Subgroups — C₅₃×#2742 #20271679

Order 6784.

416 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#400	C₂	1	2	yes	maximal, complemented
2	C₅₃×#399	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#395	C₂	1	2	yes	maximal, complemented
4	C₅₃×#393	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#389	C₂	1	2	yes	maximal, complemented
6	C₅₃×#388	C₂	1	2	yes	maximal
7	C₅₃×#387	C₂	1	2	yes	maximal, complemented
8	C₅₃×#385	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#380	C₂	1	2	yes	maximal
10	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
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×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
23	#95×C₅₃	C₂²	1	2	yes	—
24	#95×C₅₃	C₂²	1	2	yes	—
25	#95×C₅₃	C₂²	1	2	yes	—
26	#95×C₅₃	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
31	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×Q₈×C₅₃	—	4	2	no	—
33	C₂×Q₈×C₅₃	—	4	1	no	—
34	C₂²×C₄×C₅₃	—	2	4	no	—
35	C₂²×C₄×C₅₃	—	4	1	no	—
36	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
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	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
44	C₄⋊C₄×C₅₃	—	2	4	no	—
45	C₄⋊C₄×C₅₃	—	2	4	no	—
46	C₄⋊C₄×C₅₃	—	4	2	no	—
47	C₄⋊C₄×C₅₃	—	4	2	no	—
48	C₄⋊C₄×C₅₃	—	4	2	no	—
49	C₄⋊C₄×C₅₃	—	4	1	no	—
50	C₂²⋊C₄×C₅₃	—	4	2	no	—
51	C₂²⋊C₄×C₅₃	—	4	1	no	—
52	C₂²⋊C₄×C₅₃	—	4	2	no	—
53	C₂²⋊C₄×C₅₃	Q₈	1	4	yes	complemented
54	C₄²×C₅₃	—	4	2	no	—
55	C₂³×C₅₃	—	2	2	no	—
56	C₂³×C₅₃	—	2	1	no	—
57	C₂³×C₅₃	—	2	2	no	—
58	C₂³×C₅₃	—	2	1	no	—
59	C₂³×C₅₃	C₂×Q₈	1	2	yes	—
60	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
61	Q₈×C₅₃	—	4	4	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	2	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	1	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	1	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	4	4	no	—
83	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
84	C₂×C₄×C₅₃	#41	1	4	yes	—
85	C₂²×C₅₃	—	2	2	no	—
86	C₂²×C₅₃	—	2	4	no	—
87	C₂²×C₅₃	—	4	2	no	—
88	C₂²×C₅₃	—	4	1	no	—
89	C₂²×C₅₃	—	4	1	no	—
90	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
91	C₂²×C₅₃	C₂×#41	1	2	yes	—
92	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
93	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
94	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
95	C₄×C₅₃	—	2	4	no	—
96	C₄×C₅₃	—	2	4	no	—
97	C₄×C₅₃	—	4	2	no	—
98	C₄×C₅₃	—	4	2	no	—
99	C₄×C₅₃	—	4	1	no	—
100	C₄×C₅₃	—	4	2	no	—
101	C₄×C₅₃	—	4	2	no	—
102	C₄×C₅₃	—	4	1	no	—
103	#2742	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂×C₅₃	—	4	2	no	—
105	C₂×C₅₃	#542	1	2	yes	—
106	C₂×C₅₃	#544	1	1	yes	characteristic
107	C₂×C₅₃	#551	1	1	yes	characteristic
108	C₂×C₅₃	#554	1	2	yes	—
109	C₂×C₅₃	#560	1	1	yes	characteristic
110	#400	C₂×C₅₃	1	2	yes	complemented
111	#399	C₂×C₅₃	1	1	yes	characteristic
112	#395	C₂×C₅₃	1	2	yes	complemented
113	#393	C₂×C₅₃	1	1	yes	characteristic
114	#389	C₂×C₅₃	1	2	yes	complemented
115	#388	C₂×C₅₃	1	2	yes	—
116	#387	C₂×C₅₃	1	2	yes	complemented
117	#385	C₂×C₅₃	1	1	yes	characteristic
118	#380	C₂×C₅₃	1	2	yes	—
119	C₅₃	#2742	1	1	yes	characteristic, direct product factor, Sylow
120	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
121	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
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	2	yes	—
126	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
127	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
128	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
129	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
130	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
131	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
132	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
133	#95	C₂²×C₅₃	1	2	yes	—
134	#95	C₂²×C₅₃	1	2	yes	—
135	#95	C₂²×C₅₃	1	2	yes	—
136	#95	C₂²×C₅₃	1	1	yes	characteristic
137	#95	C₂²×C₅₃	1	2	yes	—
138	#95	C₂²×C₅₃	1	2	yes	—
139	#95	C₂²×C₅₃	1	2	yes	—
140	#95	C₂²×C₅₃	1	1	yes	characteristic
141	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
142	C₂×Q₈	—	4	2	no	—
143	C₂×Q₈	—	4	1	no	—
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	—	4	1	no	—
146	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
147	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
148	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
150	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
151	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
152	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
153	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	1	no	—
160	C₂²⋊C₄	—	4	2	no	—
161	C₂²⋊C₄	—	4	2	no	—
162	C₂²⋊C₄	—	4	1	no	—
163	C₂²⋊C₄	Q₈×C₅₃	1	4	yes	complemented
164	C₄²	—	4	2	no	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	1	no	—
168	C₂³	—	2	1	no	—
169	C₂³	C₂×Q₈×C₅₃	1	2	yes	—
170	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
171	Q₈	—	4	4	no	—
172	C₂×C₄	—	2	2	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	1	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	C₂×Q₈×C₅₃	1	4	yes	—
194	C₂×C₄	#41×C₅₃	1	4	yes	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	4	no	—
197	C₂²	—	4	2	no	—
198	C₂²	—	4	1	no	—
199	C₂²	—	4	1	no	—
200	C₂²	C₂²×Q₈×C₅₃	1	1	yes	characteristic
201	C₂²	C₂×#41×C₅₃	1	2	yes	—
202	C₂²	H(5,𝔽₂)×C₅₃	1	2	yes	—
203	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
204	C₂²	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
205	C₄	—	2	4	no	—
206	C₄	—	2	4	no	—
207	C₄	—	4	2	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	1	no	—
210	C₄	—	4	2	no	—
211	C₄	—	4	2	no	—
212	C₄	—	4	1	no	—
213	C₂	—	4	2	no	—
214	C₂	C₅₃×#542	1	2	yes	—
215	C₂	C₅₃×#544	1	1	yes	characteristic
216	C₂	C₅₃×#551	1	1	yes	characteristic
217	C₂	C₅₃×#554	1	2	yes	—
218	C₂	C₅₃×#560	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅₃
prime radical	C₂⁴×C₅₃	C₂³
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
1. C₅₃×#400	C₂	semidirect
3. C₅₃×#395	C₂	semidirect
5. C₅₃×#389	C₂	semidirect
7. C₅₃×#387	C₂	semidirect
53. C₂²⋊C₄×C₅₃	Q₈	semidirect
103. #2742	C₅₃	direct (N × H)
110. #400	C₂×C₅₃	semidirect
112. #395	C₂×C₅₃	semidirect
114. #389	C₂×C₅₃	semidirect
116. #387	C₂×C₅₃	semidirect
119. C₅₃	#2742	direct (N × H)
163. C₂²⋊C₄	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#400	2
C₅₃×#399	2
C₅₃×#395	2
C₅₃×#393	2
C₅₃×#389	2
C₅₃×#388	2
C₅₃×#387	2
C₅₃×#385	2
C₅₃×#380	2
#2742	53

Subgroups — C53×#2742 #20271679

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2742 #20271679