Subgroups — #20272599

Order 6784.

188 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	#16810493	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#88322	—	2	1	no	—
5	C₂×C₄⋊Dic₅₃	—	2	1	no	—
6	C₂×C₄⋊Dic₅₃	—	2	1	no	—
7	C₂×C₄⋊Dic₅₃	C₄	1	2	yes	—
8	C₄⋊#88322	—	2	1	no	—
9	C₄⋊#88322	C₄	1	2	yes	complemented
10	C₂²×C₈×C₅₃	—	2	1	no	—
11	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₅₃	—	2	1	no	—
13	C₄⋊C₈×C₅₃	C₄	1	2	yes	complemented
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×Dic₅₃	—	2	2	no	—
18	C₂×C₈×C₅₃	—	2	2	no	—
19	C₂×C₈×C₅₃	—	4	1	no	—
20	C₂×C₈×C₅₃	—	4	1	no	—
21	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₅₃	D₄	1	1	yes	characteristic
24	C₄²×C₅₃	Q₈	1	1	yes	characteristic
25	C₄⋊Dic₅₃	—	4	1	no	—
26	C₄⋊Dic₅₃	—	4	1	no	—
27	C₄⋊Dic₅₃	—	4	1	no	—
28	C₄⋊Dic₅₃	—	4	1	no	—
29	C₄⋊Dic₅₃	—	4	2	no	—
30	C₂×#88322	—	2	2	no	—
31	C₂×#88322	—	4	1	no	—
32	C₂×#88322	—	4	1	no	—
33	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	—	2	1	no	—
35	C₂×C₄×C₅₃	—	2	1	no	—
36	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
43	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
44	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
45	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
46	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
47	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
48	C₂×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
49	C₂×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
50	C₂×Dic₅₃	—	4	2	no	—
51	C₂×Dic₅₃	—	4	2	no	—
52	C₂×Dic₅₃	—	4	2	no	—
53	C₈×C₅₃	—	4	2	no	—
54	#88322	—	4	2	no	—
55	C₂²×C₅₃	#95	1	1	yes	characteristic
56	C₂²×C₅₃	#100	1	1	yes	characteristic
57	C₂²×C₅₃	#101	1	1	yes	characteristic
58	C₂²×C₅₃	#102	1	1	yes	characteristic
59	C₂²×C₅₃	#102	1	1	yes	characteristic
60	C₂²×C₅₃	#103	1	1	yes	characteristic
61	C₂²×C₅₃	#103	1	1	yes	characteristic
62	C₄×C₅₃	—	2	1	no	—
63	C₄×C₅₃	—	2	1	no	—
64	C₄×C₅₃	#102	1	1	yes	characteristic
65	C₄×C₅₃	#102	1	1	yes	characteristic
66	C₄×C₅₃	#103	1	1	yes	characteristic
67	C₄×C₅₃	#103	1	1	yes	characteristic
68	C₄×C₅₃	#106	1	1	yes	characteristic
69	C₄×C₅₃	#106	1	1	yes	characteristic
70	C₄×C₅₃	#107	1	1	yes	characteristic
71	C₄×C₅₃	#107	1	1	yes	characteristic
72	Dic₅₃	—	8	2	no	—
73	#1296	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	#331	1	1	yes	characteristic
75	C₂×C₅₃	#332	1	1	yes	characteristic
76	C₂×C₅₃	#332	1	1	yes	characteristic
77	C₂×C₅₃	#333	1	1	yes	characteristic
78	C₂×C₅₃	#340	1	1	yes	characteristic
79	C₂×C₅₃	#340	1	1	yes	characteristic
80	C₂×C₅₃	#343	1	1	yes	characteristic
81	C₂×C₄⋊C₈	—	53	1	no	—
82	C₂×C₄⋊C₈	D₅₃	1	1	yes	characteristic
83	#384	—	53	1	no	—
84	C₅₃	#1296	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₈	—	2	1	no	—
86	C₂²×C₈	—	106	1	no	—
87	C₂×C₄⋊C₄	—	53	2	no	—
88	C₂×C₄⋊C₄	—	106	1	no	—
89	C₂×C₄⋊C₄	—	106	1	no	—
90	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
91	C₄⋊C₈	—	2	1	no	—
92	C₄⋊C₈	—	53	2	no	—
93	C₄⋊C₈	—	106	1	no	—
94	C₄⋊C₈	Dic₅₃	1	2	yes	complemented
95	C₂²×C₄	—	106	2	no	—
96	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
97	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	4	1	no	—
101	C₂×C₈	—	4	1	no	—
102	C₂×C₈	—	106	2	no	—
103	C₂×C₈	—	212	1	no	—
104	C₂×C₈	—	212	1	no	—
105	C₄⋊C₄	—	212	2	no	—
106	C₄⋊C₄	—	212	1	no	—
107	C₄⋊C₄	—	212	1	no	—
108	C₄⋊C₄	—	212	1	no	—
109	C₄⋊C₄	—	212	1	no	—
110	C₄²	Dic₁₀₆	1	1	yes	characteristic
111	C₄²	C₄×D₅₃	1	1	yes	characteristic
112	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
113	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
114	C₂³	#11719921	1	1	yes	characteristic
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	212	2	no	—
118	C₂×C₄	—	212	2	no	—
119	C₂×C₄	—	212	2	no	—
120	C₂×C₄	#11719913	1	1	yes	characteristic
121	C₂×C₄	#11719913	1	1	yes	characteristic
122	C₂×C₄	D₄₂₄	1	1	yes	characteristic
123	C₂×C₄	Dic₂₁₂	1	1	yes	characteristic
124	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
125	C₂×C₄	#11719919	1	1	yes	characteristic
126	C₂×C₄	#11719919	1	1	yes	characteristic
127	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
128	C₂×C₄	#11719921	1	1	yes	characteristic
129	C₂×C₄	#11719922	1	1	yes	characteristic
130	C₂×C₄	#11719923	1	1	yes	characteristic
131	C₂×C₄	#11719924	1	1	yes	characteristic
132	C₂×C₄	#11719925	1	1	yes	characteristic
133	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
134	C₈	—	4	2	no	—
135	C₈	—	212	2	no	—
136	C₂²	#13133124	1	1	yes	characteristic
137	C₂²	#13133133	1	1	yes	characteristic
138	C₂²	#13133135	1	1	yes	characteristic
139	C₂²	#13133137	1	1	yes	characteristic
140	C₂²	#13133139	1	1	yes	characteristic
141	C₂²	#13133140	1	1	yes	characteristic
142	C₂²	#13133142	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	424	2	no	—
146	C₄	#13133138	1	1	yes	characteristic
147	C₄	#13133139	1	1	yes	characteristic
148	C₄	#13133141	1	1	yes	characteristic
149	C₄	#13133142	1	1	yes	characteristic
150	C₄	#13133148	1	1	yes	characteristic
151	C₄	#13133149	1	1	yes	characteristic
152	C₄	#13133150	1	1	yes	characteristic
153	C₄	#13133151	1	1	yes	characteristic
154	C₂	#16810366	1	1	yes	characteristic
155	C₂	#16810368	1	1	yes	characteristic
156	C₂	#16810370	1	1	yes	characteristic
157	C₂	#16810371	1	1	yes	characteristic
158	C₂	#16810385	1	1	yes	characteristic
159	C₂	#16810386	1	1	yes	characteristic
160	C₂	#16810391	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#11719921
commutator	C₂×C₄×C₅₃	C₄²
Frattini	C₂×C₄²	C₂×D₅₃
Fitting	C₂×C₄⋊C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂²⋊C₄
hypercenter	C₂×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#1296
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
9. C₄⋊#88322	C₄	semidirect
13. C₄⋊C₈×C₅₃	C₄	semidirect
84. C₅₃	#1296	semidirect
94. C₄⋊C₈	Dic₅₃	semidirect