Group #153372 — subgroups

Subgroups — #153372

Order 512.

189 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 189 conjugacy classes of subgroups — too many to draw. Only its 62 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	#13636	C₂	1	1	yes	maximal, characteristic, complemented
2	#13553	C₂	1	1	yes	maximal, characteristic
3	#12837	C₂	1	2	yes	maximal
4	#7479	C₂	1	2	yes	maximal, complemented
5	#7476	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×M₆₄	C₂²	1	1	yes	characteristic
8	C₂×M₆₄	C₂²	1	1	yes	characteristic
9	C₂×#368	C₂²	1	2	yes	—
10	C₂×#368	C₂²	1	2	yes	—
11	#1810	—	2	2	no	—
12	#1425	—	2	2	no	—
13	C₂²⋊C₃₂	—	2	2	no	—
14	#1381	—	2	2	no	—
15	C₂³×C₈	D₄	1	1	yes	characteristic
16	C₂²×C₁₆	—	2	1	no	—
17	C₂²×C₁₆	—	2	1	no	—
18	C₂²×C₁₆	—	2	2	no	—
19	C₂²×C₁₆	—	2	2	no	—
20	C₂²×C₁₆	D₄	1	1	yes	characteristic
21	C₂²×C₁₆	C₂³	1	1	yes	characteristic
22	C₂×#108	—	2	2	no	—
23	C₂×#108	—	2	2	no	—
24	#407	—	4	2	no	—
25	M₆₄	—	2	2	no	—
26	M₆₄	—	2	2	no	—
27	C₂×C₃₂	—	2	1	no	—
28	C₂×C₃₂	D₄	1	2	yes	—
29	#368	—	2	2	no	—
30	#368	—	2	2	no	—
31	#368	—	2	2	no	—
32	#368	—	2	2	no	—
33	#341	—	4	2	no	—
34	C₂³×C₄	D₈	1	1	yes	characteristic
35	C₂×M₁₆	—	4	2	no	—
36	C₂×M₁₆	—	4	2	no	—
37	C₂²×C₈	—	2	1	no	—
38	C₂²×C₈	—	2	2	no	—
39	C₂²×C₈	—	2	1	no	—
40	C₂²×C₈	—	2	2	no	—
41	C₂²×C₈	D₈	1	1	yes	characteristic
42	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
43	C₂×C₁₆	—	2	1	no	—
44	C₂×C₁₆	—	2	2	no	—
45	C₂×C₁₆	—	4	1	no	—
46	C₂×C₁₆	—	4	1	no	—
47	C₂×C₁₆	—	4	2	no	—
48	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
49	C₂×C₁₆	#41	1	2	yes	—
50	C₂×C₁₆	#41	1	2	yes	—
51	C₂×C₁₆	#41	1	1	yes	characteristic
52	#108	—	4	2	no	—
53	#108	—	4	2	no	—
54	#108	—	4	2	no	—
55	#108	—	4	2	no	—
56	C₂²⋊C₈	—	8	4	no	—
57	C₃₂	—	2	2	no	—
58	C₃₂	—	4	1	no	—
59	C₂⁴	Q₃₂	1	1	yes	characteristic
60	C₂²×C₄	—	2	1	no	—
61	C₂²×C₄	—	2	2	no	—
62	C₂²×C₄	—	2	1	no	—
63	C₂²×C₄	—	2	2	no	—
64	C₂²×C₄	Q₃₂	1	1	yes	characteristic
65	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
66	M₁₆	—	8	2	no	—
67	M₁₆	—	8	2	no	—
68	M₁₆	—	8	2	no	—
69	M₁₆	—	8	2	no	—
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	4	1	no	—
73	C₂×C₈	—	4	1	no	—
74	C₂×C₈	—	4	2	no	—
75	C₂×C₈	—	8	2	no	—
76	C₂×C₈	—	8	2	no	—
77	C₂×C₈	#121	1	1	yes	characteristic
78	C₂×C₈	#123	1	2	yes	—
79	C₂×C₈	#135	1	1	yes	characteristic
80	C₂×C₈	#137	1	2	yes	—
81	C₁₆	—	4	1	no	—
82	C₁₆	#123	1	1	yes	characteristic
83	C₁₆	#124	1	1	yes	characteristic
84	C₁₆	#126	1	2	yes	—
85	C₂³	—	2	1	no	—
86	C₂³	—	2	2	no	—
87	C₂³	—	4	2	no	—
88	C₂³	C₂×Q₃₂	1	1	yes	characteristic
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	—	4	1	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	#466	1	1	yes	characteristic
95	C₂×C₄	#480	1	2	yes	—
96	C₂×C₄	#508	1	1	yes	characteristic
97	C₂×C₄	#509	1	2	yes	—
98	C₈	—	4	1	no	—
99	C₈	—	16	2	no	—
100	C₈	—	16	2	no	—
101	C₈	#397	1	1	yes	characteristic
102	C₈	#470	1	1	yes	characteristic
103	C₈	#485	1	2	yes	—
104	C₂²	—	2	2	no	—
105	C₂²	—	2	4	no	—
106	C₂²	—	4	1	no	—
107	C₂²	—	4	1	no	—
108	C₂²	—	4	2	no	—
109	C₂²	—	4	1	no	—
110	C₂²	#2216	1	1	yes	characteristic
111	C₂²	#2236	1	2	yes	—
112	C₄	—	4	1	no	—
113	C₄	#2075	1	1	yes	characteristic
114	C₄	#2218	1	1	yes	characteristic
115	C₄	#2234	1	2	yes	—
116	C₂	—	2	1	no	—
117	C₂	—	2	2	no	—
118	C₂	—	4	2	no	—
119	C₂	#13493	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13636

C₂

yes

maximal, characteristic, complemented

#13553

C₂

yes

maximal, characteristic

#12837

C₂

yes

maximal

#7479

C₂

yes

maximal, complemented

#7476

C₂

yes

maximal, characteristic, complemented

C₂³×C₁₆

C₂²

yes

characteristic

C₂×M₆₄

C₂²

yes

characteristic

C₂×M₆₄

C₂²

yes

characteristic

C₂×#368

C₂²

yes

—

C₂×#368

C₂²

yes

—

#1810

—

#1425

—

C₂²⋊C₃₂

—

#1381

—

C₂³×C₈

D₄

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

M₆₄

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

D₄

yes

—

—

—

—

—

—

C₂³×C₄

D₈

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

#41

yes

—

C₂×C₁₆

#41

yes

—

C₂×C₁₆

yes

—

—

—

—

C₂²⋊C₈

—

C₃₂

—

C₃₂

—

C₂⁴

Q₃₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Q₃₂

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#121

yes

characteristic

C₂×C₈

#123

yes

—

C₂×C₈

#135

yes

characteristic

C₂×C₈

#137

yes

—

C₁₆

—

C₁₆

yes

C₁₆

yes

C₁₆

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×Q₃₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#466

yes

characteristic

C₂×C₄

#480

yes

—

C₂×C₄

#508

yes

characteristic

C₂×C₄

#509

yes

—

C₈

—

C₈

—

100

C₈

—

101

C₈

#397

yes

characteristic

102

C₈

#470

yes

characteristic

103

C₈

#485

yes

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

#2216

yes

characteristic

111

C₂²

#2236

yes

—

112

C₄

—

113

C₄

#2075

yes

characteristic

114

C₄

#2218

yes

characteristic

115

C₄

#2234

yes

—

116

C₂

—

117

C₂

—

118

C₂

—

119

C₂

#13493

yes

characteristic

	subgroup	G/H
center	C₄	#2218
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13493
prime radical	C₂⁴	Q₃₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

#2218

commutator, Frattini, cosocle

C₂²×C₁₆

C₂³

socle

C₂

#13493

prime radical

C₂⁴

Q₃₂

LCS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #13636	C₂	semidirect
4. #7479	C₂	semidirect
5. #7476	C₂	semidirect

H ≅ G/N

type

1. #13636

C₂

semidirect

4. #7479

C₂

semidirect

5. #7476

C₂

semidirect

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

subgroup	index
#13636	2
#13553	2
#12837	2
#7479	2
#7476	2

subgroup

index

Subgroups — #153372

Conjugacy classes of subgroups

Complements

Maximal subgroups