Subgroups — #27536733

Order 13568.

826 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133157	C₂	1	2	yes	maximal
2	#20276312	C₂	1	8	yes	maximal, complemented
3	C₄×#13133157	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₄	1	16	yes	complemented
6	C₂×#13133157	C₂²	1	4	yes	—
7	C₂×#13133157	C₂²	1	4	yes	—
8	C₂²×#11719908	C₂²	1	4	yes	—
9	#16810399	C₂²	1	8	yes	complemented
10	C₄×#11719908	C₂²	1	8	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	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
18	#13133157	C₂×C₄	1	16	yes	—
19	#13133157	C₂×C₄	1	16	yes	complemented
20	C₂×#11719908	C₂×C₄	1	16	yes	complemented
21	C₂×#11719908	C₂³	1	8	yes	—
22	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×M₁₆×C₅₃	C₈	1	8	yes	—
24	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
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	2	yes	—
29	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
30	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	#117×C₅₃	C₈	1	4	yes	—
32	#117×C₅₃	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₅₃	C₈	1	2	yes	—
34	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
36	#97×C₅₃	C₂×C₄	1	8	yes	—
37	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₅₃	C₂³	1	4	yes	—
39	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
41	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
42	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
47	M₁₆×C₅₃	C₂×C₈	1	16	yes	—
48	C₂×C₈×C₅₃	C₄²	1	4	yes	—
49	C₂×C₈×C₅₃	C₄²	1	8	yes	—
50	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	—
51	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
54	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
55	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
56	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
58	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
59	C₂²⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
60	C₄²×C₅₃	C₂×C₈	1	4	yes	—
61	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
62	#11719908	C₂²×C₄	1	16	yes	—
63	C₂³×C₅₃	—	4	2	no	—
64	C₂³×C₅₃	C₄×C₈	1	4	yes	—
65	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
66	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
67	C₂×C₄×C₅₃	—	4	4	no	—
68	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
69	C₂×C₄×C₅₃	C₄×C₈	1	8	yes	—
70	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
71	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
72	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
73	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
74	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
75	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
78	C₈×C₅₃	C₂×C₄²	1	2	yes	—
79	C₈×C₅₃	C₂×C₄²	1	4	yes	—
80	C₈×C₅₃	C₂²×C₈	1	8	yes	—
81	C₈×C₅₃	C₂³×C₄	1	2	yes	—
82	#33324	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	—	2	2	no	—
84	C₂²×C₅₃	—	2	4	no	—
85	C₂²×C₅₃	—	4	4	no	—
86	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
87	C₂²×C₅₃	C₂×C₄×C₈	1	4	yes	—
88	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
89	C₄×C₅₃	—	2	8	no	—
90	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
91	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	—
92	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
93	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
94	C₂²×M₃₂	—	53	2	no	—
95	C₂×#405	D₅₃	1	1	yes	characteristic
96	#2108	—	53	8	no	—
97	C₄×M₃₂	—	53	4	no	—
98	C₂×C₅₃	—	2	2	no	—
99	C₂×C₅₃	—	2	4	no	—
100	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
101	C₂×C₅₃	#3407	1	2	yes	—
102	C₂²×M₁₆	Dic₅₃	1	1	yes	characteristic
103	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
104	C₂×#117	Dic₅₃	1	1	yes	characteristic
105	C₂×M₃₂	—	53	16	no	—
106	C₂×M₃₂	—	53	4	no	—
107	C₂×M₃₂	—	53	4	no	—
108	C₂²×C₁₆	—	53	4	no	—
109	#405	Dic₅₃	1	4	yes	—
110	#405	C₂×D₅₃	1	4	yes	—
111	C₂×#97	Dic₅₃	1	2	yes	—
112	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
113	#346	—	53	8	no	—
114	C₄×C₁₆	—	53	8	no	—
115	C₅₃	#33324	1	1	yes	characteristic, complemented, Sylow
116	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
117	C₂×M₁₆	#88322	1	8	yes	—
118	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
119	C₂²×C₈	C₄×D₅₃	1	8	yes	—
120	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
121	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
122	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
123	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
124	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
125	#117	#88322	1	4	yes	—
126	#117	C₂×Dic₅₃	1	4	yes	—
127	C₂×C₄⋊C₄	#88322	1	2	yes	—
128	C₂×C₂²⋊C₄	#88322	1	2	yes	—
129	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
130	M₃₂	—	53	16	no	—
131	M₃₂	—	53	16	no	—
132	C₂×C₁₆	—	53	16	no	—
133	C₂×C₁₆	—	53	8	no	—
134	#97	C₂×Dic₅₃	1	8	yes	—
135	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
136	C₄×C₈	C₂²×D₅₃	1	4	yes	—
137	C₂⁴	C₂×#88322	1	1	yes	characteristic
138	C₂²×C₄	C₂×#88322	1	4	yes	—
139	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
140	C₂²×C₄	C₂×#88322	1	2	yes	—
141	C₂²×C₄	C₂×#88322	1	4	yes	—
142	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
143	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
144	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
145	M₁₆	C₂×#88322	1	16	yes	—
146	C₂×C₈	C₂×#88322	1	8	yes	—
147	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
148	C₂×C₈	C₄×Dic₅₃	1	8	yes	—
149	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
150	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
151	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
152	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
153	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
154	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
155	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
156	C₄⋊C₄	C₂×#88322	1	8	yes	—
157	C₂²⋊C₄	C₂×#88322	1	8	yes	—
158	C₄²	C₂×#88322	1	4	yes	—
159	C₄²	C₂²×Dic₅₃	1	4	yes	—
160	C₁₆	—	53	16	no	—
161	C₂³	—	4	2	no	—
162	C₂³	C₄×#88322	1	4	yes	—
163	C₂³	C₂²×#88322	1	2	yes	—
164	C₂³	C₂²×#88322	1	1	yes	characteristic
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	C₄×#88322	1	4	yes	—
167	C₂×C₄	C₄×#88322	1	8	yes	—
168	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
169	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
170	C₂×C₄	C₂²×#88322	1	8	yes	—
171	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
172	C₂×C₄	C₂²×#88322	1	2	yes	—
173	C₂×C₄	C₂²×#88322	1	2	yes	—
174	C₂×C₄	C₂²×#88322	1	4	yes	—
175	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
176	C₈	C₂×C₄×Dic₅₃	1	4	yes	—
177	C₈	C₂×C₄×Dic₅₃	1	2	yes	—
178	C₈	C₂²×#88322	1	8	yes	—
179	C₈	C₂²×C₄×D₅₃	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	—	2	4	no	—
182	C₂²	—	4	4	no	—
183	C₂²	C₂×C₄×#88322	1	2	yes	—
184	C₂²	C₂×C₄×#88322	1	4	yes	—
185	C₂²	C₂³×#88322	1	1	yes	characteristic
186	C₄	—	2	8	no	—
187	C₄	C₂×C₄×#88322	1	2	yes	—
188	C₄	C₂×C₄×#88322	1	4	yes	—
189	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
190	C₄	C₂³×#88322	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	2	4	no	—
193	C₂	C₂²×C₄×#88322	1	1	yes	characteristic
194	C₂	#20287818	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Fitting	C₂×C₅₃×#405	C₂
socle	C₂²×C₅₃	C₂³×C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#33324
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #20276312	C₂	semidirect
3. C₄×#13133157	C₂	semidirect
5. C₂×#13133157	C₄	semidirect
9. #16810399	C₂²	semidirect
10. C₄×#11719908	C₂²	semidirect
19. #13133157	C₂×C₄	semidirect
20. C₂×#11719908	C₂×C₄	semidirect
115. C₅₃	#33324	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133157	2
#20276312	2
C₄×#13133157	2
C₂×C₅₃×#405	2
#33324	53