Subgroups — #27537758

Order 13568.

444 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276624	C₂	1	8	yes	maximal, complemented
2	#20276522	C₂	1	4	yes	maximal
3	#20276518	C₂	1	2	yes	maximal
4	C₅₃×#2960	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#11719908	C₂²	1	4	yes	—
6	C₄⋊#11719908	C₂²	1	8	yes	—
7	C₂²⋊#11719908	C₂²	1	4	yes	—
8	#16810399	C₂²	1	4	yes	—
9	#16810399	C₂²	1	4	yes	—
10	C₄×#11719908	C₂²	1	4	yes	—
11	C₅₃×#533	C₄	1	1	yes	characteristic
12	C₅₃×#446	C₄	1	4	yes	—
13	C₈×Q₈×C₅₃	C₂²	1	4	yes	—
14	C₅₃×#424	C₄	1	2	yes	—
15	C₅₃×#423	C₄	1	1	yes	characteristic
16	C₅₃×#405	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×#11719908	C₂³	1	8	yes	—
19	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
22	C₄⋊Q₈×C₅₃	C₈	1	2	yes	—
23	#125×C₅₃	C₈	1	2	yes	—
24	C₂²⋊Q₈×C₅₃	C₈	1	4	yes	—
25	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
26	#117×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
29	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
30	#97×C₅₃	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₅₃	C₂³	1	4	yes	—
33	C₄×C₈×C₅₃	C₂³	1	2	yes	—
34	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
35	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	M₁₆×C₅₃	—	4	2	no	—
38	C₂×C₈×C₅₃	—	2	2	no	—
39	C₂×C₈×C₅₃	—	4	1	no	—
40	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
42	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
43	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
44	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
45	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
46	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
47	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
48	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
51	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
52	#11719908	—	2	8	no	—
53	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
54	Q₈×C₅₃	—	4	4	no	—
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	—	2	1	no	—
57	C₂×C₄×C₅₃	—	4	2	no	—
58	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
59	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
60	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
62	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
63	C₈×C₅₃	—	2	1	no	—
64	C₈×C₅₃	—	2	2	no	—
65	C₈×C₅₃	—	2	4	no	—
66	C₈×C₅₃	2₋¹⁺⁴	1	2	yes	—
67	#33501	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	—	2	2	no	—
69	C₂²×C₅₃	—	2	1	no	—
70	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
71	C₄×C₅₃	—	2	2	no	—
72	C₄×C₅₃	—	2	2	no	—
73	C₄×C₅₃	—	2	8	no	—
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	#519	1	1	yes	characteristic
76	C₄×C₅₃	#568	1	1	yes	characteristic
77	#2960	D₅₃	1	1	yes	characteristic
78	#2183	—	53	8	no	—
79	#2164	—	53	4	no	—
80	#2163	—	53	2	no	—
81	C₂×C₅₃	—	4	1	no	—
82	C₂×C₅₃	#2982	1	1	yes	characteristic
83	C₂×C₅₃	#3407	1	2	yes	—
84	#533	Dic₅₃	1	1	yes	characteristic
85	#446	Dic₅₃	1	4	yes	—
86	C₈×Q₈	C₂×D₅₃	1	4	yes	—
87	#424	Dic₅₃	1	2	yes	—
88	#423	Dic₅₃	1	1	yes	characteristic
89	#405	C₂×D₅₃	1	2	yes	—
90	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
91	C₄⋊C₁₆	—	53	4	no	—
92	C₄⋊C₁₆	—	53	8	no	—
93	C₂²⋊C₁₆	—	53	4	no	—
94	#346	—	53	4	no	—
95	#346	—	53	4	no	—
96	C₄×C₁₆	—	53	4	no	—
97	C₅₃	#33501	1	1	yes	characteristic, complemented, Sylow
98	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
99	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
100	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
101	C₄⋊Q₈	#88322	1	2	yes	—
102	#125	#88322	1	2	yes	—
103	C₂²⋊Q₈	#88322	1	4	yes	—
104	C₄×Q₈	C₂×Dic₅₃	1	4	yes	—
105	#117	C₂×Dic₅₃	1	2	yes	—
106	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
107	C₂×C₁₆	—	53	8	no	—
108	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
109	C₄⋊C₈	C₂×Dic₅₃	1	8	yes	—
110	#97	C₂×Dic₅₃	1	4	yes	—
111	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
112	C₄×C₈	C₂²×D₅₃	1	4	yes	—
113	C₄×C₈	C₂²×D₅₃	1	2	yes	—
114	C₂×Q₈	C₂×#88322	1	4	yes	—
115	C₂²×C₄	C₂×#88322	1	2	yes	—
116	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
117	M₁₆	—	4	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
121	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
122	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
123	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
124	C₄⋊C₄	C₂×#88322	1	4	yes	—
125	C₄⋊C₄	C₂×#88322	1	8	yes	—
126	C₄⋊C₄	C₂×#88322	1	4	yes	—
127	C₂²⋊C₄	C₂×#88322	1	4	yes	—
128	C₄²	C₂×#88322	1	1	yes	characteristic
129	C₄²	C₂×#88322	1	1	yes	characteristic
130	C₄²	C₂²×Dic₅₃	1	2	yes	—
131	C₄²	C₂²×Dic₅₃	1	4	yes	—
132	C₁₆	—	106	8	no	—
133	C₂³	C₂²×#88322	1	1	yes	characteristic
134	Q₈	—	4	4	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	C₂²×#88322	1	2	yes	—
139	C₂×C₄	C₂²×#88322	1	2	yes	—
140	C₂×C₄	C₂²×#88322	1	8	yes	—
141	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
142	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
143	C₈	—	2	1	no	—
144	C₈	—	2	2	no	—
145	C₈	—	2	4	no	—
146	C₈	#13133265	1	2	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	—	2	1	no	—
149	C₂²	C₂³×#88322	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	2	8	no	—
153	C₄	—	2	1	no	—
154	C₄	#16811101	1	1	yes	characteristic
155	C₄	#16811390	1	1	yes	characteristic
156	C₂	—	4	1	no	—
157	C₂	#20283796	1	1	yes	characteristic
158	C₂	#20287818	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2960	C₂
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2960	D₅₃
hypocenter	C₅₃	#33501
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
1. #20276624	C₂	semidirect
97. C₅₃	#33501	semidirect

Maximal subgroups

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

subgroup	index
#20276624	2
#20276522	2
#20276518	2
C₅₃×#2960	2
#33501	53