Subgroups — #27537179

Order 13568.

590 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276522	C₂	1	8	yes	maximal, complemented
2	#20276471	C₂	1	2	yes	maximal, complemented
3	#20276326	C₂	1	2	yes	maximal
4	C₄×#13133157	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	4	yes	—
7	C₄⋊#11719908	C₂²	1	8	yes	complemented
8	C₂²⋊#11719908	C₂²	1	8	yes	—
9	#16810399	C₂²	1	4	yes	complemented
10	C₄×#11719908	C₂²	1	4	yes	complemented
11	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
12	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
14	C₅₃×#405	C₄	1	4	yes	—
15	C₅₃×#405	C₂²	1	4	yes	—
16	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
17	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	#13133157	—	2	8	no	—
21	C₂×#11719908	C₂³	1	8	yes	—
22	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×M₁₆×C₅₃	—	2	4	no	—
24	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₅₃	—	2	4	no	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
31	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	#117×C₅₃	C₈	1	4	yes	—
33	#117×C₅₃	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊C₄×C₅₃	C₈	1	2	yes	—
35	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	#97×C₅₃	C₂×C₄	1	4	yes	—
39	#97×C₅₃	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₅₃	C₂³	1	2	yes	—
43	C₄×C₈×C₅₃	C₂³	1	2	yes	—
44	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	—	2	2	no	—
46	C₂²×C₄×C₅₃	—	2	2	no	—
47	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	M₁₆×C₅₃	—	2	8	no	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	—	2	4	no	—
57	C₂×C₈×C₅₃	—	2	4	no	—
58	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
59	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₅₃	#41	1	4	yes	—
64	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
65	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
66	C₂²⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
67	C₄²×C₅₃	C₂×C₈	1	2	yes	—
68	C₄²×C₅₃	C₂×C₈	1	2	yes	—
69	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
70	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
71	#11719908	—	2	8	no	—
72	C₂³×C₅₃	—	2	2	no	—
73	C₂³×C₅₃	—	4	2	no	—
74	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
75	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	4	no	—
80	C₂×C₄×C₅₃	—	4	2	no	—
81	C₂×C₄×C₅₃	—	4	2	no	—
82	C₂×C₄×C₅₃	#117	1	2	yes	—
83	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
84	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
85	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
86	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
88	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
89	C₈×C₅₃	—	2	2	no	—
90	C₈×C₅₃	—	2	2	no	—
91	C₈×C₅₃	#117	1	2	yes	—
92	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
93	C₈×C₅₃	C₂×#41	1	2	yes	—
94	#33419	—	53	1	no	maximal, Sylow
95	C₂²×C₅₃	—	2	2	no	—
96	C₂²×C₅₃	—	2	4	no	—
97	C₂²×C₅₃	—	2	2	no	—
98	C₂²×C₅₃	—	4	4	no	—
99	C₂²×C₅₃	#431	1	2	yes	—
100	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
101	C₄×C₅₃	—	2	4	no	—
102	C₄×C₅₃	—	2	4	no	—
103	C₄×C₅₃	—	2	2	no	—
104	C₄×C₅₃	#431	1	2	yes	—
105	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
106	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
107	C₂×#405	D₅₃	1	1	yes	characteristic
108	#2164	—	53	8	no	—
109	#2150	—	53	2	no	—
110	#2112	—	53	2	no	—
111	C₄×M₃₂	—	53	2	no	—
112	C₂×C₅₃	—	2	2	no	—
113	C₂×C₅₃	—	4	2	no	—
114	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
115	C₂×C₅₃	#3407	1	2	yes	—
116	C₂²×M₁₆	Dic₅₃	1	1	yes	characteristic
117	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
118	C₂×#117	Dic₅₃	1	1	yes	characteristic
119	C₂×M₃₂	—	53	4	no	—
120	#405	Dic₅₃	1	4	yes	—
121	#405	C₂×D₅₃	1	4	yes	—
122	C₂×#97	Dic₅₃	1	1	yes	characteristic
123	C₂×#97	Dic₅₃	1	1	yes	characteristic
124	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
125	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
126	C₄⋊C₁₆	—	53	8	no	—
127	C₂²⋊C₁₆	—	53	8	no	—
128	#346	—	53	4	no	—
129	C₄×C₁₆	—	53	4	no	—
130	C₅₃	#33419	1	1	yes	characteristic, complemented, Sylow
131	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
132	C₂×M₁₆	—	2	4	no	—
133	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
134	C₂²×C₈	—	2	4	no	—
135	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
136	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
137	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
138	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
139	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
140	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
141	#117	#88322	1	4	yes	—
142	#117	C₂×Dic₅₃	1	4	yes	—
143	C₂×C₄⋊C₄	#88322	1	2	yes	—
144	C₂×C₂²⋊C₄	#88322	1	2	yes	—
145	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
146	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
147	M₃₂	—	106	8	no	—
148	C₂×C₁₆	—	53	8	no	—
149	#97	C₂×Dic₅₃	1	4	yes	—
150	#97	C₂×Dic₅₃	1	4	yes	—
151	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
152	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
153	C₄×C₈	C₂²×D₅₃	1	2	yes	—
154	C₄×C₈	C₂²×D₅₃	1	2	yes	—
155	C₂⁴	C₂×#88322	1	1	yes	characteristic
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
159	C₂²×C₄	C₂×#88322	1	2	yes	—
160	C₂²×C₄	C₂×#88322	1	2	yes	—
161	C₂²×C₄	C₂×#88322	1	2	yes	—
162	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
163	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
164	M₁₆	—	2	8	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	#11719917	1	4	yes	—
170	C₂×C₈	#11719945	1	4	yes	—
171	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
172	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
173	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
174	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
175	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
176	C₄⋊C₄	C₂×#88322	1	8	yes	—
177	C₂²⋊C₄	C₂×#88322	1	8	yes	—
178	C₄²	C₂×#88322	1	2	yes	—
179	C₄²	C₂×#88322	1	2	yes	—
180	C₄²	C₂²×Dic₅₃	1	2	yes	—
181	C₄²	C₂²×Dic₅₃	1	2	yes	—
182	C₁₆	—	106	8	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	4	2	no	—
185	C₂³	C₂²×#88322	1	2	yes	—
186	C₂³	C₂²×#88322	1	1	yes	characteristic
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	#13133176	1	2	yes	—
194	C₂×C₄	C₂²×#88322	1	4	yes	—
195	C₂×C₄	C₂²×#88322	1	4	yes	—
196	C₂×C₄	C₂²×#88322	1	2	yes	—
197	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
198	C₂×C₄	C₂×#11719917	1	2	yes	—
199	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
200	C₈	—	2	2	no	—
201	C₈	—	2	2	no	—
202	C₈	#13133176	1	2	yes	—
203	C₈	C₂×#11719917	1	4	yes	—
204	C₈	C₂×#11719945	1	2	yes	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	4	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	4	4	no	—
209	C₂²	#16810676	1	2	yes	—
210	C₂²	C₂³×#88322	1	1	yes	characteristic
211	C₄	—	2	4	no	—
212	C₄	—	2	4	no	—
213	C₄	—	2	2	no	—
214	C₄	#16810676	1	2	yes	—
215	C₄	C₂×#13133176	1	1	yes	characteristic
216	C₄	C₂²×#11719917	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	—	4	2	no	—
219	C₂	C₂×#16810676	1	1	yes	characteristic
220	C₂	#20287818	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₂×C₅₃×#405	C₂
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#33419
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
1. #20276522	C₂	semidirect
2. #20276471	C₂	semidirect
4. C₄×#13133157	C₂	semidirect
7. C₄⋊#11719908	C₂²	semidirect
9. #16810399	C₂²	semidirect
10. C₄×#11719908	C₂²	semidirect
130. C₅₃	#33419	semidirect

Maximal subgroups

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

subgroup	index
#20276522	2
#20276471	2
#20276326	2
C₄×#13133157	2
C₂×C₅₃×#405	2
#33419	53