Subgroups — #27270979

Order 13568.

200 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810444	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133157	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133157	—	2	1	no	—
5	C₂×#13133157	C₄	1	1	yes	characteristic
6	C₂×#13133157	C₄	1	1	yes	characteristic
7	#16810444	—	2	1	no	—
8	#16810444	C₄	1	2	yes	—
9	#16810399	—	2	1	no	—
10	C₄×#11719908	—	2	1	no	—
11	C₂×M₃₂×C₅₃	—	2	1	no	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#364	—	2	1	no	—
14	C₅₃×#364	C₄	1	2	yes	—
15	#13133157	—	2	1	no	—
16	#13133157	—	2	1	no	—
17	#13133157	—	2	2	no	—
18	#13133157	—	4	1	no	—
19	#13133157	C₈	1	2	yes	—
20	#13133157	C₈	1	2	yes	—
21	C₂×#11719908	—	2	1	no	—
22	C₂×#11719908	—	2	1	no	—
23	C₂×#11719908	—	4	1	no	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
27	M₃₂×C₅₃	—	2	2	no	—
28	M₃₂×C₅₃	—	4	1	no	—
29	C₂×C₁₆×C₅₃	—	4	1	no	—
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	—	2	1	no	—
35	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
43	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
44	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
45	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
48	C₄²×C₅₃	—	2	1	no	—
49	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₁₆×C₅₃	—	4	2	no	—
52	#11719908	—	2	2	no	—
53	#11719908	—	2	2	no	—
54	#11719908	—	4	2	no	—
55	C₂³×C₅₃	#108	1	1	yes	characteristic
56	C₂×C₄×C₅₃	—	2	1	no	—
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	4	2	no	—
59	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
65	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
66	C₈×C₅₃	#97	1	1	yes	characteristic
67	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
68	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
69	C₈×C₅₃	#102	1	1	yes	characteristic
70	C₈×C₅₃	#102	1	1	yes	characteristic
71	C₈×C₅₃	#103	1	1	yes	characteristic
72	C₈×C₅₃	#103	1	1	yes	characteristic
73	#7162	—	53	1	no	maximal, Sylow
74	C₂²×C₅₃	—	4	1	no	—
75	C₂²×C₅₃	#334	1	1	yes	characteristic
76	C₂²×C₅₃	#335	1	1	yes	characteristic
77	C₂²×C₅₃	#341	1	1	yes	characteristic
78	C₄×C₅₃	—	2	2	no	—
79	C₄×C₅₃	—	4	1	no	—
80	C₄×C₅₃	#334	1	1	yes	characteristic
81	C₄×C₅₃	#335	1	1	yes	characteristic
82	C₄×C₅₃	#336	1	1	yes	characteristic
83	C₄×C₅₃	#340	1	1	yes	characteristic
84	C₂×#364	—	53	1	no	—
85	C₂×#364	D₅₃	1	1	yes	characteristic
86	C₄×M₃₂	—	53	1	no	—
87	C₂×C₅₃	—	2	1	no	—
88	C₂×C₅₃	—	2	1	no	—
89	C₂×C₅₃	—	2	1	no	—
90	C₂×C₅₃	#1277	1	1	yes	characteristic
91	C₂×M₃₂	—	2	1	no	—
92	C₂×M₃₂	—	53	1	no	—
93	C₂×M₃₂	—	53	1	no	—
94	C₂×M₃₂	—	106	1	no	—
95	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
96	#364	—	2	1	no	—
97	#364	—	53	2	no	—
98	#364	—	106	1	no	—
99	#364	Dic₅₃	1	2	yes	—
100	#346	—	106	1	no	—
101	C₄×C₁₆	—	106	1	no	—
102	C₅₃	#7162	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
104	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
105	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	—	4	1	no	—
108	M₃₂	—	53	2	no	—
109	M₃₂	—	53	2	no	—
110	M₃₂	—	106	1	no	—
111	M₃₂	—	106	1	no	—
112	M₃₂	—	106	2	no	—
113	M₃₂	—	212	1	no	—
114	C₂×C₁₆	—	4	1	no	—
115	C₂×C₁₆	—	106	1	no	—
116	C₂×C₁₆	—	106	1	no	—
117	C₂×C₁₆	—	212	1	no	—
118	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
119	C₄×C₈	D₂₁₂	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
121	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	#11719919	1	1	yes	characteristic
124	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
125	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
126	C₂×C₈	#11719912	1	1	yes	characteristic
127	C₂×C₈	#11719912	1	1	yes	characteristic
128	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
129	C₂×C₈	#11719921	1	1	yes	characteristic
130	C₂×C₈	#11719921	1	1	yes	characteristic
131	C₂×C₈	#11719922	1	1	yes	characteristic
132	C₂×C₈	#11719923	1	1	yes	characteristic
133	C₂×C₈	#11719924	1	1	yes	characteristic
134	C₂×C₈	#11719925	1	1	yes	characteristic
135	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
136	C₄²	—	2	1	no	—
137	C₄²	#11719919	1	1	yes	characteristic
138	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
139	C₁₆	—	4	2	no	—
140	C₁₆	—	106	2	no	—
141	C₁₆	—	106	2	no	—
142	C₁₆	—	212	2	no	—
143	C₂³	#13133152	1	1	yes	characteristic
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	#13133124	1	1	yes	characteristic
148	C₂×C₄	#13133146	1	1	yes	characteristic
149	C₂×C₄	#13133146	1	1	yes	characteristic
150	C₂×C₄	#13133148	1	1	yes	characteristic
151	C₂×C₄	#13133150	1	1	yes	characteristic
152	C₂×C₄	#13133152	1	1	yes	characteristic
153	C₈	C₈×Dic₅₃	1	1	yes	characteristic
154	C₈	#13133127	1	1	yes	characteristic
155	C₈	#13133129	1	1	yes	characteristic
156	C₈	#13133129	1	1	yes	characteristic
157	C₈	#13133137	1	1	yes	characteristic
158	C₈	#13133138	1	1	yes	characteristic
159	C₈	#13133140	1	1	yes	characteristic
160	C₈	#13133141	1	1	yes	characteristic
161	C₂²	—	4	1	no	—
162	C₂²	#16810373	1	1	yes	characteristic
163	C₂²	#16810375	1	1	yes	characteristic
164	C₂²	#16810387	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	4	1	no	—
167	C₄	#16810373	1	1	yes	characteristic
168	C₄	#16810375	1	1	yes	characteristic
169	C₄	#16810377	1	1	yes	characteristic
170	C₄	#16810385	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	#20272549	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13133138
commutator	C₈×C₅₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₂×C₅₃×#364	C₂
socle	C₂×C₅₃	#1277
prime radical	C₂³×C₅₃	#108
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	#7162
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
102. C₅₃	#7162	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810444	2
C₄×#13133157	2
C₂×C₅₃×#364	2
#7162	53