Subgroups — #27289792

Order 13568.

658 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274448	C₂	1	2	yes	maximal
2	#20274432	C₂	1	2	yes	maximal
3	C₂×#16810380	C₂	1	2	yes	maximal
4	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133176	C₂²	1	4	yes	—
6	#16810390	—	2	4	no	—
7	#16810386	—	2	4	no	—
8	#16810380	C₄	1	8	yes	complemented
9	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#405	—	2	4	no	—
13	C₂×#97×C₅₃	C₄	1	2	yes	complemented
14	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	complemented
15	#13133176	—	2	4	no	—
16	#13133176	—	2	4	no	—
17	#13133176	—	2	4	no	—
18	#13133176	C₂×C₄	1	8	yes	—
19	C₂×C₄⋊Dic₅₃	—	2	4	no	—
20	C₂×C₂²⋊Dic₅₃	—	2	4	no	—
21	C₂×C₄×Dic₅₃	—	2	4	no	—
22	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₅₃	—	2	2	no	—
24	C₂×M₁₆×C₅₃	—	2	2	no	—
25	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₅₃	—	2	2	no	—
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	D₄	1	4	yes	—
32	#117×C₅₃	—	2	2	no	—
33	#117×C₅₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₅₃	Q₈	1	2	yes	—
35	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
37	#97×C₅₃	—	2	4	no	—
38	C₄×C₈×C₅₃	—	2	4	no	—
39	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	#41	1	2	yes	—
46	C₂²×C₄×C₅₃	#41	1	2	yes	—
47	C₂²×C₄×C₅₃	#41	1	2	yes	—
48	C₂²×C₄×C₅₃	#41	1	2	yes	—
49	C₂²×Dic₅₃	—	4	4	no	—
50	M₁₆×C₅₃	—	2	8	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	2	1	no	—
53	C₂×C₈×C₅₃	—	2	1	no	—
54	C₂×C₈×C₅₃	—	2	1	no	—
55	C₂×C₈×C₅₃	—	2	1	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	2	8	no	—
58	C₂×C₈×C₅₃	C₄²	1	4	yes	—
59	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₄⋊C₄×C₅₃	—	2	4	no	—
62	C₂²⋊C₄×C₅₃	—	2	4	no	—
63	C₄²×C₅₃	—	2	2	no	—
64	C₄²×C₅₃	—	2	2	no	—
65	C₂²⋊Dic₅₃	—	2	8	no	—
66	C₂²⋊Dic₅₃	—	4	4	no	—
67	C₄⋊Dic₅₃	—	2	8	no	—
68	C₄⋊Dic₅₃	—	4	4	no	—
69	C₄×Dic₅₃	—	2	8	no	—
70	C₄×Dic₅₃	—	4	4	no	—
71	C₂³×C₅₃	—	4	2	no	—
72	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂³×C₅₃	C₄×D₄	1	2	yes	—
74	C₂³×C₅₃	#121	1	2	yes	—
75	C₂³×C₅₃	#124	1	2	yes	—
76	C₂×C₄×C₅₃	—	2	4	no	—
77	C₂×C₄×C₅₃	—	4	4	no	—
78	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#117	1	4	yes	—
81	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
86	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
87	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
88	C₂×C₄×C₅₃	#123	1	4	yes	—
89	C₂×C₄×C₅₃	#125	1	2	yes	—
90	C₂×C₄×C₅₃	#126	1	4	yes	—
91	C₂×Dic₅₃	—	4	8	no	—
92	C₂×Dic₅₃	—	8	8	no	—
93	C₈×C₅₃	—	2	4	no	—
94	C₈×C₅₃	—	2	2	no	—
95	C₈×C₅₃	—	2	2	no	—
96	#10814	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	—	2	4	no	—
98	C₂²×C₅₃	—	2	2	no	—
99	C₂²×C₅₃	—	4	2	no	—
100	C₂²×C₅₃	—	4	2	no	—
101	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₅₃	#380	1	1	yes	characteristic
103	C₂²×C₅₃	#382	1	1	yes	characteristic
104	C₂²×C₅₃	#388	1	4	yes	—
105	C₄×C₅₃	—	4	4	no	—
106	C₄×C₅₃	#376	1	1	yes	characteristic
107	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
108	C₄×C₅₃	#381	1	1	yes	characteristic
109	C₄×C₅₃	#383	1	1	yes	characteristic
110	C₄×C₅₃	#387	1	4	yes	—
111	Dic₅₃	—	8	8	no	—
112	C₂×#405	D₅₃	1	1	yes	characteristic
113	#1740	—	53	2	no	—
114	#1735	—	53	2	no	—
115	C₂×H(3,ℤ/4)	—	53	2	no	—
116	C₂×C₅₃	—	2	4	no	—
117	C₂×C₅₃	—	2	1	no	—
118	C₂×C₅₃	—	2	1	no	—
119	C₂×C₅₃	#1434	1	1	yes	characteristic
120	C₂×C₅₃	#1812	1	2	yes	—
121	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
122	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
123	C₂×#117	—	53	4	no	—
124	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
125	#405	—	2	4	no	—
126	C₂×#97	Dic₅₃	1	2	yes	complemented
127	C₂×C₄×C₈	Dic₅₃	1	2	yes	complemented
128	#342	—	106	4	no	—
129	#340	—	106	4	no	—
130	H(3,ℤ/4)	—	53	8	no	—
131	C₅₃	#10814	1	1	yes	characteristic, complemented, Sylow
132	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
133	C₂×M₁₆	—	2	2	no	—
134	C₂×M₁₆	—	2	2	no	—
135	C₂×M₁₆	C₄×D₅₃	1	4	yes	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	2	2	no	—
138	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
139	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
140	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
141	C₂²×C₈	C₂²⋊D₅₃	1	4	yes	—
142	#117	—	2	2	no	—
143	#117	—	2	2	no	—
144	#117	—	53	8	no	—
145	#117	—	106	4	no	—
146	#117	—	106	4	no	—
147	#117	—	106	4	no	—
148	C₂×C₄⋊C₄	—	106	4	no	—
149	C₂×C₄⋊C₄	Dic₁₀₆	1	2	yes	—
150	C₂×C₂²⋊C₄	—	106	4	no	—
151	C₂×C₂²⋊C₄	D₂₁₂	1	2	yes	—
152	C₂×C₄²	—	106	4	no	—
153	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
154	#97	—	2	4	no	—
155	C₄×C₈	—	2	4	no	—
156	C₂⁴	C₂×D₂₁₂	1	1	yes	characteristic
157	C₂²×C₄	—	212	4	no	—
158	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
159	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
160	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
161	C₂²×C₄	#11719945	1	2	yes	—
162	C₂²×C₄	#11719945	1	2	yes	—
163	C₂²×C₄	#11719945	1	2	yes	—
164	C₂²×C₄	#11719945	1	2	yes	—
165	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
167	M₁₆	—	2	8	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	8	no	—
175	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
176	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
177	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	106	8	no	—
180	C₄⋊C₄	—	212	4	no	—
181	C₂²⋊C₄	—	2	4	no	—
182	C₂²⋊C₄	—	106	8	no	—
183	C₂²⋊C₄	—	212	4	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	106	8	no	—
187	C₄²	—	212	4	no	—
188	C₂³	—	4	2	no	—
189	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
190	C₂³	C₄×D₂₁₂	1	2	yes	—
191	C₂³	#13133191	1	2	yes	—
192	C₂³	#13133205	1	2	yes	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	424	8	no	—
196	C₂×C₄	—	212	8	no	—
197	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
198	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
199	C₂×C₄	#13133174	1	4	yes	—
200	C₂×C₄	#13133176	1	1	yes	characteristic
201	C₂×C₄	#13133176	1	1	yes	characteristic
202	C₂×C₄	#13133176	1	1	yes	characteristic
203	C₂×C₄	#13133176	1	1	yes	characteristic
204	C₂×C₄	C₄×C₂²⋊D₅₃	1	4	yes	—
205	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
206	C₂×C₄	#13133196	1	2	yes	—
207	C₂×C₄	#13133200	1	4	yes	—
208	C₂×C₄	#13133209	1	2	yes	—
209	C₂×C₄	#13133211	1	4	yes	—
210	C₈	—	2	4	no	—
211	C₈	—	2	2	no	—
212	C₈	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
218	C₂²	#16810483	1	1	yes	characteristic
219	C₂²	#16810489	1	1	yes	characteristic
220	C₂²	#16810514	1	4	yes	—
221	C₄	—	4	4	no	—
222	C₄	—	424	8	no	—
223	C₄	#16810469	1	1	yes	characteristic
224	C₄	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
225	C₄	#16810486	1	1	yes	characteristic
226	C₄	#16810491	1	1	yes	characteristic
227	C₄	#16810507	1	4	yes	—
228	C₂	—	2	4	no	—
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	#20272942	1	1	yes	characteristic
232	C₂	#20274761	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133176
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#10814
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
8. #16810380	C₄	semidirect
13. C₂×#97×C₅₃	C₄	semidirect
14. C₂×C₄×C₈×C₅₃	C₄	semidirect
126. C₂×#97	Dic₅₃	semidirect
127. C₂×C₄×C₈	Dic₅₃	semidirect
131. C₅₃	#10814	semidirect

Maximal subgroups

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

subgroup	index
#20274448	2
#20274432	2
C₂×#16810380	2
C₂×C₅₃×#405	2
#10814	53