Subgroups — C₂×#3996 #65723

Order 320.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#700	C₂	1	2	yes	maximal
2	C₂×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	#3996	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
5	C₂×#170	—	2	2	no	—
6	#700	—	2	2	no	—
7	#700	C₂²	1	4	yes	—
8	C₂×C₅×C₈	C₄	1	1	yes	characteristic
9	C₂×#168	C₄	1	1	yes	characteristic
10	C₈×D₅	C₄	1	2	yes	—
11	C₈×D₅	C₂²	1	2	yes	—
12	C₂×#108	—	5	1	no	maximal, Sylow
13	C₂²×D₅	Q₈	1	1	yes	characteristic
14	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
15	C₂×Dic₅	D₄	1	1	yes	characteristic
16	C₄×D₅	C₂×C₄	1	1	yes	characteristic
17	C₄×D₅	D₄	1	1	yes	characteristic
18	C₄×D₅	Q₈	1	1	yes	characteristic
19	C₄×D₅	C₂³	1	1	yes	characteristic
20	#170	—	2	4	no	—
21	C₅×C₈	C₂×C₄	1	2	yes	—
22	#168	C₂×C₄	1	2	yes	—
23	C₂×M₁₆	—	5	2	no	—
24	C₂²×C₈	—	5	1	no	—
25	#108	—	5	4	no	—
26	C₂²×C₅	C₄⋊C₄	1	1	yes	characteristic
27	C₂×D₅	—	2	2	no	—
28	C₂×D₅	C₄⋊C₄	1	1	yes	characteristic
29	C₂×D₅	C₂×Q₈	1	1	yes	characteristic
30	C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
31	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
32	Dic₅	C₄⋊C₄	1	1	yes	characteristic
33	Dic₅	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄	—	5	1	no	—
35	M₁₆	—	5	4	no	—
36	M₁₆	—	10	2	no	—
37	C₂×C₈	—	5	1	no	—
38	C₂×C₈	—	5	2	no	—
39	C₂×C₈	—	5	2	no	—
40	C₂×C₈	—	10	2	no	—
41	C₂×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
42	C₂×C₅	#108	1	2	yes	complemented
43	C₂×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
44	D₅	—	2	1	no	—
45	D₅	—	2	1	no	—
46	C₂³	—	5	1	no	—
47	C₂×C₄	—	5	1	no	—
48	C₂×C₄	—	5	1	no	—
49	C₂×C₄	—	5	1	no	—
50	C₂×C₄	—	5	1	no	—
51	C₂×C₄	—	5	1	no	—
52	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
53	C₈	—	5	2	no	—
54	C₈	—	10	4	no	—
55	C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
56	C₅	C₂×#108	1	1	yes	characteristic, complemented, Sylow
57	C₂²	—	5	1	no	—
58	C₂²	—	5	1	no	—
59	C₂²	—	10	2	no	—
60	C₂²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
61	C₄	—	5	1	no	—
62	C₄	—	5	1	no	—
63	C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
64	C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
65	C₂	—	10	1	no	—
66	C₂	—	10	1	no	—
67	C₂	#3996	1	2	yes	direct product factor
68	C₂	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₂²×C₄
Frattini	C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₅×C₈	C₄
socle	C₂²×C₅	C₄⋊C₄
prime radical	C₂²×D₅	Q₈
cosocle	C₄×D₅	C₂³
hypercenter	C₂×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₂×#108
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
3. #3996	C₂	direct (N × H)
42. C₂×C₅	#108	semidirect
56. C₅	C₂×#108	semidirect
67. C₂	#3996	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×#700	2
C₂×C₈×D₅	2
#3996	2
C₂×#108	5

Subgroups — C2×#3996 #65723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3996 #65723