Subgroups — C₂×#16810580 #20283129

Order 6784.

592 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11719912	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133174	C₂	1	1	yes	maximal, characteristic
4	#16810580	C₂	1	8	yes	maximal, direct product factor
5	C₂×#13133127	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#97×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
10	C₂×#11719912	C₄	1	8	yes	complemented
11	C₂×#11719912	C₂²	1	4	yes	—
12	C₂²×#88322	C₂²	1	1	yes	characteristic
13	C₂²×#88322	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₅₃	C₄	1	8	yes	complemented
15	C₂×C₈×D₅₃	C₂²	1	4	yes	—
16	#13133174	C₄	1	4	yes	—
17	#13133174	C₂²	1	4	yes	—
18	C₂×#11719919	C₄	1	2	yes	—
19	C₂×#11719921	C₄	1	2	yes	—
20	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
21	#13133127	C₂²	1	4	yes	complemented
22	C₄×#88322	C₂²	1	4	yes	complemented
23	C₈×Dic₅₃	C₂²	1	4	yes	complemented
24	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
27	#97×C₅₃	C₂²	1	4	yes	complemented
28	C₂³×D₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
31	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
32	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₅₃	C₂×C₄	1	4	yes	—
35	C₂×C₄×D₅₃	C₂×C₄	1	4	yes	—
36	C₂×C₄×D₅₃	C₂³	1	2	yes	—
37	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
38	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
39	C₂×C₈×C₅₃	C₂³	1	2	yes	—
40	C₂×C₈×C₅₃	C₂³	1	2	yes	—
41	C₄²×C₅₃	C₂×C₄	1	2	yes	—
42	C₄²×C₅₃	C₂³	1	2	yes	—
43	#11719921	C₂×C₄	1	8	yes	—
44	#11719919	C₂×C₄	1	8	yes	—
45	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
46	C₄×Dic₅₃	C₂³	1	2	yes	—
47	C₂×#88322	C₂×C₄	1	4	yes	—
48	C₂×#88322	C₂×C₄	1	4	yes	—
49	C₂×#88322	C₂³	1	2	yes	—
50	C₂×#88322	C₂³	1	2	yes	—
51	#11719912	C₂×C₄	1	16	yes	complemented
52	C₈×D₅₃	C₂×C₄	1	16	yes	complemented
53	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×D₅₃	—	2	4	no	—
55	C₂²×D₅₃	C₄²	1	4	yes	—
56	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₅₃	—	2	4	no	—
58	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
63	C₂×Dic₅₃	—	2	4	no	—
64	C₂×Dic₅₃	C₄²	1	4	yes	—
65	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
66	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
67	C₄×D₅₃	C₄²	1	8	yes	—
68	C₄×D₅₃	C₂²×C₄	1	4	yes	—
69	C₄×D₅₃	C₂²×C₄	1	4	yes	—
70	C₈×C₅₃	C₂²×C₄	1	4	yes	—
71	C₈×C₅₃	C₂²×C₄	1	4	yes	—
72	#88322	C₂²×C₄	1	4	yes	—
73	#88322	C₂²×C₄	1	4	yes	—
74	C₂²×C₅₃	C₂×C₄²	1	2	yes	—
75	C₂²×C₅₃	#131	1	4	yes	—
76	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
77	C₂×D₅₃	—	2	4	no	—
78	C₂×D₅₃	—	2	8	no	—
79	C₂×D₅₃	C₂×C₄²	1	4	yes	—
80	C₄×C₅₃	—	2	4	no	—
81	C₄×C₅₃	C₂×C₄²	1	2	yes	—
82	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
83	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
84	Dic₅₃	—	2	4	no	—
85	Dic₅₃	C₂×C₄²	1	4	yes	—
86	C₂×#405	—	53	1	no	maximal, Sylow
87	C₂×C₅₃	#405	1	4	yes	complemented
88	C₂×C₅₃	C₂²×C₄²	1	1	yes	characteristic
89	C₂×C₅₃	C₂×#131	1	2	yes	—
90	D₅₃	—	2	4	no	—
91	C₂²×M₁₆	—	53	1	no	—
92	C₂³×C₈	—	53	1	no	—
93	C₂×#117	—	53	1	no	—
94	#405	—	53	8	no	—
95	C₂×#97	—	53	1	no	—
96	C₂×#97	D₅₃	1	1	yes	characteristic, complemented
97	C₂×C₄×C₈	—	53	1	no	—
98	C₂×C₄×C₈	—	53	1	no	—
99	C₅₃	C₂×#405	1	1	yes	characteristic, complemented, Sylow
100	C₂³×C₄	—	53	1	no	—
101	C₂×M₁₆	—	53	8	no	—
102	C₂×M₁₆	—	53	4	no	—
103	C₂²×C₈	—	53	1	no	—
104	C₂²×C₈	—	53	1	no	—
105	C₂²×C₈	—	53	8	no	—
106	C₂²×C₈	—	53	4	no	—
107	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
108	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
109	#117	—	53	4	no	—
110	#117	—	53	4	no	—
111	C₂×C₄⋊C₄	—	53	2	no	—
112	C₂×C₂²⋊C₄	—	53	2	no	—
113	C₂×C₄²	—	53	1	no	—
114	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
115	#97	—	53	4	no	—
116	#97	C₂×D₅₃	1	4	yes	complemented
117	C₄×C₈	—	53	4	no	—
118	C₄×C₈	—	53	4	no	—
119	C₂⁴	—	53	1	no	—
120	C₂²×C₄	—	53	1	no	—
121	C₂²×C₄	—	53	2	no	—
122	C₂²×C₄	—	53	4	no	—
123	C₂²×C₄	—	53	4	no	—
124	C₂²×C₄	—	53	2	no	—
125	C₂²×C₄	—	53	2	no	—
126	C₂²×C₄	C₄×D₅₃	1	2	yes	—
127	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
128	M₁₆	—	53	16	no	—
129	C₂×C₈	—	53	4	no	—
130	C₂×C₈	—	53	4	no	—
131	C₂×C₈	—	53	2	no	—
132	C₂×C₈	—	53	2	no	—
133	C₂×C₈	—	53	16	no	—
134	C₂×C₈	C₄×D₅₃	1	4	yes	—
135	C₂×C₈	C₄×D₅₃	1	4	yes	—
136	C₂×C₈	C₂²×D₅₃	1	2	yes	—
137	C₂×C₈	C₂²×D₅₃	1	2	yes	—
138	C₄⋊C₄	—	53	8	no	—
139	C₂²⋊C₄	—	53	8	no	—
140	C₄²	—	53	2	no	—
141	C₄²	—	53	2	no	—
142	C₄²	C₄×D₅₃	1	2	yes	—
143	C₄²	C₂²×D₅₃	1	2	yes	—
144	C₂³	—	53	4	no	—
145	C₂³	—	53	2	no	—
146	C₂³	—	106	4	no	—
147	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	53	4	no	—
150	C₂×C₄	—	53	2	no	—
151	C₂×C₄	—	53	8	no	—
152	C₂×C₄	—	53	4	no	—
153	C₂×C₄	—	53	4	no	—
154	C₂×C₄	—	53	4	no	—
155	C₂×C₄	—	106	4	no	—
156	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
157	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
159	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
160	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
161	C₈	—	53	4	no	—
162	C₈	—	53	4	no	—
163	C₈	C₂×C₄×D₅₃	1	4	yes	—
164	C₈	C₂×C₄×D₅₃	1	4	yes	—
165	C₂²	—	53	4	no	—
166	C₂²	—	106	8	no	—
167	C₂²	—	106	4	no	—
168	C₂²	C₄²×D₅₃	1	2	yes	—
169	C₂²	#13133225	1	4	yes	—
170	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
171	C₄	—	2	4	no	—
172	C₄	—	53	4	no	—
173	C₄	—	106	4	no	—
174	C₄	C₄²×D₅₃	1	2	yes	—
175	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
176	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
177	C₂	—	106	4	no	—
178	C₂	#16810580	1	4	yes	direct product factor
179	C₂	C₂×C₄²×D₅₃	1	1	yes	characteristic
180	C₂	C₂×#13133225	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×#97×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₂×#97	D₅₃
hypocenter	C₅₃	C₂×#405
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
4. #16810580	C₂	direct (N × H)
5. C₂×#13133127	C₂	semidirect
6. C₂×C₄×#88322	C₂	semidirect
7. C₂×C₈×Dic₅₃	C₂	semidirect
8. C₂×#97×C₅₃	C₂	semidirect
10. C₂×#11719912	C₄	semidirect
14. C₂×C₈×D₅₃	C₄	semidirect
21. #13133127	C₂²	semidirect
22. C₄×#88322	C₂²	semidirect
23. C₈×Dic₅₃	C₂²	semidirect
27. #97×C₅₃	C₂²	semidirect
51. #11719912	C₂×C₄	semidirect
52. C₈×D₅₃	C₂×C₄	semidirect
87. C₂×C₅₃	#405	semidirect
96. C₂×#97	D₅₃	semidirect
99. C₅₃	C₂×#405	semidirect
116. #97	C₂×D₅₃	semidirect
178. C₂	#16810580	direct (N × H)

Subgroups — C2×#16810580 #20283129

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810580 #20283129