Subgroups — C₅₃×#2604 #20271541

Order 6784.

470 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#126×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#400	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#399	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#395	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#393	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#389	C₂	1	2	yes	maximal, complemented
8	C₅₃×#388	C₂	1	2	yes	maximal
9	C₅₃×#388	C₂	1	2	yes	maximal
10	C₅₃×#387	C₂	1	2	yes	maximal, complemented
11	C₅₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	#126×C₅₃	—	2	4	no	—
14	C₄×Q₈×C₅₃	—	2	4	no	—
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
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	2	yes	—
22	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
30	#95×C₅₃	C₂²	1	2	yes	—
31	#95×C₅₃	C₂²	1	1	yes	characteristic
32	#95×C₅₃	C₂²	1	1	yes	characteristic
33	#95×C₅₃	C₂²	1	2	yes	—
34	#95×C₅₃	C₂²	1	2	yes	—
35	#95×C₅₃	C₂²	1	2	yes	—
36	#95×C₅₃	C₂²	1	2	yes	—
37	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
38	C₂×Q₈×C₅₃	—	2	2	no	—
39	C₂×Q₈×C₅₃	—	2	4	no	—
40	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
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	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
50	C₄⋊C₄×C₅₃	—	2	4	no	—
51	C₄⋊C₄×C₅₃	—	2	2	no	—
52	C₄⋊C₄×C₅₃	—	2	2	no	—
53	C₄⋊C₄×C₅₃	—	2	2	no	—
54	C₄⋊C₄×C₅₃	—	2	2	no	—
55	C₄⋊C₄×C₅₃	—	2	4	no	—
56	C₄⋊C₄×C₅₃	—	2	4	no	—
57	C₂²⋊C₄×C₅₃	—	2	2	no	—
58	C₂²⋊C₄×C₅₃	—	2	2	no	—
59	C₂²⋊C₄×C₅₃	—	2	2	no	—
60	C₂²⋊C₄×C₅₃	—	2	4	no	—
61	C₂²⋊C₄×C₅₃	—	2	4	no	—
62	C₄²×C₅₃	—	2	4	no	—
63	C₄²×C₅₃	—	2	4	no	—
64	C₄²×C₅₃	D₄	1	4	yes	—
65	C₂³×C₅₃	—	2	1	no	—
66	C₂³×C₅₃	—	2	2	no	—
67	C₂³×C₅₃	—	2	2	no	—
68	C₂³×C₅₃	—	2	1	no	—
69	C₂³×C₅₃	—	2	1	no	—
70	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
71	Q₈×C₅₃	—	2	8	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	1	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	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	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₅₃	#41	1	4	yes	—
97	C₂×C₄×C₅₃	#41	1	4	yes	—
98	C₂²×C₅₃	—	4	1	no	—
99	C₂²×C₅₃	—	4	1	no	—
100	C₂²×C₅₃	—	4	1	no	—
101	C₂²×C₅₃	—	4	2	no	—
102	C₂²×C₅₃	—	4	2	no	—
103	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
104	C₂²×C₅₃	C₂×#41	1	2	yes	—
105	C₂²×C₅₃	C₂×#41	1	2	yes	—
106	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
107	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
108	C₄×C₅₃	—	2	2	no	—
109	C₄×C₅₃	—	2	4	no	—
110	C₄×C₅₃	—	2	2	no	—
111	C₄×C₅₃	—	2	2	no	—
112	C₄×C₅₃	—	2	4	no	—
113	C₄×C₅₃	—	4	1	no	—
114	C₄×C₅₃	—	4	2	no	—
115	C₄×C₅₃	—	4	1	no	—
116	C₄×C₅₃	—	4	1	no	—
117	C₄×C₅₃	—	4	2	no	—
118	#2604	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂×C₅₃	—	8	1	no	—
120	C₂×C₅₃	#525	1	1	yes	characteristic
121	C₂×C₅₃	#532	1	1	yes	characteristic
122	C₂×C₅₃	#536	1	1	yes	characteristic
123	C₂×C₅₃	#552	1	2	yes	—
124	C₂×C₅₃	#556	1	2	yes	—
125	C₂×#126	C₂×C₅₃	1	1	yes	characteristic
126	C₂×C₄×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
127	#400	C₂×C₅₃	1	1	yes	characteristic, complemented
128	#399	C₂×C₅₃	1	1	yes	characteristic
129	#395	C₂×C₅₃	1	1	yes	characteristic, complemented
130	#393	C₂×C₅₃	1	1	yes	characteristic
131	#389	C₂×C₅₃	1	2	yes	complemented
132	#388	C₂×C₅₃	1	2	yes	—
133	#388	C₂×C₅₃	1	2	yes	—
134	#387	C₂×C₅₃	1	2	yes	complemented
135	#376	C₂×C₅₃	1	1	yes	characteristic, complemented
136	C₅₃	#2604	1	1	yes	characteristic, direct product factor, Sylow
137	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
138	#126	—	2	4	no	—
139	C₄×Q₈	—	2	4	no	—
140	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
141	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
142	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
143	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
144	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
145	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
146	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
147	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
148	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
149	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
150	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
151	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
152	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
153	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
154	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
155	#95	C₂²×C₅₃	1	2	yes	—
156	#95	C₂²×C₅₃	1	1	yes	characteristic
157	#95	C₂²×C₅₃	1	1	yes	characteristic
158	#95	C₂²×C₅₃	1	2	yes	—
159	#95	C₂²×C₅₃	1	2	yes	—
160	#95	C₂²×C₅₃	1	2	yes	—
161	#95	C₂²×C₅₃	1	2	yes	—
162	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	2	4	no	—
165	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
166	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
167	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
168	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
172	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
173	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
174	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
175	C₄⋊C₄	—	2	4	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	4	no	—
181	C₄⋊C₄	—	2	4	no	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	4	no	—
187	C₄²	—	2	4	no	—
188	C₄²	—	2	4	no	—
189	C₄²	D₄×C₅₃	1	4	yes	—
190	C₂³	—	2	1	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	1	no	—
194	C₂³	—	2	1	no	—
195	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
196	Q₈	—	2	8	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	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	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
219	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
220	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
221	C₂×C₄	#41×C₅₃	1	4	yes	—
222	C₂×C₄	#41×C₅₃	1	4	yes	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
229	C₂²	C₂×#41×C₅₃	1	2	yes	—
230	C₂²	C₂×#41×C₅₃	1	2	yes	—
231	C₂²	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
232	C₂²	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
233	C₄	—	2	2	no	—
234	C₄	—	2	4	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	2	no	—
237	C₄	—	2	4	no	—
238	C₄	—	4	1	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	1	no	—
241	C₄	—	4	1	no	—
242	C₄	—	4	2	no	—
243	C₂	—	8	1	no	—
244	C₂	C₅₃×#525	1	1	yes	characteristic
245	C₂	C₅₃×#532	1	1	yes	characteristic
246	C₂	C₅₃×#536	1	1	yes	characteristic
247	C₂	C₅₃×#552	1	2	yes	—
248	C₂	C₅₃×#556	1	2	yes	—

	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
2. C₂×C₄×Q₈×C₅₃	C₂	semidirect
3. C₅₃×#400	C₂	semidirect
5. C₅₃×#395	C₂	semidirect
7. C₅₃×#389	C₂	semidirect
10. C₅₃×#387	C₂	semidirect
11. C₅₃×#376	C₂	semidirect
118. #2604	C₅₃	direct (N × H)
126. C₂×C₄×Q₈	C₂×C₅₃	semidirect
127. #400	C₂×C₅₃	semidirect
129. #395	C₂×C₅₃	semidirect
131. #389	C₂×C₅₃	semidirect
134. #387	C₂×C₅₃	semidirect
135. #376	C₂×C₅₃	semidirect
136. C₅₃	#2604	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×#126×C₅₃	2
C₂×C₄×Q₈×C₅₃	2
C₅₃×#400	2
C₅₃×#399	2
C₅₃×#395	2
C₅₃×#393	2
C₅₃×#389	2
C₅₃×#388	2
C₅₃×#387	2
C₅₃×#376	2
#2604	53

Subgroups — C53×#2604 #20271541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2604 #20271541