Subgroups — C₅₃×#13239 #27220751

Order 13568.

502 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	C₂²×D₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#367	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#357	C₂	1	2	yes	maximal, complemented
4	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1970	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₅₃	C₂²	1	2	yes	—
7	C₂×D₁₆×C₅₃	—	2	2	no	—
8	C₂×D₁₆×C₅₃	—	2	2	no	—
9	C₂×D₁₆×C₅₃	C₄	1	4	yes	complemented
10	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#107×C₅₃	C₂²	1	2	yes	—
13	C₂×#102×C₅₃	—	2	2	no	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#390	—	2	2	no	—
16	C₅₃×#367	—	2	2	no	—
17	C₅₃×#357	—	2	4	no	—
18	C₅₃×#346	—	2	2	no	—
19	C₂²×D₄×C₅₃	—	2	2	no	—
20	C₂×D₈×C₅₃	—	2	2	no	—
21	C₂×D₈×C₅₃	—	2	2	no	—
22	C₂×D₈×C₅₃	—	2	2	no	—
23	C₂×D₈×C₅₃	—	2	2	no	—
24	C₂×D₈×C₅₃	—	2	2	no	—
25	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂×C₂²⋊C₄×C₅₃	—	4	4	no	—
30	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
31	D₁₆×C₅₃	—	2	8	no	—
32	C₂×C₁₆×C₅₃	—	2	1	no	—
33	C₂×C₁₆×C₅₃	—	2	1	no	—
34	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
36	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
37	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
38	#107×C₅₃	—	2	2	no	—
39	#107×C₅₃	—	2	2	no	—
40	#102×C₅₃	—	4	4	no	—
41	C₄×C₈×C₅₃	—	2	1	no	—
42	C₄×C₈×C₅₃	—	2	1	no	—
43	C₂⁴×C₅₃	—	4	2	no	—
44	C₂×D₄×C₅₃	—	2	4	no	—
45	C₂×D₄×C₅₃	—	4	2	no	—
46	C₂×D₄×C₅₃	—	4	2	no	—
47	C₂×D₄×C₅₃	—	4	2	no	—
48	C₂×D₄×C₅₃	—	4	2	no	—
49	C₂×D₄×C₅₃	—	4	2	no	—
50	C₂²×C₄×C₅₃	—	4	2	no	—
51	C₂²×C₄×C₅₃	D₈	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	D₈×C₅₃	—	2	4	no	—
54	D₈×C₅₃	—	2	4	no	—
55	D₈×C₅₃	—	4	2	no	—
56	D₈×C₅₃	—	4	2	no	—
57	C₂×C₈×C₅₃	—	2	1	no	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	D₈	1	2	yes	—
60	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
65	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
66	C₄⋊C₄×C₅₃	—	4	2	no	—
67	C₄⋊C₄×C₅₃	—	4	2	no	—
68	C₂²⋊C₄×C₅₃	—	8	8	no	—
69	C₄²×C₅₃	—	2	1	no	—
70	C₄²×C₅₃	—	2	1	no	—
71	C₁₆×C₅₃	—	2	2	no	—
72	C₁₆×C₅₃	C₂²⋊C₄	1	4	yes	complemented
73	C₂³×C₅₃	—	4	4	no	—
74	C₂³×C₅₃	—	8	2	no	—
75	C₂³×C₅₃	—	8	2	no	—
76	C₂³×C₅₃	—	8	4	no	—
77	C₂³×C₅₃	—	8	4	no	—
78	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
79	D₄×C₅₃	—	4	4	no	—
80	D₄×C₅₃	—	4	4	no	—
81	D₄×C₅₃	—	8	2	no	—
82	D₄×C₅₃	—	8	2	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	4	no	—
85	C₂×C₄×C₅₃	—	8	2	no	—
86	C₂×C₄×C₅₃	—	8	4	no	—
87	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
93	C₈×C₅₃	—	2	2	no	—
94	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
96	C₈×C₅₃	#121	1	1	yes	characteristic
97	C₈×C₅₃	#124	1	1	yes	characteristic
98	#13239	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₅₃	—	8	8	no	—
100	C₂²×C₅₃	—	8	4	no	—
101	C₂²×C₅₃	—	16	4	no	—
102	C₂²×C₅₃	—	16	2	no	—
103	C₂²×C₅₃	—	16	2	no	—
104	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
105	C₂²×C₅₃	#466	1	1	yes	characteristic
106	C₂²×C₅₃	#493	1	1	yes	characteristic
107	C₂²×C₅₃	#509	1	4	yes	—
108	C₄×C₅₃	—	4	2	no	—
109	C₄×C₅₃	—	16	2	no	—
110	C₄×C₅₃	#390	1	1	yes	characteristic
111	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
112	C₄×C₅₃	#466	1	1	yes	characteristic
113	C₄×C₅₃	#495	1	1	yes	characteristic
114	C₂²×D₁₆	C₂×C₅₃	1	1	yes	characteristic
115	C₂×#367	C₂×C₅₃	1	1	yes	characteristic, complemented
116	C₂×#357	C₂×C₅₃	1	2	yes	complemented
117	C₂×#346	C₂×C₅₃	1	1	yes	characteristic, complemented
118	#1970	C₂×C₅₃	1	2	yes	complemented
119	C₂×C₅₃	—	16	4	no	—
120	C₂×C₅₃	#1970	1	1	yes	characteristic
121	C₂×C₅₃	#2177	1	2	yes	—
122	C₂×C₅₃	#2218	1	2	yes	—
123	C₂×C₅₃	#2250	1	2	yes	—
124	C₂²×D₈	C₂²×C₅₃	1	2	yes	—
125	C₂×D₁₆	—	2	2	no	—
126	C₂×D₁₆	—	2	2	no	—
127	C₂×D₁₆	C₄×C₅₃	1	4	yes	complemented
128	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
129	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
130	C₂×#107	C₂²×C₅₃	1	2	yes	—
131	C₂×#102	—	2	2	no	—
132	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
133	#390	—	2	2	no	—
134	#367	—	2	2	no	—
135	#357	—	2	4	no	—
136	#346	—	2	2	no	—
137	C₅₃	#13239	1	1	yes	characteristic, direct product factor, Sylow
138	C₂²×D₄	—	2	2	no	—
139	C₂×D₈	—	2	2	no	—
140	C₂×D₈	—	2	2	no	—
141	C₂×D₈	—	2	2	no	—
142	C₂×D₈	—	2	2	no	—
143	C₂×D₈	—	2	2	no	—
144	C₂×D₈	C₂×C₄×C₅₃	1	4	yes	—
145	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
146	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₂²⋊C₄	—	4	4	no	—
149	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
150	D₁₆	—	2	8	no	—
151	C₂×C₁₆	—	2	1	no	—
152	C₂×C₁₆	—	2	1	no	—
153	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
154	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
155	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
156	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
157	#107	—	2	2	no	—
158	#107	—	2	2	no	—
159	#102	—	4	4	no	—
160	C₄×C₈	—	2	1	no	—
161	C₄×C₈	—	2	1	no	—
162	C₂⁴	—	4	2	no	—
163	C₂×D₄	—	2	4	no	—
164	C₂×D₄	—	4	2	no	—
165	C₂×D₄	—	4	2	no	—
166	C₂×D₄	—	4	2	no	—
167	C₂×D₄	—	4	2	no	—
168	C₂×D₄	—	4	2	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	D₈×C₅₃	1	2	yes	—
171	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
172	D₈	—	2	4	no	—
173	D₈	—	2	4	no	—
174	D₈	—	4	2	no	—
175	D₈	—	4	2	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	D₈×C₅₃	1	2	yes	—
179	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
180	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
181	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
182	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
183	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
184	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₂²⋊C₄	—	8	8	no	—
188	C₄²	—	2	1	no	—
189	C₄²	—	2	1	no	—
190	C₁₆	—	2	2	no	—
191	C₁₆	C₂²⋊C₄×C₅₃	1	4	yes	complemented
192	C₂³	—	4	4	no	—
193	C₂³	—	8	2	no	—
194	C₂³	—	8	4	no	—
195	C₂³	—	8	4	no	—
196	C₂³	—	8	2	no	—
197	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
198	D₄	—	4	4	no	—
199	D₄	—	4	4	no	—
200	D₄	—	8	2	no	—
201	D₄	—	8	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	8	2	no	—
205	C₂×C₄	—	8	4	no	—
206	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
207	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
208	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
209	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
210	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
211	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
214	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
215	C₈	#121×C₅₃	1	1	yes	characteristic
216	C₈	#124×C₅₃	1	1	yes	characteristic
217	C₂²	—	8	8	no	—
218	C₂²	—	8	4	no	—
219	C₂²	—	16	4	no	—
220	C₂²	—	16	2	no	—
221	C₂²	—	16	2	no	—
222	C₂²	C₄×D₈×C₅₃	1	1	yes	characteristic
223	C₂²	C₅₃×#466	1	1	yes	characteristic
224	C₂²	C₅₃×#493	1	1	yes	characteristic
225	C₂²	C₅₃×#509	1	4	yes	—
226	C₄	—	4	2	no	—
227	C₄	—	16	2	no	—
228	C₄	C₅₃×#390	1	1	yes	characteristic
229	C₄	C₄×D₈×C₅₃	1	1	yes	characteristic
230	C₄	C₅₃×#466	1	1	yes	characteristic
231	C₄	C₅₃×#495	1	1	yes	characteristic
232	C₂	—	16	4	no	—
233	C₂	C₅₃×#1970	1	1	yes	characteristic
234	C₂	C₅₃×#2177	1	2	yes	—
235	C₂	C₅₃×#2218	1	2	yes	—
236	C₂	C₅₃×#2250	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂²×D₁₆×C₅₃	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#367	C₂	semidirect
3. C₂×C₅₃×#357	C₂	semidirect
4. C₂×C₅₃×#346	C₂	semidirect
5. C₅₃×#1970	C₂	semidirect
9. C₂×D₁₆×C₅₃	C₄	semidirect
72. C₁₆×C₅₃	C₂²⋊C₄	semidirect
98. #13239	C₅₃	direct (N × H)
115. C₂×#367	C₂×C₅₃	semidirect
116. C₂×#357	C₂×C₅₃	semidirect
117. C₂×#346	C₂×C₅₃	semidirect
118. #1970	C₂×C₅₃	semidirect
127. C₂×D₁₆	C₄×C₅₃	semidirect
137. C₅₃	#13239	direct (N × H)
191. C₁₆	C₂²⋊C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₆×C₅₃	2
C₂×C₅₃×#367	2
C₂×C₅₃×#357	2
C₂×C₅₃×#346	2
C₅₃×#1970	2
#13239	53

Subgroups — C53×#13239 #27220751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13239 #27220751