Subgroups — C₂×#20272851 #27304277

Order 13568.

222 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810457	C₂	1	2	yes	maximal
2	#20272851	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
4	#16810457	C₄	1	4	yes	complemented
5	#16810457	C₂²	1	4	yes	—
6	C₂×#13133123	C₄	1	4	yes	complemented
7	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
10	C₅₃×#346	C₄	1	2	yes	—
11	C₅₃×#346	C₂²	1	2	yes	—
12	#13133123	C₂×C₄	1	8	yes	complemented
13	C₂²×C₈×C₅₃	C₈	1	1	yes	characteristic
14	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₅₃	C₈	1	1	yes	characteristic
16	C₂×C₁₆×C₅₃	C₈	1	2	yes	—
17	C₂×C₁₆×C₅₃	C₈	1	2	yes	—
18	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₅₃	C₈	1	2	yes	—
24	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₅₃	—	2	1	no	—
26	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈×C₅₃	—	2	1	no	—
28	C₂×C₈×C₅₃	—	2	1	no	—
29	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
30	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
32	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₅₃	—	2	1	no	—
37	C₄²×C₅₃	C₂×C₈	1	2	yes	—
38	C₁₆×C₅₃	C₄²	1	2	yes	—
39	C₁₆×C₅₃	C₂×C₈	1	4	yes	—
40	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
41	C₂³×C₅₃	M₃₂	1	1	yes	characteristic
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	—	4	2	no	—
44	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
46	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
47	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
48	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
50	C₈×C₅₃	—	2	2	no	—
51	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
52	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
53	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
54	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
55	C₂×#1398	—	53	1	no	maximal, Sylow
56	C₂²×C₅₃	—	2	2	no	—
57	C₂²×C₅₃	#346	1	1	yes	characteristic
58	C₂²×C₅₃	#346	1	1	yes	characteristic
59	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
60	C₄×C₅₃	—	4	2	no	—
61	C₄×C₅₃	#346	1	1	yes	characteristic
62	C₄×C₅₃	#346	1	1	yes	characteristic
63	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
64	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
65	C₂×M₆₄	—	53	2	no	—
66	C₂×#346	D₅₃	1	1	yes	characteristic
67	#1398	—	53	4	no	—
68	C₂×C₅₃	—	2	1	no	—
69	C₂×C₅₃	—	2	1	no	—
70	C₂×C₅₃	#1398	1	2	yes	complemented
71	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
72	C₂²×C₁₆	Dic₅₃	1	1	yes	characteristic
73	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
74	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
75	M₆₄	—	53	4	no	—
76	M₆₄	—	53	4	no	—
77	C₂×C₃₂	—	53	4	no	—
78	#346	Dic₅₃	1	2	yes	—
79	#346	C₂×D₅₃	1	2	yes	—
80	C₅₃	C₂×#1398	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	#88322	1	1	yes	characteristic
82	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
83	C₂×C₄²	#88322	1	1	yes	characteristic
84	C₂×C₁₆	#88322	1	2	yes	—
85	C₂×C₁₆	#88322	1	2	yes	—
86	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
87	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
88	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
89	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
90	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
91	C₄×C₈	#88322	1	2	yes	—
92	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
93	C₃₂	—	53	8	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	C₂×#88322	1	2	yes	—
99	C₂×C₈	C₂×#88322	1	1	yes	characteristic
100	C₂×C₈	C₂×#88322	1	1	yes	characteristic
101	C₂×C₈	C₂×#88322	1	1	yes	characteristic
102	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
103	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
104	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
105	C₄²	—	2	1	no	—
106	C₄²	C₂×#88322	1	2	yes	—
107	C₁₆	C₂×#88322	1	4	yes	—
108	C₁₆	C₄×Dic₅₃	1	2	yes	—
109	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
110	C₂³	#13133157	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₄×#88322	1	1	yes	characteristic
114	C₂×C₄	C₄×#88322	1	1	yes	characteristic
115	C₂×C₄	#13133157	1	1	yes	characteristic
116	C₂×C₄	#13133157	1	1	yes	characteristic
117	C₂×C₄	#13133157	1	1	yes	characteristic
118	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	C₄×#88322	1	1	yes	characteristic
121	C₈	C₄×#88322	1	1	yes	characteristic
122	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
123	C₈	C₂²×#88322	1	1	yes	characteristic
124	C₂²	—	2	2	no	—
125	C₂²	#16810399	1	1	yes	characteristic
126	C₂²	#16810399	1	1	yes	characteristic
127	C₂²	C₂×#13133157	1	1	yes	characteristic
128	C₄	—	4	2	no	—
129	C₄	#16810399	1	1	yes	characteristic
130	C₄	#16810399	1	1	yes	characteristic
131	C₄	C₂×C₄×#88322	1	1	yes	characteristic
132	C₄	C₂×#13133157	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	2	1	no	—
135	C₂	#20272851	1	2	yes	direct product factor
136	C₂	C₂×#16810399	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×Dic₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₂×C₅₃×#346	C₂
socle	C₂²×C₅₃	#346
prime radical	C₂³×C₅₃	M₃₂
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	C₂×#1398
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. #20272851	C₂	direct (N × H)
4. #16810457	C₄	semidirect
6. C₂×#13133123	C₄	semidirect
12. #13133123	C₂×C₄	semidirect
70. C₂×C₅₃	#1398	semidirect
80. C₅₃	C₂×#1398	semidirect
135. C₂	#20272851	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×#16810457	2
#20272851	2
C₂×C₅₃×#346	2
C₂×#1398	53

Subgroups — C2×#20272851 #27304277

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272851 #27304277