Subgroups — #20277674

Order 6784.

480 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	#16810528	C₂	1	2	yes	maximal
3	#16810504	C₂	1	4	yes	maximal
4	#16810488	C₂	1	2	yes	maximal
5	C₄⋊C₄×Dic₅₃	C₂	1	2	yes	maximal
6	C₄⋊C₄×Dic₅₃	C₂	1	2	yes	maximal
7	C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#384	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
10	C₄×Dic₁₀₆	C₄	1	8	yes	complemented
11	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂×#11719919	C₂²	1	2	yes	—
15	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
16	C₂×C₄×Dic₅₃	C₂²	1	4	yes	—
17	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
18	#13133124	C₂²	1	2	yes	—
19	#13133124	C₂²	1	4	yes	—
20	#13133124	C₂²	1	4	yes	—
21	#13133124	C₂²	1	4	yes	—
22	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
27	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
30	C₂²×Dic₅₃	C₂³	1	2	yes	—
31	C₂²×Dic₅₃	C₂³	1	2	yes	—
32	C₂²×Dic₅₃	C₂³	1	2	yes	—
33	C₂²×Dic₅₃	C₂³	1	2	yes	—
34	C₂×Dic₁₀₆	—	2	2	no	—
35	C₂×Dic₁₀₆	—	2	2	no	—
36	C₂×Dic₁₀₆	C₂×C₄	1	4	yes	—
37	C₄⋊C₄×C₅₃	—	2	4	no	—
38	C₄⋊C₄×C₅₃	—	2	4	no	—
39	C₄⋊C₄×C₅₃	—	4	2	no	—
40	C₄²×C₅₃	C₂×C₄	1	4	yes	—
41	C₄⋊Dic₅₃	—	2	2	no	—
42	C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
43	C₄⋊Dic₅₃	Q₈	1	4	yes	complemented
44	#11719919	C₂×C₄	1	8	yes	—
45	C₄×Dic₅₃	—	2	4	no	—
46	C₄×Dic₅₃	—	4	4	no	—
47	C₄×Dic₅₃	C₂×C₄	1	8	yes	—
48	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
49	C₂×C₄×C₅₃	—	2	1	no	—
50	C₂×C₄×C₅₃	—	2	1	no	—
51	C₂×C₄×C₅₃	—	2	4	no	—
52	C₂×C₄×C₅₃	—	2	4	no	—
53	C₂×C₄×C₅₃	—	2	4	no	—
54	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₅₃	#41	1	2	yes	—
59	C₂×C₄×C₅₃	#41	1	2	yes	—
60	C₂×C₄×C₅₃	#41	1	2	yes	—
61	C₂×Dic₅₃	—	2	2	no	—
62	C₂×Dic₅₃	—	2	2	no	—
63	C₂×Dic₅₃	—	2	2	no	—
64	C₂×Dic₅₃	—	2	2	no	—
65	C₂×Dic₅₃	—	2	2	no	—
66	C₂×Dic₅₃	—	2	2	no	—
67	C₂×Dic₅₃	—	2	2	no	—
68	C₂×Dic₅₃	—	2	2	no	—
69	C₂×Dic₅₃	—	2	2	no	—
70	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
71	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
72	C₂×Dic₅₃	C₂×Q₈	1	4	yes	—
73	Dic₁₀₆	—	2	8	no	—
74	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
75	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
76	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
77	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
78	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
79	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
80	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
81	C₄×C₅₃	—	2	2	no	—
82	C₄×C₅₃	—	4	4	no	—
83	C₄×C₅₃	#117	1	4	yes	—
84	C₄×C₅₃	C₄×Q₈	1	4	yes	—
85	Dic₅₃	—	2	4	no	—
86	Dic₅₃	—	2	4	no	—
87	Dic₅₃	—	2	4	no	—
88	Dic₅₃	—	4	2	no	—
89	#2355	—	53	1	no	maximal, Sylow
90	C₂×C₅₃	C₂×#117	1	1	yes	characteristic
91	C₂×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
92	C₂×C₅₃	#520	1	1	yes	characteristic
93	C₂×C₅₃	#554	1	1	yes	characteristic
94	C₂×C₅₃	#556	1	1	yes	characteristic
95	C₂×C₅₃	#557	1	1	yes	characteristic
96	C₂×C₅₃	#559	1	1	yes	characteristic
97	C₂×C₄×Q₈	—	53	1	no	—
98	#391	—	53	2	no	—
99	#387	—	53	4	no	—
100	#384	D₅₃	1	1	yes	characteristic
101	#382	—	53	2	no	—
102	C₄×C₄⋊C₄	—	53	2	no	—
103	C₄×C₄⋊C₄	—	53	2	no	—
104	C₄×C₄⋊C₄	—	53	1	no	—
105	C₅₃	#2355	1	1	yes	characteristic, complemented, Sylow
106	C₂²×Q₈	—	53	1	no	—
107	C₄×Q₈	—	53	8	no	—
108	C₂×C₄⋊C₄	—	53	2	no	—
109	C₂×C₄⋊C₄	—	53	1	no	—
110	C₂×C₄⋊C₄	—	53	1	no	—
111	C₂×C₄⋊C₄	—	53	1	no	—
112	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
113	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
114	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
115	C₂×C₄²	—	53	2	no	—
116	C₂×C₄²	—	53	2	no	—
117	C₂×C₄²	—	53	4	no	—
118	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
119	#95	—	53	4	no	—
120	#95	—	53	2	no	—
121	#95	—	53	4	no	—
122	#95	—	53	4	no	—
123	C₂×Q₈	—	53	4	no	—
124	C₂×Q₈	—	106	2	no	—
125	C₂×Q₈	—	106	2	no	—
126	C₂²×C₄	—	53	2	no	—
127	C₂²×C₄	—	53	2	no	—
128	C₂²×C₄	—	53	2	no	—
129	C₂²×C₄	—	53	2	no	—
130	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
131	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
132	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
134	C₄⋊C₄	—	2	4	no	—
135	C₄⋊C₄	—	2	4	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	53	8	no	—
138	C₄⋊C₄	—	53	4	no	—
139	C₄⋊C₄	—	53	4	no	—
140	C₄⋊C₄	—	106	2	no	—
141	C₄²	—	53	8	no	—
142	C₄²	—	106	4	no	—
143	C₄²	—	212	4	no	—
144	C₄²	C₄×D₅₃	1	4	yes	—
145	C₂³	C₂³×D₅₃	1	1	yes	characteristic
146	Q₈	—	106	8	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	53	4	no	—
153	C₂×C₄	—	53	4	no	—
154	C₂×C₄	—	53	4	no	—
155	C₂×C₄	—	106	2	no	—
156	C₂×C₄	—	106	2	no	—
157	C₂×C₄	—	106	2	no	—
158	C₂×C₄	—	106	2	no	—
159	C₂×C₄	—	106	2	no	—
160	C₂×C₄	—	106	2	no	—
161	C₂×C₄	—	106	2	no	—
162	C₂×C₄	—	106	2	no	—
163	C₂×C₄	—	106	2	no	—
164	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
165	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
166	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
167	C₂×C₄	#11719947	1	2	yes	—
168	C₂×C₄	#11719947	1	2	yes	—
169	C₂×C₄	Q₈×D₅₃	1	2	yes	—
170	C₂×C₄	#11719949	1	2	yes	—
171	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
172	C₂²	C₂×Q₈×D₅₃	1	1	yes	characteristic
173	C₂²	C₂×#11719947	1	1	yes	characteristic
174	C₂²	C₂×#11719947	1	1	yes	characteristic
175	C₂²	C₂×#11719949	1	1	yes	characteristic
176	C₂²	#13133262	1	1	yes	characteristic
177	C₂²	#13133265	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	4	4	no	—
180	C₄	—	106	4	no	—
181	C₄	—	106	4	no	—
182	C₄	—	106	4	no	—
183	C₄	—	212	2	no	—
184	C₄	#13133175	1	4	yes	—
185	C₄	#13133182	1	4	yes	—
186	C₂	C₂×#13133175	1	1	yes	characteristic
187	C₂	C₂×#13133182	1	1	yes	characteristic
188	C₂	#16811104	1	1	yes	characteristic
189	C₂	#16811334	1	1	yes	characteristic
190	C₂	#16811347	1	1	yes	characteristic
191	C₂	#16811348	1	1	yes	characteristic
192	C₂	#16811359	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#384	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	#384	D₅₃
hypocenter	C₅₃	#2355
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #384

Complements

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

N	H ≅ G/N	type
10. C₄×Dic₁₀₆	C₄	semidirect
43. C₄⋊Dic₅₃	Q₈	semidirect
105. C₅₃	#2355	semidirect