Subgroups — #27299489

Order 13568.

310 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276499	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810398	C₂	1	1	yes	maximal, characteristic
3	#20274597	C₂	1	2	yes	maximal
4	#20272812	C₂	1	2	yes	maximal
5	C₂×C₅₃×#368	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	1	yes	characteristic
7	#16810661	C₂²	1	2	yes	—
8	#16810581	—	2	2	no	—
9	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
10	#16810451	C₄	1	2	yes	complemented
11	#16810449	C₄	1	2	yes	complemented
12	#16810398	C₄	1	4	yes	complemented
13	#16810384	—	2	2	no	—
14	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#108×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#368	C₄	1	4	yes	complemented
17	C₂²×#88322	D₄	1	1	yes	characteristic
18	#13133176	—	2	2	no	—
19	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
20	C₂×#11719908	—	2	1	no	—
21	C₂×#11719908	—	2	1	no	—
22	C₂×#11719908	C₂×C₄	1	2	yes	—
23	#13133151	C₂×C₄	1	4	yes	—
24	#13133149	—	2	2	no	—
25	#13133127	—	4	2	no	—
26	C₈×Dic₅₃	—	4	2	no	—
27	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
28	C₈×Dic₅₃	Q₈	1	2	yes	—
29	C₂×M₁₆×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
35	#108×C₅₃	—	2	2	no	—
36	#108×C₅₃	C₂×C₄	1	4	yes	—
37	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₅₃	—	2	1	no	—
39	M₁₆×C₅₃	—	2	4	no	—
40	M₁₆×C₅₃	—	4	2	no	—
41	C₂×C₈×C₅₃	—	4	2	no	—
42	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
47	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
48	C₂²⋊Dic₅₃	—	4	2	no	—
49	C₄⋊Dic₅₃	—	2	4	no	—
50	C₄×Dic₅₃	—	2	1	no	—
51	C₄×Dic₅₃	—	4	2	no	—
52	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
53	C₂×#88322	C₄⋊C₄	1	2	yes	—
54	C₂×#88322	D₈	1	2	yes	—
55	C₂×#88322	Q₁₆	1	2	yes	—
56	C₁₆×C₅₃	C₄²	1	2	yes	—
57	C₁₆×C₅₃	C₄²	1	2	yes	—
58	#11719908	—	2	2	no	—
59	C₂³×C₅₃	#135	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
66	C₂×Dic₅₃	—	2	2	no	—
67	C₂×Dic₅₃	—	4	2	no	—
68	C₂×Dic₅₃	—	4	4	no	—
69	C₈×C₅₃	—	4	4	no	—
70	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
71	C₈×C₅₃	#117	1	1	yes	characteristic
72	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
73	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
74	#88322	#107	1	4	yes	—
75	#12671	—	53	1	no	maximal, Sylow
76	C₂²×C₅₃	—	2	2	no	—
77	C₂²×C₅₃	#427	1	1	yes	characteristic
78	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
79	C₂²×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
80	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
81	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
82	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
83	C₄×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
84	Dic₅₃	—	4	2	no	—
85	Dic₅₃	—	8	4	no	—
86	C₂×#368	D₅₃	1	1	yes	characteristic
87	#2158	—	53	1	no	—
88	C₂×#346	—	53	1	no	—
89	#1772	—	53	2	no	—
90	#1382	—	53	2	no	—
91	C₂×C₅₃	—	2	1	no	—
92	C₂×C₅₃	—	2	1	no	—
93	C₂×C₅₃	C₄×#107	1	1	yes	characteristic
94	C₂×C₅₃	#2179	1	2	yes	—
95	C₂²×C₁₆	—	53	1	no	—
96	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
97	C₂×#108	C₂×D₅₃	1	2	yes	—
98	#427	—	53	2	no	—
99	#405	—	106	2	no	—
100	C₂×C₄×C₈	—	53	1	no	—
101	#368	Dic₅₃	1	4	yes	complemented
102	#367	—	53	2	no	—
103	#366	—	53	2	no	—
104	#346	—	53	4	no	—
105	#339	—	106	2	no	—
106	C₅₃	#12671	1	1	yes	characteristic, complemented, Sylow
107	C₂×M₁₆	—	2	2	no	—
108	C₂²×C₈	—	53	1	no	—
109	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
110	#117	—	106	2	no	—
111	C₂×C₄²	—	53	1	no	—
112	C₂×C₁₆	—	53	2	no	—
113	C₂×C₁₆	—	106	1	no	—
114	C₂×C₁₆	—	106	1	no	—
115	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
116	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
117	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
118	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
119	#108	—	2	2	no	—
120	#108	C₂×Dic₅₃	1	4	yes	—
121	#107	—	53	4	no	—
122	#106	—	106	2	no	—
123	#97	—	212	2	no	—
124	C₄×C₈	—	53	2	no	—
125	C₄×C₈	—	53	2	no	—
126	C₄×C₈	—	212	2	no	—
127	C₂²×C₄	—	106	1	no	—
128	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
129	M₁₆	—	2	4	no	—
130	M₁₆	—	4	2	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	—	53	2	no	—
133	C₂×C₈	—	53	2	no	—
134	C₂×C₈	—	53	2	no	—
135	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
137	C₂×C₈	#11719947	1	1	yes	characteristic
138	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
139	C₂×C₈	#11719949	1	1	yes	characteristic
140	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
141	C₄⋊C₄	—	106	4	no	—
142	C₂²⋊C₄	—	212	2	no	—
143	C₄²	—	53	2	no	—
144	C₄²	—	106	1	no	—
145	C₄²	—	212	2	no	—
146	C₁₆	—	106	2	no	—
147	C₁₆	C₄×Dic₅₃	1	2	yes	—
148	C₁₆	C₄×Dic₅₃	1	2	yes	—
149	C₂³	#13133240	1	1	yes	characteristic
150	C₂×C₄	—	106	2	no	—
151	C₂×C₄	—	212	4	no	—
152	C₂×C₄	—	212	2	no	—
153	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
154	C₂×C₄	#13133179	1	1	yes	characteristic
155	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
156	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
157	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
158	C₂×C₄	#13133236	1	1	yes	characteristic
159	C₈	—	4	4	no	—
160	C₈	—	53	4	no	—
161	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
162	C₈	#13133175	1	1	yes	characteristic
163	C₈	#13133182	1	1	yes	characteristic
164	C₈	Q₈×Dic₅₃	1	1	yes	characteristic
165	C₂²	—	2	2	no	—
166	C₂²	#16810660	1	1	yes	characteristic
167	C₂²	#16810705	1	1	yes	characteristic
168	C₂²	Q₁₆×Dic₅₃	1	1	yes	characteristic
169	C₄	—	424	4	no	—
170	C₄	—	212	2	no	—
171	C₄	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
172	C₄	#107×D₅₃	1	1	yes	characteristic
173	C₄	D₈×Dic₅₃	1	1	yes	characteristic
174	C₄	#16810717	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	#107×Dic₅₃	1	1	yes	characteristic
178	C₂	#20276600	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#368	C₂
socle	C₂²×C₅₃	C₄×D₈
prime radical	C₂³×C₅₃	#135
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#368	D₅₃
hypocenter	C₅₃	#12671
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
10. #16810451	C₄	semidirect
11. #16810449	C₄	semidirect
12. #16810398	C₄	semidirect
16. C₅₃×#368	C₄	semidirect
101. #368	Dic₅₃	semidirect
106. C₅₃	#12671	semidirect

Maximal subgroups

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

subgroup	index
#20276499	2
C₂×#16810398	2
#20274597	2
#20272812	2
C₂×C₅₃×#368	2
#12671	53