Subgroups — #20279750

Order 6784.

452 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133209	C₂	1	2	yes	maximal
2	#16810557	C₂	1	4	yes	maximal
3	#16810521	C₂	1	2	yes	maximal
4	#16810519	C₂	1	2	yes	maximal
5	#16810493	C₂	1	1	yes	maximal, characteristic
6	#16810492	C₂	1	2	yes	maximal
7	C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
9	#13133209	—	2	8	no	—
10	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂×#11719919	C₂²	1	4	yes	—
16	C₂×#11719919	C₂²	1	4	yes	—
17	C₂×#11719919	C₂²	1	4	yes	—
18	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
19	#13133124	C₂²	1	4	yes	—
20	#13133124	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
25	#95×C₅₃	C₂²	1	2	yes	—
26	C₂²×C₄×C₅₃	C₂³	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	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
31	C₂²×Dic₅₃	C₂³	1	4	yes	—
32	C₂²×Dic₅₃	C₂³	1	2	yes	—
33	C₂²×Dic₅₃	C₂³	1	2	yes	—
34	C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₄²×C₅₃	—	2	4	no	—
37	C₄²×C₅₃	Q₈	1	4	yes	—
38	C₄⋊Dic₅₃	—	2	4	no	—
39	C₄⋊Dic₅₃	—	2	4	no	—
40	C₄⋊Dic₅₃	—	4	2	no	—
41	C₄⋊Dic₅₃	D₄	1	4	yes	—
42	C₄⋊Dic₅₃	Q₈	1	4	yes	—
43	#11719919	—	2	8	no	—
44	#11719919	—	2	8	no	—
45	#11719919	—	4	4	no	—
46	C₄×Dic₅₃	—	2	4	no	—
47	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
48	C₂×C₄×C₅₃	—	2	1	no	—
49	C₂×C₄×C₅₃	—	2	1	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
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	2	no	—
55	C₂×C₄×C₅₃	—	2	2	no	—
56	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
59	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
60	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₅₃	#41	1	2	yes	—
62	C₂×C₄×C₅₃	#41	1	2	yes	—
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	4	no	—
68	C₂×Dic₅₃	—	2	4	no	—
69	C₂×Dic₅₃	—	2	4	no	—
70	C₂×Dic₅₃	C₂×D₄	1	4	yes	—
71	C₂×Dic₅₃	C₂×Q₈	1	4	yes	—
72	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
73	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
74	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
75	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
76	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
77	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
78	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
79	C₄×C₅₃	—	2	2	no	—
80	C₄×C₅₃	—	4	2	no	—
81	C₄×C₅₃	—	4	2	no	—
82	C₄×C₅₃	#125	1	4	yes	complemented
83	C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
84	Dic₅₃	—	2	4	no	—
85	Dic₅₃	—	2	4	no	—
86	Dic₅₃	—	4	4	no	—
87	#2552	—	53	1	no	maximal, Sylow
88	C₂×C₅₃	C₂×#125	1	1	yes	characteristic
89	C₂×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
90	C₂×C₅₃	#544	1	1	yes	characteristic
91	C₂×C₅₃	#547	1	1	yes	characteristic
92	C₂×C₅₃	#550	1	1	yes	characteristic
93	C₂×C₅₃	#554	1	1	yes	characteristic
94	C₂×C₅₃	#557	1	1	yes	characteristic
95	C₂×#125	—	53	2	no	—
96	#398	—	53	4	no	—
97	#389	—	53	2	no	—
98	#389	—	53	2	no	—
99	#384	—	53	2	no	—
100	#384	—	53	1	no	—
101	C₄×C₄⋊C₄	—	53	1	no	—
102	C₄×C₄⋊C₄	D₅₃	1	1	yes	characteristic
103	C₅₃	#2552	1	1	yes	characteristic, complemented, Sylow
104	#125	—	106	8	no	—
105	C₂×C₄⋊C₄	—	53	1	no	—
106	C₂×C₄⋊C₄	—	53	1	no	—
107	C₂×C₄⋊C₄	—	53	4	no	—
108	C₂×C₄⋊C₄	—	53	4	no	—
109	C₂×C₄⋊C₄	—	53	2	no	—
110	C₂×C₄⋊C₄	—	53	2	no	—
111	C₂×C₄⋊C₄	—	53	4	no	—
112	C₂×C₄⋊C₄	—	53	2	no	—
113	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
115	C₂×C₄²	—	53	2	no	—
116	C₂×C₄²	C₂×D₅₃	1	2	yes	—
117	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
118	#95	—	53	2	no	—
119	#95	—	53	4	no	—
120	#95	C₂×D₅₃	1	2	yes	—
121	C₂²×C₄	—	53	2	no	—
122	C₂²×C₄	—	53	2	no	—
123	C₂²×C₄	—	53	4	no	—
124	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
125	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
126	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	53	4	no	—
132	C₄⋊C₄	—	53	4	no	—
133	C₄⋊C₄	—	106	8	no	—
134	C₄⋊C₄	—	106	8	no	—
135	C₄⋊C₄	—	106	4	no	—
136	C₄⋊C₄	—	106	4	no	—
137	C₄⋊C₄	—	212	4	no	—
138	C₄⋊C₄	—	212	2	no	—
139	C₄²	—	2	4	no	—
140	C₄²	—	106	4	no	—
141	C₄²	Dic₁₀₆	1	4	yes	—
142	C₂³	C₂³×D₅₃	1	1	yes	characteristic
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	53	4	no	—
152	C₂×C₄	—	53	4	no	—
153	C₂×C₄	—	106	2	no	—
154	C₂×C₄	—	106	2	no	—
155	C₂×C₄	—	106	2	no	—
156	C₂×C₄	—	106	2	no	—
157	C₂×C₄	—	106	4	no	—
158	C₂×C₄	—	106	4	no	—
159	C₂×C₄	—	106	4	no	—
160	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
161	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
162	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
163	C₂×C₄	#11719945	1	2	yes	—
164	C₂×C₄	#11719945	1	2	yes	—
165	C₂×C₄	D₄×D₅₃	1	2	yes	—
166	C₂×C₄	Q₈×D₅₃	1	2	yes	—
167	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
168	C₂²	C₂×Q₈×D₅₃	1	1	yes	characteristic
169	C₂²	C₂²×Dic₁₀₆	1	1	yes	characteristic
170	C₂²	C₂×#11719945	1	1	yes	characteristic
171	C₂²	C₂×#11719945	1	1	yes	characteristic
172	C₂²	#13133263	1	1	yes	characteristic
173	C₂²	#13133264	1	1	yes	characteristic
174	C₄	—	2	2	no	—
175	C₄	—	4	2	no	—
176	C₄	—	4	2	no	—
177	C₄	—	106	4	no	—
178	C₄	—	106	4	no	—
179	C₄	—	212	4	no	—
180	C₄	#13133209	1	4	yes	complemented
181	C₄	#13133216	1	4	yes	—
182	C₂	C₂×#13133209	1	1	yes	characteristic
183	C₂	C₂×#13133216	1	1	yes	characteristic
184	C₂	#16811251	1	1	yes	characteristic
185	C₂	#16811278	1	1	yes	characteristic
186	C₂	#16811294	1	1	yes	characteristic
187	C₂	#16811333	1	1	yes	characteristic
188	C₂	#16811351	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	#2552
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
82. C₄×C₅₃	#125	semidirect
103. C₅₃	#2552	semidirect
180. C₄	#13133209	semidirect