Subgroups — #27300440

Order 13568.

322 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810441	C₂	1	2	yes	maximal
2	#20274401	C₂	1	1	yes	maximal, characteristic
3	#20272793	C₂	1	1	yes	maximal, characteristic
4	#20272792	C₂	1	2	yes	maximal
5	C₅₃×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	2	yes	—
7	C₂×#13133128	C₄	1	1	yes	characteristic
8	C₂×#13133127	C₄	1	1	yes	characteristic
9	C₂×#13133127	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
12	#16810469	C₄	1	1	yes	characteristic
13	#16810441	—	2	4	no	—
14	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
17	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×#11719908	—	2	2	no	—
21	C₂×#11719908	—	2	2	no	—
22	C₂×#11719908	—	2	2	no	—
23	#13133128	—	2	2	no	—
24	#13133127	—	2	2	no	—
25	#13133127	—	2	2	no	—
26	C₈×Dic₅₃	—	2	2	no	—
27	C₈×Dic₅₃	D₄	1	2	yes	—
28	C₈×Dic₅₃	Q₈	1	2	yes	—
29	#13133124	C₈	1	2	yes	—
30	#13133124	C₈	1	2	yes	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₅₃	—	2	2	no	—
35	C₂×C₁₆×C₅₃	—	2	2	no	—
36	C₂×C₁₆×C₅₃	—	2	2	no	—
37	C₄×C₈×C₅₃	—	2	2	no	—
38	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₅₃	—	2	1	no	—
44	C₂×C₈×C₅₃	—	2	1	no	—
45	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
46	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₅₃	#41	1	2	yes	—
49	C₂×C₈×C₅₃	#41	1	2	yes	—
50	C₄²×C₅₃	—	2	1	no	—
51	C₄²×C₅₃	—	2	1	no	—
52	C₄×Dic₅₃	—	2	1	no	—
53	C₄×Dic₅₃	—	2	1	no	—
54	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
55	C₄×Dic₅₃	M₁₆	1	2	yes	—
56	C₂×#88322	—	2	1	no	—
57	C₂×#88322	—	2	1	no	—
58	C₂×#88322	—	2	1	no	—
59	C₂×#88322	—	2	1	no	—
60	C₂×#88322	C₄⋊C₄	1	2	yes	—
61	C₂×#88322	M₁₆	1	2	yes	—
62	C₁₆×C₅₃	—	4	2	no	—
63	#11719908	—	4	2	no	—
64	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
74	C₂×Dic₅₃	—	2	2	no	—
75	C₂×Dic₅₃	—	2	2	no	—
76	C₂×Dic₅₃	—	2	1	no	—
77	C₂×Dic₅₃	—	2	1	no	—
78	C₂×Dic₅₃	—	2	2	no	—
79	C₂×Dic₅₃	—	2	2	no	—
80	C₂×Dic₅₃	C₄⋊C₈	1	4	yes	—
81	C₈×C₅₃	—	2	2	no	—
82	C₈×C₅₃	#124	1	2	yes	—
83	C₈×C₅₃	#125	1	2	yes	—
84	#88322	—	2	2	no	—
85	#88322	—	2	2	no	—
86	#12880	—	53	1	no	maximal, Sylow
87	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₅₃	#431	1	1	yes	characteristic
89	C₂²×C₅₃	#431	1	1	yes	characteristic
90	C₂²×C₅₃	#504	1	2	yes	—
91	C₂²×C₅₃	#504	1	2	yes	—
92	C₄×C₅₃	—	4	2	no	—
93	C₄×C₅₃	#382	1	1	yes	characteristic
94	C₄×C₅₃	#423	1	1	yes	characteristic
95	C₄×C₅₃	#432	1	1	yes	characteristic
96	C₄×C₅₃	#432	1	1	yes	characteristic
97	Dic₅₃	—	2	4	no	—
98	Dic₅₃	—	4	1	no	—
99	Dic₅₃	—	4	1	no	—
100	C₂×C₄⋊C₁₆	—	53	2	no	—
101	#1726	—	53	1	no	—
102	#1374	—	53	2	no	—
103	#1374	—	53	1	no	—
104	#1374	D₅₃	1	1	yes	characteristic
105	C₂×C₅₃	#1831	1	1	yes	characteristic
106	C₂×C₅₃	#2151	1	2	yes	—
107	C₂×C₅₃	#2164	1	2	yes	—
108	C₂×C₅₃	#2164	1	1	yes	characteristic
109	C₂×C₅₃	#2164	1	1	yes	characteristic
110	C₂²×C₁₆	—	53	2	no	—
111	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
112	C₂×#97	—	53	1	no	—
113	C₂×#97	—	53	1	no	—
114	C₂×#97	—	53	1	no	—
115	C₂×C₄×C₈	—	53	1	no	—
116	C₂×C₄×C₈	—	53	1	no	—
117	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
118	#376	—	53	1	no	—
119	C₄⋊C₁₆	—	106	4	no	—
120	C₅₃	#12880	1	1	yes	characteristic, complemented, Sylow
121	C₂²×C₈	—	53	1	no	—
122	C₂²×C₈	—	53	1	no	—
123	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
124	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
125	C₂×C₄²	—	53	1	no	—
126	C₂×C₄²	—	53	1	no	—
127	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
128	C₂×C₁₆	—	2	2	no	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	106	2	no	—
132	C₂×C₁₆	—	106	2	no	—
133	C₂×C₁₆	—	106	2	no	—
134	#97	—	106	2	no	—
135	#97	—	106	2	no	—
136	#97	—	106	2	no	—
137	C₄×C₈	—	2	2	no	—
138	C₄×C₈	—	53	2	no	—
139	C₄×C₈	—	53	2	no	—
140	C₄×C₈	—	106	2	no	—
141	#95	—	53	2	no	—
142	#95	—	53	2	no	—
143	C₂²×C₄	—	53	2	no	—
144	C₂²×C₄	—	53	1	no	—
145	C₂²×C₄	—	53	1	no	—
146	C₂²×C₄	C₈×D₅₃	1	2	yes	—
147	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	53	2	no	—
151	C₂×C₈	—	53	2	no	—
152	C₂×C₈	—	106	1	no	—
153	C₂×C₈	—	106	1	no	—
154	C₂×C₈	—	106	1	no	—
155	C₂×C₈	—	106	1	no	—
156	C₂×C₈	#11719912	1	2	yes	—
157	C₂×C₈	#11719945	1	2	yes	—
158	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
159	C₂×C₈	#11719947	1	2	yes	—
160	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
161	C₄²	—	2	1	no	—
162	C₄²	—	2	1	no	—
163	C₄²	—	53	2	no	—
164	C₄²	—	53	2	no	—
165	C₄²	—	106	1	no	—
166	C₄²	—	106	1	no	—
167	C₁₆	—	4	2	no	—
168	C₁₆	—	212	2	no	—
169	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	53	4	no	—
174	C₂×C₄	—	106	2	no	—
175	C₂×C₄	—	106	2	no	—
176	C₂×C₄	—	106	2	no	—
177	C₂×C₄	—	106	2	no	—
178	C₂×C₄	—	106	1	no	—
179	C₂×C₄	—	106	1	no	—
180	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
181	C₂×C₄	#13133173	1	1	yes	characteristic
182	C₂×C₄	#13133174	1	1	yes	characteristic
183	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
184	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
185	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
186	C₈	—	2	2	no	—
187	C₈	—	106	2	no	—
188	C₈	—	106	2	no	—
189	C₈	#13133203	1	2	yes	—
190	C₈	#13133207	1	2	yes	—
191	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
192	C₂²	#16810673	1	1	yes	characteristic
193	C₂²	#16810674	1	1	yes	characteristic
194	C₂²	#16811039	1	2	yes	—
195	C₂²	#16811041	1	2	yes	—
196	C₄	—	4	2	no	—
197	C₄	—	106	4	no	—
198	C₄	—	212	1	no	—
199	C₄	—	212	1	no	—
200	C₄	#16810487	1	1	yes	characteristic
201	C₄	#16810645	1	1	yes	characteristic
202	C₄	#16810677	1	1	yes	characteristic
203	C₄	#16810678	1	1	yes	characteristic
204	C₂	#20274830	1	1	yes	characteristic
205	C₂	#20276472	1	2	yes	—
206	C₂	#20276519	1	2	yes	—
207	C₂	#20276520	1	1	yes	characteristic
208	C₂	#20276523	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂²×C₈
Fitting	C₅₃×#1374	C₂
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1374	D₅₃
hypocenter	C₅₃	#12880
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
120. C₅₃	#12880	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810441	2
#20274401	2
#20272793	2
#20272792	2
C₅₃×#1374	2
#12880	53