Group #66002 — subgroups

Subgroups — #66002

Order 320.

310 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 107 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	C₂×#713	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#711	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	#4090	C₂	1	1	yes	maximal, characteristic, complemented
5	#4086	C₂	1	1	yes	maximal, characteristic, complemented
6	#4030	C₂	1	4	yes	maximal, complemented
7	#4027	C₂	1	4	yes	maximal, complemented
8	#4021	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄²×D₅	C₂	1	1	yes	maximal, characteristic, 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	4	yes	—
14	C₂²×Dic₅	C₂²	1	2	yes	—
15	#713	—	2	4	no	—
16	#711	—	2	4	no	—
17	C₂×D₂₀	C₂²	1	1	yes	characteristic
18	C₂×D₂₀	C₂²	1	2	yes	—
19	C₂×C₄×D₅	C₂²	1	2	yes	—
20	C₂×C₄×D₅	C₂²	1	2	yes	—
21	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
22	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
23	C₅×C₂²⋊C₄	C₂²	1	4	yes	—
24	C₄²×C₅	C₂²	1	1	yes	characteristic
25	#685	C₂²	1	4	yes	—
26	#683	C₂²	1	4	yes	—
27	C₄×Dic₅	C₂²	1	2	yes	—
28	C₄×Dic₅	C₂²	1	1	yes	characteristic
29	#532	—	5	1	no	maximal, Sylow
30	C₂³×C₅	C₂³	1	2	yes	—
31	C₂²×D₅	C₂³	1	2	yes	—
32	C₂²×D₅	C₂³	1	1	yes	characteristic
33	C₅×Q₈	—	2	2	no	—
34	C₅×D₄	—	2	2	no	—
35	C₂×C₄×C₅	C₂³	1	2	yes	—
36	C₂×C₄×C₅	C₂³	1	2	yes	—
37	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₂²⋊D₅	—	2	4	no	—
39	C₂²⋊D₅	—	2	8	no	—
40	C₂×Dic₅	—	2	4	no	—
41	C₂×Dic₅	C₂³	1	2	yes	—
42	C₂×Dic₅	C₂³	1	1	yes	characteristic
43	C₂×Dic₅	C₂³	1	2	yes	—
44	D₂₀	—	2	4	no	—
45	D₂₀	—	2	2	no	—
46	C₄×D₅	—	2	4	no	—
47	C₄×D₅	—	2	4	no	—
48	C₄×D₅	D₄	1	4	yes	complemented
49	Dic₁₀	—	2	2	no	—
50	C₂×#41	—	5	1	no	—
51	C₂×#41	—	5	1	no	—
52	C₄⋊Q₈	—	5	1	no	—
53	#127	—	5	1	no	—
54	#124	—	5	1	no	—
55	#124	D₅	1	1	yes	characteristic, complemented
56	#121	—	5	4	no	—
57	C₄×D₄	—	5	4	no	—
58	C₂×C₄²	—	5	1	no	—
59	C₂²×C₅	—	2	2	no	—
60	C₂²×C₅	—	2	4	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	4	no	—
65	C₂×D₅	C₂×D₄	1	2	yes	—
66	C₄×C₅	—	2	2	no	—
67	C₄×C₅	—	2	2	no	—
68	C₄×C₅	C₂×D₄	1	2	yes	—
69	Dic₅	—	2	2	no	—
70	Dic₅	C₂×D₄	1	2	yes	—
71	Dic₅	#41	1	4	yes	complemented
72	#41	—	10	4	no	—
73	#41	—	10	4	no	—
74	C₂×Q₈	—	5	1	no	—
75	C₂×Q₈	C₂×D₅	1	1	yes	characteristic
76	C₂×D₄	—	5	4	no	—
77	C₂×D₄	—	5	2	no	—
78	C₂×D₄	—	5	2	no	—
79	C₂×D₄	—	5	1	no	—
80	C₂×D₄	C₂×D₅	1	1	yes	characteristic
81	C₂²×C₄	—	5	1	no	—
82	C₂²×C₄	—	5	2	no	—
83	C₂²×C₄	—	5	2	no	—
84	C₂²×C₄	—	5	2	no	—
85	C₄⋊C₄	—	5	4	no	—
86	C₂²⋊C₄	—	5	4	no	—
87	C₂²⋊C₄	C₂×D₅	1	4	yes	—
88	C₄²	—	5	2	no	—
89	C₄²	—	5	1	no	—
90	C₄²	C₂×D₅	1	1	yes	characteristic
91	C₂×C₅	—	4	2	no	—
92	C₂×C₅	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₅	C₂×#41	1	2	yes	—
94	D₅	—	2	2	no	—
95	D₅	—	4	2	no	—
96	C₂³	—	5	1	no	—
97	C₂³	—	5	2	no	—
98	C₂³	C₂²×D₅	1	2	yes	—
99	Q₈	—	2	2	no	—
100	Q₈	—	10	2	no	—
101	D₄	—	2	2	no	—
102	D₄	—	10	4	no	—
103	D₄	—	10	8	no	—
104	D₄	—	10	4	no	—
105	D₄	—	10	2	no	—
106	C₂×C₄	—	5	2	no	—
107	C₂×C₄	—	5	1	no	—
108	C₂×C₄	—	5	4	no	—
109	C₂×C₄	—	5	2	no	—
110	C₂×C₄	—	10	4	no	—
111	C₂×C₄	—	10	4	no	—
112	C₂×C₄	—	10	4	no	—
113	C₂×C₄	C₂²×D₅	1	2	yes	—
114	C₂×C₄	C₂²×D₅	1	2	yes	—
115	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
116	C₅	#532	1	1	yes	characteristic, complemented, Sylow
117	C₂²	—	2	2	no	—
118	C₂²	—	2	4	no	—
119	C₂²	—	5	2	no	—
120	C₂²	—	10	2	no	—
121	C₂²	—	10	2	no	—
122	C₂²	—	10	4	no	—
123	C₂²	C₂³×D₅	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	—	5	2	no	—
127	C₄	—	5	4	no	—
128	C₄	—	10	2	no	—
129	C₄	D₄×D₅	1	2	yes	—
130	C₂	—	4	2	no	—
131	C₂	—	10	2	no	—
132	C₂	—	20	2	no	—
133	C₂	C₂×D₄×D₅	1	1	yes	characteristic
134	C₂	D₅×#41	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#713

C₂

yes

maximal, characteristic, complemented

C₂×#711

C₂

yes

maximal, characteristic, complemented

C₅×#124

C₂

yes

maximal, characteristic, complemented

#4090

C₂

yes

maximal, characteristic, complemented

#4086

C₂

yes

maximal, characteristic, complemented

#4030

C₂

yes

maximal, complemented

#4027

C₂

yes

maximal, complemented

#4021

C₂

yes

maximal, characteristic, complemented

C₄²×D₅

C₂

yes

maximal, characteristic, 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

—

#713

—

#711

—

C₂×D₂₀

C₂²

yes

characteristic

C₂×D₂₀

C₂²

yes

—

C₂×C₄×D₅

C₂²

yes

—

C₂×C₄×D₅

C₂²

yes

—

C₂×C₄×D₅

C₂²

yes

characteristic

C₂×Dic₁₀

C₂²

yes

characteristic

C₅×C₂²⋊C₄

C₂²

yes

—

C₄²×C₅

C₂²

yes

characteristic

#685

C₂²

yes

—

#683

C₂²

yes

—

C₄×Dic₅

C₂²

yes

—

C₄×Dic₅

C₂²

yes

characteristic

#532

—

maximal, Sylow

C₂³×C₅

C₂³

yes

—

C₂²×D₅

C₂³

yes

—

C₂²×D₅

C₂³

yes

characteristic

C₅×Q₈

—

C₅×D₄

—

C₂×C₄×C₅

C₂³

yes

—

C₂×C₄×C₅

C₂³

yes

—

C₂×C₄×C₅

C₂³

yes

characteristic

C₂²⋊D₅

—

C₂²⋊D₅

—

C₂×Dic₅

—

C₂×Dic₅

C₂³

yes

—

C₂×Dic₅

C₂³

yes

characteristic

C₂×Dic₅

C₂³

yes

—

D₂₀

—

D₂₀

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

D₄

yes

complemented

Dic₁₀

—

C₂×#41

—

C₂×#41

—

C₄⋊Q₈

—

#127

—

#124

—

#124

D₅

yes

characteristic, complemented

#121

—

C₄×D₄

—

C₂×C₄²

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂⁴

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

C₂×D₄

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂×D₄

yes

—

Dic₅

—

Dic₅

C₂×D₄

yes

—

Dic₅

yes

—

—

C₂×Q₈

—

C₂×Q₈

C₂×D₅

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₅

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₅

yes

—

C₄²

—

C₄²

—

C₄²

C₂×D₅

yes

characteristic

C₂×C₅

—

C₂×C₅

C₂²×D₄

yes

characteristic

C₂×C₅

C₂×#41

yes

—

D₅

—

D₅

—

C₂³

—

C₂³

—

C₂³

C₂²×D₅

yes

—

Q₈

—

100

Q₈

—

101

D₄

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²×D₅

yes

—

114

C₂×C₄

C₂²×D₅

yes

—

115

C₂×C₄

C₂²×D₅

yes

characteristic

116

C₅

#532

yes

characteristic, complemented, Sylow

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

C₂³×D₅

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

D₄×D₅

yes

—

130

C₂

—

131

C₂

—

132

C₂

—

133

C₂

C₂×D₄×D₅

yes

characteristic

134

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	#532
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. C₂×#713	C₂	semidirect
2. C₂×#711	C₂	semidirect
3. C₅×#124	C₂	semidirect
4. #4090	C₂	semidirect
5. #4086	C₂	semidirect
6. #4030	C₂	semidirect
7. #4027	C₂	semidirect
8. #4021	C₂	semidirect
9. C₄²×D₅	C₂	semidirect
48. C₄×D₅	D₄	semidirect
55. #124	D₅	semidirect
71. Dic₅	#41	semidirect
116. C₅	#532	semidirect

H ≅ G/N

type

1. C₂×#713

C₂

semidirect

2. C₂×#711

C₂

semidirect

3. C₅×#124

C₂

semidirect

4. #4090

C₂

semidirect

5. #4086

C₂

semidirect

6. #4030

C₂

semidirect

7. #4027

C₂

semidirect

8. #4021

C₂

semidirect

9. C₄²×D₅

C₂

semidirect

48. C₄×D₅

D₄

semidirect

55. #124

D₅

semidirect

71. Dic₅

#41

semidirect

116. C₅

#532

semidirect

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

subgroup	index
C₂×#713	2
C₂×#711	2
C₅×#124	2
#4090	2
#4086	2
#4030	2
#4027	2
#4021	2
C₄²×D₅	2
#532	5

subgroup

index

C₄²×D₅

Subgroups — #66002

Conjugacy classes of subgroups

Complements

Maximal subgroups