅₃	1	2	yes	—
185	C₂×C₄	#100×C₅₃	1	2	yes	—
186	C₂×C₄	#101×C₅₃	1	2	yes	—
187	C₂×C₄	#104×C₅₃	1	8	yes	—
188	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
190	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
191	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
192	C₈	—	2	16	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
198	C₂²	C₂×#100×C₅₃	1	1	yes	characteristic
199	C₂²	C₂×#101×C₅₃	1	1	yes	characteristic
200	C₂²	C₂×#104×C₅₃	1	4	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	8	4	no	—
206	C₄	C₅₃×#339	1	8	yes	—
207	C₄	C₅₃×#343	1	4	yes	—
208	C₂	—	4	2	no	—
209	C₂	C₅₃×#1525	1	2	yes	—
210	C₂	C₅₃×#1526	1	2	yes	—
211	C₂	C₂×C₅₃×#339	1	2	yes	—
212	C₂	C₂×C₅₃×#343	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1295	C₂	semidirect
8. C₄×M₁₆×C₅₃	C₄	semidirect
86. #8028	C₅₃	direct (N × H)
104. #1295	C₂×C₅₃	semidirect
114. C₄×M₁₆	C₄×C₅₃	semidirect
121. C₅₃	#8028	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×C₄×M₁₆×C₅₃	2
C₅₃×#2285	2
C₅₃×#1295	2
#8028	53

Subgroups — C53×#8028 #27215540

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8028 #27215540