Subgroups — #20279741

Order 6784.

556 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133217	C₂	1	2	yes	maximal
2	C₂×#13133216	C₂	1	2	yes	maximal
3	#16810544	C₂	1	4	yes	maximal
4	#16810522	C₂	1	2	yes	maximal
5	#16810521	C₂	1	2	yes	maximal
6	#16810493	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₀₆	C₂²	1	4	yes	—
10	#13133217	—	2	8	no	—
11	#13133216	—	2	8	no	—
12	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
14	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
18	C₂×#11719919	C₂²	1	4	yes	—
19	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
20	#13133124	C₂²	1	4	yes	—
21	#13133124	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
26	#95×C₅₃	C₂²	1	2	yes	—
27	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
28	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
30	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
32	C₂²×Dic₅₃	C₂³	1	4	yes	—
33	C₂²×Dic₅₃	C₂³	1	2	yes	—
34	C₂²×Dic₅₃	C₂³	1	2	yes	—
35	C₂×Dic₁₀₆	—	2	8	no	—
36	C₂×Dic₁₀₆	—	2	8	no	—
37	C₂×Dic₁₀₆	—	4	4	no	—
38	C₄⋊C₄×C₅₃	D₄	1	4	yes	—
39	C₄⋊C₄×C₅₃	Q₈	1	4	yes	complemented
40	C₄²×C₅₃	—	2	4	no	—
41	C₄²×C₅₃	Q₈	1	4	yes	complemented
42	C₄⋊Dic₅₃	—	2	4	no	—
43	C₄⋊Dic₅₃	—	2	4	no	—
44	C₄⋊Dic₅₃	—	2	8	no	—
45	C₄⋊Dic₅₃	—	4	2	no	—
46	C₄⋊Dic₅₃	D₄	1	4	yes	—
47	C₄⋊Dic₅₃	Q₈	1	4	yes	complemented
48	#11719919	—	2	8	no	—
49	C₄×Dic₅₃	—	2	4	no	—
50	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	—	2	2	no	—
54	C₂×C₄×C₅₃	—	2	1	no	—
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
58	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
62	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
65	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
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	—	2	4	no	—
71	C₂×Dic₅₃	—	2	4	no	—
72	C₂×Dic₅₃	—	2	4	no	—
73	C₂×Dic₅₃	C₂×D₄	1	4	yes	—
74	C₂×Dic₅₃	C₂×Q₈	1	4	yes	—
75	Dic₁₀₆	—	4	16	no	—
76	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
77	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
78	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
79	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
80	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
81	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
82	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
83	C₄×C₅₃	—	2	2	no	—
84	C₄×C₅₃	—	2	4	no	—
85	C₄×C₅₃	—	2	4	no	—
86	C₄×C₅₃	C₄⋊Q₈	1	4	yes	complemented
87	C₄×C₅₃	C₄⋊Q₈	1	4	yes	complemented
88	Dic₅₃	—	2	4	no	—
89	Dic₅₃	—	2	4	no	—
90	Dic₅₃	—	4	4	no	—
91	#2551	—	53	1	no	maximal, Sylow
92	C₂×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
93	C₂×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
94	C₂×C₅₃	#544	1	1	yes	characteristic
95	C₂×C₅₃	D₄²	1	1	yes	characteristic
96	C₂×C₅₃	D₄×Q₈	1	1	yes	characteristic
97	C₂×C₅₃	D₄×Q₈	1	1	yes	characteristic
98	C₂×C₅₃	Q₈²	1	1	yes	characteristic
99	C₂×C₄⋊Q₈	—	53	2	no	—
100	C₂×C₄⋊Q₈	—	53	2	no	—
101	#395	—	53	4	no	—
102	#389	—	53	2	no	—
103	#389	—	53	2	no	—
104	#384	—	53	1	no	—
105	C₄×C₄⋊C₄	—	53	1	no	—
106	C₄×C₄⋊C₄	D₅₃	1	1	yes	characteristic
107	C₅₃	#2551	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₈	—	53	4	no	—
109	C₄⋊Q₈	—	106	8	no	—
110	C₄⋊Q₈	—	106	8	no	—
111	C₂×C₄⋊C₄	—	53	1	no	—
112	C₂×C₄⋊C₄	—	53	1	no	—
113	C₂×C₄⋊C₄	—	53	4	no	—
114	C₂×C₄⋊C₄	—	53	2	no	—
115	C₂×C₄⋊C₄	—	53	2	no	—
116	C₂×C₄⋊C₄	—	53	4	no	—
117	C₂×C₄⋊C₄	—	53	2	no	—
118	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
120	C₂×C₄²	—	53	2	no	—
121	C₂×C₄²	C₂×D₅₃	1	2	yes	—
122	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
123	#95	—	53	2	no	—
124	#95	—	53	4	no	—
125	#95	C₂×D₅₃	1	2	yes	—
126	C₂×Q₈	—	106	8	no	—
127	C₂×Q₈	—	106	8	no	—
128	C₂×Q₈	—	212	4	no	—
129	C₂²×C₄	—	53	2	no	—
130	C₂²×C₄	—	53	2	no	—
131	C₂²×C₄	—	53	4	no	—
132	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
134	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
135	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
137	C₄⋊C₄	—	53	4	no	—
138	C₄⋊C₄	—	53	4	no	—
139	C₄⋊C₄	—	106	8	no	—
140	C₄⋊C₄	—	106	8	no	—
141	C₄⋊C₄	—	106	4	no	—
142	C₄⋊C₄	—	106	4	no	—
143	C₄⋊C₄	—	212	2	no	—
144	C₄⋊C₄	Dic₁₀₆	1	4	yes	complemented
145	C₄⋊C₄	D₂₁₂	1	4	yes	—
146	C₄²	—	2	4	no	—
147	C₄²	—	106	4	no	—
148	C₄²	Dic₁₀₆	1	4	yes	complemented
149	C₂³	C₂³×D₅₃	1	1	yes	characteristic
150	Q₈	—	212	16	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	53	4	no	—
158	C₂×C₄	—	53	4	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	4	no	—
164	C₂×C₄	—	106	4	no	—
165	C₂×C₄	—	106	4	no	—
166	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
167	C₂×C₄	C₂×Dic₁₀₆	1	4	yes	—
168	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
169	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
170	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
171	C₂×C₄	C₂×D₂₁₂	1	4	yes	—
172	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
173	C₂×C₄	D₄×D₅₃	1	2	yes	—
174	C₂×C₄	Q₈×D₅₃	1	2	yes	—
175	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
176	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
177	C₂²	C₂×Q₈×D₅₃	1	1	yes	characteristic
178	C₂²	C₂²×Dic₁₀₆	1	1	yes	characteristic
179	C₂²	C₂²×Dic₁₀₆	1	1	yes	characteristic
180	C₂²	#13133263	1	1	yes	characteristic
181	C₂²	#13133264	1	1	yes	characteristic
182	C₄	—	2	4	no	—
183	C₄	—	2	2	no	—
184	C₄	—	2	4	no	—
185	C₄	—	106	4	no	—
186	C₄	—	106	4	no	—
187	C₄	—	212	4	no	—
188	C₄	#13133216	1	4	yes	complemented
189	C₄	#13133217	1	4	yes	complemented
190	C₂	C₂×#13133216	1	1	yes	characteristic
191	C₂	C₂×#13133217	1	1	yes	characteristic
192	C₂	#16811251	1	1	yes	characteristic
193	C₂	D₄×D₂₁₂	1	1	yes	characteristic
194	C₂	D₄×Dic₁₀₆	1	1	yes	characteristic
195	C₂	Q₈×D₂₁₂	1	1	yes	characteristic
196	C₂	Q₈×Dic₁₀₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₄×C₄⋊C₄×C₅₃	C₂
hypercenter	C₄×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#2551
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
39. C₄⋊C₄×C₅₃	Q₈	semidirect
41. C₄²×C₅₃	Q₈	semidirect
47. C₄⋊Dic₅₃	Q₈	semidirect
86. C₄×C₅₃	C₄⋊Q₈	semidirect
87. C₄×C₅₃	C₄⋊Q₈	semidirect
107. C₅₃	#2551	semidirect
144. C₄⋊C₄	Dic₁₀₆	semidirect
148. C₄²	Dic₁₀₆	semidirect
188. C₄	#13133216	semidirect
189. C₄	#13133217	semidirect