Subgroups — C₂×#20276994 #27539574

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811025	C₂	1	2	yes	maximal
2	#20276994	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810451	C₂	1	2	yes	maximal
4	C₂×#16810449	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133209	—	2	2	no	—
7	#16811025	—	2	4	no	—
8	#16811025	C₂²	1	8	yes	—
9	C₂×#13133151	C₂²	1	4	yes	—
10	C₂×#13133149	—	2	2	no	—
11	#16810451	C₂²	1	8	yes	—
12	#16810449	C₂²	1	8	yes	—
13	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₁₆×C₅₃	C₂²	1	4	yes	—
17	#13133209	—	2	8	no	—
18	#13133209	—	4	4	no	—
19	C₂×C₄⋊Dic₅₃	—	2	4	no	—
20	C₂×#11719919	—	4	4	no	—
21	#13133151	—	2	4	no	—
22	#13133151	C₂³	1	8	yes	—
23	#13133149	—	2	8	no	—
24	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₅₃	Q₈	1	4	yes	—
28	C₂×C₁₆×C₅₃	Q₈	1	4	yes	—
29	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
30	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
31	C₄×C₈×C₅₃	D₄	1	2	yes	—
32	C₄×C₈×C₅₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₅₃	D₈	1	2	yes	—
34	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×Dic₅₃	—	4	4	no	—
36	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
37	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
39	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
40	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
41	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
42	C₄²×C₅₃	D₈	1	2	yes	—
43	C₄²×C₅₃	C₂×D₄	1	2	yes	—
44	C₄⋊Dic₅₃	—	2	8	no	—
45	C₄⋊Dic₅₃	—	4	4	no	—
46	#11719919	—	4	16	no	—
47	C₁₆×C₅₃	C₂×Q₈	1	4	yes	—
48	C₁₆×C₅₃	C₂×Q₈	1	4	yes	—
49	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	—	2	4	no	—
51	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
52	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
53	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
55	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
56	C₂×Dic₅₃	—	4	8	no	—
57	C₂×Dic₅₃	—	8	8	no	—
58	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
59	C₈×C₅₃	C₂×Q₁₆	1	4	yes	—
60	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
61	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
62	C₂×#2253	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	#500	1	2	yes	—
64	C₂²×C₅₃	#508	1	4	yes	—
65	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
66	C₄×C₅₃	—	2	4	no	—
67	C₄×C₅₃	#500	1	2	yes	—
68	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
69	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
70	Dic₅₃	—	8	8	no	—
71	C₂×#499	—	53	2	no	—
72	#2253	—	53	8	no	—
73	C₂×#367	—	53	2	no	—
74	C₂×#366	—	53	2	no	—
75	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
76	C₂×C₅₃	#2253	1	4	yes	complemented
77	C₂×C₅₃	C₂×#500	1	1	yes	characteristic
78	C₂×C₅₃	C₂×#508	1	2	yes	—
79	C₂×#125	—	106	2	no	—
80	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
81	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
82	#499	—	53	8	no	—
83	#499	—	106	4	no	—
84	C₂×#107	—	53	4	no	—
85	C₂×#106	—	106	2	no	—
86	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
87	#367	—	53	8	no	—
88	#366	—	53	8	no	—
89	C₄×C₁₆	C₂×D₅₃	1	4	yes	—
90	C₅₃	C₂×#2253	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
92	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
93	#125	—	106	8	no	—
94	#125	—	212	4	no	—
95	C₂×C₄⋊C₄	—	106	4	no	—
96	C₂×C₄⋊C₄	—	212	4	no	—
97	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
98	C₂×C₁₆	Dic₁₀₆	1	4	yes	—
99	C₂×C₁₆	Dic₁₀₆	1	4	yes	—
100	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
101	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
102	#107	—	53	8	no	—
103	#107	—	106	4	no	—
104	#106	—	106	8	no	—
105	C₄×C₈	D₂₁₂	1	2	yes	—
106	C₄×C₈	C₂²×D₅₃	1	2	yes	—
107	C₂²×C₄	—	212	4	no	—
108	C₂²×C₄	D₄₂₄	1	2	yes	—
109	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
110	C₂×C₈	Dic₂₁₂	1	4	yes	—
111	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
112	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
113	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
114	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
115	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
116	C₄⋊C₄	—	106	8	no	—
117	C₄⋊C₄	—	212	4	no	—
118	C₄⋊C₄	—	212	16	no	—
119	C₄²	D₄₂₄	1	2	yes	—
120	C₄²	C₂×D₂₁₂	1	2	yes	—
121	C₁₆	C₂×Dic₁₀₆	1	4	yes	—
122	C₁₆	C₂×Dic₁₀₆	1	4	yes	—
123	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	424	8	no	—
126	C₂×C₄	—	212	8	no	—
127	C₂×C₄	#13133217	1	2	yes	—
128	C₂×C₄	C₂×D₄₂₄	1	4	yes	—
129	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
130	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
131	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
132	C₈	#13133217	1	2	yes	—
133	C₈	C₂×Dic₂₁₂	1	4	yes	—
134	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
135	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
136	C₂²	#16811028	1	2	yes	—
137	C₂²	#16811055	1	4	yes	—
138	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
139	C₄	—	2	4	no	—
140	C₄	—	424	8	no	—
141	C₄	#16811028	1	2	yes	—
142	C₄	C₂×#13133217	1	1	yes	characteristic
143	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
144	C₂	#20276994	1	4	yes	direct product factor
145	C₂	C₂×#16811028	1	1	yes	characteristic
146	C₂	C₂×#16811055	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₂₄
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	C₂×#2253
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20276994	C₂	direct (N × H)
76. C₂×C₅₃	#2253	semidirect
90. C₅₃	C₂×#2253	semidirect
144. C₂	#20276994	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16811025	2
#20276994	2
C₂×#16810451	2
C₂×#16810449	2
C₂×C₄×C₁₆×C₅₃	2
C₂×#2253	53

Subgroups — C2×#20276994 #27539574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276994 #27539574