Subgroups — C₅₃×#21995 #27229507

Order 13568.

658 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2921	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#433	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2905	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1950	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1855	C₂	1	4	yes	maximal
6	C₅₃×#1849	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1838	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1837	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
11	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#433	—	2	4	no	—
13	C₅₃×#433	C₄	1	8	yes	—
14	C₂×#108×C₅₃	C₂²	1	4	yes	—
15	C₂×#108×C₅₃	C₂²	1	4	yes	—
16	C₅₃×#424	—	2	4	no	—
17	C₅₃×#423	C₂²	1	2	yes	—
18	C₅₃×#423	C₂²	1	1	yes	characteristic
19	C₅₃×#423	C₂²	1	1	yes	characteristic
20	C₅₃×#423	C₂²	1	1	yes	characteristic
21	C₅₃×#423	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
23	C₅₃×#407	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
25	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
26	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#341	C₂²	1	4	yes	—
29	C₅₃×#341	C₂²	1	4	yes	—
30	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₅₃	—	2	2	no	—
32	C₂×M₁₆×C₅₃	—	4	4	no	—
33	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
34	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
35	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
36	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₅₃	—	2	2	no	—
38	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
40	#117×C₅₃	C₂×C₄	1	4	yes	—
41	#117×C₅₃	D₄	1	4	yes	—
42	C₂×C₄⋊C₄×C₅₃	Q₈	1	2	yes	—
43	C₂×C₂²⋊C₄×C₅₃	Q₈	1	2	yes	—
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	#108×C₅₃	—	4	4	no	—
47	#108×C₅₃	—	4	4	no	—
48	C₄⋊C₈×C₅₃	—	2	4	no	—
49	C₄⋊C₈×C₅₃	—	2	2	no	—
50	C₄⋊C₈×C₅₃	—	2	2	no	—
51	C₄⋊C₈×C₅₃	—	2	2	no	—
52	C₄⋊C₈×C₅₃	—	2	2	no	—
53	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
54	C₂²⋊C₈×C₅₃	—	2	4	no	—
55	C₂²⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
56	#97×C₅₃	—	2	2	no	—
57	#97×C₅₃	C₂×C₄	1	4	yes	—
58	C₄×C₈×C₅₃	—	2	2	no	—
59	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
60	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	—	2	2	no	—
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
69	M₁₆×C₅₃	—	2	4	no	—
70	M₁₆×C₅₃	—	2	4	no	—
71	M₁₆×C₅₃	—	2	4	no	—
72	M₁₆×C₅₃	—	2	4	no	—
73	M₁₆×C₅₃	—	4	4	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	—	2	2	no	—
79	C₂×C₈×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	—	2	2	no	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	4	4	no	—
83	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
85	C₄⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
86	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
87	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
88	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
89	C₄²×C₅₃	C₂×D₄	1	2	yes	—
90	C₄²×C₅₃	C₂×D₄	1	2	yes	—
91	C₂³×C₅₃	—	2	2	no	—
92	C₂³×C₅₃	—	4	2	no	—
93	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
94	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	4	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	4	2	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
103	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
107	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
108	C₈×C₅₃	—	2	4	no	—
109	C₈×C₅₃	—	2	4	no	—
110	C₈×C₅₃	—	4	2	no	—
111	C₈×C₅₃	—	4	2	no	—
112	C₈×C₅₃	—	4	2	no	—
113	C₈×C₅₃	—	4	2	no	—
114	#21995	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂²×C₅₃	—	2	2	no	—
116	C₂²×C₅₃	—	2	2	no	—
117	C₂²×C₅₃	—	2	4	no	—
118	C₂²×C₅₃	—	8	2	no	—
119	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₅₃	#544	1	2	yes	—
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	—	2	4	no	—
123	C₄×C₅₃	—	2	4	no	—
124	C₄×C₅₃	#520	1	2	yes	—
125	C₄×C₅₃	#535	1	1	yes	characteristic
126	C₄×C₅₃	#536	1	1	yes	characteristic
127	#2921	C₂×C₅₃	1	1	yes	characteristic
128	C₂×#433	C₂×C₅₃	1	1	yes	characteristic
129	#2905	C₂×C₅₃	1	1	yes	characteristic
130	#1950	C₂×C₅₃	1	4	yes	complemented
131	#1855	C₂×C₅₃	1	4	yes	—
132	#1849	C₂×C₅₃	1	1	yes	characteristic, complemented
133	#1848	C₂×C₅₃	1	1	yes	characteristic, complemented
134	#1838	C₂×C₅₃	1	1	yes	characteristic, complemented
135	#1837	C₂×C₅₃	1	1	yes	characteristic, complemented
136	C₂×C₅₃	—	2	2	no	—
137	C₂×C₅₃	—	4	2	no	—
138	C₂×C₅₃	#2317	1	1	yes	characteristic
139	C₂×C₅₃	#2916	1	2	yes	—
140	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
141	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
142	#433	—	2	4	no	—
143	#433	C₄×C₅₃	1	8	yes	—
144	C₂×#108	C₂²×C₅₃	1	4	yes	—
145	C₂×#108	C₂²×C₅₃	1	4	yes	—
146	#424	—	2	4	no	—
147	#423	C₂²×C₅₃	1	1	yes	characteristic
148	#423	C₂²×C₅₃	1	1	yes	characteristic
149	#423	C₂²×C₅₃	1	1	yes	characteristic
150	#423	C₂²×C₅₃	1	1	yes	characteristic
151	#423	C₂²×C₅₃	1	2	yes	—
152	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
153	#407	C₂²×C₅₃	1	2	yes	—
154	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
155	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
156	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
157	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
158	#341	C₂²×C₅₃	1	4	yes	—
159	#341	C₂²×C₅₃	1	4	yes	—
160	C₅₃	#21995	1	1	yes	characteristic, direct product factor, Sylow
161	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	4	4	no	—
164	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
165	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
166	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
167	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
170	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
171	#117	C₂×C₄×C₅₃	1	4	yes	—
172	#117	D₄×C₅₃	1	4	yes	—
173	C₂×C₄⋊C₄	Q₈×C₅₃	1	2	yes	—
174	C₂×C₂²⋊C₄	Q₈×C₅₃	1	2	yes	—
175	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
176	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
177	#108	—	4	4	no	—
178	#108	—	4	4	no	—
179	C₄⋊C₈	—	2	4	no	—
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	—	2	2	no	—
184	C₄⋊C₈	C₂×C₄×C₅₃	1	8	yes	—
185	C₂²⋊C₈	—	2	4	no	—
186	C₂²⋊C₈	C₂×C₄×C₅₃	1	8	yes	—
187	#97	—	2	2	no	—
188	#97	C₂×C₄×C₅₃	1	4	yes	—
189	C₄×C₈	—	2	2	no	—
190	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
191	C₂⁴	C₂×Q₈×C₅₃	1	1	yes	characteristic
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
195	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
196	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
197	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
198	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
199	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
200	M₁₆	—	2	4	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	—	2	4	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	4	4	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	4	4	no	—
214	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
215	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
216	C₄⋊C₄	C₄⋊C₄×C₅₃	1	8	yes	—
217	C₂²⋊C₄	C₄⋊C₄×C₅₃	1	8	yes	—
218	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
219	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
220	C₄²	C₂×D₄×C₅₃	1	2	yes	—
221	C₄²	C₂×D₄×C₅₃	1	2	yes	—
222	C₂³	—	2	2	no	—
223	C₂³	—	4	2	no	—
224	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
225	C₂³	C₂²×Q₈×C₅₃	1	1	yes	characteristic
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
233	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
234	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
235	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
237	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
238	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
239	C₈	—	2	4	no	—
240	C₈	—	2	4	no	—
241	C₈	—	4	2	no	—
242	C₈	—	4	2	no	—
243	C₈	—	4	2	no	—
244	C₈	—	4	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	4	no	—
248	C₂²	—	8	2	no	—
249	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
250	C₂²	C₅₃×#544	1	2	yes	—
251	C₄	—	2	2	no	—
252	C₄	—	2	4	no	—
253	C₄	—	2	4	no	—
254	C₄	C₅₃×#520	1	2	yes	—
255	C₄	C₅₃×#535	1	1	yes	characteristic
256	C₄	C₅₃×#536	1	1	yes	characteristic
257	C₂	—	2	2	no	—
258	C₂	—	4	2	no	—
259	C₂	C₅₃×#2317	1	1	yes	characteristic
260	C₂	C₅₃×#2916	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×#117×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1950	C₂	semidirect
6. C₅₃×#1849	C₂	semidirect
7. C₅₃×#1848	C₂	semidirect
8. C₅₃×#1838	C₂	semidirect
9. C₅₃×#1837	C₂	semidirect
114. #21995	C₅₃	direct (N × H)
130. #1950	C₂×C₅₃	semidirect
132. #1849	C₂×C₅₃	semidirect
133. #1848	C₂×C₅₃	semidirect
134. #1838	C₂×C₅₃	semidirect
135. #1837	C₂×C₅₃	semidirect
160. C₅₃	#21995	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×#2921	2
C₂×C₅₃×#433	2
C₅₃×#2905	2
C₅₃×#1950	2
C₅₃×#1855	2
C₅₃×#1849	2
C₅₃×#1848	2
C₅₃×#1838	2
C₅₃×#1837	2
#21995	53

Subgroups — C53×#21995 #27229507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21995 #27229507