Subgroups — C₅₃×#10837 #27218349

Order 13568.

842 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2285	C₂	1	2	yes	maximal
3	C₂×C₅₃×H(3,ℤ/4)	C₂	1	4	yes	maximal
4	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
5	C₂×#117×C₅₃	C₂²	1	4	yes	—
6	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
7	C₄×M₁₆×C₅₃	—	2	2	no	—
8	C₄×M₁₆×C₅₃	—	2	2	no	—
9	C₂×#97×C₅₃	C₄	1	2	yes	complemented
10	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	complemented
11	C₅₃×#384	—	2	2	no	—
12	C₅₃×#382	—	2	2	no	—
13	C₅₃×#380	—	2	4	no	—
14	C₄×C₄⋊C₄×C₅₃	C₄	1	8	yes	complemented
15	C₅₃×H(3,ℤ/4)	—	2	8	no	—
16	C₂³×C₄×C₅₃	D₄	1	2	yes	—
17	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₅₃	—	2	4	no	—
19	C₂×M₁₆×C₅₃	—	2	4	no	—
20	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
21	C₂×M₁₆×C₅₃	Q₈	1	4	yes	—
22	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
23	#117×C₅₃	—	2	8	no	—
24	#117×C₅₃	—	2	8	no	—
25	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
27	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	D₄	1	2	yes	—
33	C₂×C₄²×C₅₃	D₄	1	2	yes	—
34	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
35	#97×C₅₃	—	2	4	no	—
36	C₄×C₈×C₅₃	—	2	2	no	—
37	C₄×C₈×C₅₃	—	2	2	no	—
38	#95×C₅₃	—	2	8	no	—
39	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	—	2	8	no	—
41	C₂²×C₄×C₅₃	—	4	4	no	—
42	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
43	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
44	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
52	C₂²×C₄×C₅₃	#41	1	4	yes	—
53	M₁₆×C₅₃	—	2	8	no	—
54	M₁₆×C₅₃	—	2	8	no	—
55	C₂×C₈×C₅₃	—	2	4	no	—
56	C₂×C₈×C₅₃	—	2	4	no	—
57	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	complemented
58	C₄⋊C₄×C₅₃	—	4	16	no	—
59	C₄⋊C₄×C₅₃	—	4	8	no	—
60	C₂²⋊C₄×C₅₃	—	4	4	no	—
61	C₂²⋊C₄×C₅₃	—	4	4	no	—
62	C₄²×C₅₃	—	2	16	no	—
63	C₄²×C₅₃	C₄²	1	2	yes	—
64	C₄²×C₅₃	C₄²	1	2	yes	—
65	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
69	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
70	C₂³×C₅₃	—	4	2	no	—
71	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
72	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂³×C₅₃	#124	1	2	yes	—
74	C₂³×C₅₃	#127	1	1	yes	characteristic
75	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	—	2	8	no	—
77	C₂×C₄×C₅₃	—	2	4	no	—
78	C₂×C₄×C₅₃	—	4	4	no	—
79	C₂×C₄×C₅₃	—	4	2	no	—
80	C₂×C₄×C₅₃	—	4	16	no	—
81	C₂×C₄×C₅₃	—	4	8	no	—
82	C₂×C₄×C₅₃	#95	1	4	yes	—
83	C₂×C₄×C₅₃	#95	1	2	yes	—
84	C₂×C₄×C₅₃	#95	1	2	yes	—
85	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
88	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
89	C₂×C₄×C₅₃	#117	1	2	yes	—
90	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
91	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
92	C₂×C₄×C₅₃	#121	1	4	yes	—
93	C₂×C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
94	C₂×C₄×C₅₃	#125	1	2	yes	—
95	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
96	C₈×C₅₃	—	2	8	no	—
97	#10837	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂²×C₅₃	—	2	4	no	—
99	C₂²×C₅₃	—	2	2	no	—
100	C₂²×C₅₃	—	4	2	no	—
101	C₂²×C₅₃	—	4	1	no	—
102	C₂²×C₅₃	—	4	1	no	—
103	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
104	C₂²×C₅₃	#382	1	1	yes	characteristic
105	C₂²×C₅₃	#384	1	1	yes	characteristic
106	C₂²×C₅₃	#390	1	2	yes	—
107	C₂²×C₅₃	#391	1	2	yes	—
108	C₄×C₅₃	—	2	4	no	—
109	C₄×C₅₃	—	2	2	no	—
110	C₄×C₅₃	—	8	8	no	—
111	C₄×C₅₃	H(3,ℤ/4)	1	4	yes	complemented
112	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
113	C₄×C₅₃	#380	1	2	yes	—
114	C₄×C₅₃	#389	1	4	yes	—
115	C₂×C₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic
116	#2285	C₂×C₅₃	1	2	yes	—
117	C₂×H(3,ℤ/4)	C₂×C₅₃	1	4	yes	—
118	C₂×C₅₃	—	2	2	no	—
119	C₂×C₅₃	—	2	1	no	—
120	C₂×C₅₃	—	2	1	no	—
121	C₂×C₅₃	—	2	2	no	—
122	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
123	C₂×C₅₃	C₂×H(3,ℤ/4)	1	2	yes	—
124	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
125	C₂×#117	C₂²×C₅₃	1	4	yes	—
126	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
127	C₄×M₁₆	—	2	2	no	—
128	C₄×M₁₆	—	2	2	no	—
129	C₂×#97	C₄×C₅₃	1	2	yes	complemented
130	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	complemented
131	#384	—	2	2	no	—
132	#382	—	2	2	no	—
133	#380	—	2	4	no	—
134	C₄×C₄⋊C₄	C₄×C₅₃	1	8	yes	complemented
135	H(3,ℤ/4)	—	2	8	no	—
136	C₅₃	#10837	1	1	yes	characteristic, direct product factor, Sylow
137	C₂³×C₄	D₄×C₅₃	1	2	yes	—
138	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
139	C₂×M₁₆	—	2	4	no	—
140	C₂×M₁₆	—	2	4	no	—
141	C₂×M₁₆	D₄×C₅₃	1	4	yes	—
142	C₂×M₁₆	Q₈×C₅₃	1	4	yes	—
143	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
144	#117	—	2	8	no	—
145	#117	—	2	8	no	—
146	C₂×C₄⋊C₄	—	2	4	no	—
147	C₂×C₄⋊C₄	—	2	8	no	—
148	C₂×C₂²⋊C₄	—	2	4	no	—
149	C₂×C₄²	C₂×C₄×C₅₃	1	8	yes	—
150	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
151	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
152	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
153	C₂×C₄²	D₄×C₅₃	1	2	yes	—
154	C₂×C₄²	D₄×C₅₃	1	2	yes	—
155	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
156	#97	—	2	4	no	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	2	2	no	—
159	#95	—	2	8	no	—
160	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
161	C₂²×C₄	—	2	8	no	—
162	C₂²×C₄	—	4	4	no	—
163	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
164	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
165	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
166	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
167	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
168	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
169	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
170	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
171	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
172	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
173	C₂²×C₄	#41×C₅₃	1	4	yes	—
174	M₁₆	—	2	8	no	—
175	M₁₆	—	2	8	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	C₄⋊C₄×C₅₃	1	8	yes	complemented
179	C₄⋊C₄	—	4	16	no	—
180	C₄⋊C₄	—	4	8	no	—
181	C₂²⋊C₄	—	4	4	no	—
182	C₂²⋊C₄	—	4	4	no	—
183	C₄²	—	2	16	no	—
184	C₄²	C₄²×C₅₃	1	2	yes	—
185	C₄²	C₄²×C₅₃	1	2	yes	—
186	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
187	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
188	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
189	C₄²	C₄⋊C₄×C₅₃	1	4	yes	—
190	C₄²	C₄⋊C₄×C₅₃	1	4	yes	—
191	C₂³	—	4	2	no	—
192	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
193	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
194	C₂³	#124×C₅₃	1	2	yes	—
195	C₂³	#127×C₅₃	1	1	yes	characteristic
196	C₂³	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	16	no	—
202	C₂×C₄	—	4	8	no	—
203	C₂×C₄	#95×C₅₃	1	2	yes	—
204	C₂×C₄	#95×C₅₃	1	4	yes	—
205	C₂×C₄	#95×C₅₃	1	2	yes	—
206	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
207	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
208	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
209	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
210	C₂×C₄	#117×C₅₃	1	2	yes	—
211	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
212	C₂×C₄	C₄×Q₈×C₅₃	1	4	yes	—
213	C₂×C₄	#121×C₅₃	1	4	yes	—
214	C₂×C₄	C₂²⋊Q₈×C₅₃	1	4	yes	—
215	C₂×C₄	#125×C₅₃	1	2	yes	—
216	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
217	C₈	—	2	8	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	1	no	—
223	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
224	C₂²	C₅₃×#382	1	1	yes	characteristic
225	C₂²	C₅₃×#384	1	1	yes	characteristic
226	C₂²	C₅₃×#390	1	2	yes	—
227	C₂²	C₅₃×#391	1	2	yes	—
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	—	8	8	no	—
231	C₄	C₅₃×H(3,ℤ/4)	1	4	yes	complemented
232	C₄	C₄×C₄⋊C₄×C₅₃	1	2	yes	—
233	C₄	C₅₃×#380	1	2	yes	—
234	C₄	C₅₃×#389	1	4	yes	—
235	C₂	—	2	2	no	—
236	C₂	—	2	1	no	—
237	C₂	—	2	1	no	—
238	C₂	—	2	2	no	—
239	C₂	C₅₃×C₄⋊#95	1	1	yes	characteristic
240	C₂	C₂×C₅₃×H(3,ℤ/4)	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₂×#97×C₅₃	C₄	semidirect
10. C₂×C₄×C₈×C₅₃	C₄	semidirect
14. C₄×C₄⋊C₄×C₅₃	C₄	semidirect
57. C₂×C₈×C₅₃	C₄⋊C₄	semidirect
97. #10837	C₅₃	direct (N × H)
111. C₄×C₅₃	H(3,ℤ/4)	semidirect
129. C₂×#97	C₄×C₅₃	semidirect
130. C₂×C₄×C₈	C₄×C₅₃	semidirect
134. C₄×C₄⋊C₄	C₄×C₅₃	semidirect
136. C₅₃	#10837	direct (N × H)
178. C₂×C₈	C₄⋊C₄×C₅₃	semidirect
231. C₄	C₅₃×H(3,ℤ/4)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×M₁₆×C₅₃	2
C₅₃×#2285	2
C₂×C₅₃×H(3,ℤ/4)	2
#10837	53

Subgroups — C53×#10837 #27218349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10837 #27218349