Subgroups — C₅₃×#2540 #20271477

Order 6784.

760 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#124×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊Q₈×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₂×#121×C₅₃	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#399	C₂	1	2	yes	maximal, complemented
6	C₅₃×#392	C₂	1	2	yes	maximal, complemented
7	C₅₃×#390	C₂	1	4	yes	maximal, complemented
8	C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₅₃	C₂²	1	4	yes	—
13	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
14	#124×C₅₃	—	2	4	no	—
15	C₂²⋊Q₈×C₅₃	—	2	8	no	—
16	#121×C₅₃	—	2	8	no	—
17	C₄×D₄×C₅₃	—	2	4	no	—
18	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
23	C₂×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
24	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
26	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
28	#95×C₅₃	C₂²	1	2	yes	—
29	C₂⁴×C₅₃	C₂³	1	2	yes	—
30	C₂⁴×C₅₃	C₂³	1	2	yes	—
31	C₂×Q₈×C₅₃	—	2	4	no	—
32	C₂×Q₈×C₅₃	—	2	2	no	—
33	C₂×D₄×C₅₃	—	2	4	no	—
34	C₂×D₄×C₅₃	—	2	8	no	—
35	C₂×D₄×C₅₃	—	2	2	no	—
36	C₂×D₄×C₅₃	—	2	8	no	—
37	C₂×D₄×C₅₃	—	2	4	no	—
38	C₂×D₄×C₅₃	—	2	2	no	—
39	C₂×D₄×C₅₃	—	4	4	no	—
40	C₂²×C₄×C₅₃	—	2	4	no	—
41	C₂²×C₄×C₅₃	—	2	4	no	—
42	C₂²×C₄×C₅₃	—	2	4	no	—
43	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
47	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₅₃	—	2	2	no	—
50	C₄⋊C₄×C₅₃	—	2	8	no	—
51	C₄⋊C₄×C₅₃	D₄	1	8	yes	complemented
52	C₂²⋊C₄×C₅₃	—	2	4	no	—
53	C₂²⋊C₄×C₅₃	—	2	8	no	—
54	C₂²⋊C₄×C₅₃	—	2	8	no	—
55	C₂²⋊C₄×C₅₃	—	2	4	no	—
56	C₂²⋊C₄×C₅₃	—	2	4	no	—
57	C₄²×C₅₃	—	4	2	no	—
58	C₄²×C₅₃	D₄	1	4	yes	complemented
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	4	no	—
66	C₂³×C₅₃	—	2	2	no	—
67	C₂³×C₅₃	—	2	4	no	—
68	C₂³×C₅₃	—	2	2	no	—
69	C₂³×C₅₃	—	2	2	no	—
70	C₂³×C₅₃	#41	1	4	yes	—
71	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
72	Q₈×C₅₃	—	4	4	no	—
73	D₄×C₅₃	—	2	8	no	—
74	D₄×C₅₃	—	4	16	no	—
75	D₄×C₅₃	—	4	4	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	4	no	—
78	C₂×C₄×C₅₃	—	2	4	no	—
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	1	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	4	no	—
86	C₂×C₄×C₅₃	—	4	8	no	—
87	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₅₃	C₂×D₄	1	8	yes	—
89	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
92	C₂²×C₅₃	—	2	4	no	—
93	C₂²×C₅₃	—	2	4	no	—
94	C₂²×C₅₃	—	2	4	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	—	4	2	no	—
101	C₂²×C₅₃	—	4	2	no	—
102	C₂²×C₅₃	—	4	4	no	—
103	C₂²×C₅₃	—	4	4	no	—
104	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
105	C₂²×C₅₃	C₂²×D₄	1	2	yes	—
106	C₂²×C₅₃	C₂×#41	1	2	yes	—
107	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
108	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
109	C₄×C₅₃	—	2	2	no	—
110	C₄×C₅₃	—	2	2	no	—
111	C₄×C₅₃	—	2	8	no	—
112	C₄×C₅₃	—	4	2	no	—
113	C₄×C₅₃	—	4	2	no	—
114	C₄×C₅₃	#127	1	4	yes	—
115	#2540	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂×C₅₃	—	4	4	no	—
117	C₂×C₅₃	—	8	2	no	—
118	C₂×C₅₃	#525	1	1	yes	characteristic
119	C₂×C₅₃	C₂×#127	1	1	yes	characteristic
120	C₂×C₅₃	#537	1	1	yes	characteristic
121	C₂×C₅₃	#546	1	2	yes	—
122	C₂×C₅₃	#548	1	2	yes	—
123	C₂×#124	C₂×C₅₃	1	1	yes	characteristic, complemented
124	C₂×C₂²⋊Q₈	C₂×C₅₃	1	2	yes	complemented
125	C₂×#121	C₂×C₅₃	1	2	yes	complemented
126	C₂×C₄×D₄	C₂×C₅₃	1	1	yes	characteristic, complemented
127	#399	C₂×C₅₃	1	2	yes	complemented
128	#392	C₂×C₅₃	1	2	yes	complemented
129	#390	C₂×C₅₃	1	4	yes	complemented
130	C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
131	C₅₃	#2540	1	1	yes	characteristic, direct product factor, Sylow
132	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
133	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
134	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
135	C₂²×D₄	C₂²×C₅₃	1	4	yes	—
136	C₂³×C₄	C₂²×C₅₃	1	2	yes	—
137	#124	—	2	4	no	—
138	C₂²⋊Q₈	—	2	8	no	—
139	#121	—	2	8	no	—
140	C₄×D₄	—	2	4	no	—
141	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
142	C₂×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
143	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
144	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
145	C₂×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
146	C₂×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
147	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
148	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
149	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
150	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
151	#95	C₂²×C₅₃	1	2	yes	—
152	C₂⁴	C₂³×C₅₃	1	2	yes	—
153	C₂⁴	C₂³×C₅₃	1	2	yes	—
154	C₂×Q₈	—	2	4	no	—
155	C₂×Q₈	—	2	2	no	—
156	C₂×D₄	—	2	4	no	—
157	C₂×D₄	—	2	8	no	—
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	—	2	8	no	—
160	C₂×D₄	—	2	4	no	—
161	C₂×D₄	—	2	2	no	—
162	C₂×D₄	—	4	4	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
167	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
168	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
170	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	8	no	—
174	C₄⋊C₄	D₄×C₅₃	1	8	yes	complemented
175	C₂²⋊C₄	—	2	4	no	—
176	C₂²⋊C₄	—	2	8	no	—
177	C₂²⋊C₄	—	2	8	no	—
178	C₂²⋊C₄	—	2	4	no	—
179	C₂²⋊C₄	—	2	4	no	—
180	C₄²	—	4	2	no	—
181	C₄²	D₄×C₅₃	1	4	yes	complemented
182	C₂³	—	2	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	4	no	—
190	C₂³	—	2	4	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	2	no	—
193	C₂³	#41×C₅₃	1	4	yes	—
194	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
195	Q₈	—	4	4	no	—
196	D₄	—	2	8	no	—
197	D₄	—	4	16	no	—
198	D₄	—	4	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	8	no	—
210	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
211	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
212	C₂×C₄	C₂×D₄×C₅₃	1	8	yes	—
213	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
214	C₂×C₄	C₂×D₄×C₅₃	1	4	yes	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	4	no	—
217	C₂²	—	2	4	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	—	4	2	no	—
225	C₂²	—	4	4	no	—
226	C₂²	—	4	4	no	—
227	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
228	C₂²	C₂²×D₄×C₅₃	1	2	yes	—
229	C₂²	C₂×#41×C₅₃	1	2	yes	—
230	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
231	C₂²	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	8	no	—
235	C₄	—	4	2	no	—
236	C₄	—	4	2	no	—
237	C₄	#127×C₅₃	1	4	yes	—
238	C₂	—	4	4	no	—
239	C₂	—	8	2	no	—
240	C₂	C₅₃×#525	1	1	yes	characteristic
241	C₂	C₂×#127×C₅₃	1	1	yes	characteristic
242	C₂	C₅₃×#537	1	1	yes	characteristic
243	C₂	C₅₃×#546	1	2	yes	—
244	C₂	C₅₃×#548	1	2	yes	—

	subgroup	G/H
center, socle	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
1. C₂×#124×C₅₃	C₂	semidirect
2. C₂×C₂²⋊Q₈×C₅₃	C₂	semidirect
3. C₂×#121×C₅₃	C₂	semidirect
4. C₂×C₄×D₄×C₅₃	C₂	semidirect
5. C₅₃×#399	C₂	semidirect
6. C₅₃×#392	C₂	semidirect
7. C₅₃×#390	C₂	semidirect
8. C₄×C₄⋊C₄×C₅₃	C₂	semidirect
51. C₄⋊C₄×C₅₃	D₄	semidirect
58. C₄²×C₅₃	D₄	semidirect
115. #2540	C₅₃	direct (N × H)
123. C₂×#124	C₂×C₅₃	semidirect
124. C₂×C₂²⋊Q₈	C₂×C₅₃	semidirect
125. C₂×#121	C₂×C₅₃	semidirect
126. C₂×C₄×D₄	C₂×C₅₃	semidirect
127. #399	C₂×C₅₃	semidirect
128. #392	C₂×C₅₃	semidirect
129. #390	C₂×C₅₃	semidirect
130. C₄×C₄⋊C₄	C₂×C₅₃	semidirect
131. C₅₃	#2540	direct (N × H)
174. C₄⋊C₄	D₄×C₅₃	semidirect
181. C₄²	D₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#124×C₅₃	2
C₂×C₂²⋊Q₈×C₅₃	2
C₂×#121×C₅₃	2
C₂×C₄×D₄×C₅₃	2
C₅₃×#399	2
C₅₃×#392	2
C₅₃×#390	2
C₄×C₄⋊C₄×C₅₃	2
#2540	53

Subgroups — C53×#2540 #20271477

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2540 #20271477