Subgroups — #66032

Order 320.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#4041	C₂	1	4	yes	maximal, complemented
3	#4041	C₂	1	2	yes	maximal, complemented
4	#4040	C₂	1	4	yes	maximal, complemented
5	#4039	C₂	1	2	yes	maximal, complemented
6	#4020	C₂	1	1	yes	maximal, characteristic, complemented
7	#4018	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₂₀	C₂²	1	2	yes	—
9	C₂×D₂₀	C₂²	1	4	yes	—
10	C₂×D₂₀	C₂²	1	4	yes	—
11	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₅	C₂²	1	4	yes	—
13	C₅×C₄⋊C₄	C₂²	1	2	yes	—
14	C₅×C₄⋊C₄	C₂²	1	4	yes	—
15	C₄²×C₅	C₂²	1	1	yes	characteristic
16	#685	C₂²	1	2	yes	—
17	#685	C₂²	1	4	yes	—
18	#685	C₂²	1	4	yes	—
19	#683	C₂²	1	2	yes	—
20	C₄×Dic₅	C₂²	1	1	yes	characteristic
21	#540	—	5	1	no	maximal, Sylow
22	C₂²×D₅	C₂³	1	1	yes	characteristic
23	C₂²×D₅	C₂³	1	4	yes	—
24	C₂×C₄×C₅	C₂³	1	2	yes	—
25	C₂×C₄×C₅	C₂³	1	4	yes	—
26	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂×Dic₅	C₂³	1	2	yes	—
28	C₂×Dic₅	C₂³	1	1	yes	characteristic
29	D₂₀	—	2	4	no	—
30	D₂₀	—	4	4	no	—
31	D₂₀	—	4	4	no	—
32	C₄×D₅	—	2	2	no	—
33	C₄×D₅	—	4	4	no	—
34	#127	—	5	1	no	—
35	#125	D₅	1	1	yes	characteristic, complemented
36	#123	—	5	4	no	—
37	#121	—	5	4	no	—
38	#121	—	5	2	no	—
39	C₄×D₄	—	5	2	no	—
40	#117	—	5	1	no	—
41	C₂²×C₅	C₂⁴	1	1	yes	characteristic
42	C₂×D₅	—	2	1	no	—
43	C₂×D₅	—	2	4	no	—
44	C₂×D₅	—	2	4	no	—
45	C₂×D₅	—	2	2	no	—
46	C₂×D₅	—	2	4	no	—
47	C₄×C₅	—	2	4	no	—
48	C₄×C₅	—	2	2	no	—
49	C₄×C₅	#41	1	2	yes	—
50	Dic₅	—	2	1	no	—
51	Dic₅	—	2	2	no	—
52	C₂×D₄	—	5	2	no	—
53	C₂×D₄	—	5	4	no	—
54	C₂×D₄	—	5	4	no	—
55	C₂²×C₄	—	5	1	no	—
56	C₂²×C₄	—	5	4	no	—
57	C₄⋊C₄	—	5	2	no	—
58	C₄⋊C₄	C₂×D₅	1	2	yes	—
59	C₄⋊C₄	C₂×D₅	1	4	yes	—
60	C₂²⋊C₄	—	5	2	no	—
61	C₂²⋊C₄	—	5	4	no	—
62	C₂²⋊C₄	—	5	4	no	—
63	C₄²	—	5	1	no	—
64	C₄²	C₂×D₅	1	1	yes	characteristic
65	C₂×C₅	C₂×#41	1	1	yes	characteristic
66	C₂×C₅	H(5,𝔽₂)	1	2	yes	—
67	D₅	—	4	1	no	—
68	D₅	—	4	4	no	—
69	C₂³	—	5	1	no	—
70	C₂³	—	5	4	no	—
71	D₄	—	10	4	no	—
72	D₄	—	20	4	no	—
73	D₄	—	20	4	no	—
74	C₂×C₄	—	5	2	no	—
75	C₂×C₄	—	5	1	no	—
76	C₂×C₄	—	10	2	no	—
77	C₂×C₄	—	20	4	no	—
78	C₂×C₄	C₂²×D₅	1	2	yes	—
79	C₂×C₄	C₂²×D₅	1	4	yes	—
80	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
81	C₅	#540	1	1	yes	characteristic, complemented, Sylow
82	C₂²	—	10	1	no	—
83	C₂²	—	10	4	no	—
84	C₂²	—	10	4	no	—
85	C₂²	—	10	2	no	—
86	C₂²	—	10	4	no	—
87	C₂²	C₂³×D₅	1	1	yes	characteristic
88	C₄	—	2	4	no	—
89	C₄	—	2	2	no	—
90	C₄	—	10	1	no	—
91	C₄	—	10	2	no	—
92	C₄	#713	1	2	yes	—
93	C₂	—	20	1	no	—
94	C₂	—	20	4	no	—
95	C₂	C₂×#713	1	1	yes	characteristic
96	C₂	#4149	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×#125	C₂
hypercenter	#125	D₅
hypocenter	C₅	#540
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. C₅×#125	C₂	semidirect
2. #4041	C₂	semidirect
3. #4041	C₂	semidirect
4. #4040	C₂	semidirect
5. #4039	C₂	semidirect
6. #4020	C₂	semidirect
7. #4018	C₂	semidirect
35. #125	D₅	semidirect
81. C₅	#540	semidirect

Maximal subgroups

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

subgroup	index
C₅×#125	2
#4041	2
#4040	2
#4039	2
#4020	2
#4018	2
#540	5