Subgroups — C₅₃×#2808 #20271745

Order 6784.

428 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#397	C₂	1	2	yes	maximal, complemented
2	C₅₃×#394	C₂	1	2	yes	maximal
3	C₅₃×#393	C₂	1	2	yes	maximal, complemented
4	C₅₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#390	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#388	C₂	1	2	yes	maximal, complemented
8	C₅₃×#388	C₂	1	2	yes	maximal, complemented
9	C₅₃×#387	C₂	1	2	yes	maximal, complemented
10	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
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	2	yes	—
21	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
25	#95×C₅₃	C₂²	1	2	yes	—
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	C₂⁴×C₅₃	C₂³	1	2	yes	—
32	C₂×Q₈×C₅₃	—	2	2	no	—
33	C₂×Q₈×C₅₃	—	4	1	no	—
34	C₂×Q₈×C₅₃	—	4	1	no	—
35	C₂×D₄×C₅₃	—	2	2	no	—
36	C₂×D₄×C₅₃	—	4	1	no	—
37	C₂×D₄×C₅₃	—	4	1	no	—
38	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₅₃	—	2	4	no	—
52	C₂²⋊C₄×C₅₃	—	2	4	no	—
53	C₂²⋊C₄×C₅₃	—	4	2	no	—
54	C₂²⋊C₄×C₅₃	—	4	2	no	—
55	C₂²⋊C₄×C₅₃	—	4	2	no	—
56	C₂²⋊C₄×C₅₃	—	4	2	no	—
57	C₄²×C₅₃	—	4	1	no	—
58	C₄²×C₅₃	—	4	2	no	—
59	C₂³×C₅₃	—	2	2	no	—
60	C₂³×C₅₃	—	2	2	no	—
61	C₂³×C₅₃	—	2	2	no	—
62	C₂³×C₅₃	—	2	2	no	—
63	C₂³×C₅₃	—	2	2	no	—
64	C₂³×C₅₃	—	2	2	no	—
65	C₂³×C₅₃	—	2	2	no	—
66	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
67	Q₈×C₅₃	—	4	4	no	—
68	D₄×C₅₃	—	4	4	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	2	1	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	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	#41	1	2	yes	—
91	C₂×C₄×C₅₃	#41	1	2	yes	—
92	C₂×C₄×C₅₃	#41	1	2	yes	—
93	C₂²×C₅₃	—	4	2	no	—
94	C₂²×C₅₃	—	4	2	no	—
95	C₂²×C₅₃	—	4	2	no	—
96	C₂²×C₅₃	—	4	2	no	—
97	C₂²×C₅₃	—	4	2	no	—
98	C₂²×C₅₃	—	4	2	no	—
99	C₂²×C₅₃	—	4	2	no	—
100	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
101	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
102	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
103	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
104	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
105	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
106	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
107	C₄×C₅₃	—	2	2	no	—
108	C₄×C₅₃	—	2	2	no	—
109	C₄×C₅₃	—	2	2	no	—
110	C₄×C₅₃	—	4	2	no	—
111	C₄×C₅₃	—	4	2	no	—
112	C₄×C₅₃	—	4	2	no	—
113	C₄×C₅₃	—	4	2	no	—
114	C₄×C₅₃	—	4	2	no	—
115	#2808	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂×C₅₃	—	8	2	no	—
117	C₂×C₅₃	#538	1	1	yes	characteristic
118	C₂×C₅₃	#538	1	1	yes	characteristic
119	C₂×C₅₃	#539	1	1	yes	characteristic
120	C₂×C₅₃	#555	1	1	yes	characteristic
121	C₂×C₅₃	#556	1	1	yes	characteristic
122	C₂×C₅₃	#559	1	1	yes	characteristic
123	C₂×C₅₃	#562	1	1	yes	characteristic
124	#397	C₂×C₅₃	1	2	yes	complemented
125	#394	C₂×C₅₃	1	2	yes	—
126	#393	C₂×C₅₃	1	2	yes	complemented
127	#391	C₂×C₅₃	1	1	yes	characteristic, complemented
128	#390	C₂×C₅₃	1	1	yes	characteristic
129	#389	C₂×C₅₃	1	1	yes	characteristic, complemented
130	#388	C₂×C₅₃	1	2	yes	complemented
131	#388	C₂×C₅₃	1	2	yes	complemented
132	#387	C₂×C₅₃	1	2	yes	complemented
133	C₅₃	#2808	1	1	yes	characteristic, direct product factor, Sylow
134	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
135	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
136	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
137	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
138	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
139	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
140	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
141	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
142	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
143	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
144	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
145	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
146	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
147	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
148	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
149	#95	C₂²×C₅₃	1	2	yes	—
150	#95	C₂²×C₅₃	1	2	yes	—
151	#95	C₂²×C₅₃	1	2	yes	—
152	#95	C₂²×C₅₃	1	2	yes	—
153	#95	C₂²×C₅₃	1	2	yes	—
154	#95	C₂²×C₅₃	1	2	yes	—
155	C₂⁴	C₂³×C₅₃	1	2	yes	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	—	4	1	no	—
158	C₂×Q₈	—	4	1	no	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	—	4	1	no	—
161	C₂×D₄	—	4	1	no	—
162	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
163	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
164	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
165	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
166	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
167	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	4	1	no	—
173	C₄⋊C₄	—	4	1	no	—
174	C₄⋊C₄	—	4	1	no	—
175	C₂²⋊C₄	—	2	4	no	—
176	C₂²⋊C₄	—	2	4	no	—
177	C₂²⋊C₄	—	4	2	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₂²⋊C₄	—	4	2	no	—
180	C₂²⋊C₄	—	4	2	no	—
181	C₄²	—	4	1	no	—
182	C₄²	—	4	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	2	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	2	no	—
189	C₂³	—	2	2	no	—
190	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
191	Q₈	—	4	4	no	—
192	D₄	—	4	4	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	#41×C₅₃	1	2	yes	—
215	C₂×C₄	#41×C₅₃	1	2	yes	—
216	C₂×C₄	#41×C₅₃	1	2	yes	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
225	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
226	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
227	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
228	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
229	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
230	C₂²	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	4	2	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	2	no	—
238	C₄	—	4	2	no	—
239	C₂	—	8	2	no	—
240	C₂	C₅₃×#538	1	1	yes	characteristic
241	C₂	C₅₃×#538	1	1	yes	characteristic
242	C₂	C₅₃×#539	1	1	yes	characteristic
243	C₂	C₅₃×#555	1	1	yes	characteristic
244	C₂	C₅₃×#556	1	1	yes	characteristic
245	C₂	C₅₃×#559	1	1	yes	characteristic
246	C₂	C₅₃×#562	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅₃
prime radical	C₂²×D₄×C₅₃	C₂²
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×#397	C₂	semidirect
3. C₅₃×#393	C₂	semidirect
4. C₅₃×#391	C₂	semidirect
6. C₅₃×#389	C₂	semidirect
7. C₅₃×#388	C₂	semidirect
8. C₅₃×#388	C₂	semidirect
9. C₅₃×#387	C₂	semidirect
115. #2808	C₅₃	direct (N × H)
124. #397	C₂×C₅₃	semidirect
126. #393	C₂×C₅₃	semidirect
127. #391	C₂×C₅₃	semidirect
129. #389	C₂×C₅₃	semidirect
130. #388	C₂×C₅₃	semidirect
131. #388	C₂×C₅₃	semidirect
132. #387	C₂×C₅₃	semidirect
133. C₅₃	#2808	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×#397	2
C₅₃×#394	2
C₅₃×#393	2
C₅₃×#391	2
C₅₃×#390	2
C₅₃×#389	2
C₅₃×#388	2
C₅₃×#387	2
#2808	53

Subgroups — C53×#2808 #20271745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2808 #20271745