Group #13409 — subgroups

Subgroups — #13409

Order 256.

231 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 69 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×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×#358	C₂	1	2	yes	maximal
3	C₂×#358	C₂	1	2	yes	maximal
4	#1942	C₂	1	1	yes	maximal, characteristic
5	#1374	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆	—	2	1	no	—
7	C₂²×Q₁₆	C₂²	1	2	yes	—
8	C₂×C₄⋊Q₈	—	2	1	no	—
9	C₂²×C₁₆	C₂²	1	2	yes	—
10	#494	—	2	4	no	—
11	C₂×#107	C₂²	1	1	yes	characteristic
12	C₂×#107	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	#389	—	2	1	no	—
15	#358	—	2	4	no	—
16	#358	—	2	4	no	—
17	#340	—	2	1	no	—
18	C₂²×Q₈	—	2	2	no	—
19	C₂×Q₁₆	—	2	4	no	—
20	C₂×Q₁₆	—	2	2	no	—
21	C₂×Q₁₆	—	2	2	no	—
22	C₂×Q₁₆	—	2	2	no	—
23	C₂×Q₁₆	—	2	2	no	—
24	C₂×Q₁₆	—	2	2	no	—
25	C₂×Q₁₆	—	4	2	no	—
26	C₂×Q₁₆	—	4	2	no	—
27	C₂×Q₁₆	D₄	1	4	yes	—
28	C₂²×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₈	C₂³	1	1	yes	characteristic
30	C₄⋊Q₈	—	2	4	no	—
31	C₄⋊Q₈	—	4	2	no	—
32	C₂×C₄⋊C₄	—	2	1	no	—
33	C₂×C₄⋊C₄	—	2	1	no	—
34	C₂×C₄⋊C₄	—	4	1	no	—
35	C₂×C₄⋊C₄	—	4	1	no	—
36	C₂×C₄⋊C₄	—	4	1	no	—
37	C₂×C₄²	D₄	1	1	yes	characteristic
38	C₂×C₁₆	—	2	2	no	—
39	C₂×C₁₆	—	2	2	no	—
40	C₂×C₁₆	—	2	2	no	—
41	#107	—	2	1	no	—
42	#107	—	2	2	no	—
43	#107	D₄	1	2	yes	—
44	C₄×C₈	—	2	2	no	—
45	#95	—	4	1	no	—
46	C₂×Q₈	—	2	4	no	—
47	C₂×Q₈	—	4	2	no	—
48	C₂×Q₈	—	4	2	no	—
49	C₂×Q₈	—	4	2	no	—
50	C₂×Q₈	—	4	2	no	—
51	C₂×Q₈	—	4	2	no	—
52	C₂²×C₄	—	4	1	no	—
53	C₂²×C₄	—	4	2	no	—
54	C₂²×C₄	—	4	1	no	—
55	C₂²×C₄	D₈	1	1	yes	characteristic
56	C₂²×C₄	D₈	1	1	yes	characteristic
57	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
58	Q₁₆	—	2	4	no	—
59	Q₁₆	—	2	4	no	—
60	Q₁₆	—	4	8	no	—
61	Q₁₆	—	4	2	no	—
62	Q₁₆	—	4	2	no	—
63	C₂×C₈	—	2	1	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	SD₁₆	1	2	yes	—
66	C₂×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈	C₂×D₄	1	2	yes	—
68	C₂×C₈	#41	1	2	yes	—
69	C₂×C₈	#41	1	1	yes	characteristic
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	—	4	4	no	—
72	C₄⋊C₄	—	4	4	no	—
73	C₄⋊C₄	—	4	1	no	—
74	C₄⋊C₄	—	4	1	no	—
75	C₄⋊C₄	—	4	1	no	—
76	C₄⋊C₄	—	8	2	no	—
77	C₄²	—	2	1	no	—
78	C₄²	—	2	1	no	—
79	C₁₆	—	4	2	no	—
80	C₂³	C₂×D₈	1	1	yes	characteristic
81	Q₈	—	4	4	no	—
82	Q₈	—	4	4	no	—
83	Q₈	—	8	2	no	—
84	Q₈	—	8	2	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	4	4	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	8	2	no	—
92	C₂×C₄	—	8	2	no	—
93	C₂×C₄	—	8	1	no	—
94	C₂×C₄	—	8	1	no	—
95	C₂×C₄	—	8	1	no	—
96	C₂×C₄	—	8	1	no	—
97	C₂×C₄	—	8	1	no	—
98	C₂×C₄	SD₃₂	1	2	yes	—
99	C₂×C₄	Q₃₂	1	2	yes	—
100	C₂×C₄	#120	1	1	yes	characteristic
101	C₂×C₄	#121	1	1	yes	characteristic
102	C₂×C₄	#124	1	1	yes	characteristic
103	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
106	C₈	—	2	2	no	—
107	C₈	#121	1	2	yes	—
108	C₈	#123	1	2	yes	—
109	C₂²	#447	1	1	yes	characteristic
110	C₂²	#459	1	1	yes	characteristic
111	C₂²	#486	1	1	yes	characteristic
112	C₂²	C₂×SD₃₂	1	1	yes	characteristic
113	C₂²	C₂×Q₃₂	1	1	yes	characteristic
114	C₂²	#510	1	1	yes	characteristic
115	C₂²	#510	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	8	2	no	—
119	C₄	—	8	4	no	—
120	C₄	—	16	1	no	—
121	C₄	#394	1	1	yes	characteristic
122	C₄	#450	1	1	yes	characteristic
123	C₄	#460	1	1	yes	characteristic
124	C₄	#490	1	1	yes	characteristic
125	C₂	#2041	1	1	yes	characteristic
126	C₂	#2185	1	1	yes	characteristic
127	C₂	#2188	1	1	yes	characteristic
128	C₂	#2208	1	1	yes	characteristic
129	C₂	#2209	1	1	yes	characteristic
130	C₂	#2241	1	1	yes	characteristic
131	C₂	#2245	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#494

C₂

yes

maximal, characteristic

C₂×#358

C₂

yes

maximal

C₂×#358

C₂

yes

maximal

#1942

C₂

yes

maximal, characteristic

#1374

C₂

yes

maximal, characteristic

C₂²×Q₁₆

—

C₂²×Q₁₆

C₂²

yes

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

C₂²

yes

—

#494

—

C₂×#107

C₂²

yes

characteristic

C₂×#107

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

—

—

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

D₄

yes

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#107

—

#107

D₄

yes

—

C₄×C₈

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₈

yes

characteristic

C₂²×C₄

D₈

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

SD₁₆

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

#41

yes

—

C₂×C₈

#41

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₁₆

—

C₂³

C₂×D₈

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

SD₃₂

yes

—

C₂×C₄

Q₃₂

yes

—

100

C₂×C₄

#120

yes

characteristic

101

C₂×C₄

#121

yes

characteristic

102

C₂×C₄

#124

yes

characteristic

103

C₂×C₄

C₂×SD₁₆

yes

characteristic

104

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

105

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

106

C₈

—

107

C₈

#121

yes

—

108

C₈

#123

yes

—

109

C₂²

#447

yes

characteristic

110

C₂²

#459

yes

characteristic

111

C₂²

#486

yes

characteristic

112

C₂²

C₂×SD₃₂

yes

characteristic

113

C₂²

C₂×Q₃₂

yes

characteristic

114

C₂²

#510

yes

characteristic

115

C₂²

#510

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#394

yes

characteristic

122

C₄

#450

yes

characteristic

123

C₄

#460

yes

characteristic

124

C₄

#490

yes

characteristic

125

C₂

#2041

yes

characteristic

126

C₂

#2185

yes

characteristic

127

C₂

#2188

yes

characteristic

128

C₂

#2208

yes

characteristic

129

C₂

#2209

yes

characteristic

130

C₂

#2241

yes

characteristic

131

C₂

#2245

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×D₈

commutator, Frattini, cosocle

C₂²×C₈

C₂³

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
C₂×#494	2
C₂×#358	2
#1942	2
#1374	2

subgroup

index

Subgroups — #13409

Conjugacy classes of subgroups

Maximal subgroups