Subgroups — C₂²×#3998 #10617455

Order 640.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#168	C₂	1	1	yes	maximal, characteristic
2	C₂×#3998	C₂	1	12	yes	maximal, direct product factor
3	C₂²×#674	C₂	1	2	yes	maximal, complemented
4	C₂³×C₄×C₅	C₄	1	1	yes	characteristic
5	C₂²×#168	C₄	1	6	yes	—
6	C₂²×#168	C₄	1	1	yes	characteristic
7	C₂²×#168	C₂²	1	1	yes	characteristic
8	C₂²×#168	C₂²	1	6	yes	—
9	#3998	C₂²	1	16	yes	direct product factor
10	C₂×#674	C₂²	1	12	yes	complemented
11	C₂²×M₃₂	—	5	1	no	maximal, Sylow
12	C₂⁴×C₅	C₈	1	1	yes	characteristic
13	C₂²×C₄×C₅	C₈	1	6	yes	—
14	C₂²×C₄×C₅	C₈	1	1	yes	characteristic
15	C₂²×C₄×C₅	C₂×C₄	1	6	yes	—
16	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂×#168	C₂×C₄	1	3	yes	—
18	C₂×#168	C₂×C₄	1	12	yes	—
19	C₂×#168	C₂×C₄	1	6	yes	—
20	C₂×#168	C₂³	1	3	yes	—
21	C₂×#168	C₂³	1	4	yes	—
22	#674	C₂³	1	8	yes	complemented
23	C₂³×C₈	—	5	1	no	—
24	C₂×M₃₂	—	5	12	no	—
25	C₂²×C₁₆	—	5	2	no	—
26	C₂³×C₅	—	2	4	no	—
27	C₂³×C₅	C₂×C₈	1	6	yes	—
28	C₂³×C₅	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₅	C₂×C₈	1	12	yes	—
30	C₂×C₄×C₅	C₂×C₈	1	6	yes	—
31	C₂×C₄×C₅	C₂×C₈	1	3	yes	—
32	C₂×C₄×C₅	C₂²×C₄	1	4	yes	—
33	C₂×C₄×C₅	C₂²×C₄	1	3	yes	—
34	#168	C₂²×C₄	1	3	yes	—
35	#168	C₂²×C₄	1	4	yes	—
36	#168	C₂⁴	1	1	yes	characteristic
37	C₂³×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
38	C₂²×C₈	—	5	6	no	—
39	C₂²×C₈	—	5	6	no	—
40	C₂²×C₈	—	5	1	no	—
41	C₂²×C₈	—	5	1	no	—
42	M₃₂	—	5	16	no	—
43	C₂×C₁₆	—	5	12	no	—
44	C₂²×C₅	—	2	12	no	—
45	C₂²×C₅	M₃₂	1	4	yes	complemented
46	C₂²×C₅	C₂²×C₈	1	4	yes	—
47	C₂²×C₅	C₂²×C₈	1	3	yes	—
48	C₄×C₅	C₂²×C₈	1	4	yes	—
49	C₄×C₅	C₂²×C₈	1	3	yes	—
50	C₄×C₅	C₂³×C₄	1	1	yes	characteristic
51	C₂⁴	#170	1	1	yes	characteristic
52	C₂²×C₄	#170	1	1	yes	characteristic
53	C₂²×C₄	#170	1	6	yes	—
54	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	6	yes	—
55	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
56	C₂×C₈	—	5	3	no	—
57	C₂×C₈	—	5	4	no	—
58	C₂×C₈	—	5	12	no	—
59	C₂×C₈	—	5	3	no	—
60	C₂×C₈	—	5	6	no	—
61	C₁₆	—	5	8	no	—
62	C₂×C₅	—	2	4	no	—
63	C₂×C₅	C₂×M₃₂	1	6	yes	complemented
64	C₂×C₅	C₂³×C₈	1	1	yes	characteristic
65	C₂³	—	2	4	no	—
66	C₂³	C₂×#170	1	6	yes	—
67	C₂³	C₂×#170	1	1	yes	characteristic
68	C₂×C₄	C₂×#170	1	12	yes	—
69	C₂×C₄	C₂×#170	1	6	yes	—
70	C₂×C₄	C₂×#170	1	3	yes	—
71	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	4	yes	—
72	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	3	yes	—
73	C₈	—	5	1	no	—
74	C₈	—	5	3	no	—
75	C₈	—	5	4	no	—
76	C₅	C₂²×M₃₂	1	1	yes	characteristic, complemented, Sylow
77	C₂²	—	2	12	no	—
78	C₂²	#3998	1	4	yes	direct product factor
79	C₂²	C₂²×#170	1	4	yes	—
80	C₂²	C₂²×#170	1	3	yes	—
81	C₄	C₂²×#170	1	4	yes	—
82	C₄	C₂²×#170	1	3	yes	—
83	C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
84	C₂	—	2	4	no	—
85	C₂	C₂×#3998	1	6	yes	direct product factor
86	C₂	C₂³×#170	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂³×C₈
Frattini	C₄	C₂³×AGL(1,𝔽₅)
Fitting	C₂³×C₄×C₅	C₄
socle	C₂³×C₅	C₂×C₈
prime radical	C₂⁴×C₅	C₈
cosocle	#168	C₂⁴
hypercenter	C₂³×C₄	AGL(1,𝔽₅)
hypocenter	C₅	C₂²×M₃₂
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#3998	C₂	direct (N × H)
3. C₂²×#674	C₂	semidirect
9. #3998	C₂²	direct (N × H)
10. C₂×#674	C₂²	semidirect
22. #674	C₂³	semidirect
45. C₂²×C₅	M₃₂	semidirect
63. C₂×C₅	C₂×M₃₂	semidirect
76. C₅	C₂²×M₃₂	semidirect
78. C₂²	#3998	direct (N × H)
85. C₂	C₂×#3998	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#168	2
C₂×#3998	2
C₂²×#674	2
C₂²×M₃₂	5

Subgroups — C22×#3998 #10617455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#3998 #10617455