Subgroups — C₅₃×#2726 #20271663

Order 6784.

416 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#125×C₅₃	C₂	1	2	yes	maximal
2	C₅₃×#398	C₂	1	2	yes	maximal
3	C₅₃×#398	C₂	1	2	yes	maximal
4	C₅₃×#398	C₂	1	2	yes	maximal
5	C₅₃×#389	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#389	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#389	C₂	1	2	yes	maximal
8	C₅₃×#389	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#387	C₂	1	2	yes	maximal
10	#125×C₅₃	—	2	8	no	—
11	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
26	#95×C₅₃	C₂²	1	2	yes	—
27	#95×C₅₃	C₂²	1	2	yes	—
28	#95×C₅₃	C₂²	1	2	yes	—
29	#95×C₅₃	C₂²	1	2	yes	—
30	#95×C₅₃	C₂²	1	2	yes	—
31	#95×C₅₃	C₂²	1	2	yes	—
32	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
34	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
35	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₅₃	—	2	4	no	—
42	C₄⋊C₄×C₅₃	—	2	4	no	—
43	C₄⋊C₄×C₅₃	—	2	4	no	—
44	C₄⋊C₄×C₅₃	—	2	4	no	—
45	C₄⋊C₄×C₅₃	—	2	4	no	—
46	C₄⋊C₄×C₅₃	—	2	4	no	—
47	C₄⋊C₄×C₅₃	—	2	2	no	—
48	C₄⋊C₄×C₅₃	—	4	1	no	—
49	C₄⋊C₄×C₅₃	—	4	1	no	—
50	C₄⋊C₄×C₅₃	—	4	1	no	—
51	C₄⋊C₄×C₅₃	—	4	2	no	—
52	C₄⋊C₄×C₅₃	D₄	1	4	yes	—
53	C₄⋊C₄×C₅₃	Q₈	1	4	yes	—
54	C₄²×C₅₃	—	2	4	no	—
55	C₄²×C₅₃	—	4	1	no	—
56	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	1	no	—
60	C₂×C₄×C₅₃	—	2	1	no	—
61	C₂×C₄×C₅₃	—	2	1	no	—
62	C₂×C₄×C₅₃	—	2	1	no	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
80	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
82	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
83	C₂×C₄×C₅₃	#41	1	2	yes	—
84	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
85	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
87	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
88	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
89	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
90	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
91	C₄×C₅₃	—	2	2	no	—
92	C₄×C₅₃	—	2	2	no	—
93	C₄×C₅₃	—	2	2	no	—
94	C₄×C₅₃	—	2	4	no	—
95	C₄×C₅₃	—	2	4	no	—
96	C₄×C₅₃	—	4	2	no	—
97	C₄×C₅₃	—	4	2	no	—
98	C₄×C₅₃	—	4	2	no	—
99	C₄×C₅₃	—	4	2	no	—
100	#2726	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂×C₅₃	#537	1	1	yes	characteristic
102	C₂×C₅₃	#539	1	1	yes	characteristic
103	C₂×C₅₃	#544	1	1	yes	characteristic
104	C₂×C₅₃	#547	1	1	yes	characteristic
105	C₂×C₅₃	D₄×Q₈	1	1	yes	characteristic
106	C₂×C₅₃	#557	1	1	yes	characteristic
107	C₂×C₅₃	#564	1	1	yes	characteristic
108	C₂×#125	C₂×C₅₃	1	2	yes	—
109	#398	C₂×C₅₃	1	2	yes	—
110	#398	C₂×C₅₃	1	2	yes	—
111	#398	C₂×C₅₃	1	2	yes	—
112	#389	C₂×C₅₃	1	1	yes	characteristic
113	#389	C₂×C₅₃	1	1	yes	characteristic
114	#389	C₂×C₅₃	1	2	yes	—
115	#389	C₂×C₅₃	1	1	yes	characteristic
116	#387	C₂×C₅₃	1	2	yes	—
117	C₅₃	#2726	1	1	yes	characteristic, direct product factor, Sylow
118	#125	—	2	8	no	—
119	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
120	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
121	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
122	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
123	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
124	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
125	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
126	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
127	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
128	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
129	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
130	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
131	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
132	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
133	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
134	#95	C₂²×C₅₃	1	2	yes	—
135	#95	C₂²×C₅₃	1	2	yes	—
136	#95	C₂²×C₅₃	1	2	yes	—
137	#95	C₂²×C₅₃	1	2	yes	—
138	#95	C₂²×C₅₃	1	2	yes	—
139	#95	C₂²×C₅₃	1	2	yes	—
140	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
141	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
142	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
143	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
144	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
145	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
146	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
147	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
148	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	4	1	no	—
157	C₄⋊C₄	—	4	1	no	—
158	C₄⋊C₄	—	4	1	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	D₄×C₅₃	1	4	yes	—
161	C₄⋊C₄	Q₈×C₅₃	1	4	yes	—
162	C₄²	—	2	4	no	—
163	C₄²	—	4	1	no	—
164	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	C₂×D₄×C₅₃	1	4	yes	—
188	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
189	C₂×C₄	C₂×Q₈×C₅₃	1	2	yes	—
190	C₂×C₄	C₂×Q₈×C₅₃	1	4	yes	—
191	C₂×C₄	#41×C₅₃	1	2	yes	—
192	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
193	C₂²	C₂²×Q₈×C₅₃	1	1	yes	characteristic
194	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
195	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
196	C₂²	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
197	C₂²	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
198	C₂²	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	2	no	—
202	C₄	—	2	4	no	—
203	C₄	—	2	4	no	—
204	C₄	—	4	2	no	—
205	C₄	—	4	2	no	—
206	C₄	—	4	2	no	—
207	C₄	—	4	2	no	—
208	C₂	C₅₃×#537	1	1	yes	characteristic
209	C₂	C₅₃×#539	1	1	yes	characteristic
210	C₂	C₅₃×#544	1	1	yes	characteristic
211	C₂	C₅₃×#547	1	1	yes	characteristic
212	C₂	D₄×Q₈×C₅₃	1	1	yes	characteristic
213	C₂	C₅₃×#557	1	1	yes	characteristic
214	C₂	C₅₃×#564	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
100. #2726	C₅₃	direct (N × H)
117. C₅₃	#2726	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×#125×C₅₃	2
C₅₃×#398	2
C₅₃×#389	2
C₅₃×#387	2
#2726	53

Subgroups — C53×#2726 #20271663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2726 #20271663