Group #151739 — subgroups

Subgroups — #151739

Order 512.

191 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 191 conjugacy classes of subgroups — too many to draw. Only its 58 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	#13727	C₂	1	1	yes	maximal, characteristic
2	#13726	C₂	1	1	yes	maximal, characteristic, complemented
3	#13720	C₂	1	1	yes	maximal, characteristic, complemented
4	#13719	C₂	1	1	yes	maximal, characteristic
5	#12518	C₂	1	1	yes	maximal, characteristic
6	#12504	C₂	1	1	yes	maximal, characteristic, complemented
7	#7331	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₆₄	—	2	2	no	—
9	C₂×SD₆₄	—	2	2	no	—
10	#2252	C₂²	1	1	yes	characteristic
11	#2248	—	2	1	no	—
12	#2247	C₂²	1	1	yes	characteristic
13	#2247	C₂²	1	1	yes	characteristic
14	#2246	C₂²	1	1	yes	characteristic
15	#2241	—	2	1	no	—
16	#1709	—	2	1	no	—
17	#1707	—	2	1	no	—
18	#1416	—	2	2	no	—
19	#1415	—	2	2	no	—
20	C₄×C₃₂	C₂²	1	1	yes	characteristic
21	C₄×C₃₂	C₂²	1	1	yes	characteristic
22	C₈×C₁₆	C₂²	1	1	yes	characteristic
23	C₂×Q₃₂	—	2	1	no	—
24	C₂×Q₃₂	—	2	1	no	—
25	C₂×Q₃₂	—	2	1	no	—
26	C₂×Q₃₂	—	2	1	no	—
27	C₂×SD₃₂	—	4	2	no	—
28	C₂×D₁₆	—	2	1	no	—
29	C₂×D₁₆	—	2	1	no	—
30	#500	—	2	1	no	—
31	#498	—	4	1	no	—
32	#494	—	2	1	no	—
33	#494	—	2	1	no	—
34	#493	—	2	1	no	—
35	#487	—	4	1	no	—
36	#486	—	4	1	no	—
37	#486	—	4	1	no	—
38	Q₆₄	—	4	4	no	—
39	SD₆₄	—	4	4	no	—
40	C₂×C₃₂	D₄	1	2	yes	—
41	C₂×C₃₂	D₄	1	2	yes	—
42	#366	—	2	1	no	—
43	#366	—	2	1	no	—
44	#358	—	4	2	no	—
45	C₄×C₁₆	D₄	1	1	yes	characteristic
46	C₄×C₁₆	C₂³	1	1	yes	characteristic
47	C₈²	D₄	1	1	yes	characteristic
48	C₂×Q₁₆	—	4	1	no	—
49	C₂×Q₁₆	—	4	1	no	—
50	C₂×Q₁₆	—	4	1	no	—
51	C₂×Q₁₆	—	4	1	no	—
52	C₂×D₈	—	4	1	no	—
53	C₂×D₈	—	4	1	no	—
54	C₄⋊Q₈	—	4	1	no	—
55	C₄⋊Q₈	—	4	1	no	—
56	C₄⋊Q₈	—	4	1	no	—
57	#127	—	4	1	no	—
58	Q₃₂	—	4	2	no	—
59	Q₃₂	—	4	2	no	—
60	Q₃₂	—	4	2	no	—
61	Q₃₂	—	4	2	no	—
62	SD₃₂	—	8	4	no	—
63	D₁₆	—	4	2	no	—
64	D₁₆	—	4	2	no	—
65	C₂×C₁₆	D₈	1	2	yes	—
66	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
67	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
68	#107	—	4	1	no	—
69	#107	—	4	1	no	—
70	#106	—	8	2	no	—
71	#103	—	8	2	no	—
72	#102	—	8	2	no	—
73	#102	—	8	2	no	—
74	C₄×C₈	D₈	1	1	yes	characteristic
75	C₄×C₈	D₈	1	1	yes	characteristic
76	C₄×C₈	C₂×D₄	1	1	yes	characteristic
77	C₃₂	—	2	2	no	—
78	C₃₂	—	2	2	no	—
79	C₂×Q₈	—	8	1	no	—
80	C₂×Q₈	—	8	1	no	—
81	C₂×Q₈	—	8	1	no	—
82	C₂×D₄	—	8	1	no	—
83	C₂×D₄	—	8	1	no	—
84	C₂×D₄	—	8	1	no	—
85	Q₁₆	—	8	1	no	—
86	Q₁₆	—	8	1	no	—
87	Q₁₆	—	8	1	no	—
88	Q₁₆	—	8	1	no	—
89	Q₁₆	—	8	2	no	—
90	Q₁₆	—	8	2	no	—
91	D₈	—	8	1	no	—
92	D₈	—	8	1	no	—
93	D₈	—	8	2	no	—
94	C₂×C₈	D₁₆	1	2	yes	—
95	C₂×C₈	D₁₆	1	2	yes	—
96	C₂×C₈	#127	1	1	yes	characteristic
97	C₂×C₈	C₂×D₈	1	1	yes	characteristic
98	C₄⋊C₄	—	8	1	no	—
99	C₄⋊C₄	—	8	1	no	—
100	C₄⋊C₄	—	8	1	no	—
101	C₄⋊C₄	—	8	1	no	—
102	C₄⋊C₄	—	8	1	no	—
103	C₄⋊C₄	—	8	1	no	—
104	C₄²	C₂×D₈	1	1	yes	characteristic
105	C₁₆	—	2	2	no	—
106	C₁₆	AGL(1,ℤ/8)	1	2	yes	—
107	C₁₆	AGL(1,ℤ/8)	1	2	yes	—
108	C₂³	—	16	1	no	—
109	Q₈	—	16	1	no	—
110	Q₈	—	16	1	no	—
111	Q₈	—	16	1	no	—
112	Q₈	—	16	1	no	—
113	Q₈	—	16	2	no	—
114	D₄	—	16	1	no	—
115	D₄	—	16	1	no	—
116	D₄	—	16	2	no	—
117	D₄	—	16	2	no	—
118	C₂×C₄	—	16	1	no	—
119	C₂×C₄	—	16	1	no	—
120	C₂×C₄	—	16	1	no	—
121	C₂×C₄	#493	1	1	yes	characteristic
122	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
123	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
124	C₈	—	2	2	no	—
125	C₈	—	2	2	no	—
126	C₈	#496	1	1	yes	characteristic
127	C₈	#496	1	1	yes	characteristic
128	C₈	#509	1	2	yes	—
129	C₂²	—	32	1	no	—
130	C₂²	—	32	1	no	—
131	C₂²	—	32	1	no	—
132	C₂²	#2246	1	1	yes	characteristic
133	C₄	—	32	1	no	—
134	C₄	—	32	1	no	—
135	C₄	—	32	1	no	—
136	C₄	#1721	1	1	yes	characteristic
137	C₄	#2250	1	1	yes	characteristic
138	C₄	#2262	1	2	yes	—
139	C₄	#2264	1	2	yes	—
140	C₂	—	64	1	no	—
141	C₂	#12525	1	1	yes	characteristic
142	C₂	#13728	1	1	yes	characteristic
143	C₂	#13730	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13727

C₂

yes

maximal, characteristic

#13726

C₂

yes

maximal, characteristic, complemented

#13720

C₂

yes

maximal, characteristic, complemented

#13719

C₂

yes

maximal, characteristic

#12518

C₂

yes

maximal, characteristic

#12504

C₂

yes

maximal, characteristic, complemented

#7331

C₂

yes

maximal, characteristic, complemented

C₂×Q₆₄

—

C₂×SD₆₄

—

#2252

C₂²

yes

characteristic

#2248

—

#2247

C₂²

yes

characteristic

#2247

C₂²

yes

characteristic

#2246

C₂²

yes

—

—

—

—

—

C₄×C₃₂

C₂²

yes

characteristic

C₄×C₃₂

C₂²

yes

characteristic

C₈×C₁₆

C₂²

yes

characteristic

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×SD₃₂

—

C₂×D₁₆

—

C₂×D₁₆

—

—

—

—

—

—

—

—

—

Q₆₄

—

SD₆₄

—

C₂×C₃₂

D₄

yes

—

C₂×C₃₂

D₄

yes

—

#366

—

#366

—

#358

—

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

C₂³

yes

characteristic

C₈²

D₄

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

SD₃₂

—

D₁₆

—

D₁₆

—

C₂×C₁₆

D₈

yes

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

—

—

—

—

—

—

C₄×C₈

D₈

yes

characteristic

C₄×C₈

D₈

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₃₂

—

C₃₂

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

D₈

—

D₈

—

D₈

—

C₂×C₈

D₁₆

yes

—

C₂×C₈

D₁₆

yes

—

C₂×C₈

#127

yes

characteristic

C₂×C₈

C₂×D₈

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄²

C₂×D₈

yes

characteristic

105

C₁₆

—

106

C₁₆

AGL(1,ℤ/8)

yes

—

107

C₁₆

AGL(1,ℤ/8)

yes

—

108

C₂³

—

109

Q₈

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

D₄

—

115

D₄

—

116

D₄

—

117

D₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

#493

yes

characteristic

122

C₂×C₄

C₂×D₁₆

yes

characteristic

123

C₂×C₄

C₂×D₁₆

yes

characteristic

124

C₈

—

125

C₈

—

126

C₈

#496

yes

characteristic

127

C₈

#496

yes

characteristic

128

C₈

#509

yes

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#2246

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#1721

yes

characteristic

137

C₄

#2250

yes

characteristic

138

C₄

#2262

yes

—

139

C₄

#2264

yes

—

140

C₂

—

141

C₂

#12525

yes

characteristic

142

C₂

#13728

yes

characteristic

143

C₂

#13730

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#2246
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	#2246	C₂²
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2246

commutator, Frattini, cosocle

C₄×C₁₆

C₂³

prime radical

#2246

C₂²

LCS

G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆ ◁ G

LECS

G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #13726	C₂	semidirect
3. #13720	C₂	semidirect
6. #12504	C₂	semidirect
7. #7331	C₂	semidirect

H ≅ G/N

type

2. #13726

C₂

semidirect

3. #13720

C₂

semidirect

6. #12504

C₂

semidirect

7. #7331

C₂

semidirect

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

subgroup	index
#13727	2
#13726	2
#13720	2
#13719	2
#12518	2
#12504	2
#7331	2

subgroup

index

Subgroups — #151739

Conjugacy classes of subgroups

Complements

Maximal subgroups