Subgroups — C₂×#16811346 #20288264

Order 6784.

1112 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811346	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13133217	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13133212	C₂	1	4	yes	maximal, complemented
4	C₂×#13133205	C₂	1	2	yes	maximal
5	C₂×#13133193	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×Dic₁₀₆	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×D₂₁₂	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133175	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₁₀₆	C₂²	1	2	yes	—
11	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
12	C₂²×C₄×D₅₃	C₂²	1	2	yes	—
13	#13133217	C₂²	1	8	yes	complemented
14	#13133212	C₂²	1	32	yes	complemented
15	#13133205	C₂²	1	16	yes	—
16	#13133193	C₂²	1	16	yes	complemented
17	C₄×Dic₁₀₆	C₂²	1	16	yes	complemented
18	C₄×D₂₁₂	C₂²	1	8	yes	—
19	#13133175	C₂²	1	16	yes	complemented
20	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
21	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
23	C₂×#11719919	C₂²	1	4	yes	—
24	C₂×#11719921	C₂²	1	4	yes	—
25	C₂×#11719921	C₂²	1	2	yes	—
26	C₂×#11719921	C₂²	1	4	yes	—
27	C₂×C₄×Dic₅₃	C₂²	1	4	yes	—
28	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₄×Q₈×C₅₃	C₂²	1	8	yes	complemented
30	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
34	C₂³×D₅₃	C₂³	1	2	yes	—
35	C₂×Q₈×C₅₃	—	2	4	no	—
36	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
37	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂²×Dic₅₃	C₂³	1	2	yes	—
43	C₂²×Dic₅₃	C₂³	1	4	yes	—
44	C₂×D₂₁₂	—	2	4	no	—
45	C₂×D₂₁₂	C₂³	1	4	yes	—
46	C₂×C₄×D₅₃	—	2	8	no	—
47	C₂×C₄×D₅₃	C₂³	1	8	yes	—
48	C₂×Dic₁₀₆	—	2	8	no	—
49	C₂×Dic₁₀₆	C₂³	1	8	yes	—
50	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
51	C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
52	C₄²×C₅₃	C₂³	1	4	yes	—
53	C₄²×C₅₃	C₂³	1	8	yes	—
54	#11719921	C₂³	1	16	yes	—
55	#11719921	C₂³	1	8	yes	—
56	#11719921	C₂³	1	16	yes	—
57	C₄⋊Dic₅₃	C₂³	1	8	yes	—
58	C₄⋊Dic₅₃	C₂³	1	4	yes	—
59	C₄⋊Dic₅₃	C₂³	1	8	yes	—
60	#11719919	C₂³	1	16	yes	—
61	C₄×Dic₅₃	C₂³	1	16	yes	—
62	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
63	C₂²×D₅₃	—	2	4	no	—
64	C₂²×D₅₃	—	2	4	no	—
65	C₂²×D₅₃	—	2	4	no	—
66	C₂²×D₅₃	C₂⁴	1	4	yes	—
67	Q₈×C₅₃	—	2	8	no	—
68	C₂×C₄×C₅₃	—	2	4	no	—
69	C₂×C₄×C₅₃	—	2	4	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	#41	1	4	yes	—
72	C₂×C₄×C₅₃	#41	1	4	yes	—
73	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
74	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
75	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
76	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
77	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
78	C₂×Dic₅₃	—	2	8	no	—
79	C₂×Dic₅₃	—	2	4	no	—
80	C₂×Dic₅₃	C₂⁴	1	8	yes	—
81	C₂×Dic₅₃	C₂⁴	1	4	yes	—
82	D₂₁₂	—	2	8	no	—
83	C₄×D₅₃	—	2	16	no	—
84	Dic₁₀₆	—	2	16	no	—
85	C₂²×C₅₃	C₂×#41	1	2	yes	—
86	C₂²×C₅₃	C₂×#41	1	2	yes	—
87	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
88	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
89	C₂×D₅₃	—	2	4	no	—
90	C₂×D₅₃	—	2	4	no	—
91	C₂×D₅₃	—	2	4	no	—
92	C₂×D₅₃	—	4	8	no	—
93	C₄×C₅₃	—	2	4	no	—
94	C₄×C₅₃	—	2	4	no	—
95	C₄×C₅₃	—	2	2	no	—
96	C₄×C₅₃	C₂×#41	1	4	yes	—
97	C₄×C₅₃	C₂×#41	1	4	yes	—
98	Dic₅₃	—	2	8	no	—
99	Dic₅₃	—	2	4	no	—
100	C₂×#556	—	53	1	no	maximal, Sylow
101	C₂×C₅₃	#556	1	4	yes	complemented
102	C₂×C₅₃	C₂²×#41	1	1	yes	characteristic
103	C₂×C₅₃	C₂²×#41	1	1	yes	characteristic
104	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
105	D₅₃	—	4	4	no	—
106	#556	—	53	16	no	—
107	C₂×C₄⋊Q₈	—	53	1	no	—
108	C₂×#126	—	53	4	no	—
109	C₂×#124	—	53	2	no	—
110	C₂×C₂²⋊Q₈	—	53	2	no	—
111	C₂×C₄×Q₈	—	53	2	no	—
112	C₂×C₄×Q₈	D₅₃	1	1	yes	characteristic, complemented
113	C₂×C₄×D₄	—	53	1	no	—
114	C₂×#117	—	53	2	no	—
115	C₅₃	C₂×#556	1	1	yes	characteristic, complemented, Sylow
116	C₂²×Q₈	—	53	2	no	—
117	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
118	C₂²×D₄	—	53	1	no	—
119	C₂³×C₄	—	53	2	no	—
120	C₄⋊Q₈	—	53	8	no	—
121	#126	—	53	32	no	—
122	#124	—	53	16	no	—
123	C₂²⋊Q₈	—	53	16	no	—
124	C₄×Q₈	—	53	16	no	—
125	C₄×Q₈	C₂×D₅₃	1	8	yes	complemented
126	C₄×D₄	—	53	8	no	—
127	#117	—	53	16	no	—
128	C₂×C₄⋊C₄	—	53	2	no	—
129	C₂×C₄⋊C₄	—	53	1	no	—
130	C₂×C₄⋊C₄	—	53	4	no	—
131	C₂×C₄⋊C₄	—	53	2	no	—
132	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
133	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
134	C₂×C₂²⋊C₄	—	53	4	no	—
135	C₂×C₂²⋊C₄	—	53	2	no	—
136	C₂×C₂²⋊C₄	—	53	4	no	—
137	C₂×C₄²	—	53	4	no	—
138	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
139	C₂×C₄²	C₂×D₅₃	1	2	yes	—
140	C₂⁴	—	53	2	no	—
141	C₂×Q₈	—	2	4	no	—
142	C₂×Q₈	—	53	8	no	—
143	C₂×Q₈	—	106	8	no	—
144	C₂×Q₈	C₂²×D₅₃	1	4	yes	—
145	C₂×D₄	—	53	4	no	—
146	C₂×D₄	—	106	4	no	—
147	C₂²×C₄	—	53	2	no	—
148	C₂²×C₄	—	53	4	no	—
149	C₂²×C₄	—	53	8	no	—
150	C₂²×C₄	—	106	8	no	—
151	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
152	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
153	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
154	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
155	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
156	C₄⋊C₄	—	53	8	no	—
157	C₄⋊C₄	—	53	4	no	—
158	C₄⋊C₄	—	53	16	no	—
159	C₄⋊C₄	—	53	8	no	—
160	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
161	C₄⋊C₄	C₂²×D₅₃	1	8	yes	—
162	C₂²⋊C₄	—	53	16	no	—
163	C₂²⋊C₄	—	53	8	no	—
164	C₂²⋊C₄	—	53	16	no	—
165	C₄²	—	53	16	no	—
166	C₄²	C₂²×D₅₃	1	4	yes	—
167	C₄²	C₂²×D₅₃	1	8	yes	—
168	C₂³	—	53	4	no	—
169	C₂³	—	106	4	no	—
170	C₂³	—	106	4	no	—
171	C₂³	—	106	4	no	—
172	C₂³	C₂³×D₅₃	1	1	yes	characteristic
173	Q₈	—	2	8	no	—
174	Q₈	—	106	16	no	—
175	D₄	—	106	8	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	53	4	no	—
180	C₂×C₄	—	53	8	no	—
181	C₂×C₄	—	106	4	no	—
182	C₂×C₄	—	106	8	no	—
183	C₂×C₄	—	106	16	no	—
184	C₂×C₄	#11719945	1	4	yes	—
185	C₂×C₄	#11719949	1	4	yes	—
186	C₂×C₄	C₂³×D₅₃	1	4	yes	—
187	C₂×C₄	C₂³×D₅₃	1	2	yes	—
188	C₂×C₄	C₂³×D₅₃	1	4	yes	—
189	C₂×C₄	C₂³×D₅₃	1	2	yes	—
190	C₂×C₄	C₂³×D₅₃	1	2	yes	—
191	C₂²	—	106	4	no	—
192	C₂²	—	106	4	no	—
193	C₂²	—	106	4	no	—
194	C₂²	—	212	8	no	—
195	C₂²	C₂×#11719945	1	2	yes	—
196	C₂²	C₂×#11719949	1	2	yes	—
197	C₂²	#13133264	1	2	yes	—
198	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	2	4	no	—
201	C₄	—	2	2	no	—
202	C₄	—	106	4	no	—
203	C₄	—	106	8	no	—
204	C₄	C₂×#11719945	1	4	yes	—
205	C₄	C₂×#11719949	1	4	yes	—
206	C₂	—	212	4	no	—
207	C₂	#16811346	1	4	yes	direct product factor
208	C₂	C₂²×#11719945	1	1	yes	characteristic
209	C₂	C₂²×#11719949	1	1	yes	characteristic
210	C₂	C₂×#13133264	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂⁵
Frattini	C₂²	C₂⁴×D₅₃
Fitting	C₂×C₄×Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂×C₄×Q₈	D₅₃
hypocenter	C₅₃	C₂×#556
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #16811346	C₂	direct (N × H)
2. C₂×#13133217	C₂	semidirect
3. C₂×#13133212	C₂	semidirect
5. C₂×#13133193	C₂	semidirect
6. C₂×C₄×Dic₁₀₆	C₂	semidirect
8. C₂×#13133175	C₂	semidirect
9. C₂×C₄×Q₈×C₅₃	C₂	semidirect
13. #13133217	C₂²	semidirect
14. #13133212	C₂²	semidirect
16. #13133193	C₂²	semidirect
17. C₄×Dic₁₀₆	C₂²	semidirect
19. #13133175	C₂²	semidirect
29. C₄×Q₈×C₅₃	C₂²	semidirect
101. C₂×C₅₃	#556	semidirect
112. C₂×C₄×Q₈	D₅₃	semidirect
115. C₅₃	C₂×#556	semidirect
125. C₄×Q₈	C₂×D₅₃	semidirect
207. C₂	#16811346	direct (N × H)

Subgroups — C2×#16811346 #20288264

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811346 #20288264