Group #20010 — subgroups

Subgroups — #20010

Order 256.

283 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 111 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	#3395	C₂	1	2	yes	maximal
2	#3383	C₂	1	2	yes	maximal
3	#1713	C₂	1	1	yes	maximal, characteristic
4	#1674	C₂	1	2	yes	maximal
5	#1670	C₂	1	2	yes	maximal
6	#1648	C₂	1	4	yes	maximal
7	C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic
8	C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic
9	#557	C₂²	1	4	yes	—
10	#500	C₂²	1	2	yes	—
11	#500	C₂²	1	2	yes	—
12	#499	—	2	2	no	—
13	#494	—	2	2	no	—
14	#494	C₂²	1	2	yes	—
15	#479	—	2	4	no	—
16	#475	C₂²	1	4	yes	—
17	#462	—	2	4	no	—
18	#462	C₂²	1	4	yes	—
19	C₈×Q₈	C₂²	1	2	yes	—
20	C₈×Q₈	C₂²	1	2	yes	—
21	C₄×Q₁₆	—	2	2	no	—
22	C₄×Q₁₆	—	2	2	no	—
23	C₄×Q₁₆	C₂²	1	1	yes	characteristic
24	C₄×Q₁₆	C₂²	1	1	yes	characteristic
25	#335	C₂²	1	1	yes	characteristic
26	#335	C₂²	1	1	yes	characteristic
27	#332	C₂²	1	4	yes	—
28	Q₈⋊C₈	C₂²	1	2	yes	—
29	Q₈⋊C₈	C₂²	1	2	yes	—
30	C₈²	C₂²	1	1	yes	characteristic
31	C₂×Q₁₆	—	2	4	no	—
32	C₂×Q₁₆	—	2	2	no	—
33	C₂×Q₁₆	—	2	2	no	—
34	C₂×Q₁₆	D₄	1	1	yes	characteristic
35	C₂×Q₁₆	D₄	1	1	yes	characteristic
36	C₄⋊Q₈	—	2	4	no	—
37	C₄⋊Q₈	C₂³	1	4	yes	—
38	#125	—	2	4	no	—
39	#125	—	4	4	no	—
40	C₄×Q₈	—	2	4	no	—
41	C₄×Q₈	—	2	4	no	—
42	C₄×Q₈	C₂³	1	2	yes	—
43	C₄×Q₈	C₂³	1	2	yes	—
44	#107	—	2	2	no	—
45	#107	—	2	2	no	—
46	#107	—	4	2	no	—
47	#107	D₄	1	1	yes	characteristic
48	#107	D₄	1	1	yes	characteristic
49	#106	—	2	4	no	—
50	C₄⋊C₈	—	2	2	no	—
51	C₄⋊C₈	—	2	2	no	—
52	C₄⋊C₈	C₂³	1	2	yes	—
53	C₄⋊C₈	C₂³	1	2	yes	—
54	#103	—	2	4	no	—
55	#103	—	2	4	no	—
56	#103	D₄	1	2	yes	—
57	#103	D₄	1	2	yes	—
58	C₄×C₈	—	2	2	no	—
59	C₄×C₈	—	2	2	no	—
60	C₄×C₈	C₂³	1	1	yes	characteristic
61	C₄×C₈	C₂³	1	1	yes	characteristic
62	C₄×C₈	C₂³	1	1	yes	characteristic
63	C₂×Q₈	—	2	4	no	—
64	C₂×Q₈	C₂×D₄	1	2	yes	—
65	C₂×Q₈	C₂×D₄	1	2	yes	—
66	Q₁₆	—	2	2	no	—
67	Q₁₆	—	4	4	no	—
68	Q₁₆	—	8	2	no	—
69	Q₁₆	—	8	4	no	—
70	Q₁₆	Q₁₆	1	4	yes	complemented
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈	C₂×D₄	1	1	yes	characteristic
78	C₄⋊C₄	—	2	2	no	—
79	C₄⋊C₄	—	2	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	4	no	—
82	C₄⋊C₄	—	4	4	no	—
83	C₄⋊C₄	—	4	4	no	—
84	C₄⋊C₄	—	4	4	no	—
85	C₄⋊C₄	C₂×D₄	1	2	yes	—
86	C₄⋊C₄	C₂×D₄	1	2	yes	—
87	C₄²	—	2	2	no	—
88	C₄²	—	2	2	no	—
89	C₄²	—	4	4	no	—
90	C₄²	C₂⁴	1	1	yes	characteristic
91	Q₈	—	2	2	no	—
92	Q₈	—	2	2	no	—
93	Q₈	—	2	2	no	—
94	Q₈	—	4	4	no	—
95	Q₈	—	4	4	no	—
96	Q₈	C₂×Q₁₆	1	4	yes	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	4	no	—
102	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	—	2	1	no	—
107	C₈	—	2	1	no	—
108	C₈	—	4	2	no	—
109	C₈	—	4	2	no	—
110	C₈	C₂×Q₁₆	1	2	yes	—
111	C₈	#135	1	2	yes	—
112	C₂²	D₄²	1	1	yes	characteristic
113	C₄	—	2	4	no	—
114	C₄	—	2	1	no	—
115	C₄	—	4	2	no	—
116	C₄	—	4	2	no	—
117	C₄	—	4	2	no	—
118	C₄	—	8	4	no	—
119	C₄	C₂²×Q₁₆	1	1	yes	characteristic
120	C₄	C₂×#135	1	1	yes	characteristic
121	C₄	#576	1	1	yes	characteristic
122	C₄	#578	1	1	yes	characteristic
123	C₂	D₄×Q₁₆	1	1	yes	characteristic
124	C₂	#3287	1	1	yes	characteristic
125	C₂	#3293	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3395

C₂

yes

maximal

#3383

C₂

yes

maximal

#1713

C₂

yes

maximal, characteristic

#1674

C₂

yes

maximal

#1670

C₂

yes

maximal

#1648

C₂

yes

maximal

C₈×Q₁₆

C₂

yes

maximal, characteristic

C₈×Q₁₆

C₂

yes

maximal, characteristic

#557

C₂²

yes

—

#500

C₂²

yes

—

#500

C₂²

yes

—

#499

—

#494

—

#494

C₂²

yes

—

#479

—

#475

C₂²

yes

—

#462

—

#462

C₂²

yes

—

C₈×Q₈

C₂²

yes

—

C₈×Q₈

C₂²

yes

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₄×Q₁₆

C₂²

yes

characteristic

C₄×Q₁₆

C₂²

yes

characteristic

#335

C₂²

yes

characteristic

#335

C₂²

yes

characteristic

#332

C₂²

yes

—

Q₈⋊C₈

C₂²

yes

—

Q₈⋊C₈

C₂²

yes

—

C₈²

C₂²

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

D₄

yes

characteristic

C₂×Q₁₆

D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

C₂³

yes

—

#125

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂³

yes

—

C₄×Q₈

C₂³

yes

—

—

—

—

D₄

yes

characteristic

#107

D₄

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂³

yes

—

C₄⋊C₈

C₂³

yes

—

#103

—

#103

—

#103

D₄

yes

—

#103

D₄

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₂×D₄

yes

—

C₂×Q₈

C₂×D₄

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

yes

complemented

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄

yes

—

C₄⋊C₄

C₂×D₄

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂⁴

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

C₂×Q₁₆

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₂²×D₄

yes

characteristic

103

C₂×C₄

C₂²×D₄

yes

characteristic

104

C₂×C₄

H(5,𝔽₂)

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

C₂×Q₁₆

yes

—

111

C₈

#135

yes

—

112

C₂²

D₄²

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

C₂²×Q₁₆

yes

characteristic

120

C₄

C₂×#135

yes

characteristic

121

C₄

#576

yes

characteristic

122

C₄

#578

yes

characteristic

123

C₂

D₄×Q₁₆

yes

characteristic

124

C₂

#3287

yes

characteristic

125

C₂

#3293

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂²

D₄²

commutator, Frattini, cosocle

C₄²

C₂⁴

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
70. Q₁₆	Q₁₆	semidirect

H ≅ G/N

type

70. Q₁₆

Q₁₆

semidirect

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

subgroup	index
#3395	2
#3383	2
#1713	2
#1674	2
#1670	2
#1648	2
C₈×Q₁₆	2

subgroup

index

C₈×Q₁₆

Subgroups — #20010

Conjugacy classes of subgroups

Complements

Maximal subgroups