Subgroups — #27298917

Order 13568.

478 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810398	C₂	1	2	yes	maximal
2	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×Dic₅₃	C₂	1	2	yes	maximal
4	#20274401	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	1	yes	characteristic
7	C₂²×#11719908	C₂²	1	1	yes	characteristic
8	C₂×#13133128	C₄	1	1	yes	characteristic
9	C₂×#13133127	C₄	1	2	yes	—
10	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
11	#16810469	C₄	1	1	yes	characteristic
12	#16810398	C₄	1	8	yes	complemented
13	C₄×#11719908	C₄	1	4	yes	complemented
14	C₁₆×Dic₅₃	C₄	1	8	yes	complemented
15	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#346	C₄	1	4	yes	complemented
18	C₂²×#88322	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
20	C₂×#11719908	C₂×C₄	1	4	yes	—
21	C₂×#11719908	C₂×C₄	1	2	yes	—
22	C₂×#11719908	C₂×C₄	1	2	yes	—
23	C₂×#11719908	C₂×C₄	1	4	yes	—
24	#13133128	C₈	1	4	yes	—
25	#13133127	C₈	1	8	yes	—
26	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
27	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
28	#13133124	C₈	1	4	yes	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
33	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
34	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
35	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×Dic₅₃	C₂×C₈	1	4	yes	—
40	C₂×C₈×C₅₃	C₄²	1	2	yes	—
41	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
42	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
44	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
45	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₅₃	C₄²	1	2	yes	—
47	C₄²×C₅₃	C₂×C₈	1	2	yes	—
48	C₄×Dic₅₃	C₄²	1	4	yes	—
49	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
50	C₂×#88322	C₄²	1	4	yes	—
51	C₂×#88322	C₂×C₈	1	4	yes	—
52	C₂×#88322	C₂×C₈	1	4	yes	—
53	C₁₆×C₅₃	C₄²	1	8	yes	—
54	#11719908	C₄²	1	4	yes	—
55	#11719908	C₄²	1	4	yes	—
56	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
57	C₂×C₄×C₅₃	—	2	4	no	—
58	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
59	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
60	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
63	C₂×Dic₅₃	—	2	8	no	—
64	C₂×Dic₅₃	C₄×C₈	1	8	yes	—
65	C₈×C₅₃	C₄×C₈	1	4	yes	—
66	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
67	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
68	C₈×C₅₃	C₂×C₄²	1	2	yes	—
69	#88322	C₄×C₈	1	8	yes	—
70	#12539	—	53	1	no	maximal, Sylow
71	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
72	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
73	C₂²×C₅₃	#504	1	4	yes	—
74	C₄×C₅₃	—	2	4	no	—
75	C₄×C₅₃	C₄³	1	1	yes	characteristic
76	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
77	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
78	Dic₅₃	—	2	8	no	—
79	C₂×#346	—	53	2	no	—
80	C₂×#346	D₅₃	1	1	yes	characteristic
81	C₂×C₄×C₁₆	—	53	2	no	—
82	C₂×C₄×C₁₆	—	53	1	no	—
83	#1726	—	53	1	no	—
84	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
85	C₂×C₅₃	#2108	1	4	yes	—
86	C₂×C₅₃	#2108	1	2	yes	—
87	C₂²×C₁₆	—	53	1	no	—
88	C₂²×C₁₆	—	53	1	no	—
89	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
90	C₂×#97	—	53	2	no	—
91	C₂×#97	—	53	1	no	—
92	C₂×C₄×C₈	—	53	2	no	—
93	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
94	#376	—	53	1	no	—
95	#346	—	53	8	no	—
96	#346	Dic₅₃	1	4	yes	complemented
97	C₄×C₁₆	—	53	8	no	—
98	C₄×C₁₆	—	53	4	no	—
99	C₅₃	#12539	1	1	yes	characteristic, complemented, Sylow
100	C₂²×C₈	—	53	2	no	—
101	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
103	C₂×C₄²	—	53	2	no	—
104	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
105	C₂×C₁₆	—	53	2	no	—
106	C₂×C₁₆	—	53	2	no	—
107	C₂×C₁₆	—	53	4	no	—
108	C₂×C₁₆	—	53	4	no	—
109	C₂×C₁₆	C₄×D₅₃	1	4	yes	—
110	C₂×C₁₆	C₄×D₅₃	1	4	yes	—
111	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
112	#97	—	53	8	no	—
113	#97	—	53	4	no	—
114	C₄×C₈	—	53	4	no	—
115	C₄×C₈	—	53	4	no	—
116	C₄×C₈	C₄×D₅₃	1	2	yes	—
117	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
118	#95	—	53	4	no	—
119	C₂²×C₄	—	53	4	no	—
120	C₂²×C₄	C₈×D₅₃	1	2	yes	—
121	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
122	C₂×C₈	—	53	4	no	—
123	C₂×C₈	—	53	4	no	—
124	C₂×C₈	—	53	4	no	—
125	C₂×C₈	C₈×D₅₃	1	4	yes	—
126	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
127	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
128	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
129	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
130	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
131	C₄²	—	53	4	no	—
132	C₄²	—	53	4	no	—
133	C₄²	C₈×D₅₃	1	2	yes	—
134	C₄²	C₄×Dic₅₃	1	2	yes	—
135	C₁₆	—	53	4	no	—
136	C₁₆	—	53	4	no	—
137	C₁₆	C₄×Dic₅₃	1	8	yes	—
138	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	53	8	no	—
141	C₂×C₄	—	106	8	no	—
142	C₂×C₄	C₈×Dic₅₃	1	4	yes	—
143	C₂×C₄	C₄²×D₅₃	1	2	yes	—
144	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
145	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
146	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
147	C₈	—	53	8	no	—
148	C₈	C₈×Dic₅₃	1	4	yes	—
149	C₈	C₄²×D₅₃	1	1	yes	characteristic
150	C₈	C₄²×D₅₃	1	1	yes	characteristic
151	C₈	C₂×C₄×Dic₅₃	1	2	yes	—
152	C₂²	C₄×C₈×D₅₃	1	2	yes	—
153	C₂²	C₂×C₈×Dic₅₃	1	1	yes	characteristic
154	C₂²	#16811041	1	4	yes	—
155	C₄	—	2	4	no	—
156	C₄	—	106	8	no	—
157	C₄	C₄²×Dic₅₃	1	1	yes	characteristic
158	C₄	C₄×C₈×D₅₃	1	2	yes	—
159	C₄	C₂×C₈×Dic₅₃	1	1	yes	characteristic
160	C₂	C₄×C₈×Dic₅₃	1	1	yes	characteristic
161	C₂	#20276313	1	4	yes	—
162	C₂	#20276314	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂×C₅₃	C₄²×C₈
Fitting	C₂×C₅₃×#346	C₂
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	#12539
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
12. #16810398	C₄	semidirect
13. C₄×#11719908	C₄	semidirect
14. C₁₆×Dic₅₃	C₄	semidirect
17. C₅₃×#346	C₄	semidirect
96. #346	Dic₅₃	semidirect
99. C₅₃	#12539	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810398	2
C₂×C₄×#11719908	2
C₂×C₁₆×Dic₅₃	2
#20274401	2
C₂×C₅₃×#346	2
#12539	53