Subgroups — C₂×#20277009 #27542201

Order 13568.

470 conjugacy classes of subgroups.

The subgroup lattice has 118 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20277009	C₂	1	8	yes	maximal, direct product factor
2	C₂×#16810457	C₂	1	3	yes	maximal, complemented
3	C₂²×#13133123	C₂	1	3	yes	maximal, complemented
4	C₂×C₅₃×#504	C₂	1	1	yes	maximal, characteristic
5	#16810457	C₂²	1	12	yes	complemented
6	C₂×#13133123	C₂²	1	1	yes	characteristic
7	C₂×#13133123	C₂²	1	3	yes	—
8	C₂×#13133123	C₂²	1	12	yes	complemented
9	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
10	C₅₃×#504	C₄	1	4	yes	—
11	C₅₃×#504	C₂²	1	4	yes	—
12	C₂×M₃₂×C₅₃	C₄	1	3	yes	—
13	C₂²×C₁₆×C₅₃	C₂²	1	3	yes	—
14	#13133123	C₂³	1	2	yes	—
15	#13133123	C₂³	1	6	yes	—
16	C₂×#41×C₅₃	C₈	1	1	yes	characteristic
17	#131×C₅₃	C₈	1	4	yes	—
18	#131×C₅₃	C₂×C₄	1	4	yes	—
19	C₂×M₁₆×C₅₃	C₈	1	3	yes	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
21	M₃₂×C₅₃	C₂×C₄	1	12	yes	—
22	C₂×C₁₆×C₅₃	C₂×C₄	1	3	yes	—
23	C₂×C₁₆×C₅₃	C₂×C₄	1	6	yes	—
24	C₂×C₁₆×C₅₃	C₂³	1	6	yes	—
25	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
26	#41×C₅₃	C₁₆	1	4	yes	—
27	#41×C₅₃	C₂×C₈	1	4	yes	—
28	C₂×Q₈×C₅₃	C₁₆	1	1	yes	characteristic
29	C₂×D₄×C₅₃	C₁₆	1	3	yes	—
30	C₂²×C₄×C₅₃	C₂×C₈	1	3	yes	—
31	M₁₆×C₅₃	C₂×C₈	1	12	yes	—
32	C₂×C₈×C₅₃	C₂×C₈	1	3	yes	—
33	C₂×C₈×C₅₃	C₂×C₈	1	6	yes	—
34	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
36	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₁₆×C₅₃	C₂²×C₄	1	6	yes	—
38	C₁₆×C₅₃	C₂⁴	1	1	yes	characteristic
39	C₂³×C₅₃	C₂×C₁₆	1	3	yes	—
40	Q₈×C₅₃	C₂×C₁₆	1	4	yes	—
41	D₄×C₅₃	C₂×C₁₆	1	12	yes	—
42	C₂×C₄×C₅₃	C₂×C₁₆	1	3	yes	—
43	C₂×C₄×C₅₃	C₂×C₁₆	1	6	yes	—
44	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
46	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
47	C₈×C₅₃	C₂²×C₈	1	6	yes	—
48	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
49	C₂×#2258	—	53	1	no	maximal, Sylow
50	C₂²×C₅₃	—	2	6	no	—
51	C₂²×C₅₃	C₂²×C₁₆	1	6	yes	—
52	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
53	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
54	C₄×C₅₃	C₂²×C₁₆	1	6	yes	—
55	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
56	C₂×#504	D₅₃	1	1	yes	characteristic
57	#2258	—	53	8	no	—
58	C₂×M₆₄	—	53	3	no	—
59	C₂²×C₃₂	—	53	3	no	—
60	C₂×C₅₃	—	2	6	no	—
61	C₂×C₅₃	#2258	1	2	yes	complemented
62	C₂×C₅₃	C₂³×C₁₆	1	1	yes	characteristic
63	C₂×#131	Dic₅₃	1	1	yes	characteristic
64	#504	Dic₅₃	1	4	yes	—
65	#504	C₂×D₅₃	1	4	yes	—
66	C₂×M₃₂	Dic₅₃	1	3	yes	—
67	C₂²×C₁₆	C₂×D₅₃	1	3	yes	—
68	M₆₄	—	53	12	no	—
69	C₂×C₃₂	—	53	12	no	—
70	C₂×C₃₂	—	53	1	no	—
71	C₂×C₃₂	—	53	3	no	—
72	C₅₃	C₂×#2258	1	1	yes	characteristic, complemented, Sylow
73	C₂×#41	#88322	1	1	yes	characteristic
74	#131	#88322	1	4	yes	—
75	#131	C₂×Dic₅₃	1	4	yes	—
76	C₂×M₁₆	#88322	1	3	yes	—
77	C₂²×C₈	C₂×Dic₅₃	1	3	yes	—
78	M₃₂	C₂×Dic₅₃	1	12	yes	—
79	C₂×C₁₆	C₂×Dic₅₃	1	3	yes	—
80	C₂×C₁₆	C₂×Dic₅₃	1	6	yes	—
81	C₂×C₁₆	C₂²×D₅₃	1	6	yes	—
82	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
83	C₃₂	—	53	2	no	—
84	C₃₂	—	53	6	no	—
85	#41	#11719908	1	4	yes	—
86	#41	C₂×#88322	1	4	yes	—
87	C₂×Q₈	#11719908	1	1	yes	characteristic
88	C₂×D₄	#11719908	1	3	yes	—
89	C₂²×C₄	C₂×#88322	1	3	yes	—
90	M₁₆	C₂×#88322	1	12	yes	—
91	C₂×C₈	C₂×#88322	1	3	yes	—
92	C₂×C₈	C₂×#88322	1	6	yes	—
93	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
94	C₂×C₈	C₂²×Dic₅₃	1	6	yes	—
95	C₁₆	C₂²×Dic₅₃	1	1	yes	characteristic
96	C₁₆	C₂²×Dic₅₃	1	6	yes	—
97	C₁₆	C₂³×D₅₃	1	1	yes	characteristic
98	C₂³	C₂×#11719908	1	3	yes	—
99	Q₈	C₂×#11719908	1	4	yes	—
100	D₄	C₂×#11719908	1	12	yes	—
101	C₂×C₄	C₂×#11719908	1	3	yes	—
102	C₂×C₄	C₂×#11719908	1	6	yes	—
103	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
104	C₂×C₄	C₂²×#88322	1	6	yes	—
105	C₈	C₂²×#88322	1	1	yes	characteristic
106	C₈	C₂²×#88322	1	6	yes	—
107	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
108	C₂²	—	2	6	no	—
109	C₂²	C₂²×#11719908	1	6	yes	—
110	C₂²	C₂²×#11719908	1	1	yes	characteristic
111	C₄	C₂²×#11719908	1	1	yes	characteristic
112	C₄	C₂²×#11719908	1	6	yes	—
113	C₄	C₂³×#88322	1	1	yes	characteristic
114	C₂	—	2	6	no	—
115	C₂	#20277009	1	2	yes	direct product factor
116	C₂	C₂³×#11719908	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₅₃
commutator	C₂×C₅₃	C₂³×C₁₆
Frattini	C₁₆	C₂³×D₅₃
Fitting	C₂×C₅₃×#504	C₂
socle	C₂²×C₅₃	C₂²×C₁₆
prime radical	C₂×#41×C₅₃	C₈
cosocle	C₁₆×C₅₃	C₂⁴
hypercenter	C₂×#504	D₅₃
hypocenter	C₅₃	C₂×#2258
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. #20277009	C₂	direct (N × H)
2. C₂×#16810457	C₂	semidirect
3. C₂²×#13133123	C₂	semidirect
5. #16810457	C₂²	semidirect
8. C₂×#13133123	C₂²	semidirect
61. C₂×C₅₃	#2258	semidirect
72. C₅₃	C₂×#2258	semidirect
115. C₂	#20277009	direct (N × H)

Maximal subgroups

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

subgroup	index
#20277009	2
C₂×#16810457	2
C₂²×#13133123	2
C₂×C₅₃×#504	2
C₂×#2258	53

Subgroups — C2×#20277009 #27542201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20277009 #27542201