Group #135797 — subgroups

Subgroups — #135797

Order 512.

228 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 126 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	#12023	C₂	1	2	yes	maximal
2	#12022	C₂	1	2	yes	maximal
3	C₂×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#8824	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
7	#1837	—	2	2	no	—
8	#1567	C₂²	1	4	yes	—
9	#1566	—	2	2	no	—
10	C₂×C₈²	C₂²	1	1	yes	characteristic
11	#1372	—	2	4	no	—
12	#1371	—	2	4	no	—
13	C₈×C₁₆	C₄	1	1	yes	characteristic
14	C₈×C₁₆	C₄	1	1	yes	characteristic
15	C₈×C₁₆	C₄	1	1	yes	characteristic
16	C₈×C₁₆	C₄	1	1	yes	characteristic
17	C₂²×C₁₆	Q₈	1	2	yes	—
18	C₂²×C₁₆	Q₈	1	2	yes	—
19	#423	—	2	4	no	—
20	C₂×C₄×C₈	D₄	1	1	yes	characteristic
21	C₂×C₄×C₈	D₄	1	1	yes	characteristic
22	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
23	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₄×C₁₆	D₄	1	2	yes	—
28	C₄×C₁₆	D₄	1	2	yes	—
29	#341	—	4	4	no	—
30	#335	—	2	4	no	—
31	#335	—	2	4	no	—
32	#334	—	4	4	no	—
33	C₈²	C₂×C₄	1	1	yes	characteristic
34	C₈²	C₂×C₄	1	1	yes	characteristic
35	C₈²	Q₈	1	1	yes	characteristic
36	C₈²	Q₈	1	1	yes	characteristic
37	C₂×M₁₆	—	4	4	no	—
38	C₂²×C₈	D₈	1	2	yes	—
39	C₂²×C₈	Q₁₆	1	2	yes	—
40	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
43	C₂×C₁₆	—	2	4	no	—
44	C₂×C₁₆	—	2	4	no	—
45	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
46	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
47	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
48	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
49	C₄⋊C₈	—	4	4	no	—
50	C₄⋊C₈	—	4	4	no	—
51	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₈	D₈	1	2	yes	—
56	C₄×C₈	Q₁₆	1	2	yes	—
57	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
58	C₄×C₈	C₂×D₄	1	1	yes	characteristic
59	C₄×C₈	C₂×D₄	1	1	yes	characteristic
60	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
62	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
63	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
64	M₁₆	—	8	8	no	—
65	C₂×C₈	—	2	4	no	—
66	C₂×C₈	—	2	4	no	—
67	C₂×C₈	—	2	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	8	4	no	—
70	C₂×C₈	#107	1	2	yes	—
71	C₂×C₈	#107	1	2	yes	—
72	C₂×C₈	#107	1	2	yes	—
73	C₂×C₈	#107	1	2	yes	—
74	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
78	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₄²	#127	1	1	yes	characteristic
80	C₄²	C₂×D₈	1	1	yes	characteristic
81	C₄²	C₂×Q₁₆	1	1	yes	characteristic
82	C₁₆	—	2	2	no	—
83	C₁₆	—	2	2	no	—
84	C₁₆	#108	1	4	yes	—
85	C₁₆	#108	1	4	yes	—
86	C₂³	#500	1	1	yes	characteristic
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	#384	1	1	yes	characteristic
91	C₂×C₄	C₂×#107	1	1	yes	characteristic
92	C₂×C₄	C₂×#107	1	1	yes	characteristic
93	C₂×C₄	#493	1	1	yes	characteristic
94	C₂×C₄	#494	1	1	yes	characteristic
95	C₂×C₄	#500	1	1	yes	characteristic
96	C₂×C₄	#508	1	4	yes	—
97	C₈	—	2	1	no	—
98	C₈	—	2	1	no	—
99	C₈	—	2	2	no	—
100	C₈	—	2	2	no	—
101	C₈	—	2	2	no	—
102	C₈	—	16	4	no	—
103	C₈	#368	1	4	yes	—
104	C₈	C₂×#108	1	2	yes	—
105	C₈	C₂×#108	1	2	yes	—
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	#1845	1	1	yes	characteristic
109	C₂²	#2253	1	2	yes	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	#1848	1	1	yes	characteristic
113	C₄	#1848	1	1	yes	characteristic
114	C₄	#2158	1	2	yes	—
115	C₄	C₂×#368	1	2	yes	—
116	C₄	#2249	1	2	yes	—
117	C₂	—	2	2	no	—
118	C₂	#8823	1	1	yes	characteristic
119	C₂	#12907	1	1	yes	characteristic
120	C₂	#12909	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12023

C₂

yes

maximal

#12022

C₂

yes

maximal

C₂×C₈×C₁₆

C₂

yes

maximal, characteristic

#8824

C₂

yes

maximal

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1837

—

#1567

C₂²

yes

—

#1566

—

C₂×C₈²

C₂²

yes

characteristic

#1372

—

#1371

—

C₈×C₁₆

C₄

yes

characteristic

C₈×C₁₆

C₄

yes

characteristic

C₈×C₁₆

C₄

yes

characteristic

C₈×C₁₆

C₄

yes

characteristic

C₂²×C₁₆

Q₈

yes

—

C₂²×C₁₆

Q₈

yes

—

#423

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

D₄

yes

—

C₄×C₁₆

D₄

yes

—

—

—

—

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

Q₈

yes

characteristic

C₈²

Q₈

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

Q₁₆

yes

—

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

D₈

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₂²×C₄

C₄⋊Q₈

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

C₂×Q₁₆

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#107

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₄²

C₂×C₄⋊C₄

yes

characteristic

C₄²

#127

yes

characteristic

C₄²

C₂×D₈

yes

characteristic

C₄²

C₂×Q₁₆

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

#108

yes

—

C₁₆

#108

yes

—

C₂³

#500

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₈

—

C₈

—

C₈

—

100

C₈

—

101

C₈

—

102

C₈

—

103

C₈

#368

yes

—

104

C₈

C₂×#108

yes

—

105

C₈

C₂×#108

yes

—

106

C₂²

—

107

C₂²

—

108

C₂²

#1845

yes

characteristic

109

C₂²

#2253

yes

—

110

C₄

—

111

C₄

—

112

C₄

#1848

yes

characteristic

113

C₄

#1848

yes

characteristic

114

C₄

#2158

yes

—

115

C₄

C₂×#368

yes

—

116

C₄

#2249

yes

—

117

C₂

—

118

C₂

#8823

yes

characteristic

119

C₂

#12907

yes

characteristic

120

C₂

#12909

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#493
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1845
prime radical	C₂³	#500
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#493

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#12023	2
#12022	2
C₂×C₈×C₁₆	2
#8824	2

subgroup

index

#12023

#12022

C₂×C₈×C₁₆

#8824

Subgroups — #135797

Conjugacy classes of subgroups

Maximal subgroups