Subgroups — C₂×#20289884 #28312670

Order 13568.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289884	C₂	1	4	yes	maximal, direct product factor
2	C₂×#16811462	C₂	1	2	yes	maximal, complemented
3	C₂²×#13133123	C₂	1	1	yes	maximal, characteristic
4	#16811462	C₂²	1	4	yes	complemented
5	C₂×#13133123	C₄	1	2	yes	—
6	C₂×#13133123	C₄	1	1	yes	characteristic
7	C₂×#13133123	C₂²	1	1	yes	characteristic
8	C₂×#13133123	C₂²	1	2	yes	—
9	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
10	#13133123	C₂×C₄	1	1	yes	characteristic
11	#13133123	C₂×C₄	1	2	yes	—
12	#13133123	C₂³	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	C₈	1	1	yes	characteristic
14	C₂×C₁₆×C₅₃	C₈	1	1	yes	characteristic
15	C₂×C₁₆×C₅₃	C₈	1	2	yes	—
16	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₁₆	1	1	yes	characteristic
19	C₂×C₈×C₅₃	C₁₆	1	2	yes	—
20	C₂×C₈×C₅₃	C₁₆	1	1	yes	characteristic
21	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
22	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
23	C₁₆×C₅₃	C₂×C₈	1	2	yes	—
24	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
25	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
26	C₂³×C₅₃	C₃₂	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₃₂	1	2	yes	—
28	C₂×C₄×C₅₃	C₃₂	1	1	yes	characteristic
29	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
30	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
31	C₈×C₅₃	C₂×C₁₆	1	2	yes	—
32	C₈×C₅₃	C₂×C₁₆	1	1	yes	characteristic
33	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
34	C₂×M₁₂₈	—	53	1	no	maximal, Sylow
35	C₂²×C₅₃	—	2	2	no	—
36	C₂²×C₅₃	C₂×C₃₂	1	2	yes	—
37	C₂²×C₅₃	C₂×C₃₂	1	1	yes	characteristic
38	C₄×C₅₃	C₂×C₃₂	1	1	yes	characteristic
39	C₄×C₅₃	C₂×C₃₂	1	2	yes	—
40	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
41	C₂²×C₃₂	—	53	1	no	—
42	M₁₂₈	—	53	4	no	—
43	C₂×C₆₄	—	53	2	no	—
44	C₂×C₅₃	—	2	2	no	—
45	C₂×C₅₃	M₁₂₈	1	2	yes	complemented
46	C₂×C₅₃	C₂²×C₃₂	1	1	yes	characteristic
47	C₂²×C₁₆	#6157	1	1	yes	characteristic
48	C₂×C₃₂	—	53	2	no	—
49	C₂×C₃₂	—	53	2	no	—
50	C₂×C₃₂	—	53	1	no	—
51	C₂×C₃₂	—	53	1	no	—
52	C₆₄	—	53	4	no	—
53	C₅₃	C₂×M₁₂₈	1	1	yes	characteristic, complemented, Sylow
54	C₂²×C₈	#88324	1	1	yes	characteristic
55	C₂×C₁₆	#88324	1	1	yes	characteristic
56	C₂×C₁₆	#88324	1	2	yes	—
57	C₂×C₁₆	C₂×#6157	1	2	yes	—
58	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
59	C₃₂	—	53	1	no	—
60	C₃₂	—	53	2	no	—
61	C₃₂	—	53	1	no	—
62	C₂²×C₄	#11719910	1	1	yes	characteristic
63	C₂×C₈	#11719910	1	1	yes	characteristic
64	C₂×C₈	#11719910	1	2	yes	—
65	C₂×C₈	C₂×#88324	1	2	yes	—
66	C₂×C₈	C₂×#88324	1	1	yes	characteristic
67	C₁₆	C₂×#88324	1	2	yes	—
68	C₁₆	C₂×#88324	1	1	yes	characteristic
69	C₁₆	C₂²×#6157	1	1	yes	characteristic
70	C₂³	#13133267	1	1	yes	characteristic
71	C₂×C₄	#13133267	1	2	yes	—
72	C₂×C₄	#13133267	1	1	yes	characteristic
73	C₂×C₄	C₂×#11719910	1	2	yes	—
74	C₂×C₄	C₂×#11719910	1	1	yes	characteristic
75	C₈	C₂×#11719910	1	2	yes	—
76	C₈	C₂×#11719910	1	1	yes	characteristic
77	C₈	C₂²×#88324	1	1	yes	characteristic
78	C₂²	—	2	2	no	—
79	C₂²	C₂×#13133267	1	2	yes	—
80	C₂²	C₂×#13133267	1	1	yes	characteristic
81	C₄	C₂×#13133267	1	1	yes	characteristic
82	C₄	C₂×#13133267	1	2	yes	—
83	C₄	C₂²×#11719910	1	1	yes	characteristic
84	C₂	—	2	2	no	—
85	C₂	#20289884	1	2	yes	direct product factor
86	C₂	C₂²×#13133267	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂×#6157
commutator	C₂×C₅₃	C₂²×C₃₂
Frattini	C₁₆	C₂²×#6157
Fitting	C₂²×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₃₂
prime radical	C₂³×C₅₃	C₃₂
cosocle	#13133123	C₂³
hypercenter	C₂²×C₁₆	#6157
hypocenter	C₅₃	C₂×M₁₂₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
1. #20289884	C₂	direct (N × H)
2. C₂×#16811462	C₂	semidirect
4. #16811462	C₂²	semidirect
45. C₂×C₅₃	M₁₂₈	semidirect
53. C₅₃	C₂×M₁₂₈	semidirect
85. C₂	#20289884	direct (N × H)

Maximal subgroups

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

subgroup	index
#20289884	2
C₂×#16811462	2
C₂²×#13133123	2
C₂×M₁₂₈	53

Subgroups — C2×#20289884 #28312670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20289884 #28312670