Group #153316 — subgroups

Subgroups — #153316

Order 512.

250 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 68 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	C₂²×Q₆₄	C₂	1	1	yes	maximal, characteristic
2	#13628	C₂	1	1	yes	maximal, characteristic
3	C₂×#1420	C₂	1	2	yes	maximal
4	#13244	C₂	1	2	yes	maximal
5	#7472	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₃₂	C₂²	1	2	yes	—
7	C₂×Q₆₄	—	2	4	no	—
8	C₂×Q₆₄	—	2	2	no	—
9	C₂×M₆₄	C₂²	1	1	yes	characteristic
10	C₂²×C₃₂	C₂²	1	1	yes	characteristic
11	C₂×#368	C₂²	1	2	yes	—
12	C₂×#362	—	2	2	no	—
13	#2108	C₂²	1	1	yes	characteristic
14	#1979	—	2	2	no	—
15	C₄⋊C₃₂	—	2	2	no	—
16	#1420	—	2	4	no	—
17	C₂²×Q₁₆	—	2	2	no	—
18	C₂×Q₃₂	—	2	2	no	—
19	C₂×Q₃₂	—	2	2	no	—
20	C₂×Q₃₂	—	2	2	no	—
21	C₂×Q₃₂	—	2	2	no	—
22	C₂×Q₃₂	—	2	2	no	—
23	C₂×Q₃₂	D₄	1	4	yes	—
24	C₂×M₃₂	D₄	1	1	yes	characteristic
25	C₂²×C₁₆	C₂³	1	1	yes	characteristic
26	C₂×#108	—	2	2	no	—
27	#419	—	4	2	no	—
28	C₂×#101	—	4	2	no	—
29	#405	D₄	1	1	yes	characteristic
30	Q₆₄	—	2	8	no	—
31	M₆₄	—	2	2	no	—
32	C₂×C₃₂	—	2	2	no	—
33	C₂×C₃₂	—	2	1	no	—
34	C₂×C₃₂	D₄	1	2	yes	—
35	#368	—	2	2	no	—
36	#368	—	2	2	no	—
37	#362	—	4	4	no	—
38	#346	—	2	1	no	—
39	C₄×C₁₆	—	2	1	no	—
40	C₂²×Q₈	—	4	2	no	—
41	C₂×Q₁₆	—	2	4	no	—
42	C₂×Q₁₆	—	4	2	no	—
43	C₂×Q₁₆	—	4	2	no	—
44	C₂×Q₁₆	—	4	2	no	—
45	C₂×Q₁₆	—	4	2	no	—
46	C₂×Q₁₆	—	4	2	no	—
47	C₂×M₁₆	—	4	2	no	—
48	C₂×M₁₆	D₈	1	1	yes	characteristic
49	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
50	#117	D₈	1	1	yes	characteristic
51	Q₃₂	—	2	4	no	—
52	Q₃₂	—	2	4	no	—
53	Q₃₂	—	4	2	no	—
54	Q₃₂	—	4	2	no	—
55	M₃₂	—	2	2	no	—
56	C₂×C₁₆	—	2	1	no	—
57	C₂×C₁₆	C₂×D₄	1	2	yes	—
58	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
59	C₂×C₁₆	#41	1	1	yes	characteristic
60	C₂×C₁₆	#41	1	2	yes	—
61	#108	—	4	2	no	—
62	#108	—	4	2	no	—
63	#103	—	8	4	no	—
64	#101	—	8	4	no	—
65	#97	—	2	1	no	—
66	C₄×C₈	—	2	1	no	—
67	C₃₂	—	2	2	no	—
68	C₃₂	—	4	1	no	—
69	C₂×Q₈	—	4	4	no	—
70	C₂×Q₈	—	8	2	no	—
71	C₂×Q₈	—	8	2	no	—
72	C₂×Q₈	—	8	2	no	—
73	C₂×Q₈	—	8	2	no	—
74	C₂×Q₈	—	8	2	no	—
75	C₂²×C₄	—	8	2	no	—
76	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
77	Q₁₆	—	4	4	no	—
78	Q₁₆	—	4	4	no	—
79	Q₁₆	—	8	2	no	—
80	Q₁₆	—	8	2	no	—
81	M₁₆	—	2	2	no	—
82	M₁₆	—	8	2	no	—
83	M₁₆	—	8	2	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	8	2	no	—
86	C₂×C₈	#121	1	2	yes	—
87	C₂×C₈	#121	1	1	yes	characteristic
88	C₂×C₈	#135	1	1	yes	characteristic
89	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
90	C₄⋊C₄	D₁₆	1	2	yes	—
91	C₂²⋊C₄	—	2	1	no	—
92	C₄²	—	2	1	no	—
93	C₁₆	—	4	1	no	—
94	C₁₆	#120	1	1	yes	characteristic
95	C₁₆	#123	1	2	yes	—
96	C₁₆	#124	1	1	yes	characteristic
97	C₂³	#508	1	1	yes	characteristic
98	Q₈	—	8	4	no	—
99	Q₈	—	8	4	no	—
100	Q₈	—	16	2	no	—
101	Q₈	—	16	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	8	4	no	—
104	C₂×C₄	—	16	2	no	—
105	C₂×C₄	—	16	2	no	—
106	C₂×C₄	#459	1	2	yes	—
107	C₂×C₄	#466	1	1	yes	characteristic
108	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
109	C₂×C₄	#509	1	2	yes	—
110	C₈	—	4	1	no	—
111	C₈	—	16	2	no	—
112	C₈	#394	1	1	yes	characteristic
113	C₈	#463	1	2	yes	—
114	C₈	#469	1	1	yes	characteristic
115	C₂²	—	2	2	no	—
116	C₂²	#2210	1	2	yes	—
117	C₂²	#2215	1	1	yes	characteristic
118	C₄	—	4	2	no	—
119	C₄	—	16	4	no	—
120	C₄	#2054	1	1	yes	characteristic
121	C₄	#2206	1	2	yes	—
122	C₄	#2218	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	#13448	1	1	yes	characteristic
125	C₂	#13701	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₆₄

C₂

yes

maximal, characteristic

#13628

C₂

yes

maximal, characteristic

C₂×#1420

C₂

yes

maximal

#13244

C₂

yes

maximal

#7472

C₂

yes

maximal, characteristic

C₂²×Q₃₂

C₂²

yes

—

C₂×Q₆₄

—

C₂×Q₆₄

—

C₂×M₆₄

C₂²

yes

characteristic

C₂²×C₃₂

C₂²

yes

characteristic

C₂×#368

C₂²

yes

—

C₂×#362

—

#2108

C₂²

yes

characteristic

#1979

—

C₄⋊C₃₂

—

#1420

—

C₂²×Q₁₆

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

D₄

yes

—

C₂×M₃₂

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

D₄

yes

characteristic

Q₆₄

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

D₄

yes

—

—

—

—

—

C₄×C₁₆

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

D₈

yes

characteristic

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂×D₄

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

—

—

—

—

—

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#121

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₄⋊C₄

D₁₆

yes

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

#120

yes

characteristic

C₁₆

#123

yes

—

C₁₆

yes

C₂³

yes

Q₈

—

Q₈

—

100

Q₈

—

101

Q₈

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

#459

yes

—

107

C₂×C₄

#466

yes

characteristic

108

C₂×C₄

C₂×D₁₆

yes

characteristic

109

C₂×C₄

#509

yes

—

110

C₈

—

111

C₈

—

112

C₈

#394

yes

characteristic

113

C₈

#463

yes

—

114

C₈

#469

yes

characteristic

115

C₂²

—

116

C₂²

#2210

yes

—

117

C₂²

#2215

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

#2054

yes

characteristic

121

C₄

#2206

yes

—

122

C₄

#2218

yes

characteristic

123

C₂

—

124

C₂

#13448

yes

characteristic

125

C₂

#13701

yes

—

	subgroup	G/H
center, socle	C₂²	#2215
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#508
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2215

commutator, Frattini, cosocle

C₂²×C₁₆

C₂³

prime radical

C₂³

#508

LCS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
C₂²×Q₆₄	2
#13628	2
C₂×#1420	2
#13244	2
#7472	2

subgroup

index

C₂²×Q₆₄

Subgroups — #153316

Conjugacy classes of subgroups

Maximal subgroups