Subgroups — #65706

Order 320.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4038	C₂	1	1	yes	maximal, characteristic
2	#4004	C₂	1	1	yes	maximal, characteristic, complemented
3	#4003	C₂	1	1	yes	maximal, characteristic
4	#4003	C₂	1	1	yes	maximal, characteristic
5	#4002	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄⋊#170	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#170	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
9	C₂×#170	C₂²	1	1	yes	characteristic
10	C₂×#170	C₂²	1	1	yes	characteristic
11	C₂×#170	C₂²	1	1	yes	characteristic
12	C₂×#170	C₂²	1	1	yes	characteristic
13	C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
14	#685	C₄	1	1	yes	characteristic
15	#685	C₄	1	1	yes	characteristic
16	C₄⋊Dic₅	C₄	1	1	yes	characteristic
17	C₄×Dic₅	C₂²	1	1	yes	characteristic
18	C₄×Dic₅	C₂²	1	1	yes	characteristic
19	#433	—	5	1	no	maximal, Sylow
20	C₂²×D₅	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₅	C₂×C₄	1	1	yes	characteristic
25	C₂×Dic₅	C₂×C₄	1	1	yes	characteristic
26	C₂×Dic₅	C₂³	1	1	yes	characteristic
27	C₄×D₅	—	2	1	no	—
28	C₄×D₅	—	2	1	no	—
29	#170	—	2	1	no	—
30	#170	—	2	1	no	—
31	#170	—	2	1	no	—
32	#170	—	2	1	no	—
33	#117	—	5	1	no	—
34	C₄⋊C₈	—	5	1	no	—
35	C₄⋊C₈	—	5	1	no	—
36	C₂²⋊C₈	—	5	1	no	—
37	C₂²⋊C₈	—	5	1	no	—
38	#97	—	5	1	no	—
39	C₄×C₈	—	5	1	no	—
40	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
41	C₂×D₅	—	2	1	no	—
42	C₂×D₅	—	2	1	no	—
43	C₂×D₅	—	2	1	no	—
44	C₄×C₅	—	2	1	no	—
45	C₄×C₅	—	2	1	no	—
46	C₄×C₅	—	2	1	no	—
47	Dic₅	—	2	1	no	—
48	Dic₅	—	2	1	no	—
49	Dic₅	#41	1	1	yes	characteristic
50	Dic₅	#41	1	1	yes	characteristic
51	C₂²×C₄	—	5	1	no	—
52	C₂×C₈	—	5	1	no	—
53	C₂×C₈	—	5	1	no	—
54	C₂×C₈	—	5	1	no	—
55	C₂×C₈	—	5	1	no	—
56	C₄⋊C₄	—	5	1	no	—
57	C₄⋊C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
58	C₂²⋊C₄	—	5	1	no	—
59	C₂²⋊C₄	—	5	1	no	—
60	C₄²	—	5	1	no	—
61	C₄²	—	5	1	no	—
62	C₂×C₅	#117	1	1	yes	characteristic
63	C₂×C₅	#131	1	1	yes	characteristic
64	C₂×C₅	#131	1	1	yes	characteristic
65	D₅	—	4	1	no	—
66	C₂³	—	5	1	no	—
67	C₂×C₄	—	5	1	no	—
68	C₂×C₄	—	5	1	no	—
69	C₂×C₄	—	5	1	no	—
70	C₂×C₄	—	10	1	no	—
71	C₂×C₄	—	10	1	no	—
72	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
73	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
74	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
75	C₈	—	10	1	no	—
76	C₈	—	10	1	no	—
77	C₈	—	10	1	no	—
78	C₈	—	10	1	no	—
79	C₅	#433	1	1	yes	characteristic, complemented, Sylow
80	C₂²	—	10	1	no	—
81	C₂²	—	10	1	no	—
82	C₂²	—	10	1	no	—
83	C₂²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	2	1	no	—
86	C₄	—	2	1	no	—
87	C₄	—	5	1	no	—
88	C₄	—	5	1	no	—
89	C₄	—	10	1	no	—
90	C₄	—	10	1	no	—
91	C₂	—	20	1	no	—
92	C₂	#4130	1	1	yes	characteristic
93	C₂	#4131	1	1	yes	characteristic
94	C₂	#4133	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂²×C₄
Fitting	C₅×C₄⋊C₄	C₄
prime radical	C₂²×D₅	C₂×C₄
cosocle	C₂×Dic₅	C₂³
hypercenter	C₄⋊C₄	AGL(1,𝔽₅)
hypocenter	C₅	#433
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #4004	C₂	semidirect
5. #4002	C₂	semidirect
6. C₄⋊#170	C₂	semidirect
7. C₄×#170	C₂	semidirect
79. C₅	#433	semidirect

Maximal subgroups

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

subgroup	index
#4038	2
#4004	2
#4003	2
#4002	2
C₄⋊#170	2
C₄×#170	2
#433	5