Subgroups — C₅₃×#2337 #20271274

Order 6784.

540 conjugacy classes of subgroups.

The subgroup lattice has 238 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
2	C₅₃×#391	C₂	1	2	yes	maximal, complemented
3	C₅₃×#388	C₂	1	2	yes	maximal, complemented
4	C₅₃×#388	C₂	1	2	yes	maximal, complemented
5	C₅₃×#386	C₂	1	2	yes	maximal
6	C₅₃×#385	C₂	1	2	yes	maximal, complemented
7	C₅₃×#383	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₂²⋊C₄×C₅₃	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	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
13	#124×C₅₃	C₄	1	8	yes	complemented
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
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×C₄²×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
29	#95×C₅₃	C₂²	1	2	yes	—
30	#95×C₅₃	C₂²	1	1	yes	characteristic
31	#95×C₅₃	C₂²	1	2	yes	—
32	C₂⁴×C₅₃	C₂³	1	2	yes	—
33	C₂×Q₈×C₅₃	—	4	2	no	—
34	C₂×Q₈×C₅₃	C₂×C₄	1	4	yes	complemented
35	C₂×D₄×C₅₃	—	4	2	no	—
36	C₂×D₄×C₅₃	C₂×C₄	1	4	yes	—
37	C₂²×C₄×C₅₃	—	2	4	no	—
38	C₂²×C₄×C₅₃	—	4	2	no	—
39	C₂²×C₄×C₅₃	—	4	2	no	—
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	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
48	C₄⋊C₄×C₅₃	—	2	4	no	—
49	C₄⋊C₄×C₅₃	—	4	2	no	—
50	C₂²⋊C₄×C₅₃	—	2	4	no	—
51	C₂²⋊C₄×C₅₃	—	4	2	no	—
52	C₂²⋊C₄×C₅₃	—	4	2	no	—
53	C₂²⋊C₄×C₅₃	C₂×C₄	1	8	yes	complemented
54	C₂²⋊C₄×C₅₃	C₂×C₄	1	8	yes	complemented
55	C₄²×C₅₃	—	4	1	no	—
56	C₄²×C₅₃	—	4	2	no	—
57	C₄²×C₅₃	—	4	1	no	—
58	C₄²×C₅₃	C₂×C₄	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	C₂²×C₄	1	4	yes	—
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	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	1	no	—
76	C₂×C₄×C₅₃	—	2	1	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	—	4	8	no	—
87	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
89	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₅₃	#41	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	C₂³×C₄	1	1	yes	characteristic
100	C₂²×C₅₃	C₂×#41	1	2	yes	—
101	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
102	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
103	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	—	2	2	no	—
106	C₄×C₅₃	—	2	4	no	—
107	C₄×C₅₃	—	2	2	no	—
108	C₄×C₅₃	—	2	4	no	—
109	C₄×C₅₃	—	4	2	no	—
110	C₄×C₅₃	—	4	2	no	—
111	C₄×C₅₃	—	4	2	no	—
112	#2337	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂×C₅₃	—	4	4	no	—
114	C₂×C₅₃	C₄×#41	1	1	yes	characteristic
115	C₂×C₅₃	#518	1	1	yes	characteristic
116	C₂×C₅₃	#520	1	1	yes	characteristic
117	C₂×C₅₃	#538	1	2	yes	—
118	C₂×C₅₃	#542	1	2	yes	—
119	C₂×#124	C₂×C₅₃	1	1	yes	characteristic
120	#391	C₂×C₅₃	1	2	yes	complemented
121	#388	C₂×C₅₃	1	2	yes	complemented
122	#388	C₂×C₅₃	1	2	yes	complemented
123	#386	C₂×C₅₃	1	2	yes	—
124	#385	C₂×C₅₃	1	2	yes	complemented
125	#383	C₂×C₅₃	1	1	yes	characteristic, complemented
126	C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
127	C₄×C₂²⋊C₄	C₂×C₅₃	1	2	yes	complemented
128	C₅₃	#2337	1	1	yes	characteristic, direct product factor, Sylow
129	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
130	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
131	C₂³×C₄	C₂²×C₅₃	1	2	yes	—
132	#124	C₄×C₅₃	1	8	yes	complemented
133	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
134	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
135	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
136	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
137	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
138	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
139	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
140	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
141	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
142	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
143	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
144	#95	C₂²×C₅₃	1	2	yes	—
145	#95	C₂²×C₅₃	1	2	yes	—
146	#95	C₂²×C₅₃	1	2	yes	—
147	#95	C₂²×C₅₃	1	1	yes	characteristic
148	#95	C₂²×C₅₃	1	2	yes	—
149	#95	C₂²×C₅₃	1	1	yes	characteristic
150	#95	C₂²×C₅₃	1	2	yes	—
151	C₂⁴	C₂³×C₅₃	1	2	yes	—
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	C₂×C₄×C₅₃	1	4	yes	complemented
154	C₂×D₄	—	4	2	no	—
155	C₂×D₄	C₂×C₄×C₅₃	1	4	yes	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	—	4	2	no	—
159	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
160	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
161	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
162	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
164	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
165	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
166	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₂²⋊C₄	—	2	4	no	—
170	C₂²⋊C₄	—	4	2	no	—
171	C₂²⋊C₄	—	4	2	no	—
172	C₂²⋊C₄	C₂×C₄×C₅₃	1	8	yes	complemented
173	C₂²⋊C₄	C₂×C₄×C₅₃	1	8	yes	complemented
174	C₄²	—	4	1	no	—
175	C₄²	—	4	1	no	—
176	C₄²	—	4	2	no	—
177	C₄²	C₂×C₄×C₅₃	1	4	yes	complemented
178	C₂³	—	2	2	no	—
179	C₂³	—	2	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	C₂²×C₄×C₅₃	1	4	yes	—
185	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
186	Q₈	—	4	4	no	—
187	D₄	—	4	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	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	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	8	no	—
206	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
207	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
208	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
209	C₂×C₄	C₂²×C₄×C₅₃	1	4	yes	—
210	C₂×C₄	#41×C₅₃	1	4	yes	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	4	no	—
213	C₂²	—	2	4	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	C₂³×C₄×C₅₃	1	1	yes	characteristic
219	C₂²	C₂×#41×C₅₃	1	2	yes	—
220	C₂²	H(5,𝔽₂)×C₅₃	1	2	yes	—
221	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
222	C₂²	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	2	4	no	—
226	C₄	—	2	2	no	—
227	C₄	—	2	4	no	—
228	C₄	—	4	2	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	2	no	—
231	C₂	—	4	4	no	—
232	C₂	C₄×#41×C₅₃	1	1	yes	characteristic
233	C₂	C₅₃×#518	1	1	yes	characteristic
234	C₂	C₅₃×#520	1	1	yes	characteristic
235	C₂	C₅₃×#538	1	2	yes	—
236	C₂	C₅₃×#542	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁴×C₅₃
prime radical	C₂²×D₄×C₅₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#391	C₂	semidirect
3. C₅₃×#388	C₂	semidirect
4. C₅₃×#388	C₂	semidirect
6. C₅₃×#385	C₂	semidirect
7. C₅₃×#383	C₂	semidirect
8. C₄×C₄⋊C₄×C₅₃	C₂	semidirect
9. C₄×C₂²⋊C₄×C₅₃	C₂	semidirect
13. #124×C₅₃	C₄	semidirect
34. C₂×Q₈×C₅₃	C₂×C₄	semidirect
53. C₂²⋊C₄×C₅₃	C₂×C₄	semidirect
54. C₂²⋊C₄×C₅₃	C₂×C₄	semidirect
58. C₄²×C₅₃	C₂×C₄	semidirect
112. #2337	C₅₃	direct (N × H)
120. #391	C₂×C₅₃	semidirect
121. #388	C₂×C₅₃	semidirect
122. #388	C₂×C₅₃	semidirect
124. #385	C₂×C₅₃	semidirect
125. #383	C₂×C₅₃	semidirect
126. C₄×C₄⋊C₄	C₂×C₅₃	semidirect
127. C₄×C₂²⋊C₄	C₂×C₅₃	semidirect
128. C₅₃	#2337	direct (N × H)
132. #124	C₄×C₅₃	semidirect
153. C₂×Q₈	C₂×C₄×C₅₃	semidirect
172. C₂²⋊C₄	C₂×C₄×C₅₃	semidirect
173. C₂²⋊C₄	C₂×C₄×C₅₃	semidirect
177. C₄²	C₂×C₄×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
C₅₃×#391	2
C₅₃×#388	2
C₅₃×#386	2
C₅₃×#385	2
C₅₃×#383	2
C₄×C₄⋊C₄×C₅₃	2
C₄×C₂²⋊C₄×C₅₃	2
#2337	53

Subgroups — C53×#2337 #20271274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2337 #20271274