Subgroups — C₂×#20272781 #27295644

Order 13568.

254 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810374	C₂	1	2	yes	maximal
2	#20272781	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#88322	—	2	2	no	—
5	#16810374	—	2	2	no	—
6	#16810374	C₂²	1	4	yes	—
7	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#346	C₄	1	2	yes	—
11	C₅₃×#346	C₂²	1	2	yes	—
12	C₂²×#88322	—	4	2	no	—
13	C₄⋊#88322	—	2	4	no	—
14	C₄⋊#88322	—	4	2	no	—
15	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₅₃	—	2	2	no	—
19	C₂×C₁₆×C₅₃	C₈	1	4	yes	complemented
20	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
25	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
28	C₂×C₈×C₅₃	—	2	2	no	—
29	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
32	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
33	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
35	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₄²×C₅₃	SD₁₆	1	2	yes	—
37	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×#88322	—	4	4	no	—
39	C₂×#88322	—	8	4	no	—
40	C₁₆×C₅₃	—	2	2	no	—
41	C₁₆×C₅₃	C₂×C₈	1	4	yes	complemented
42	C₂³×C₅₃	#106	1	1	yes	characteristic
43	C₂×C₄×C₅₃	—	2	4	no	—
44	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
45	C₂×C₄×C₅₃	#106	1	2	yes	—
46	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#108	1	2	yes	—
48	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
50	C₈×C₅₃	—	2	1	no	—
51	C₈×C₅₃	—	2	1	no	—
52	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
53	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
54	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
55	#88322	—	8	4	no	—
56	C₂×#1368	—	53	1	no	maximal, Sylow
57	C₂²×C₅₃	#334	1	2	yes	—
58	C₂²×C₅₃	#365	1	4	yes	—
59	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
60	C₄×C₅₃	—	2	2	no	—
61	C₄×C₅₃	—	2	2	no	—
62	C₄×C₅₃	#334	1	2	yes	—
63	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
64	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
65	C₂×#346	D₅₃	1	1	yes	characteristic
66	C₂×#334	—	53	2	no	—
67	#1368	—	53	4	no	—
68	C₂×C₅₃	#1368	1	4	yes	complemented
69	C₂×C₅₃	C₂×#334	1	1	yes	characteristic
70	C₂×C₅₃	C₂×#365	1	2	yes	—
71	C₂²×C₁₆	Dic₅₃	1	1	yes	characteristic
72	C₂²×C₁₆	Dic₅₃	1	1	yes	characteristic
73	C₂×C₄⋊C₈	—	106	2	no	—
74	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
75	#346	Dic₅₃	1	2	yes	—
76	#346	C₂×D₅₃	1	2	yes	—
77	#334	—	53	4	no	—
78	#334	—	106	2	no	—
79	C₅₃	C₂×#1368	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₈	—	212	2	no	—
81	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
82	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
83	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
84	C₂×C₁₆	—	2	2	no	—
85	C₂×C₁₆	#88322	1	4	yes	complemented
86	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
87	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
88	C₄⋊C₈	—	106	4	no	—
89	C₄⋊C₈	—	212	2	no	—
90	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
91	C₄×C₈	D₂₁₂	1	1	yes	characteristic
92	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
93	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
94	C₂²×C₄	#11719913	1	2	yes	—
95	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	424	4	no	—
98	C₂×C₈	—	212	4	no	—
99	C₂×C₈	C₂×#88322	1	2	yes	—
100	C₂×C₈	#11719917	1	2	yes	—
101	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
102	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
103	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
104	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
105	C₄²	#11719913	1	2	yes	—
106	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
107	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
108	C₁₆	—	2	2	no	—
109	C₁₆	C₂×#88322	1	4	yes	complemented
110	C₂³	#13133149	1	1	yes	characteristic
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	C₄⋊#88322	1	2	yes	—
113	C₂×C₄	#13133149	1	2	yes	—
114	C₂×C₄	#13133149	1	1	yes	characteristic
115	C₂×C₄	#13133153	1	2	yes	—
116	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
117	C₂×C₄	C₂×#11719913	1	2	yes	—
118	C₈	—	2	1	no	—
119	C₈	—	2	1	no	—
120	C₈	—	424	4	no	—
121	C₈	C₄⋊#88322	1	2	yes	—
122	C₈	C₂²×#88322	1	1	yes	characteristic
123	C₈	C₂×#11719917	1	1	yes	characteristic
124	C₂²	#16810374	1	2	yes	—
125	C₂²	#16810446	1	4	yes	—
126	C₂²	C₂×#13133149	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	2	no	—
129	C₄	#16810374	1	2	yes	—
130	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
131	C₄	C₂×#13133153	1	1	yes	characteristic
132	C₂	#20272781	1	4	yes	direct product factor
133	C₂	C₂×#16810374	1	1	yes	characteristic
134	C₂	C₂×#16810446	1	2	yes	—

	subgroup	G/H
center	C₂³	#13133149
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#346	C₂
socle, prime radical	C₂³×C₅₃	#106
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	C₂×#1368
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. #20272781	C₂	direct (N × H)
19. C₂×C₁₆×C₅₃	C₈	semidirect
41. C₁₆×C₅₃	C₂×C₈	semidirect
68. C₂×C₅₃	#1368	semidirect
79. C₅₃	C₂×#1368	semidirect
85. C₂×C₁₆	#88322	semidirect
109. C₁₆	C₂×#88322	semidirect
132. C₂	#20272781	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×#16810374	2
#20272781	2
C₂×C₅₃×#346	2
C₂×#1368	53

Subgroups — C2×#20272781 #27295644

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272781 #27295644