Group #70463 — subgroups

Subgroups — #70463

Order 384.

224 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 89 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×C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#818	C₂	1	1	yes	maximal, characteristic
3	#4928	C₂	1	2	yes	maximal
4	#4551	C₂	1	2	yes	maximal
5	#4551	C₂	1	1	yes	maximal, characteristic
6	#2087	—	3	1	no	maximal, Sylow
7	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×Dic₃⋊C₄	—	2	2	no	—
13	C₃×C₄⋊C₈	—	2	2	no	—
14	C₃⋊#95	—	2	2	no	—
15	#818	—	2	2	no	—
16	C₂×#107	—	3	1	no	—
17	C₂×C₄⋊C₈	D₃	1	1	yes	characteristic
18	#389	—	3	2	no	—
19	#340	—	3	1	no	—
20	#340	—	3	2	no	—
21	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
22	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
23	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
24	C₂²×Dic₃	—	2	2	no	—
25	C₂²×Dic₃	—	2	2	no	—
26	C₂×C₃×C₈	—	2	2	no	—
27	C₂×C₃×C₈	—	2	1	no	—
28	C₂×C₃×C₈	—	2	2	no	—
29	C₂×C₃×C₈	D₄	1	2	yes	—
30	C₃×C₄²	—	2	1	no	—
31	C₃×C₄²	—	2	1	no	—
32	C₄⋊Dic₃	—	2	2	no	—
33	C₄⋊Dic₃	—	2	2	no	—
34	C₄⋊Dic₃	—	2	2	no	—
35	C₄⋊Dic₃	Q₈	1	4	yes	—
36	Dic₃⋊C₄	—	4	4	no	—
37	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
38	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
39	C₂×C₄⋊C₄	—	3	2	no	—
40	C₂×C₄⋊C₄	—	3	2	no	—
41	C₂×C₄⋊C₄	—	6	2	no	—
42	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
43	#107	—	6	2	no	—
44	C₄⋊C₈	—	2	2	no	—
45	#95	—	6	2	no	—
46	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₃×C₄	—	2	1	no	—
48	C₂×C₃×C₄	—	2	1	no	—
49	C₂×C₃×C₄	—	2	1	no	—
50	C₂×C₃×C₄	—	2	2	no	—
51	C₂×C₃×C₄	Q₁₆	1	2	yes	—
52	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
54	C₂×C₃×C₄	#41	1	1	yes	characteristic
55	C₂×C₃×C₄	#41	1	2	yes	—
56	C₂×Dic₃	—	2	4	no	—
57	C₂×Dic₃	—	4	2	no	—
58	C₂×Dic₃	—	4	2	no	—
59	C₂×Dic₃	—	4	2	no	—
60	C₂×Dic₃	—	4	2	no	—
61	C₂×Dic₃	—	4	2	no	—
62	C₃×C₈	—	2	2	no	—
63	C₃×C₈	—	4	1	no	—
64	C₂²×C₄	—	6	2	no	—
65	C₂²×C₄	—	6	2	no	—
66	C₂²×C₄	D₁₂	1	1	yes	characteristic
67	C₂²×C₄	D₁₂	1	1	yes	characteristic
68	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	#67	1	2	yes	—
73	C₄⋊C₄	—	3	4	no	—
74	C₄⋊C₄	—	6	2	no	—
75	C₄⋊C₄	—	6	2	no	—
76	C₄⋊C₄	—	6	2	no	—
77	C₄⋊C₄	—	12	4	no	—
78	C₄²	—	2	1	no	—
79	C₄²	—	2	1	no	—
80	C₂²×C₃	#121	1	1	yes	characteristic
81	C₂²×C₃	C₂²⋊Q₈	1	2	yes	—
82	C₂²×C₃	C₂×Q₁₆	1	1	yes	characteristic
83	C₂²×C₃	#135	1	1	yes	characteristic
84	C₂²×C₃	AGL(1,ℤ/8)	1	2	yes	—
85	C₃×C₄	—	2	2	no	—
86	C₃×C₄	—	4	1	no	—
87	C₃×C₄	#123	1	1	yes	characteristic
88	C₃×C₄	#125	1	2	yes	—
89	C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
90	Dic₃	—	4	4	no	—
91	Dic₃	—	8	2	no	—
92	C₂³	C₂×D₁₂	1	1	yes	characteristic
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	6	4	no	—
98	C₂×C₄	—	12	2	no	—
99	C₂×C₄	—	12	2	no	—
100	C₂×C₄	—	12	2	no	—
101	C₂×C₄	—	12	2	no	—
102	C₂×C₄	—	12	2	no	—
103	C₂×C₄	Dic₁₂	1	2	yes	—
104	C₂×C₄	#235	1	1	yes	characteristic
105	C₂×C₄	#237	1	2	yes	—
106	C₂×C₄	D₃×Q₈	1	2	yes	—
107	C₂×C₄	C₂×#67	1	1	yes	characteristic
108	C₈	—	2	2	no	—
109	C₈	—	4	1	no	—
110	C₂×C₃	#398	1	1	yes	characteristic
111	C₂×C₃	#467	1	1	yes	characteristic
112	C₂×C₃	#469	1	1	yes	characteristic
113	C₂×C₃	#477	1	2	yes	—
114	C₂×C₃	#478	1	2	yes	—
115	C₂²	#897	1	2	yes	—
116	C₂²	#904	1	1	yes	characteristic
117	C₂²	C₂×Dic₁₂	1	1	yes	characteristic
118	C₂²	#908	1	2	yes	—
119	C₂²	#930	1	1	yes	characteristic
120	C₄	—	2	2	no	—
121	C₄	—	4	1	no	—
122	C₄	—	12	4	no	—
123	C₄	—	24	2	no	—
124	C₄	#889	1	2	yes	—
125	C₄	#935	1	1	yes	characteristic
126	C₄	#944	1	1	yes	characteristic
127	C₃	#2087	1	1	yes	characteristic, complemented, Sylow
128	C₂	#4847	1	2	yes	—
129	C₂	#4850	1	2	yes	—
130	C₂	#4981	1	1	yes	characteristic
131	C₂	#5116	1	1	yes	characteristic
132	C₂	#5135	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂×#818

C₂

yes

maximal, characteristic

#4928

C₂

yes

maximal

#4551

C₂

yes

maximal

#4551

C₂

yes

maximal, characteristic

#2087

—

maximal, Sylow

C₂²×C₃×C₈

C₂²

yes

characteristic

C₂²×C₃×C₈

C₂²

yes

characteristic

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×Dic₃⋊C₄

—

C₃×C₄⋊C₈

—

C₃⋊#95

—

#818

—

C₂×#107

—

C₂×C₄⋊C₈

D₃

yes

—

—

—

C₂²×C₃×C₄

D₄

yes

characteristic

C₂²×C₃×C₄

D₄

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

—

C₂²×Dic₃

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

D₄

yes

—

C₃×C₄²

—

C₃×C₄²

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

Q₈

yes

—

Dic₃⋊C₄

—

C₂²×C₈

C₂×D₃

yes

characteristic

C₂²×C₈

C₂×D₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃

yes

characteristic

#107

—

C₄⋊C₈

—

#95

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

Q₁₆

yes

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

#41

yes

characteristic

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃×C₈

—

C₃×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₂

yes

characteristic

C₂²×C₄

D₁₂

yes

characteristic

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#67

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₂²×C₃

#121

yes

characteristic

C₂²×C₃

C₂²⋊Q₈

yes

—

C₂²×C₃

C₂×Q₁₆

yes

C₂²×C₃

yes

C₂²×C₃

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#123

yes

characteristic

C₃×C₄

#125

yes

—

C₃×C₄

C₄⋊Q₈

yes

characteristic

Dic₃

—

Dic₃

—

C₂³

C₂×D₁₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

Dic₁₂

yes

—

104

C₂×C₄

#235

yes

characteristic

105

C₂×C₄

#237

yes

—

106

C₂×C₄

D₃×Q₈

yes

—

107

C₂×C₄

C₂×#67

yes

characteristic

108

C₈

—

109

C₈

—

110

C₂×C₃

#398

yes

characteristic

111

C₂×C₃

#467

yes

characteristic

112

C₂×C₃

#469

yes

characteristic

113

C₂×C₃

#477

yes

—

114

C₂×C₃

#478

yes

—

115

C₂²

#897

yes

—

116

C₂²

#904

yes

characteristic

117

C₂²

C₂×Dic₁₂

yes

characteristic

118

C₂²

#908

yes

—

119

C₂²

#930

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

#889

yes

—

125

C₄

#935

yes

characteristic

126

C₄

#944

yes

characteristic

127

C₃

#2087

yes

characteristic, complemented, Sylow

128

C₂

#4847

yes

—

129

C₂

#4850

yes

—

130

C₂

#4981

yes

characteristic

131

C₂

#5116

yes

characteristic

132

C₂

#5135

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₂
commutator, cosocle	C₂²×C₃×C₄	C₂³
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂×C₃×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₃
hypocenter	C₃	#2087
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₁₂

commutator, cosocle

C₂²×C₃×C₄

C₂³

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₂×C₃×C₄⋊C₈

C₂

socle, prime radical

C₂³×C₃

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₃

hypocenter

C₃

#2087

LCS

G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
127. C₃	#2087	semidirect

H ≅ G/N

type

127. C₃

#2087

semidirect

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

subgroup	index
C₂×C₃×C₄⋊C₈	2
C₂×#818	2
#4928	2
#4551	2
#2087	3

subgroup

index

C₂×C₃×C₄⋊C₈

Subgroups — #70463

Conjugacy classes of subgroups

Complements

Maximal subgroups