Group #32702 — subgroups

Subgroups — #32702

Order 256.

240 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 89 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	#2103	C₂	1	2	yes	maximal, complemented
2	#2102	C₂	1	1	yes	maximal, characteristic, complemented
3	#2080	C₂	1	2	yes	maximal, complemented
4	#2050	C₂	1	1	yes	maximal, characteristic
5	#1937	C₂	1	2	yes	maximal, complemented
6	#1881	C₂	1	2	yes	maximal
7	#1861	C₂	1	2	yes	maximal, complemented
8	#1858	C₂	1	2	yes	maximal, complemented
9	#1837	C₂	1	1	yes	maximal, characteristic, complemented
10	#575	C₂²	1	1	yes	characteristic
11	#568	C₂²	1	2	yes	—
12	#430	C₂²	1	1	yes	characteristic
13	#430	C₂²	1	2	yes	—
14	#424	C₂²	1	2	yes	—
15	#423	C₂²	1	1	yes	characteristic
16	#423	C₂²	1	1	yes	characteristic
17	#421	C₂²	1	2	yes	—
18	C₂×#104	C₂²	1	2	yes	—
19	#416	C₂²	1	2	yes	—
20	#413	C₂²	1	2	yes	—
21	#408	C₂²	1	2	yes	—
22	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
23	#344	C₂²	1	2	yes	—
24	#344	C₂²	1	2	yes	—
25	#341	C₂²	1	2	yes	—
26	#341	C₂²	1	2	yes	—
27	#341	C₂²	1	2	yes	—
28	#339	C₂²	1	2	yes	—
29	#338	C₂²	1	1	yes	characteristic
30	H(3,ℤ/4)	C₂²	1	1	yes	characteristic
31	C₂×#41	C₂³	1	2	yes	—
32	#137	—	2	2	no	—
33	AGL(1,ℤ/8)	—	2	2	no	—
34	#135	—	4	1	no	—
35	#131	—	4	2	no	—
36	#131	—	4	2	no	—
37	C₂×M₁₆	—	4	2	no	—
38	C₂×M₁₆	C₂³	1	2	yes	—
39	C₂×M₁₆	C₂³	1	2	yes	—
40	C₂×M₁₆	C₂³	1	1	yes	characteristic
41	C₂×M₁₆	C₂³	1	1	yes	characteristic
42	C₂×M₁₆	C₂³	1	2	yes	—
43	C₂²×C₈	C₂³	1	2	yes	—
44	#117	C₂³	1	2	yes	—
45	C₂×C₄²	C₂³	1	1	yes	characteristic
46	#108	—	4	2	no	—
47	#108	—	4	1	no	—
48	C₄⋊C₈	—	4	1	no	—
49	C₄⋊C₈	—	4	1	no	—
50	C₄⋊C₈	—	4	2	no	—
51	#104	—	2	2	no	—
52	#104	—	2	2	no	—
53	#104	—	4	2	no	—
54	#103	—	2	2	no	—
55	#102	—	2	2	no	—
56	#101	—	2	2	no	—
57	#100	—	2	2	no	—
58	C₂²⋊C₈	—	4	2	no	—
59	C₄×C₈	—	2	1	no	—
60	C₄×C₈	—	2	1	no	—
61	#41	—	2	2	no	—
62	#41	—	2	2	no	—
63	#41	—	2	2	no	—
64	#41	—	2	2	no	—
65	C₂×Q₈	#41	1	2	yes	—
66	C₂×D₄	—	2	2	no	—
67	C₂×D₄	#41	1	2	yes	—
68	C₂²×C₄	—	2	1	no	—
69	C₂²×C₄	—	2	2	no	—
70	C₂²×C₄	C₂⁴	1	1	yes	characteristic
71	Q₁₆	—	4	1	no	—
72	SD₁₆	—	4	2	no	—
73	D₈	—	4	1	no	—
74	M₁₆	—	2	2	no	—
75	M₁₆	—	2	2	no	—
76	M₁₆	—	2	2	no	—
77	M₁₆	—	2	2	no	—
78	M₁₆	—	2	2	no	—
79	M₁₆	—	2	2	no	—
80	M₁₆	—	2	2	no	—
81	M₁₆	—	2	2	no	—
82	M₁₆	—	4	4	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	4	4	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	—	2	1	no	—
95	C₄²	—	2	2	no	—
96	C₄²	#41	1	1	yes	characteristic
97	C₄²	#41	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	2₋¹⁺⁴	1	1	yes	characteristic
100	Q₈	—	2	2	no	—
101	Q₈	—	2	2	no	—
102	D₄	—	2	2	no	—
103	D₄	—	2	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	2	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₂×#41	1	2	yes	—
114	C₂×C₄	C₂×#41	1	1	yes	characteristic
115	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂×C₄	H(5,𝔽₂)	1	2	yes	—
117	C₈	—	4	1	no	—
118	C₈	—	4	1	no	—
119	C₈	—	4	2	no	—
120	C₈	—	4	2	no	—
121	C₈	—	4	2	no	—
122	C₈	—	4	2	no	—
123	C₂²	—	4	1	no	—
124	C₂²	—	4	2	no	—
125	C₂²	—	8	2	no	—
126	C₂²	#539	1	1	yes	characteristic
127	C₂²	#539	1	2	yes	—
128	C₄	—	2	2	no	—
129	C₄	—	4	1	no	—
130	C₄	—	4	2	no	—
131	C₄	—	8	2	no	—
132	C₄	#551	1	1	yes	characteristic
133	C₄	#553	1	2	yes	—
134	C₄	H(3,𝔽₄)	1	1	yes	characteristic
135	C₂	—	2	1	no	—
136	C₂	—	2	2	no	—
137	C₂	—	8	2	no	—
138	C₂	#2782	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2103

C₂

yes

maximal, complemented

#2102

C₂

yes

maximal, characteristic, complemented

#2080

C₂

yes

maximal, complemented

#2050

C₂

yes

maximal, characteristic

#1937

C₂

yes

maximal, complemented

#1881

C₂

yes

maximal

#1861

C₂

yes

maximal, complemented

#1858

C₂

yes

maximal, complemented

#1837

C₂

yes

maximal, characteristic, complemented

#575

C₂²

yes

characteristic

#568

C₂²

yes

—

#430

C₂²

yes

characteristic

#430

C₂²

yes

—

#424

C₂²

yes

—

#423

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

#421

C₂²

yes

—

C₂×#104

C₂²

yes

—

#416

C₂²

yes

—

#413

C₂²

yes

—

#408

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

#344

C₂²

yes

—

#344

C₂²

yes

—

#341

C₂²

yes

—

#341

C₂²

yes

—

#341

C₂²

yes

—

#339

C₂²

yes

—

#338

C₂²

yes

characteristic

H(3,ℤ/4)

C₂²

yes

characteristic

C₂×#41

C₂³

yes

—

—

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

C₂³

yes

—

C₂×M₁₆

C₂³

yes

—

C₂×M₁₆

C₂³

yes

characteristic

C₂×M₁₆

C₂³

yes

characteristic

C₂×M₁₆

C₂³

yes

—

C₂²×C₈

C₂³

yes

—

#117

C₂³

yes

—

C₂×C₄²

C₂³

yes

characteristic

#108

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

—

—

—

—

—

—

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

—

—

—

—

C₂×Q₈

#41

yes

—

C₂×D₄

—

C₂×D₄

#41

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂⁴

yes

characteristic

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

yes

C₄²

yes

C₂³

—

C₂³

2₋¹⁺⁴

yes

characteristic

100

Q₈

—

101

Q₈

—

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

C₂×#41

yes

—

114

C₂×C₄

C₂×#41

yes

characteristic

115

C₂×C₄

H(5,𝔽₂)

yes

characteristic

116

C₂×C₄

H(5,𝔽₂)

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

#539

yes

characteristic

127

C₂²

#539

yes

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#551

yes

characteristic

133

C₄

#553

yes

—

134

C₄

H(3,𝔽₄)

yes

characteristic

135

C₂

—

136

C₂

—

137

C₂

—

138

C₂

#2782

yes

characteristic

	subgroup	G/H
center	C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂	#2782
prime radical	#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

H(3,𝔽₄)

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴

C₂

C₂²

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #2103	C₂	semidirect
2. #2102	C₂	semidirect
3. #2080	C₂	semidirect
5. #1937	C₂	semidirect
7. #1861	C₂	semidirect
8. #1858	C₂	semidirect
9. #1837	C₂	semidirect

H ≅ G/N

type

1. #2103

C₂

semidirect

2. #2102

C₂

semidirect

3. #2080

C₂

semidirect

5. #1937

C₂

semidirect

7. #1861

C₂

semidirect

8. #1858

C₂

semidirect

9. #1837

C₂

semidirect

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

subgroup	index
#2103	2
#2102	2
#2080	2
#2050	2
#1937	2
#1881	2
#1861	2
#1858	2
#1837	2

subgroup

index

Subgroups — #32702

Conjugacy classes of subgroups

Complements

Maximal subgroups