Subgroups — #27298924

Order 13568.

350 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276320	C₂	1	1	yes	maximal, characteristic
2	#20276319	C₂	1	2	yes	maximal
3	#20276311	C₂	1	2	yes	maximal
4	#20274470	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#2108	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	#16810578	C₂²	1	2	yes	—
8	#16810401	C₄	1	4	yes	complemented
9	#16810400	C₄	1	8	yes	complemented
10	#16810398	C₄	1	4	yes	complemented
11	C₁₆×Dic₅₃	C₄	1	4	yes	complemented
12	#16810381	C₄	1	1	yes	characteristic
13	#16810380	C₄	1	1	yes	characteristic
14	#16810379	C₄	1	2	yes	—
15	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#405	C₂²	1	1	yes	characteristic
18	C₅₃×#346	C₄	1	2	yes	complemented
19	C₄×C₁₆×C₅₃	C₄	1	2	yes	complemented
20	C₂×#11719917	C₂×C₄	1	2	yes	—
21	#13133176	C₂×C₄	1	2	yes	—
22	#13133157	C₂×C₄	1	4	yes	—
23	#13133157	C₂×C₄	1	4	yes	—
24	C₂×#11719908	C₂×C₄	1	4	yes	—
25	#13133127	C₂×C₄	1	4	yes	—
26	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
27	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
30	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
36	#97×C₅₃	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	M₁₆×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	C₄²	1	2	yes	—
41	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
43	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
44	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₄⋊C₄×C₅₃	—	2	1	no	—
46	C₂²⋊C₄×C₅₃	—	2	1	no	—
47	C₄²×C₅₃	C₄²	1	2	yes	—
48	C₂²⋊Dic₅₃	—	2	2	no	—
49	C₄⋊Dic₅₃	—	2	2	no	—
50	C₄×Dic₅₃	C₄²	1	4	yes	—
51	#11719917	—	2	2	no	—
52	#11719917	—	2	2	no	—
53	C₂×#88322	C₄²	1	4	yes	—
54	C₁₆×C₅₃	C₄²	1	2	yes	—
55	C₁₆×C₅₃	C₄²	1	1	yes	characteristic
56	C₁₆×C₅₃	C₄²	1	4	yes	—
57	C₁₆×C₅₃	C₄²	1	1	yes	characteristic
58	#11719908	C₄²	1	8	yes	—
59	C₂³×C₅₃	#131	1	1	yes	characteristic
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
62	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#131	1	2	yes	—
65	C₂×Dic₅₃	—	2	4	no	—
66	C₈×C₅₃	—	2	2	no	—
67	C₈×C₅₃	C₂×C₄²	1	2	yes	—
68	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
69	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
70	#88322	—	2	4	no	—
71	#12542	—	53	1	no	maximal, Sylow
72	C₂²×C₅₃	—	4	1	no	—
73	C₂²×C₅₃	#405	1	2	yes	—
74	C₂²×C₅₃	#405	1	1	yes	characteristic
75	C₄×C₅₃	—	4	2	no	—
76	C₄×C₅₃	C₄³	1	1	yes	characteristic
77	C₄×C₅₃	#405	1	2	yes	—
78	C₄×C₅₃	#405	1	1	yes	characteristic
79	Dic₅₃	—	4	4	no	—
80	#2110	—	53	2	no	—
81	#2110	—	53	1	no	—
82	#2108	—	53	2	no	—
83	#2108	D₅₃	1	1	yes	characteristic
84	#1747	—	53	1	no	—
85	C₂×C₅₃	—	2	2	no	—
86	C₂×C₅₃	—	2	1	no	—
87	C₂×C₅₃	#1726	1	1	yes	characteristic
88	C₂×M₃₂	—	53	2	no	—
89	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
90	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
91	#405	—	53	2	no	—
92	#405	C₂×D₅₃	1	1	yes	characteristic
93	#347	—	53	8	no	—
94	#347	—	53	4	no	—
95	#346	—	53	4	no	—
96	#346	Dic₅₃	1	2	yes	complemented
97	C₄×C₁₆	—	53	4	no	—
98	C₄×C₁₆	Dic₅₃	1	2	yes	complemented
99	#338	—	53	1	no	—
100	H(3,ℤ/4)	—	53	1	no	—
101	H(3,ℤ/4)	—	53	2	no	—
102	C₅₃	#12542	1	1	yes	characteristic, complemented, Sylow
103	C₂×M₁₆	—	53	2	no	—
104	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
105	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
106	#117	—	53	2	no	—
107	#117	C₄×D₅₃	1	1	yes	characteristic
108	M₃₂	—	53	4	no	—
109	M₃₂	—	53	4	no	—
110	M₃₂	C₄×D₅₃	1	4	yes	—
111	C₂×C₁₆	—	53	4	no	—
112	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
113	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
114	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
115	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
116	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
117	#97	—	53	4	no	—
118	#97	C₄×D₅₃	1	2	yes	—
119	C₄×C₈	—	53	4	no	—
120	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
121	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
122	M₁₆	—	2	2	no	—
123	M₁₆	—	106	2	no	—
124	M₁₆	—	106	2	no	—
125	C₂×C₈	—	53	4	no	—
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	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	106	2	no	—
133	C₂²⋊C₄	—	2	1	no	—
134	C₂²⋊C₄	—	106	2	no	—
135	C₄²	—	53	4	no	—
136	C₄²	C₄×Dic₅₃	1	2	yes	—
137	C₁₆	—	53	8	no	—
138	C₁₆	C₄×Dic₅₃	1	1	yes	characteristic
139	C₁₆	C₄×Dic₅₃	1	2	yes	—
140	C₁₆	C₄×Dic₅₃	1	4	yes	—
141	C₁₆	C₄×Dic₅₃	1	1	yes	characteristic
142	C₂³	#13133223	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	106	4	no	—
145	C₂×C₄	C₄²×D₅₃	1	2	yes	—
146	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
147	C₂×C₄	#13133223	1	1	yes	characteristic
148	C₂×C₄	#13133223	1	2	yes	—
149	C₈	—	2	2	no	—
150	C₈	—	106	4	no	—
151	C₈	C₄²×D₅₃	1	2	yes	—
152	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
153	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
154	C₂²	—	4	1	no	—
155	C₂²	#16810577	1	2	yes	—
156	C₂²	#16810578	1	1	yes	characteristic
157	C₄	—	4	2	no	—
158	C₄	—	212	4	no	—
159	C₄	C₄²×Dic₅₃	1	1	yes	characteristic
160	C₄	#16810577	1	2	yes	—
161	C₄	#16810578	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	2	2	no	—
164	C₂	#20274401	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₄×Dic₅₃
commutator	C₄×C₅₃	C₄³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2108	C₂
socle	C₂×C₅₃	#1726
prime radical	C₂³×C₅₃	#131
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2108	D₅₃
hypocenter	C₅₃	#12542
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
8. #16810401	C₄	semidirect
9. #16810400	C₄	semidirect
10. #16810398	C₄	semidirect
11. C₁₆×Dic₅₃	C₄	semidirect
18. C₅₃×#346	C₄	semidirect
19. C₄×C₁₆×C₅₃	C₄	semidirect
96. #346	Dic₅₃	semidirect
98. C₄×C₁₆	Dic₅₃	semidirect
102. C₅₃	#12542	semidirect

Maximal subgroups

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

subgroup	index
#20276320	2
#20276319	2
#20276311	2
#20274470	2
C₅₃×#2108	2
#12542	53