Group #151742 — subgroups

Subgroups — #151742

Order 512.

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

normal subgroups only

This group has 156 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	#13733	C₂	1	1	yes	maximal, characteristic
2	#13731	C₂	1	1	yes	maximal, characteristic
3	#13720	C₂	1	1	yes	maximal, characteristic
4	#13720	C₂	1	1	yes	maximal, characteristic
5	#12522	C₂	1	1	yes	maximal, characteristic
6	#12504	C₂	1	1	yes	maximal, characteristic
7	#7331	C₂	1	1	yes	maximal, characteristic
8	#2254	—	2	1	no	—
9	#2252	C₂²	1	1	yes	characteristic
10	#2252	C₂²	1	1	yes	characteristic
11	#2252	C₂²	1	1	yes	characteristic
12	#2247	C₂²	1	1	yes	characteristic
13	#2241	—	2	1	no	—
14	#1710	—	2	1	no	—
15	#1707	—	2	1	no	—
16	#1424	—	2	2	no	—
17	#1423	—	2	2	no	—
18	#1416	—	2	2	no	—
19	#1416	—	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	#500	—	2	1	no	—
26	#500	—	2	1	no	—
27	#500	—	2	1	no	—
28	#498	—	4	1	no	—
29	#494	—	2	1	no	—
30	#487	—	4	1	no	—
31	#487	—	4	1	no	—
32	#487	—	4	1	no	—
33	C₂×C₃₂	D₄	1	2	yes	—
34	C₂×C₃₂	D₄	1	2	yes	—
35	#367	—	4	2	no	—
36	#366	—	2	1	no	—
37	#366	—	2	1	no	—
38	#366	—	2	1	no	—
39	#366	—	2	1	no	—
40	#366	—	2	1	no	—
41	#366	—	2	1	no	—
42	#358	—	4	2	no	—
43	C₄×C₁₆	D₄	1	1	yes	characteristic
44	C₄×C₁₆	C₂³	1	1	yes	characteristic
45	C₈²	D₄	1	1	yes	characteristic
46	C₂×Q₁₆	—	4	1	no	—
47	C₂×Q₁₆	—	4	1	no	—
48	C₄⋊Q₈	—	4	1	no	—
49	C₄⋊Q₈	—	4	1	no	—
50	C₄⋊Q₈	—	4	1	no	—
51	C₄⋊Q₈	—	4	1	no	—
52	Q₃₂	—	4	2	no	—
53	Q₃₂	—	4	2	no	—
54	C₂×C₁₆	D₈	1	2	yes	—
55	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
56	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
57	#107	—	4	1	no	—
58	#107	—	4	1	no	—
59	#107	—	4	1	no	—
60	#107	—	4	1	no	—
61	#107	—	4	1	no	—
62	#107	—	4	1	no	—
63	#106	—	8	2	no	—
64	#103	—	8	2	no	—
65	#103	—	8	2	no	—
66	#103	—	8	2	no	—
67	C₄×C₈	D₈	1	1	yes	characteristic
68	C₄×C₈	D₈	1	1	yes	characteristic
69	C₄×C₈	C₂×D₄	1	1	yes	characteristic
70	C₃₂	—	2	2	no	—
71	C₃₂	—	2	2	no	—
72	C₂×Q₈	—	8	1	no	—
73	C₂×Q₈	—	8	1	no	—
74	C₂×Q₈	—	8	1	no	—
75	C₂×Q₈	—	8	1	no	—
76	Q₁₆	—	8	1	no	—
77	Q₁₆	—	8	1	no	—
78	Q₁₆	—	8	2	no	—
79	C₂×C₈	D₁₆	1	2	yes	—
80	C₂×C₈	D₁₆	1	2	yes	—
81	C₂×C₈	#127	1	1	yes	characteristic
82	C₂×C₈	C₂×D₈	1	1	yes	characteristic
83	C₄⋊C₄	—	8	1	no	—
84	C₄⋊C₄	—	8	1	no	—
85	C₄⋊C₄	—	8	1	no	—
86	C₄⋊C₄	—	8	1	no	—
87	C₄⋊C₄	—	8	1	no	—
88	C₄⋊C₄	—	8	1	no	—
89	C₄⋊C₄	—	8	1	no	—
90	C₄⋊C₄	—	8	1	no	—
91	C₄²	C₂×D₈	1	1	yes	characteristic
92	C₁₆	—	2	2	no	—
93	C₁₆	#137	1	2	yes	—
94	C₁₆	#137	1	2	yes	—
95	Q₈	—	16	1	no	—
96	Q₈	—	16	1	no	—
97	Q₈	—	16	2	no	—
98	Q₈	—	16	2	no	—
99	Q₈	—	16	2	no	—
100	C₂×C₄	—	16	1	no	—
101	C₂×C₄	—	16	1	no	—
102	C₂×C₄	—	16	1	no	—
103	C₂×C₄	—	16	1	no	—
104	C₂×C₄	#493	1	1	yes	characteristic
105	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
106	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	2	2	no	—
109	C₈	#497	1	1	yes	characteristic
110	C₈	#497	1	1	yes	characteristic
111	C₈	#510	1	2	yes	—
112	C₂²	#2246	1	1	yes	characteristic
113	C₄	—	32	1	no	—
114	C₄	—	32	1	no	—
115	C₄	—	32	1	no	—
116	C₄	—	32	1	no	—
117	C₄	#1722	1	1	yes	characteristic
118	C₄	#2251	1	1	yes	characteristic
119	C₄	#2262	1	2	yes	—
120	C₄	#2264	1	2	yes	—
121	C₂	#12531	1	1	yes	characteristic
122	C₂	#13728	1	1	yes	characteristic
123	C₂	#13730	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13733

C₂

yes

maximal, characteristic

#13731

C₂

yes

maximal, characteristic

#13720

C₂

yes

maximal, characteristic

#13720

C₂

yes

maximal, characteristic

#12522

C₂

yes

maximal, characteristic

#12504

C₂

yes

maximal, characteristic

#7331

C₂

yes

maximal, characteristic

#2254

—

#2252

C₂²

yes

characteristic

#2252

C₂²

yes

characteristic

#2252

C₂²

yes

characteristic

#2247

C₂²

yes

—

—

—

—

—

—

—

C₄×C₃₂

C₂²

yes

characteristic

C₄×C₃₂

C₂²

yes

characteristic

C₈×C₁₆

C₂²

yes

characteristic

C₂×Q₃₂

—

C₂×Q₃₂

—

—

—

—

—

—

—

—

—

C₂×C₃₂

D₄

yes

—

C₂×C₃₂

D₄

yes

—

—

—

—

—

—

—

—

—

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

C₂³

yes

characteristic

C₈²

D₄

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

Q₃₂

—

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×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

D₁₆

yes

—

C₂×C₈

D₁₆

yes

—

C₂×C₈

#127

yes

characteristic

C₂×C₈

C₂×D₈

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₈

yes

characteristic

C₁₆

—

C₁₆

#137

yes

—

C₁₆

#137

yes

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

#493

yes

characteristic

105

C₂×C₄

C₂×D₁₆

yes

characteristic

106

C₂×C₄

C₂×D₁₆

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

#497

yes

characteristic

110

C₈

#497

yes

characteristic

111

C₈

#510

yes

—

112

C₂²

#2246

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

#1722

yes

characteristic

118

C₄

#2251

yes

characteristic

119

C₄

#2262

yes

—

120

C₄

#2264

yes

—

121

C₂

#12531

yes

characteristic

122

C₂

#13728

yes

characteristic

123

C₂

#13730

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	#2246
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
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, prime radical

C₂²

#2246

commutator, Frattini, cosocle

C₄×C₁₆

C₂³

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

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

subgroup	index
#13733	2
#13731	2
#13720	2
#12522	2
#12504	2
#7331	2

subgroup

index

Subgroups — #151742

Conjugacy classes of subgroups

Maximal subgroups