Group #20242176 — subgroups

Subgroups — #20242176

Order 6016.

158 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 77 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	#15689147	C₂	1	1	yes	maximal, characteristic
3	#15688875	C₂	1	2	yes	maximal, complemented
4	#15688873	C₂	1	2	yes	maximal, complemented
5	C₄₇×#431	C₂	1	1	yes	maximal, characteristic
6	C₂×#10625329	—	2	1	no	—
7	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
8	C₄⋊(C₄₇⋊C₈)	C₄	1	2	yes	—
9	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	C₂²⋊(C₄₇⋊C₈)	C₄	1	2	yes	—
11	C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²⋊C₈×C₄₇	—	2	1	no	—
16	C₄×C₈×C₄₇	—	2	1	no	—
17	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₇	Q₈	1	1	yes	characteristic
20	C₂×C₈×C₄₇	—	2	1	no	—
21	C₂×C₈×C₄₇	—	2	1	no	—
22	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₄₇	D₄	1	1	yes	characteristic
24	C₄²×C₄₇	Q₈	1	1	yes	characteristic
25	C₄²×C₄₇	C₂³	1	1	yes	characteristic
26	#10625329	—	4	2	no	—
27	C₂×C₄₇⋊C₈	—	2	2	no	—
28	C₂×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
29	C₂³×C₄₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₄₇	—	2	2	no	—
31	C₂×C₄×C₄₇	—	2	2	no	—
32	C₂×C₄×C₄₇	—	2	1	no	—
33	C₂×C₄×C₄₇	—	2	1	no	—
34	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
40	C₈×C₄₇	—	4	1	no	—
41	C₈×C₄₇	—	4	1	no	—
42	C₄₇⋊C₈	—	2	4	no	—
43	C₄₇⋊C₈	—	4	2	no	—
44	C₂²×C₄₇	—	2	1	no	—
45	C₂²×C₄₇	—	2	1	no	—
46	C₂²×C₄₇	—	2	1	no	—
47	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₄×C₄₇	—	2	1	no	—
49	C₄×C₄₇	—	2	1	no	—
50	C₄×C₄₇	—	2	1	no	—
51	C₄×C₄₇	#108	1	2	yes	—
52	C₄×C₄₇	#131	1	2	yes	—
53	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
54	C₄×C₄₇	#137	1	1	yes	characteristic
55	#1574	—	47	1	no	maximal, Sylow
56	C₂×C₄₇	—	4	1	no	—
57	C₂×C₄₇	#424	1	1	yes	characteristic
58	C₂×C₄₇	#428	1	1	yes	characteristic
59	C₂×C₄₇	C₂×#108	1	1	yes	characteristic
60	#431	—	47	1	no	—
61	#431	D₄₇	1	1	yes	characteristic
62	#423	—	47	1	no	—
63	#335	—	47	2	no	—
64	#334	—	47	2	no	—
65	C₄₇	#1574	1	1	yes	characteristic, complemented, Sylow
66	C₂×M₁₆	—	94	1	no	—
67	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
68	C₄⋊C₈	—	47	2	no	—
69	C₄⋊C₈	—	47	2	no	—
70	C₄⋊C₈	—	94	1	no	—
71	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
72	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
73	C₂²⋊C₈	—	2	1	no	—
74	C₂²⋊C₈	—	47	2	no	—
75	C₄×C₈	—	2	1	no	—
76	C₄×C₈	—	47	2	no	—
77	C₂²×C₄	Dic₉₄	1	1	yes	characteristic
78	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
80	M₁₆	—	188	2	no	—
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	47	4	no	—
84	C₂×C₈	—	94	2	no	—
85	C₄²	Dic₉₄	1	1	yes	characteristic
86	C₄²	C₄×D₄₇	1	1	yes	characteristic
87	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
88	C₄²	C₂²×D₄₇	1	1	yes	characteristic
89	C₂³	Dic₄₇⋊C₄	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
95	C₂×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
96	C₂×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
97	C₂×C₄	C₂×Dic₉₄	1	1	yes	characteristic
98	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
99	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
100	C₈	—	4	1	no	—
101	C₈	—	4	1	no	—
102	C₈	—	94	4	no	—
103	C₈	—	188	2	no	—
104	C₂²	—	2	1	no	—
105	C₂²	—	2	1	no	—
106	C₂²	—	2	1	no	—
107	C₂²	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	#13093959	1	2	yes	—
112	C₄	#13094030	1	2	yes	—
113	C₄	#13094049	1	1	yes	characteristic
114	C₄	#13094054	1	1	yes	characteristic
115	C₂	—	4	1	no	—
116	C₂	#15689149	1	1	yes	characteristic
117	C₂	#15689166	1	1	yes	characteristic
118	C₂	C₂×#13093959	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689176

C₂

yes

maximal, characteristic

#15689147

C₂

yes

maximal, characteristic

#15688875

C₂

yes

maximal, complemented

#15688873

C₂

yes

maximal, complemented

C₄₇×#431

C₂

yes

maximal, characteristic

C₂×#10625329

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

C₄

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂²⋊(C₄₇⋊C₈)

C₄

yes

—

C₄×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

Q₈

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

Q₈

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

C₂³×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#108

yes

—

C₄×C₄₇

#131

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

D₄₇

yes

—

—

—

C₄₇

#1574

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

characteristic

C₄⋊C₈

C₂×D₄₇

yes

characteristic

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

Dic₉₄

yes

characteristic

C₂²×C₄

C₄×D₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

M₁₆

—

188

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

Dic₉₄

yes

characteristic

C₄²

C₄×D₄₇

yes

characteristic

C₄²

C₂²⋊D₄₇

yes

characteristic

C₄²

C₂²×D₄₇

yes

characteristic

C₂³

Dic₄₇⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

Dic₄₇⋊C₄

yes

characteristic

C₂×C₄

Dic₄₇⋊C₄

yes

characteristic

C₂×C₄

Dic₄₇⋊C₄

yes

characteristic

C₂×C₄

C₂×Dic₉₄

yes

characteristic

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

100

C₈

—

101

C₈

—

102

C₈

—

103

C₈

—

188

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×Dic₄₇⋊C₄

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#13093959

yes

—

112

C₄

#13094030

yes

—

113

C₄

#13094049

yes

characteristic

114

C₄

#13094054

yes

characteristic

115

C₂

—

116

C₂

#15689149

yes

characteristic

117

C₂

#15689166

yes

characteristic

118

C₂

C₂×#13093959

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#431	C₂
socle	C₂²×C₄₇	C₂×C₄⋊C₄
hypercenter	#431	D₄₇
hypocenter	C₄₇	#1574
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#431

C₂

socle

C₂²×C₄₇

C₂×C₄⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #15688875	C₂	semidirect
4. #15688873	C₂	semidirect
65. C₄₇	#1574	semidirect

H ≅ G/N

type

3. #15688875

C₂

semidirect

4. #15688873

C₂

semidirect

65. C₄₇

#1574

semidirect

Subgroups — #20242176

Conjugacy classes of subgroups

Complements