Subgroups — #27299102

Order 13568.

354 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₃₂×Dic₅₃	C₂	1	2	yes	maximal
2	#97×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20272795	C₂	1	1	yes	maximal, characteristic
4	#20272794	C₂	1	2	yes	maximal
5	C₅₃×#1375	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂×#13133128	C₄	1	1	yes	characteristic
8	C₂×#13133127	C₄	1	1	yes	characteristic
9	C₂×#13133127	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₄²×Dic₅₃	C₄	1	1	yes	characteristic
13	#16810398	C₄	1	4	yes	complemented
14	C₁₆×Dic₅₃	C₄	1	4	yes	complemented
15	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
16	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
18	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₅₃	—	2	2	no	—
20	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
22	#13133157	—	2	2	no	—
23	#13133157	—	2	2	no	—
24	C₂×#11719908	C₂×C₄	1	4	yes	—
25	#13133128	C₈	1	4	yes	—
26	#13133127	—	2	1	no	—
27	#13133127	—	2	1	no	—
28	#13133127	C₈	1	4	yes	—
29	C₈×Dic₅₃	—	2	2	no	—
30	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
31	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	M₃₂×C₅₃	—	2	2	no	—
36	M₃₂×C₅₃	—	2	2	no	—
37	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
38	#97×C₅₃	—	2	2	no	—
39	C₂²×C₄×C₅₃	—	2	1	no	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×Dic₅₃	—	2	1	no	—
42	C₂²×Dic₅₃	M₁₆	1	2	yes	—
43	C₂×C₈×C₅₃	—	2	1	no	—
44	C₂×C₈×C₅₃	—	2	1	no	—
45	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
47	C₂×C₈×C₅₃	#41	1	2	yes	—
48	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
49	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
50	C₄²×C₅₃	—	2	1	no	—
51	C₄²×C₅₃	C₂×C₈	1	2	yes	—
52	C₄×Dic₅₃	—	2	1	no	—
53	C₄×Dic₅₃	—	4	4	no	—
54	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
55	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
56	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
57	C₄×Dic₅₃	M₁₆	1	2	yes	—
58	C₂×#88322	—	2	1	no	—
59	C₂×#88322	—	2	1	no	—
60	C₂×#88322	C₄²	1	2	yes	—
61	C₂×#88322	C₂×C₈	1	2	yes	—
62	C₂×#88322	C₂×C₈	1	2	yes	—
63	C₂×#88322	C₂×C₈	1	2	yes	—
64	C₁₆×C₅₃	—	2	4	no	—
65	#11719908	—	2	4	no	—
66	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	4	2	no	—
69	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
75	C₂×Dic₅₃	—	2	4	no	—
76	C₂×Dic₅₃	—	2	2	no	—
77	C₂×Dic₅₃	—	4	2	no	—
78	C₂×Dic₅₃	#97	1	2	yes	—
79	C₂×Dic₅₃	#97	1	2	yes	—
80	C₈×C₅₃	—	2	2	no	—
81	C₈×C₅₃	#117	1	2	yes	—
82	C₈×C₅₃	#117	1	2	yes	—
83	#88322	—	2	2	no	—
84	#88322	C₄×C₈	1	4	yes	—
85	#12594	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	—	2	2	no	—
87	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
88	C₂²×C₅₃	#431	1	1	yes	characteristic
89	C₂²×C₅₃	#431	1	1	yes	characteristic
90	C₄×C₅₃	—	4	2	no	—
91	C₄×C₅₃	#376	1	1	yes	characteristic
92	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
93	C₄×C₅₃	#431	1	1	yes	characteristic
94	C₄×C₅₃	#431	1	1	yes	characteristic
95	Dic₅₃	—	2	2	no	—
96	Dic₅₃	—	4	2	no	—
97	Dic₅₃	#347	1	4	yes	—
98	C₄×M₃₂	—	53	2	no	—
99	C₄×#97	—	53	1	no	—
100	#1375	—	53	2	no	—
101	#1375	—	53	1	no	—
102	#1375	D₅₃	1	1	yes	characteristic
103	C₂×C₅₃	—	2	1	no	—
104	C₂×C₅₃	—	2	1	no	—
105	C₂×C₅₃	#1744	1	1	yes	characteristic
106	C₂×C₅₃	C₂×#347	1	2	yes	—
107	C₂×M₃₂	—	53	2	no	—
108	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
109	C₂×#97	—	53	1	no	—
110	C₂×#97	—	53	1	no	—
111	C₂×#97	—	53	1	no	—
112	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
113	C₂×C₄×C₈	—	53	1	no	—
114	C₂×C₄×C₈	—	53	1	no	—
115	C₄³	—	53	1	no	—
116	#346	—	53	4	no	—
117	C₄×C₁₆	—	53	4	no	—
118	C₅₃	#12594	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₈	—	53	1	no	—
120	C₂²×C₈	—	53	1	no	—
121	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
122	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
123	C₂×C₄²	—	53	1	no	—
124	C₂×C₄²	—	53	1	no	—
125	C₂×C₄²	—	106	2	no	—
126	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
127	M₃₂	—	2	2	no	—
128	M₃₂	—	2	2	no	—
129	M₃₂	—	106	2	no	—
130	M₃₂	—	106	2	no	—
131	C₂×C₁₆	—	53	4	no	—
132	C₂×C₁₆	C₄×D₅₃	1	4	yes	—
133	#97	—	2	2	no	—
134	#97	—	53	4	no	—
135	#97	—	53	4	no	—
136	#97	—	106	1	no	—
137	#97	—	106	1	no	—
138	C₄×C₈	—	53	2	no	—
139	C₄×C₈	—	53	2	no	—
140	C₄×C₈	—	106	2	no	—
141	C₂²×C₄	—	2	1	no	—
142	C₂²×C₄	—	53	2	no	—
143	C₂²×C₄	—	106	1	no	—
144	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	53	2	no	—
148	C₂×C₈	—	53	2	no	—
149	C₂×C₈	—	53	2	no	—
150	C₂×C₈	—	53	2	no	—
151	C₂×C₈	—	106	1	no	—
152	C₂×C₈	—	106	1	no	—
153	C₂×C₈	C₈×D₅₃	1	2	yes	—
154	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
155	C₂×C₈	#11719947	1	1	yes	characteristic
156	C₂×C₈	#11719947	1	2	yes	—
157	C₂×C₈	#11719949	1	1	yes	characteristic
158	C₄²	—	2	1	no	—
159	C₄²	—	53	1	no	—
160	C₄²	—	53	1	no	—
161	C₄²	—	53	2	no	—
162	C₄²	—	53	2	no	—
163	C₄²	—	106	1	no	—
164	C₄²	—	212	4	no	—
165	C₄²	C₈×D₅₃	1	2	yes	—
166	C₁₆	—	2	4	no	—
167	C₁₆	—	106	4	no	—
168	C₂³	M₁₆×D₅₃	1	1	yes	characteristic
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	53	2	no	—
172	C₂×C₄	—	53	2	no	—
173	C₂×C₄	—	106	4	no	—
174	C₂×C₄	—	106	2	no	—
175	C₂×C₄	—	212	2	no	—
176	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
177	C₂×C₄	#13133173	1	1	yes	characteristic
178	C₂×C₄	#13133175	1	1	yes	characteristic
179	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
180	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
181	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	—	53	4	no	—
184	C₈	—	106	2	no	—
185	C₈	#13133173	1	2	yes	—
186	C₈	#13133175	1	2	yes	—
187	C₂²	—	2	2	no	—
188	C₂²	#97×D₅₃	1	1	yes	characteristic
189	C₂²	#16810673	1	1	yes	characteristic
190	C₂²	#16810675	1	1	yes	characteristic
191	C₄	—	4	2	no	—
192	C₄	—	53	4	no	—
193	C₄	—	106	2	no	—
194	C₄	—	212	2	no	—
195	C₄	#16810468	1	1	yes	characteristic
196	C₄	C₄×C₈×D₅₃	1	1	yes	characteristic
197	C₄	#16810673	1	1	yes	characteristic
198	C₄	#16810675	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#20274459	1	1	yes	characteristic
202	C₂	#347×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄²×D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1375	C₂
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1375	D₅₃
hypocenter	C₅₃	#12594
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #1375

Complements

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

N	H ≅ G/N	type
13. #16810398	C₄	semidirect
14. C₁₆×Dic₅₃	C₄	semidirect
118. C₅₃	#12594	semidirect

Maximal subgroups

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

subgroup	index
M₃₂×Dic₅₃	2
#97×Dic₅₃	2
#20272795	2
#20272794	2
C₅₃×#1375	2
#12594	53