Subgroups — C₂×#16810672 #20283322

Order 6784.

460 conjugacy classes of subgroups.

The subgroup lattice has 146 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	2	yes	maximal
2	#16810672	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13133153	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	—	2	2	no	—
6	C₂×#11719917	—	2	4	no	—
7	C₂×#11719917	C₂²	1	4	yes	—
8	C₂×#11719917	C₂²	1	8	yes	—
9	C₂²×#88322	—	2	2	no	—
10	#13133153	C₂²	1	16	yes	complemented
11	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₅₃	C₄	1	4	yes	—
13	C₂×M₁₆×C₅₃	C₄	1	4	yes	—
14	C₂×M₁₆×C₅₃	C₂²	1	4	yes	—
15	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
16	C₂⁴×C₅₃	Q₈	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₅₃	D₄	1	2	yes	—
20	C₂²×C₄×C₅₃	D₄	1	2	yes	—
21	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	Q₈	1	2	yes	—
23	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
25	M₁₆×C₅₃	C₂×C₄	1	16	yes	—
26	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
27	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₅₃	C₂³	1	4	yes	—
29	#11719917	—	2	8	no	—
30	#11719917	—	2	4	no	—
31	#11719917	C₂³	1	8	yes	—
32	C₂×#88322	—	2	8	no	—
33	C₂×#88322	—	2	4	no	—
34	C₂³×C₅₃	—	4	2	no	—
35	C₂³×C₅₃	C₄⋊C₄	1	2	yes	—
36	C₂³×C₅₃	C₄⋊C₄	1	2	yes	—
37	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	C₂³×C₅₃	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
40	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
41	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
46	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
47	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
51	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
52	C₈×C₅₃	C₂²×C₄	1	8	yes	—
53	#88322	—	2	8	no	—
54	C₂²×C₅₃	—	2	4	no	—
55	C₂²×C₅₃	—	2	2	no	—
56	C₂²×C₅₃	—	4	2	no	—
57	C₂²×C₅₃	—	4	2	no	—
58	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
60	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
62	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
64	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
67	C₂×#430	—	53	1	no	maximal, Sylow
68	C₂×C₅₃	—	2	4	no	—
69	C₂×C₅₃	—	2	1	no	—
70	C₂×C₅₃	—	2	1	no	—
71	C₂×C₅₃	#430	1	2	yes	complemented
72	C₂×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×M₁₆	—	53	2	no	—
74	C₂²×M₁₆	D₅₃	1	1	yes	characteristic
75	#430	—	53	8	no	—
76	C₂×#108	—	53	4	no	—
77	C₅₃	C₂×#430	1	1	yes	characteristic, complemented, Sylow
78	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
79	C₂×M₁₆	—	53	4	no	—
80	C₂×M₁₆	—	53	8	no	—
81	C₂×M₁₆	—	106	4	no	—
82	C₂×M₁₆	—	106	2	no	—
83	C₂×M₁₆	Dic₅₃	1	4	yes	—
84	C₂×M₁₆	Dic₅₃	1	4	yes	—
85	C₂×M₁₆	C₂×D₅₃	1	4	yes	—
86	C₂²×C₈	—	106	2	no	—
87	C₂²×C₈	C₂×D₅₃	1	2	yes	—
88	#108	—	53	16	no	—
89	C₂⁴	Dic₁₀₆	1	1	yes	characteristic
90	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
91	C₂²×C₄	Dic₁₀₆	1	2	yes	—
92	C₂²×C₄	D₂₁₂	1	2	yes	—
93	C₂²×C₄	D₂₁₂	1	2	yes	—
94	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
95	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
96	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
97	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
98	M₁₆	—	53	8	no	—
99	M₁₆	—	106	8	no	—
100	M₁₆	—	106	4	no	—
101	M₁₆	C₂×Dic₅₃	1	16	yes	—
102	C₂×C₈	—	106	8	no	—
103	C₂×C₈	—	106	4	no	—
104	C₂×C₈	C₂×Dic₅₃	1	4	yes	—
105	C₂×C₈	C₂×Dic₅₃	1	4	yes	—
106	C₂×C₈	C₂²×D₅₃	1	4	yes	—
107	C₂³	—	4	2	no	—
108	C₂³	C₄⋊Dic₅₃	1	2	yes	—
109	C₂³	C₄⋊Dic₅₃	1	2	yes	—
110	C₂³	C₂×Dic₁₀₆	1	1	yes	characteristic
111	C₂³	C₂×Dic₁₀₆	1	2	yes	—
112	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
113	C₂×C₄	C₄⋊Dic₅₃	1	4	yes	—
114	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
115	C₂×C₄	C₄⋊Dic₅₃	1	4	yes	—
116	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
117	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
118	C₂×C₄	C₂×D₂₁₂	1	4	yes	—
119	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
120	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
121	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
122	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
123	C₂×C₄	C₂²×Dic₅₃	1	4	yes	—
124	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
125	C₈	—	106	8	no	—
126	C₈	C₂²×Dic₅₃	1	8	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	—	2	4	no	—
129	C₂²	—	4	2	no	—
130	C₂²	—	4	2	no	—
131	C₂²	C₂×C₄⋊Dic₅₃	1	4	yes	—
132	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
133	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
134	C₂²	C₂²×Dic₁₀₆	1	1	yes	characteristic
135	C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
136	C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
137	C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
138	C₄	C₂³×Dic₅₃	1	1	yes	characteristic
139	C₄	C₂²×D₂₁₂	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	2	4	no	—
142	C₂	—	2	1	no	—
143	C₂	#16810672	1	2	yes	direct product factor
144	C₂	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₁₂
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂²×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₂²×M₁₆	D₅₃
hypocenter	C₅₃	C₂×#430
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
2. #16810672	C₂	direct (N × H)
3. C₂×#13133153	C₂	semidirect
10. #13133153	C₂²	semidirect
71. C₂×C₅₃	#430	semidirect
77. C₅₃	C₂×#430	semidirect
143. C₂	#16810672	direct (N × H)

Subgroups — C2×#16810672 #20283322

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810672 #20283322