Subgroups — #20283458

Order 6784.

492 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133223	C₂	1	2	yes	maximal, complemented
2	#16811173	C₂	1	1	yes	maximal, characteristic
3	#16810700	C₂	1	2	yes	maximal, complemented
4	#16810693	C₂	1	4	yes	maximal, complemented
5	#16810688	C₂	1	2	yes	maximal, complemented
6	#16810589	C₂	1	2	yes	maximal, complemented
7	C₄×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11719945	C₂²	1	2	yes	—
10	#13133223	—	2	8	no	—
11	C₂×#11719917	C₂²	1	1	yes	characteristic
12	C₂×#11719917	C₂²	1	1	yes	characteristic
13	C₂×#11719912	C₂²	1	4	yes	—
14	C₂×C₈×D₅₃	C₂²	1	4	yes	—
15	#13133217	C₄	1	1	yes	characteristic
16	#13133214	C₄	1	1	yes	characteristic
17	#13133205	C₄	1	2	yes	—
18	#13133191	C₄	1	2	yes	—
19	#13133191	C₄	1	2	yes	—
20	C₄×D₂₁₂	C₂²	1	2	yes	—
21	C₄×D₂₁₂	C₂²	1	2	yes	—
22	#13133129	C₂²	1	4	yes	—
23	#13133129	C₂²	1	4	yes	—
24	#13133128	C₂²	1	2	yes	—
25	C₄×#88322	C₂²	1	2	yes	—
26	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
30	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊D₅₃	C₂×C₄	1	4	yes	—
35	#11719945	—	2	4	no	—
36	#11719945	—	2	4	no	—
37	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
38	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
39	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₅₃	C₂³	1	4	yes	—
41	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
42	C₂×C₈×C₅₃	—	2	4	no	—
43	C₂×C₈×C₅₃	C₂³	1	2	yes	—
44	C₂×C₈×C₅₃	C₂³	1	2	yes	—
45	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	#11719921	C₂×C₄	1	4	yes	—
50	#11719921	C₂×C₄	1	4	yes	—
51	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
52	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
53	#11719917	—	2	2	no	—
54	#11719917	D₄	1	4	yes	complemented
55	C₂×#88322	C₂³	1	2	yes	—
56	C₂×#88322	C₂³	1	2	yes	—
57	#11719912	—	2	8	no	—
58	C₈×D₅₃	—	2	8	no	—
59	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×D₅₃	C₂²×C₄	1	4	yes	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	4	1	no	—
63	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₅₃	#41	1	2	yes	—
70	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
71	C₂²⋊D₅₃	—	2	8	no	—
72	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
73	D₂₁₂	—	2	4	no	—
74	D₂₁₂	—	2	4	no	—
75	D₂₁₂	—	4	2	no	—
76	C₄×D₅₃	—	2	4	no	—
77	C₄×D₅₃	—	2	4	no	—
78	Dic₁₀₆	—	2	4	no	—
79	C₈×C₅₃	—	2	2	no	—
80	C₈×C₅₃	—	2	2	no	—
81	#88322	—	2	2	no	—
82	#88322	C₂×D₄	1	4	yes	—
83	C₂²×C₅₃	—	2	1	no	—
84	C₂²×C₅₃	—	2	1	no	—
85	C₂²×C₅₃	C₄×D₄	1	2	yes	—
86	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
87	C₂×D₅₃	—	2	4	no	—
88	C₂×D₅₃	—	2	4	no	—
89	C₂×D₅₃	—	2	4	no	—
90	C₄×C₅₃	—	2	1	no	—
91	C₄×C₅₃	—	2	1	no	—
92	C₄×C₅₃	—	2	1	no	—
93	C₄×C₅₃	C₄×D₄	1	2	yes	—
94	C₄×C₅₃	#131	1	2	yes	complemented
95	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
96	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
97	Dic₅₃	—	2	4	no	—
98	#2935	—	53	1	no	maximal, Sylow
99	C₂×C₅₃	—	2	2	no	—
100	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
101	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
102	C₂×C₅₃	#568	1	1	yes	characteristic
103	D₅₃	—	4	4	no	—
104	C₂×#131	—	53	2	no	—
105	#532	—	53	1	no	—
106	#436	—	53	2	no	—
107	#435	—	53	4	no	—
108	C₈×D₄	—	53	2	no	—
109	C₂×C₄⋊C₈	D₅₃	1	1	yes	characteristic, complemented
110	#408	—	53	2	no	—
111	C₄×M₁₆	—	53	1	no	—
112	C₅₃	#2935	1	1	yes	characteristic, complemented, Sylow
113	C₂×#41	—	53	2	no	—
114	#131	—	106	8	no	—
115	C₂×M₁₆	—	53	1	no	—
116	C₂×M₁₆	—	53	1	no	—
117	C₂×M₁₆	—	53	4	no	—
118	C₂²×C₈	—	53	4	no	—
119	C₂²×C₈	C₂×D₅₃	1	2	yes	—
120	C₄⋊Q₈	—	53	1	no	—
121	#127	—	53	1	no	—
122	#124	—	53	2	no	—
123	#121	—	53	2	no	—
124	#121	—	53	2	no	—
125	C₄×D₄	—	53	2	no	—
126	C₄×D₄	—	53	2	no	—
127	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
128	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
129	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
130	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
131	C₂²⋊C₈	—	53	4	no	—
132	C₂²⋊C₈	—	53	4	no	—
133	#97	—	53	2	no	—
134	C₄×C₈	—	53	2	no	—
135	#41	—	106	4	no	—
136	#41	—	106	4	no	—
137	C₂×Q₈	—	53	2	no	—
138	C₂×D₄	—	53	4	no	—
139	C₂×D₄	—	53	2	no	—
140	C₂×D₄	—	53	2	no	—
141	C₂×D₄	—	53	2	no	—
142	C₂²×C₄	—	53	4	no	—
143	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
144	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
146	M₁₆	—	53	4	no	—
147	M₁₆	—	106	8	no	—
148	M₁₆	—	106	2	no	—
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	53	2	no	—
151	C₂×C₈	—	53	2	no	—
152	C₂×C₈	—	106	8	no	—
153	C₂×C₈	C₂²×D₅₃	1	2	yes	—
154	C₂×C₈	C₂²×D₅₃	1	2	yes	—
155	C₄⋊C₄	—	53	2	no	—
156	C₄⋊C₄	—	53	2	no	—
157	C₂²⋊C₄	—	53	4	no	—
158	C₂²⋊C₄	—	53	4	no	—
159	C₄²	C₄×D₅₃	1	1	yes	characteristic
160	C₄²	C₄×D₅₃	1	1	yes	characteristic
161	C₄²	C₂²×D₅₃	1	1	yes	characteristic
162	C₄²	C₂²×D₅₃	1	1	yes	characteristic
163	C₂³	—	53	4	no	—
164	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
165	Q₈	—	106	4	no	—
166	D₄	—	106	8	no	—
167	D₄	—	106	4	no	—
168	D₄	—	106	4	no	—
169	D₄	—	212	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	53	4	no	—
173	C₂×C₄	—	106	4	no	—
174	C₂×C₄	—	106	4	no	—
175	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
176	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
180	C₂×C₄	D₄×D₅₃	1	2	yes	—
181	C₂×C₄	#11719949	1	2	yes	—
182	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	2	2	no	—
185	C₈	—	53	4	no	—
186	C₈	—	106	2	no	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	106	4	no	—
190	C₂²	—	106	4	no	—
191	C₂²	—	106	4	no	—
192	C₂²	#13133179	1	2	yes	—
193	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	106	4	no	—
198	C₄	#13133179	1	2	yes	—
199	C₄	#13133223	1	2	yes	complemented
200	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
201	C₄	C₂×#11719949	1	1	yes	characteristic
202	C₂	—	2	2	no	—
203	C₂	—	212	4	no	—
204	C₂	C₂×#13133179	1	1	yes	characteristic
205	C₂	C₂×#13133223	1	1	yes	characteristic
206	C₂	#16811389	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₂×C₄⋊C₈×C₅₃	C₂
hypercenter	C₂×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#2935
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#13133223	C₂	semidirect
3. #16810700	C₂	semidirect
4. #16810693	C₂	semidirect
5. #16810688	C₂	semidirect
6. #16810589	C₂	semidirect
7. C₄×#11719917	C₂	semidirect
8. C₂×C₄⋊C₈×C₅₃	C₂	semidirect
54. #11719917	D₄	semidirect
94. C₄×C₅₃	#131	semidirect
109. C₂×C₄⋊C₈	D₅₃	semidirect
112. C₅₃	#2935	semidirect
199. C₄	#13133223	semidirect