Group #20242140 — subgroups

Subgroups — #20242140

Order 6016.

180 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 113 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	#15689176	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#15688876	C₂	1	2	yes	maximal, complemented
4	#15688874	C₂	1	2	yes	maximal, complemented
5	C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₈	—	2	1	no	—
7	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
8	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
9	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
12	C₄×C₄₇⋊C₈	—	2	1	no	—
13	C₂×M₁₆×C₄₇	C₄	1	1	yes	characteristic
14	C₂×M₁₆×C₄₇	C₄	1	1	yes	characteristic
15	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	#97×C₄₇	C₄	1	2	yes	—
17	C₄×C₈×C₄₇	C₂²	1	2	yes	—
18	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	Q₈	1	1	yes	characteristic
21	M₁₆×C₄₇	—	2	2	no	—
22	M₁₆×C₄₇	C₈	1	4	yes	complemented
23	C₂×C₈×C₄₇	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₄₇	C₂×C₄	1	2	yes	—
25	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₄₇	D₄	1	1	yes	characteristic
27	C₄²×C₄₇	Q₈	1	1	yes	characteristic
28	C₄²×C₄₇	C₂³	1	1	yes	characteristic
29	C₂×C₄₇⋊C₈	—	2	1	no	—
30	C₂×C₄₇⋊C₈	—	2	1	no	—
31	C₂×C₄₇⋊C₈	—	2	2	no	—
32	C₂×C₄₇⋊C₈	—	4	2	no	—
33	C₂³×C₄₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₄₇	—	2	1	no	—
35	C₂×C₄×C₄₇	—	2	1	no	—
36	C₂×C₄×C₄₇	—	4	1	no	—
37	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₄₇	C₂×C₈	1	2	yes	—
41	C₂×C₄×C₄₇	M₁₆	1	2	yes	—
42	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₄₇	—	2	2	no	—
46	C₈×C₄₇	C₂×C₈	1	4	yes	complemented
47	C₄₇⋊C₈	—	4	1	no	—
48	C₄₇⋊C₈	—	4	2	no	—
49	C₄₇⋊C₈	—	4	1	no	—
50	C₂²×C₄₇	—	2	1	no	—
51	C₂²×C₄₇	—	2	1	no	—
52	C₂²×C₄₇	C₄⋊C₈	1	2	yes	—
53	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₄₇	—	2	1	no	—
55	C₄×C₄₇	—	2	1	no	—
56	C₄×C₄₇	—	2	1	no	—
57	C₄×C₄₇	C₄⋊C₈	1	2	yes	—
58	C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
59	C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
60	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₄×C₄₇	#137	1	1	yes	characteristic
62	#1565	—	47	1	no	maximal, Sylow
63	C₂×C₄₇	—	2	2	no	—
64	C₂×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₄₇	#428	1	1	yes	characteristic
66	C₂×C₄₇	#430	1	1	yes	characteristic
67	#431	—	47	1	no	—
68	C₂×C₄⋊C₈	—	47	1	no	—
69	C₄×M₁₆	D₄₇	1	1	yes	characteristic
70	#335	—	47	2	no	—
71	#334	—	47	2	no	—
72	C₄₇	#1565	1	1	yes	characteristic, complemented, Sylow
73	C₂×M₁₆	Dic₄₇	1	1	yes	characteristic
74	C₂×M₁₆	Dic₄₇	1	1	yes	characteristic
75	C₂²×C₈	—	94	1	no	—
76	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
77	C₄⋊C₈	—	47	1	no	—
78	C₄⋊C₈	—	47	2	no	—
79	C₄⋊C₈	—	47	1	no	—
80	C₄⋊C₈	—	94	1	no	—
81	C₂²⋊C₈	—	94	1	no	—
82	#97	Dic₄₇	1	2	yes	—
83	C₄×C₈	—	94	1	no	—
84	C₄×C₈	C₂×D₄₇	1	2	yes	—
85	C₂²×C₄	Dic₉₄	1	1	yes	characteristic
86	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
87	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
88	M₁₆	—	2	2	no	—
89	M₁₆	C₄₇⋊C₈	1	4	yes	complemented
90	C₂×C₈	—	94	1	no	—
91	C₂×C₈	—	94	1	no	—
92	C₂×C₈	—	94	2	no	—
93	C₂×C₈	—	188	2	no	—
94	C₂×C₈	C₂×Dic₄₇	1	2	yes	—
95	C₂×C₈	C₂×Dic₄₇	1	2	yes	—
96	C₄²	Dic₉₄	1	1	yes	characteristic
97	C₄²	D₁₈₈	1	1	yes	characteristic
98	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
99	C₄²	C₂²×D₄₇	1	1	yes	characteristic
100	C₂³	C₄⋊Dic₄₇	1	1	yes	characteristic
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
105	C₂×C₄	#10625329	1	2	yes	—
106	C₂×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
107	C₂×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
108	C₂×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
109	C₂×C₄	C₂×Dic₉₄	1	1	yes	characteristic
110	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
111	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	188	1	no	—
114	C₈	—	188	2	no	—
115	C₈	—	188	1	no	—
116	C₈	C₂×C₄₇⋊C₈	1	4	yes	complemented
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
120	C₂²	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
125	C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
126	C₄	C₂×#10625329	1	1	yes	characteristic
127	C₄	#13094052	1	1	yes	characteristic
128	C₄	#13094057	1	1	yes	characteristic
129	C₂	—	2	2	no	—
130	C₂	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
131	C₂	#15689165	1	1	yes	characteristic
132	C₂	#15689172	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689176

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic

#15688876

C₂

yes

maximal, complemented

#15688874

C₂

yes

maximal, complemented

C₄×M₁₆×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₂×M₁₆×C₄₇

C₄

yes

characteristic

C₂×M₁₆×C₄₇

C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#97×C₄₇

C₄

yes

—

C₄×C₈×C₄₇

C₂²

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

Q₈

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

C₈

yes

complemented

C₂×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₈×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

Q₈

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×C₈

yes

—

C₂×C₄×C₄₇

M₁₆

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

C₂×C₈

yes

complemented

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₄⋊C₈

yes

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄⋊C₈

yes

—

C₄×C₄₇

C₂²×C₈

yes

characteristic

C₄×C₄₇

C₂×M₁₆

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₂×C₄₇

—

C₂×C₄₇

C₂×C₄⋊C₈

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄⋊C₈

—

C₄×M₁₆

D₄₇

yes

characteristic

#335

—

#334

—

C₄₇

#1565

yes

characteristic, complemented, Sylow

C₂×M₁₆

Dic₄₇

yes

characteristic

C₂×M₁₆

Dic₄₇

yes

characteristic

C₂²×C₈

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

Dic₄₇

yes

—

C₄×C₈

—

C₄×C₈

C₂×D₄₇

yes

—

C₂²×C₄

Dic₉₄

yes

characteristic

C₂²×C₄

D₁₈₈

yes

characteristic

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

M₁₆

—

M₁₆

C₄₇⋊C₈

yes

complemented

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

C₂×Dic₄₇

yes

—

C₂×C₈

C₂×Dic₄₇

yes

—

C₄²

Dic₉₄

yes

characteristic

C₄²

D₁₈₈

yes

characteristic

C₄²

C₂×Dic₄₇

yes

characteristic

C₄²

C₂²×D₄₇

yes

characteristic

100

C₂³

C₄⋊Dic₄₇

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×C₄₇⋊C₈

yes

—

105

C₂×C₄

#10625329

yes

—

106

C₂×C₄

C₄⋊Dic₄₇

yes

characteristic

107

C₂×C₄

C₄⋊Dic₄₇

yes

characteristic

108

C₂×C₄

C₄⋊Dic₄₇

yes

characteristic

109

C₂×C₄

C₂×Dic₉₄

yes

characteristic

110

C₂×C₄

C₂×D₁₈₈

yes

characteristic

111

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

112

C₈

—

113

C₈

—

188

—

114

C₈

—

188

—

115

C₈

—

188

—

116

C₈

C₂×C₄₇⋊C₈

yes

complemented

117

C₂²

—

118

C₂²

—

119

C₂²

C₄⋊(C₄₇⋊C₈)

yes

—

120

C₂²

C₂×C₄⋊Dic₄₇

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

C₄⋊(C₄₇⋊C₈)

yes

—

125

C₄

C₂²×C₄₇⋊C₈

yes

characteristic

126

C₄

C₂×#10625329

yes

characteristic

127

C₄

#13094052

yes

characteristic

128

C₄

#13094057

yes

characteristic

129

C₂

—

130

C₂

C₂×C₄⋊(C₄₇⋊C₈)

yes

characteristic

131

C₂

#15689165

yes

characteristic

132

C₂

#15689172

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₈₈
commutator	C₄×C₄₇	C₂²×C₈
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₄⋊C₄
hypercenter	C₄×M₁₆	D₄₇
hypocenter	C₄₇	#1565
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₁₈₈

commutator

C₄×C₄₇

C₂²×C₈

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄×M₁₆×C₄₇

C₂

socle

C₂²×C₄₇

C₂×C₄⋊C₄

hypercenter

C₄×M₁₆

D₄₇

hypocenter

C₄₇

#1565

LCS

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
3. #15688876	C₂	semidirect
4. #15688874	C₂	semidirect
22. M₁₆×C₄₇	C₈	semidirect
46. C₈×C₄₇	C₂×C₈	semidirect
72. C₄₇	#1565	semidirect
89. M₁₆	C₄₇⋊C₈	semidirect
116. C₈	C₂×C₄₇⋊C₈	semidirect

H ≅ G/N

type

3. #15688876

C₂

semidirect

4. #15688874

C₂

semidirect

22. M₁₆×C₄₇

C₈

semidirect

46. C₈×C₄₇

C₂×C₈

semidirect

72. C₄₇

#1565

semidirect

89. M₁₆

C₄₇⋊C₈

semidirect

116. C₈

C₂×C₄₇⋊C₈

semidirect

Subgroups — #20242140

Conjugacy classes of subgroups

Complements