Group #66004 — subgroups

Subgroups — #66004

Order 320.

236 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	#4093	C₂	1	1	yes	maximal, characteristic, complemented
3	#4090	C₂	1	1	yes	maximal, characteristic, complemented
4	#4085	C₂	1	2	yes	maximal, complemented
5	#4031	C₂	1	2	yes	maximal, complemented
6	#4030	C₂	1	2	yes	maximal, complemented
7	#4027	C₂	1	2	yes	maximal, complemented
8	#4023	C₂	1	2	yes	maximal, complemented
9	#4018	C₂	1	2	yes	maximal, complemented
10	C₂×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×D₄	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₅	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₅	C₂²	1	2	yes	—
14	C₂²×Dic₅	C₂²	1	2	yes	—
15	C₂×D₂₀	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₅	C₂²	1	2	yes	—
17	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
18	C₅×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₅×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₄²×C₅	C₂²	1	1	yes	characteristic
21	C₂²⋊Dic₅	C₂²	1	2	yes	—
22	#685	C₂²	1	2	yes	—
23	#685	C₂²	1	2	yes	—
24	#685	C₂²	1	2	yes	—
25	#683	C₂²	1	2	yes	—
26	#683	C₂²	1	2	yes	—
27	#683	C₂²	1	2	yes	—
28	C₄×Dic₅	C₂²	1	2	yes	—
29	C₄×Dic₅	C₂²	1	1	yes	characteristic
30	C₄×Dic₅	C₂²	1	1	yes	characteristic
31	#555	—	5	1	no	maximal, Sylow
32	C₂³×C₅	C₂³	1	2	yes	—
33	C₂²×D₅	C₂³	1	2	yes	—
34	C₅×Q₈	—	4	1	no	—
35	C₅×D₄	—	4	1	no	—
36	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₅	C₂³	1	2	yes	—
38	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂²⋊D₅	—	2	4	no	—
41	C₂²⋊D₅	—	4	2	no	—
42	C₂×Dic₅	—	2	4	no	—
43	C₂×Dic₅	C₂³	1	2	yes	—
44	C₂×Dic₅	C₂³	1	2	yes	—
45	C₂×Dic₅	C₂³	1	2	yes	—
46	D₂₀	—	4	1	no	—
47	C₄×D₅	—	2	4	no	—
48	Dic₁₀	—	4	1	no	—
49	C₄⋊Q₈	—	5	1	no	—
50	#124	—	5	1	no	—
51	#124	—	5	2	no	—
52	#124	D₅	1	1	yes	characteristic, complemented
53	#121	—	5	2	no	—
54	#121	—	5	2	no	—
55	C₄×D₄	—	5	2	no	—
56	#117	—	5	2	no	—
57	#117	—	5	2	no	—
58	C₂²×C₅	—	2	2	no	—
59	C₂²×C₅	—	2	2	no	—
60	C₂²×C₅	—	2	2	no	—
61	C₂²×C₅	C₂⁴	1	1	yes	characteristic
62	C₂×D₅	—	2	2	no	—
63	C₂×D₅	—	2	2	no	—
64	C₂×D₅	—	2	2	no	—
65	C₄×C₅	—	2	2	no	—
66	C₄×C₅	—	2	1	no	—
67	C₄×C₅	—	2	1	no	—
68	C₄×C₅	—	2	1	no	—
69	Dic₅	—	2	2	no	—
70	Dic₅	—	2	2	no	—
71	Dic₅	#41	1	4	yes	—
72	C₂×Q₈	—	5	1	no	—
73	C₂×Q₈	C₂×D₅	1	1	yes	characteristic
74	C₂×D₄	—	5	2	no	—
75	C₂×D₄	—	5	2	no	—
76	C₂×D₄	—	5	1	no	—
77	C₂×D₄	C₂×D₅	1	1	yes	characteristic
78	C₂²×C₄	—	5	2	no	—
79	C₂²×C₄	—	5	2	no	—
80	C₄⋊C₄	—	5	2	no	—
81	C₄⋊C₄	—	5	2	no	—
82	C₄⋊C₄	—	5	2	no	—
83	C₂²⋊C₄	—	5	2	no	—
84	C₂²⋊C₄	—	5	2	no	—
85	C₂²⋊C₄	—	5	2	no	—
86	C₂²⋊C₄	—	5	2	no	—
87	C₂²⋊C₄	C₂×D₅	1	2	yes	—
88	C₂²⋊C₄	C₂×D₅	1	2	yes	—
89	C₄²	—	5	1	no	—
90	C₄²	—	5	2	no	—
91	C₄²	—	5	1	no	—
92	C₄²	C₂×D₅	1	1	yes	characteristic
93	C₂×C₅	—	4	2	no	—
94	C₂×C₅	C₂×#41	1	2	yes	—
95	C₂×C₅	H(5,𝔽₂)	1	1	yes	characteristic
96	D₅	—	4	2	no	—
97	C₂³	—	5	2	no	—
98	C₂³	C₂²×D₅	1	2	yes	—
99	Q₈	—	4	1	no	—
100	Q₈	—	20	1	no	—
101	D₄	—	4	1	no	—
102	D₄	—	10	4	no	—
103	D₄	—	20	1	no	—
104	D₄	—	20	2	no	—
105	C₂×C₄	—	5	2	no	—
106	C₂×C₄	—	5	2	no	—
107	C₂×C₄	—	5	2	no	—
108	C₂×C₄	—	10	4	no	—
109	C₂×C₄	—	10	4	no	—
110	C₂×C₄	C₂²×D₅	1	2	yes	—
111	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
112	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
114	C₅	#555	1	1	yes	characteristic, complemented, Sylow
115	C₂²	—	2	2	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	10	2	no	—
119	C₂²	—	10	2	no	—
120	C₂²	—	10	2	no	—
121	C₂²	C₂³×D₅	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	2	no	—
126	C₄	—	5	4	no	—
127	C₄	—	10	2	no	—
128	C₄	—	10	2	no	—
129	C₂	—	4	2	no	—
130	C₂	—	20	2	no	—
131	C₂	#4144	1	1	yes	characteristic
132	C₂	D₅×#41	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#124

C₂

yes

maximal, characteristic, complemented

#4093

C₂

yes

maximal, characteristic, complemented

#4090

C₂

yes

maximal, characteristic, complemented

#4085

C₂

yes

maximal, complemented

#4031

C₂

yes

maximal, complemented

#4030

C₂

yes

maximal, complemented

#4027

C₂

yes

maximal, complemented

#4023

C₂

yes

maximal, complemented

#4018

C₂

yes

maximal, complemented

C₂×C₅×Q₈

C₂²

yes

characteristic

C₂×C₅×D₄

C₂²

yes

characteristic

C₂×C₂²⋊D₅

C₂²

yes

—

C₂×C₂²⋊D₅

C₂²

yes

—

C₂²×Dic₅

C₂²

yes

—

C₂×D₂₀

C₂²

yes

characteristic

C₂×C₄×D₅

C₂²

yes

—

C₂×Dic₁₀

C₂²

yes

characteristic

C₅×C₂²⋊C₄

C₂²

yes

—

C₅×C₂²⋊C₄

C₂²

yes

—

C₄²×C₅

C₂²

yes

characteristic

C₂²⋊Dic₅

C₂²

yes

—

#685

C₂²

yes

—

#685

C₂²

yes

—

#685

C₂²

yes

—

#683

C₂²

yes

—

#683

C₂²

yes

—

#683

C₂²

yes

—

C₄×Dic₅

C₂²

yes

—

C₄×Dic₅

C₂²

yes

characteristic

C₄×Dic₅

C₂²

yes

characteristic

#555

—

maximal, Sylow

C₂³×C₅

C₂³

yes

—

C₂²×D₅

C₂³

yes

—

C₅×Q₈

—

C₅×D₄

—

C₂×C₄×C₅

C₂³

yes

characteristic

C₂×C₄×C₅

C₂³

yes

—

C₂×C₄×C₅

C₂³

yes

characteristic

C₂×C₄×C₅

C₂³

yes

characteristic

C₂²⋊D₅

—

C₂²⋊D₅

—

C₂×Dic₅

—

C₂×Dic₅

C₂³

yes

—

C₂×Dic₅

C₂³

yes

—

C₂×Dic₅

C₂³

yes

—

D₂₀

—

C₄×D₅

—

Dic₁₀

—

C₄⋊Q₈

—

#124

—

#124

—

#124

D₅

yes

characteristic, complemented

#121

—

#121

—

C₄×D₄

—

#117

—

#117

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂⁴

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

Dic₅

—

Dic₅

#41

yes

—

C₂×Q₈

—

C₂×Q₈

C₂×D₅

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₅

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₅

yes

—

C₂²⋊C₄

C₂×D₅

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×D₅

yes

characteristic

C₂×C₅

—

C₂×C₅

C₂×#41

yes

—

C₂×C₅

H(5,𝔽₂)

yes

characteristic

D₅

—

C₂³

—

C₂³

C₂²×D₅

yes

—

Q₈

—

100

Q₈

—

101

D₄

—

102

D₄

—

103

D₄

—

104

D₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂²×D₅

yes

—

111

C₂×C₄

C₂²×D₅

yes

characteristic

112

C₂×C₄

C₂²×D₅

yes

characteristic

113

C₂×C₄

C₂²×D₅

yes

characteristic

114

C₅

#555

yes

characteristic, complemented, Sylow

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂³×D₅

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₂

—

130

C₂

—

131

C₂

#4144

yes

characteristic

132

C₂

D₅×#41

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×#124	C₂
hypercenter	#124	D₅
hypocenter	C₅	#555
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅

commutator, socle, cosocle

C₂²×C₅

C₂⁴

C₂

D₅

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. C₅×#124	C₂	semidirect
2. #4093	C₂	semidirect
3. #4090	C₂	semidirect
4. #4085	C₂	semidirect
5. #4031	C₂	semidirect
6. #4030	C₂	semidirect
7. #4027	C₂	semidirect
8. #4023	C₂	semidirect
9. #4018	C₂	semidirect
52. #124	D₅	semidirect
114. C₅	#555	semidirect

H ≅ G/N

type

1. C₅×#124

C₂

semidirect

2. #4093

C₂

semidirect

3. #4090

C₂

semidirect

4. #4085

C₂

semidirect

5. #4031

C₂

semidirect

6. #4030

C₂

semidirect

7. #4027

C₂

semidirect

8. #4023

C₂

semidirect

9. #4018

C₂

semidirect

52. #124

D₅

semidirect

114. C₅

#555

semidirect

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

subgroup	index
C₅×#124	2
#4093	2
#4090	2
#4085	2
#4031	2
#4030	2
#4027	2
#4023	2
#4018	2
#555	5

subgroup

index

Subgroups — #66004

Conjugacy classes of subgroups

Complements

Maximal subgroups