Subgroups — C₂×#16810578 #20283127

Order 6784.

592 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11719917	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133176	C₂	1	1	yes	maximal, characteristic
3	#16810578	C₂	1	8	yes	maximal, direct product factor
4	C₂×#13133127	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×Dic₅₃	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#11719917	C₄	1	8	yes	complemented
8	C₂×#11719917	C₂²	1	4	yes	—
9	C₂²×#88322	C₂²	1	2	yes	—
10	#13133176	C₄	1	4	yes	—
11	#13133176	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₅₃	C₄	1	2	yes	—
13	C₂×C₂²⋊Dic₅₃	C₄	1	2	yes	—
14	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
15	#13133127	C₂²	1	8	yes	complemented
16	C₈×Dic₅₃	C₂²	1	8	yes	complemented
17	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₅₃	C₄	1	8	yes	complemented
19	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂²	1	4	yes	—
21	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
29	C₂²×Dic₅₃	C₂×C₄	1	4	yes	—
30	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
32	C₂×C₈×C₅₃	C₂×C₄	1	16	yes	complemented
33	C₂×C₈×C₅₃	C₂³	1	2	yes	—
34	C₂×C₈×C₅₃	C₂³	1	2	yes	—
35	C₂²⋊Dic₅₃	C₂×C₄	1	8	yes	—
36	C₄⋊Dic₅₃	C₂×C₄	1	8	yes	—
37	C₄×Dic₅₃	C₂×C₄	1	4	yes	—
38	C₄×Dic₅₃	C₂³	1	4	yes	—
39	#11719917	C₂×C₄	1	16	yes	complemented
40	C₂×#88322	C₂×C₄	1	8	yes	—
41	C₂×#88322	C₂³	1	4	yes	—
42	C₂³×C₅₃	—	2	4	no	—
43	C₂³×C₅₃	C₄²	1	4	yes	—
44	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
45	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₄²	1	8	yes	—
47	C₂×C₄×C₅₃	C₄²	1	4	yes	—
48	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
52	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
54	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₅₃	—	2	8	no	—
56	C₂×Dic₅₃	C₂²×C₄	1	8	yes	—
57	C₈×C₅₃	C₂²×C₄	1	4	yes	—
58	C₈×C₅₃	C₂²×C₄	1	4	yes	—
59	#88322	C₂²×C₄	1	8	yes	—
60	C₂²×C₅₃	—	2	4	no	—
61	C₂²×C₅₃	—	2	8	no	—
62	C₂²×C₅₃	C₂×C₄²	1	4	yes	—
63	C₂²×C₅₃	C₂×C₄²	1	2	yes	—
64	C₂²×C₅₃	#131	1	4	yes	—
65	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₅₃	C₂×C₄²	1	2	yes	—
67	C₄×C₅₃	C₂×C₄²	1	4	yes	—
68	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
70	Dic₅₃	—	2	8	no	—
71	C₂×#405	—	53	1	no	maximal, Sylow
72	C₂×C₅₃	—	2	4	no	—
73	C₂×C₅₃	#405	1	4	yes	complemented
74	C₂×C₅₃	C₂²×C₄²	1	1	yes	characteristic
75	C₂×C₅₃	C₂×#131	1	2	yes	—
76	C₂²×M₁₆	—	53	1	no	—
77	C₂³×C₈	D₅₃	1	1	yes	characteristic
78	C₂×#117	—	53	1	no	—
79	#405	—	53	8	no	—
80	C₂×#97	—	53	2	no	—
81	C₂×C₄×C₈	—	53	2	no	—
82	C₅₃	C₂×#405	1	1	yes	characteristic, complemented, Sylow
83	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
84	C₂×M₁₆	—	53	8	no	—
85	C₂×M₁₆	—	53	4	no	—
86	C₂²×C₈	—	53	2	no	—
87	C₂²×C₈	Dic₅₃	1	8	yes	complemented
88	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
89	C₂²×C₈	C₂×D₅₃	1	4	yes	—
90	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
91	#117	—	53	4	no	—
92	#117	—	53	4	no	—
93	C₂×C₄⋊C₄	—	53	2	no	—
94	C₂×C₂²⋊C₄	—	53	2	no	—
95	C₂×C₄²	—	53	2	no	—
96	#97	—	53	8	no	—
97	C₄×C₈	—	53	8	no	—
98	C₂⁴	C₄×D₅₃	1	1	yes	characteristic
99	C₂²×C₄	—	53	4	no	—
100	C₂²×C₄	C₄×D₅₃	1	4	yes	—
101	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
102	C₂²×C₄	C₄×D₅₃	1	2	yes	—
103	C₂²×C₄	C₂×Dic₅₃	1	4	yes	—
104	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
106	M₁₆	—	53	16	no	—
107	C₂×C₈	—	53	8	no	—
108	C₂×C₈	—	53	4	no	—
109	C₂×C₈	C₂×Dic₅₃	1	4	yes	—
110	C₂×C₈	C₂×Dic₅₃	1	4	yes	—
111	C₂×C₈	C₂×Dic₅₃	1	16	yes	complemented
112	C₂×C₈	C₂²×D₅₃	1	2	yes	—
113	C₂×C₈	C₂²×D₅₃	1	2	yes	—
114	C₄⋊C₄	—	53	8	no	—
115	C₂²⋊C₄	—	53	8	no	—
116	C₄²	—	53	4	no	—
117	C₄²	—	53	4	no	—
118	C₂³	—	2	4	no	—
119	C₂³	C₄×Dic₅₃	1	4	yes	—
120	C₂³	C₂×C₄×D₅₃	1	2	yes	—
121	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
122	C₂×C₄	—	53	8	no	—
123	C₂×C₄	—	106	8	no	—
124	C₂×C₄	C₄×Dic₅₃	1	8	yes	—
125	C₂×C₄	C₄×Dic₅₃	1	4	yes	—
126	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
128	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
129	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
130	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
131	C₂×C₄	C₂²×Dic₅₃	1	4	yes	—
132	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
133	C₈	—	53	8	no	—
134	C₈	C₂²×Dic₅₃	1	4	yes	—
135	C₈	C₂²×Dic₅₃	1	4	yes	—
136	C₂²	—	2	8	no	—
137	C₂²	—	2	4	no	—
138	C₂²	C₂×C₄×Dic₅₃	1	4	yes	—
139	C₂²	C₂×C₄×Dic₅₃	1	2	yes	—
140	C₂²	#13133223	1	4	yes	—
141	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
142	C₄	—	106	8	no	—
143	C₄	C₂×C₄×Dic₅₃	1	2	yes	—
144	C₄	C₂×C₄×Dic₅₃	1	4	yes	—
145	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
146	C₄	C₂³×Dic₅₃	1	1	yes	characteristic
147	C₂	—	2	4	no	—
148	C₂	#16810578	1	4	yes	direct product factor
149	C₂	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
150	C₂	C₂×#13133223	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂³×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₂³×C₈	D₅₃
hypocenter	C₅₃	C₂×#405
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
3. #16810578	C₂	direct (N × H)
4. C₂×#13133127	C₂	semidirect
5. C₂×C₈×Dic₅₃	C₂	semidirect
7. C₂×#11719917	C₄	semidirect
15. #13133127	C₂²	semidirect
16. C₈×Dic₅₃	C₂²	semidirect
18. C₂²×C₈×C₅₃	C₄	semidirect
32. C₂×C₈×C₅₃	C₂×C₄	semidirect
39. #11719917	C₂×C₄	semidirect
73. C₂×C₅₃	#405	semidirect
82. C₅₃	C₂×#405	semidirect
87. C₂²×C₈	Dic₅₃	semidirect
111. C₂×C₈	C₂×Dic₅₃	semidirect
148. C₂	#16810578	direct (N × H)

Subgroups — C2×#16810578 #20283127

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810578 #20283127