Group #153415 — subgroups

Subgroups — #153415

Order 512.

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

normal subgroups only

This group has 167 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	#13167	C₂	1	1	yes	maximal, characteristic
2	#12910	C₂	1	1	yes	maximal, characteristic
3	#7478	C₂	1	2	yes	maximal
4	#7474	C₂	1	2	yes	maximal
5	#7468	C₂	1	1	yes	maximal, characteristic
6	C₂×M₆₄	C₂²	1	2	yes	—
7	C₂×#368	—	2	1	no	—
8	C₂×#368	C₂²	1	2	yes	—
9	C₂×#367	C₂²	1	1	yes	characteristic
10	C₂×#366	C₂²	1	1	yes	characteristic
11	C₂×#346	C₂²	1	1	yes	characteristic
12	#1942	—	2	1	no	—
13	#1848	—	2	1	no	—
14	#1380	—	2	1	no	—
15	C₂²×C₁₆	D₄	1	1	yes	characteristic
16	C₂²×C₁₆	C₂³	1	1	yes	characteristic
17	C₂×#108	—	2	2	no	—
18	C₂×#108	—	4	1	no	—
19	C₂×#107	—	2	1	no	—
20	C₂×#107	—	2	1	no	—
21	#423	—	4	1	no	—
22	C₂×C₄×C₈	D₄	1	1	yes	characteristic
23	#389	—	4	1	no	—
24	M₆₄	—	2	2	no	—
25	M₆₄	—	2	2	no	—
26	C₂×C₃₂	—	2	2	no	—
27	#368	—	2	4	no	—
28	#368	—	4	2	no	—
29	#367	—	2	1	no	—
30	#367	Q₈	1	2	yes	—
31	#366	—	2	2	no	—
32	#346	—	2	2	no	—
33	#340	—	4	1	no	—
34	C₂×M₁₆	—	4	2	no	—
35	C₂²×C₈	D₈	1	1	yes	characteristic
36	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄	—	4	1	no	—
38	C₂×C₄⋊C₄	—	4	1	no	—
39	C₂×C₄⋊C₄	—	8	1	no	—
40	C₂×C₄²	D₈	1	1	yes	characteristic
41	C₂×C₁₆	—	2	1	no	—
42	C₂×C₁₆	—	2	1	no	—
43	C₂×C₁₆	SD₁₆	1	2	yes	—
44	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₁₆	#41	1	2	yes	—
46	C₂×C₁₆	#41	1	2	yes	—
47	C₂×C₁₆	#41	1	1	yes	characteristic
48	#108	—	4	2	no	—
49	#108	—	4	2	no	—
50	#108	—	8	2	no	—
51	#107	—	2	2	no	—
52	#107	—	4	1	no	—
53	#107	—	4	1	no	—
54	#107	—	4	1	no	—
55	C₄⋊C₈	—	8	2	no	—
56	C₄×C₈	—	2	1	no	—
57	C₄×C₈	—	2	1	no	—
58	#95	—	8	1	no	—
59	C₃₂	—	4	2	no	—
60	C₂²×C₄	—	8	1	no	—
61	C₂²×C₄	—	8	1	no	—
62	C₂²×C₄	SD₃₂	1	1	yes	characteristic
63	C₂²×C₄	SD₃₂	1	1	yes	characteristic
64	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
65	M₁₆	—	8	2	no	—
66	M₁₆	—	8	2	no	—
67	C₂×C₈	—	2	1	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	8	2	no	—
70	C₂×C₈	Q₃₂	1	2	yes	—
71	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₈	#123	1	1	yes	characteristic
73	C₂×C₈	#124	1	1	yes	characteristic
74	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₈	#137	1	1	yes	characteristic
76	C₂×C₈	#137	1	1	yes	characteristic
77	C₄⋊C₄	—	4	2	no	—
78	C₄⋊C₄	—	8	1	no	—
79	C₄⋊C₄	—	8	1	no	—
80	C₄⋊C₄	—	8	1	no	—
81	C₄⋊C₄	—	16	2	no	—
82	C₄²	—	2	1	no	—
83	C₄²	—	2	1	no	—
84	C₁₆	—	2	2	no	—
85	C₁₆	#125	1	2	yes	—
86	C₁₆	#126	1	2	yes	—
87	C₂³	C₂×SD₃₂	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	4	1	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	—	8	2	no	—
93	C₂×C₄	—	16	1	no	—
94	C₂×C₄	—	16	1	no	—
95	C₂×C₄	—	16	1	no	—
96	C₂×C₄	—	16	1	no	—
97	C₂×C₄	—	16	1	no	—
98	C₂×C₄	#474	1	1	yes	characteristic
99	C₂×C₄	#480	1	1	yes	characteristic
100	C₂×C₄	#486	1	1	yes	characteristic
101	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
102	C₂×C₄	#509	1	1	yes	characteristic
103	C₂×C₄	#509	1	1	yes	characteristic
104	C₈	—	2	2	no	—
105	C₈	—	16	2	no	—
106	C₈	#400	1	1	yes	characteristic
107	C₈	#475	1	1	yes	characteristic
108	C₈	#483	1	1	yes	characteristic
109	C₈	#491	1	1	yes	characteristic
110	C₂²	—	2	1	no	—
111	C₂²	—	2	1	no	—
112	C₂²	#2226	1	1	yes	characteristic
113	C₂²	#2233	1	1	yes	characteristic
114	C₂²	#2241	1	1	yes	characteristic
115	C₄	—	4	1	no	—
116	C₄	—	4	1	no	—
117	C₄	—	16	2	no	—
118	C₄	—	32	1	no	—
119	C₄	#2099	1	1	yes	characteristic
120	C₄	#2227	1	1	yes	characteristic
121	C₄	#2236	1	1	yes	characteristic
122	C₄	#2244	1	1	yes	characteristic
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	#13538	1	1	yes	characteristic
126	C₂	#13717	1	1	yes	characteristic
127	C₂	#13718	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13167

C₂

yes

maximal, characteristic

#12910

C₂

yes

maximal, characteristic

#7478

C₂

yes

maximal

#7474

C₂

yes

maximal

#7468

C₂

yes

maximal, characteristic

C₂×M₆₄

C₂²

yes

—

C₂×#368

—

C₂×#368

C₂²

yes

—

C₂×#367

C₂²

yes

characteristic

C₂×#366

C₂²

yes

characteristic

C₂×#346

C₂²

yes

—

—

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

#389

—

M₆₄

—

M₆₄

—

C₂×C₃₂

—

—

—

—

Q₈

yes

—

#366

—

#346

—

#340

—

C₂×M₁₆

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

SD₁₆

yes

—

C₂×C₁₆

C₂×Q₈

yes

characteristic

C₂×C₁₆

#41

yes

—

C₂×C₁₆

#41

yes

—

C₂×C₁₆

yes

—

—

—

—

—

—

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

#95

—

C₃₂

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

SD₃₂

yes

characteristic

C₂²×C₄

SD₃₂

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Q₃₂

yes

—

C₂×C₈

C₂²⋊Q₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₂×SD₁₆

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

#125

yes

—

C₁₆

#126

yes

—

C₂³

C₂×SD₃₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

#486

yes

characteristic

101

C₂×C₄

C₂×Q₃₂

yes

characteristic

102

C₂×C₄

#509

yes

characteristic

103

C₂×C₄

#509

yes

characteristic

104

C₈

—

105

C₈

—

106

C₈

#400

yes

characteristic

107

C₈

#475

yes

characteristic

108

C₈

#483

yes

characteristic

109

C₈

#491

yes

characteristic

110

C₂²

—

111

C₂²

—

112

C₂²

#2226

yes

characteristic

113

C₂²

#2233

yes

characteristic

114

C₂²

#2241

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

#2099

yes

characteristic

120

C₄

#2227

yes

characteristic

121

C₄

#2236

yes

characteristic

122

C₄

#2244

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#13538

yes

characteristic

126

C₂

#13717

yes

characteristic

127

C₂

#13718

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#2233
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	C₂×SD₃₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2233

commutator, Frattini, cosocle

C₂²×C₁₆

C₂³

prime radical

C₂³

C₂×SD₃₂

LCS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ 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
#13167	2
#12910	2
#7478	2
#7474	2
#7468	2

subgroup

index

Subgroups — #153415

Conjugacy classes of subgroups

Maximal subgroups