Group #141692 — subgroups

Subgroups — #141692

Order 512.

388 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 142 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	C₂×#1713	C₂	1	1	yes	maximal, characteristic
2	#13282	C₂	1	4	yes	maximal
3	C₂×#1311	C₂	1	1	yes	maximal, characteristic
4	#8816	C₂	1	1	yes	maximal, characteristic
5	C₂×#500	—	2	1	no	—
6	C₂×#500	C₂²	1	2	yes	—
7	C₂×#494	—	2	1	no	—
8	C₂×C₄×C₁₆	C₂²	1	2	yes	—
9	#1845	C₂²	1	2	yes	—
10	#1834	—	2	1	no	—
11	#1834	—	2	1	no	—
12	#1713	—	2	2	no	—
13	#1713	C₄	1	4	yes	complemented
14	C₂×C₈²	C₂²	1	1	yes	characteristic
15	#1380	—	2	8	no	—
16	#1311	—	2	2	no	—
17	C₂²×Q₁₆	—	4	2	no	—
18	C₂×C₄⋊Q₈	—	2	2	no	—
19	C₂²×C₁₆	D₄	1	4	yes	—
20	#500	—	2	4	no	—
21	#500	—	2	2	no	—
22	#500	—	2	4	no	—
23	#500	—	4	2	no	—
24	#500	C₂×C₄	1	4	yes	—
25	#494	—	2	4	no	—
26	#494	—	4	2	no	—
27	C₂×#107	—	2	4	no	—
28	C₂×#107	—	2	4	no	—
29	C₂×#107	—	4	2	no	—
30	C₂×C₄×C₈	D₄	1	1	yes	characteristic
31	C₂×C₄×C₈	D₄	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
33	#384	—	2	2	no	—
34	C₄×C₁₆	—	2	2	no	—
35	C₄×C₁₆	—	2	2	no	—
36	#340	—	4	2	no	—
37	#340	—	4	2	no	—
38	C₈²	C₂×C₄	1	2	yes	—
39	C₈²	D₄	1	2	yes	—
40	C₂²×Q₈	—	4	2	no	—
41	C₂×Q₁₆	—	4	8	no	—
42	C₂×Q₁₆	—	8	8	no	—
43	C₂²×C₈	D₈	1	2	yes	—
44	C₂²×C₈	D₈	1	2	yes	—
45	C₂²×C₈	C₂×D₄	1	2	yes	—
46	C₄⋊Q₈	—	2	4	no	—
47	C₄⋊Q₈	—	4	2	no	—
48	C₄⋊Q₈	—	4	2	no	—
49	C₄⋊Q₈	—	4	2	no	—
50	C₂×C₄⋊C₄	—	4	2	no	—
51	C₂×C₄⋊C₄	—	4	2	no	—
52	C₂×C₄⋊C₄	—	4	2	no	—
53	C₂×C₄⋊C₄	—	4	2	no	—
54	C₂×C₄⋊C₄	—	4	2	no	—
55	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
56	C₂×C₁₆	—	2	4	no	—
57	C₂×C₁₆	—	2	4	no	—
58	C₂×C₁₆	Q₁₆	1	8	yes	complemented
59	#107	—	2	8	no	—
60	#107	—	4	8	no	—
61	#107	—	4	4	no	—
62	#107	—	4	8	no	—
63	C₄×C₈	C₂²⋊C₄	1	2	yes	—
64	C₄×C₈	C₂²⋊C₄	1	2	yes	—
65	C₄×C₈	SD₁₆	1	2	yes	—
66	C₄×C₈	SD₁₆	1	2	yes	—
67	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
68	C₄×C₈	C₂×D₄	1	1	yes	characteristic
69	C₄×C₈	#41	1	2	yes	—
70	C₂×Q₈	—	4	4	no	—
71	C₂×Q₈	—	8	2	no	—
72	C₂×Q₈	—	8	8	no	—
73	C₂²×C₄	—	8	2	no	—
74	C₂²×C₄	—	8	2	no	—
75	C₂²×C₄	#127	1	1	yes	characteristic
76	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
77	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
78	Q₁₆	—	8	16	no	—
79	C₂×C₈	—	2	4	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	#102	1	4	yes	—
82	C₂×C₈	#102	1	4	yes	—
83	C₂×C₈	C₄×D₄	1	2	yes	—
84	C₂×C₈	#121	1	2	yes	—
85	C₂×C₈	C₂×Q₁₆	1	4	yes	—
86	C₂×C₈	#135	1	4	yes	—
87	C₄⋊C₄	—	4	4	no	—
88	C₄⋊C₄	—	4	4	no	—
89	C₄⋊C₄	—	8	2	no	—
90	C₄⋊C₄	—	8	2	no	—
91	C₄⋊C₄	—	8	4	no	—
92	C₄⋊C₄	—	8	4	no	—
93	C₄⋊C₄	—	8	4	no	—
94	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₄²	#124	1	1	yes	characteristic
96	C₄²	C₂×SD₁₆	1	1	yes	characteristic
97	C₄²	C₂×SD₁₆	1	1	yes	characteristic
98	C₁₆	—	2	8	no	—
99	C₂³	#493	1	1	yes	characteristic
100	Q₈	—	8	8	no	—
101	Q₈	—	16	4	no	—
102	C₂×C₄	—	8	4	no	—
103	C₂×C₄	—	16	4	no	—
104	C₂×C₄	—	16	2	no	—
105	C₂×C₄	—	16	2	no	—
106	C₂×C₄	—	16	2	no	—
107	C₂×C₄	#390	1	1	yes	characteristic
108	C₂×C₄	C₂×#102	1	1	yes	characteristic
109	C₂×C₄	C₂×#102	1	1	yes	characteristic
110	C₂×C₄	#486	1	1	yes	characteristic
111	C₂×C₄	#486	1	1	yes	characteristic
112	C₂×C₄	#492	1	1	yes	characteristic
113	C₂×C₄	#494	1	2	yes	—
114	C₂×C₄	#495	1	2	yes	—
115	C₂×C₄	#509	1	4	yes	—
116	C₂×C₄	#510	1	4	yes	—
117	C₈	—	2	4	no	—
118	C₈	—	2	4	no	—
119	C₈	C₄×Q₁₆	1	4	yes	—
120	C₈	#467	1	4	yes	—
121	C₂²	#1709	1	1	yes	characteristic
122	C₂²	#1713	1	1	yes	characteristic
123	C₂²	#1957	1	1	yes	characteristic
124	C₂²	#2250	1	2	yes	—
125	C₂²	#2251	1	2	yes	—
126	C₄	—	16	4	no	—
127	C₄	—	32	2	no	—
128	C₄	#1707	1	2	yes	—
129	C₄	#1969	1	2	yes	—
130	C₄	#2140	1	2	yes	—
131	C₄	#2141	1	2	yes	—
132	C₄	#2244	1	2	yes	—
133	C₄	#2245	1	2	yes	—
134	C₂	#9825	1	1	yes	characteristic
135	C₂	#12512	1	2	yes	—
136	C₂	#12524	1	2	yes	—
137	C₂	#13206	1	1	yes	characteristic
138	C₂	#13207	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1713

C₂

yes

maximal, characteristic

#13282

C₂

yes

maximal

C₂×#1311

C₂

yes

maximal, characteristic

#8816

C₂

yes

maximal, characteristic

C₂×#500

—

C₂×#500

C₂²

yes

—

C₂×#494

—

C₂×C₄×C₁₆

C₂²

yes

—

#1845

C₂²

yes

—

—

—

—

C₄

yes

complemented

C₂×C₈²

C₂²

yes

characteristic

#1380

—

#1311

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

D₄

yes

—

—

—

—

—

C₂×C₄

yes

—

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

#384

—

C₄×C₁₆

—

C₄×C₁₆

—

#340

—

#340

—

C₈²

C₂×C₄

yes

—

C₈²

D₄

yes

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

C₂×D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

Q₁₆

yes

—

—

—

—

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

#41

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#102

yes

—

C₂×C₈

#102

yes

—

C₂×C₈

C₄×D₄

yes

—

C₂×C₈

#121

yes

—

C₂×C₈

C₂×Q₁₆

yes

—

C₂×C₈

#135

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊C₄

yes

characteristic

C₄²

#124

yes

characteristic

C₄²

C₂×SD₁₆

yes

characteristic

C₄²

C₂×SD₁₆

yes

characteristic

C₁₆

—

C₂³

#493

yes

characteristic

100

Q₈

—

101

Q₈

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#390

yes

characteristic

108

C₂×C₄

C₂×#102

yes

characteristic

109

C₂×C₄

C₂×#102

yes

characteristic

110

C₂×C₄

#486

yes

characteristic

111

C₂×C₄

#486

yes

characteristic

112

C₂×C₄

#492

yes

characteristic

113

C₂×C₄

#494

yes

—

114

C₂×C₄

#495

yes

—

115

C₂×C₄

#509

yes

—

116

C₂×C₄

#510

yes

—

117

C₈

—

118

C₈

—

119

C₈

C₄×Q₁₆

yes

—

120

C₈

#467

yes

—

121

C₂²

#1709

yes

characteristic

122

C₂²

#1713

yes

characteristic

123

C₂²

#1957

yes

characteristic

124

C₂²

#2250

yes

—

125

C₂²

#2251

yes

—

126

C₄

—

127

C₄

—

128

C₄

#1707

yes

—

129

C₄

#1969

yes

—

130

C₄

#2140

yes

—

131

C₄

#2141

yes

—

132

C₄

#2244

yes

—

133

C₄

#2245

yes

—

134

C₂

#9825

yes

characteristic

135

C₂

#12512

yes

—

136

C₂

#12524

yes

—

137

C₂

#13206

yes

characteristic

138

C₂

#13207

yes

characteristic

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

subgroup

G/H

center, socle, prime radical

C₂³

#493

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
13. #1713	C₄	semidirect
58. C₂×C₁₆	Q₁₆	semidirect

H ≅ G/N

type

13. #1713

C₄

semidirect

58. C₂×C₁₆

Q₁₆

semidirect

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

subgroup	index
C₂×#1713	2
#13282	2
C₂×#1311	2
#8816	2

subgroup

index

Subgroups — #141692

Conjugacy classes of subgroups

Complements

Maximal subgroups