Subgroups — #27289897

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	#20277122	C₂	1	2	yes	maximal
2	C₂×#16810380	C₂	1	4	yes	maximal
3	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133176	C₂²	1	4	yes	—
5	#16810493	—	2	2	no	—
6	#16810489	—	2	2	no	—
7	#16810483	—	2	4	no	—
8	C₄×C₄⋊Dic₅₃	C₄	1	8	yes	complemented
9	#16810380	—	2	8	no	—
10	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
11	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₄×M₁₆×C₅₃	—	2	2	no	—
13	C₄×M₁₆×C₅₃	—	2	2	no	—
14	C₂×#97×C₅₃	C₄	1	2	yes	complemented
15	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	complemented
16	#13133176	—	2	8	no	—
17	#13133176	—	2	8	no	—
18	C₂×C₄⋊Dic₅₃	—	2	8	no	—
19	C₂×C₄⋊Dic₅₃	—	2	4	no	—
20	C₂×C₂²⋊Dic₅₃	—	2	4	no	—
21	C₂×C₄×Dic₅₃	C₂×C₄	1	8	yes	—
22	#13133124	—	2	8	no	—
23	C₂³×C₄×C₅₃	D₄	1	2	yes	—
24	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂×M₁₆×C₅₃	—	2	4	no	—
26	C₂×M₁₆×C₅₃	—	2	4	no	—
27	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
28	C₂×M₁₆×C₅₃	Q₈	1	4	yes	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	D₄	1	2	yes	—
34	C₂×C₄²×C₅₃	D₄	1	2	yes	—
35	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
36	#97×C₅₃	—	2	4	no	—
37	C₄×C₈×C₅₃	—	2	2	no	—
38	C₄×C₈×C₅₃	—	2	2	no	—
39	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	—	4	4	no	—
41	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
42	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
43	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₅₃	#41	1	4	yes	—
52	C₂²×Dic₅₃	—	2	8	no	—
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	C₄²×C₅₃	C₄²	1	2	yes	—
59	C₄²×C₅₃	C₄²	1	2	yes	—
60	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
64	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
65	C₂²⋊Dic₅₃	—	4	4	no	—
66	C₂²⋊Dic₅₃	—	4	4	no	—
67	C₄⋊Dic₅₃	—	4	16	no	—
68	C₄⋊Dic₅₃	—	4	8	no	—
69	C₄×Dic₅₃	—	2	16	no	—
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	#95	1	4	yes	—
81	C₂×C₄×C₅₃	#95	1	2	yes	—
82	C₂×C₄×C₅₃	#95	1	2	yes	—
83	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
86	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
87	C₂×C₄×C₅₃	#117	1	2	yes	—
88	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
89	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
90	C₂×C₄×C₅₃	#121	1	4	yes	—
91	C₂×C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
92	C₂×C₄×C₅₃	#125	1	2	yes	—
93	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
94	C₂×Dic₅₃	—	4	16	no	—
95	C₂×Dic₅₃	—	4	8	no	—
96	C₈×C₅₃	—	2	8	no	—
97	#10837	—	53	1	no	maximal, 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	H(3,ℤ/4)	1	4	yes	complemented
111	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
112	C₄×C₅₃	#380	1	2	yes	—
113	C₄×C₅₃	#389	1	4	yes	—
114	Dic₅₃	—	8	8	no	—
115	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
116	#2285	—	53	2	no	—
117	C₂×H(3,ℤ/4)	—	53	4	no	—
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	C₂×D₅₃	1	2	yes	—
125	C₂×#117	—	53	4	no	—
126	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
127	C₄×M₁₆	—	2	2	no	—
128	C₄×M₁₆	—	2	2	no	—
129	C₂×#97	Dic₅₃	1	2	yes	complemented
130	C₂×C₄×C₈	Dic₅₃	1	2	yes	complemented
131	#384	—	106	2	no	—
132	#382	—	106	2	no	—
133	#380	—	106	4	no	—
134	C₄×C₄⋊C₄	—	53	8	no	—
135	H(3,ℤ/4)	—	106	8	no	—
136	C₅₃	#10837	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₄	D₂₁₂	1	2	yes	—
138	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
139	C₂×M₁₆	—	2	4	no	—
140	C₂×M₁₆	—	2	4	no	—
141	C₂×M₁₆	Dic₁₀₆	1	4	yes	—
142	C₂×M₁₆	D₂₁₂	1	4	yes	—
143	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
144	#117	—	106	8	no	—
145	#117	—	106	8	no	—
146	C₂×C₄⋊C₄	—	106	4	no	—
147	C₂×C₄⋊C₄	—	106	8	no	—
148	C₂×C₂²⋊C₄	—	106	4	no	—
149	C₂×C₄²	—	53	8	no	—
150	C₂×C₄²	Dic₁₀₆	1	2	yes	—
151	C₂×C₄²	C₄×D₅₃	1	4	yes	—
152	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
153	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
154	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
155	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
156	#97	—	2	4	no	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	2	2	no	—
159	#95	—	106	8	no	—
160	C₂⁴	C₂×D₂₁₂	1	1	yes	characteristic
161	C₂²×C₄	—	4	4	no	—
162	C₂²×C₄	—	106	8	no	—
163	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
164	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
165	C₂²×C₄	#11719921	1	4	yes	—
166	C₂²×C₄	#11719921	1	4	yes	—
167	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
168	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
169	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
170	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
171	C₂²×C₄	#11719945	1	4	yes	—
172	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	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⋊Dic₅₃	1	8	yes	complemented
179	C₄⋊C₄	—	212	16	no	—
180	C₄⋊C₄	—	212	8	no	—
181	C₂²⋊C₄	—	212	4	no	—
182	C₂²⋊C₄	—	212	4	no	—
183	C₄²	—	106	16	no	—
184	C₄²	C₄×Dic₅₃	1	2	yes	—
185	C₄²	C₄×Dic₅₃	1	2	yes	—
186	C₄²	#11719919	1	4	yes	—
187	C₄²	#11719919	1	4	yes	—
188	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
189	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
190	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
191	C₂³	—	4	2	no	—
192	C₂³	C₂×#11719921	1	2	yes	—
193	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
194	C₂³	#13133205	1	2	yes	—
195	C₂³	#13133214	1	1	yes	characteristic
196	C₂³	#13133217	1	1	yes	characteristic
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	212	16	no	—
202	C₂×C₄	—	212	8	no	—
203	C₂×C₄	#13133124	1	2	yes	—
204	C₂×C₄	#13133124	1	4	yes	—
205	C₂×C₄	#13133124	1	2	yes	—
206	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
207	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
208	C₂×C₄	C₂×#11719919	1	2	yes	—
209	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
210	C₂×C₄	#13133176	1	2	yes	—
211	C₂×C₄	C₄×D₂₁₂	1	4	yes	—
212	C₂×C₄	C₄×Dic₁₀₆	1	4	yes	—
213	C₂×C₄	#13133191	1	4	yes	—
214	C₂×C₄	#13133196	1	4	yes	—
215	C₂×C₄	#13133209	1	2	yes	—
216	C₂×C₄	#13133217	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×#13133124	1	1	yes	characteristic
224	C₂²	#16810489	1	1	yes	characteristic
225	C₂²	#16810493	1	1	yes	characteristic
226	C₂²	#16810525	1	2	yes	—
227	C₂²	#16810529	1	2	yes	—
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	—	424	8	no	—
231	C₄	#16810380	1	4	yes	complemented
232	C₄	C₄×C₄⋊Dic₅₃	1	2	yes	—
233	C₄	#16810483	1	2	yes	—
234	C₄	#16810521	1	4	yes	—
235	C₂	—	2	2	no	—
236	C₂	—	2	2	no	—
237	C₂	—	2	1	no	—
238	C₂	—	2	1	no	—
239	C₂	#20272945	1	1	yes	characteristic
240	C₂	C₂×#16810380	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₄×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×C₄×M₁₆	D₅₃
hypocenter	C₅₃	#10837
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
8. C₄×C₄⋊Dic₅₃	C₄	semidirect
14. C₂×#97×C₅₃	C₄	semidirect
15. C₂×C₄×C₈×C₅₃	C₄	semidirect
57. C₂×C₈×C₅₃	C₄⋊C₄	semidirect
110. C₄×C₅₃	H(3,ℤ/4)	semidirect
129. C₂×#97	Dic₅₃	semidirect
130. C₂×C₄×C₈	Dic₅₃	semidirect
136. C₅₃	#10837	semidirect
178. C₂×C₈	C₄⋊Dic₅₃	semidirect
231. C₄	#16810380	semidirect

Maximal subgroups

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

subgroup	index
#20277122	2
C₂×#16810380	2
C₂×C₄×M₁₆×C₅₃	2
#10837	53