Group #20242136 — subgroups

Subgroups — #20242136

Order 6016.

180 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 117 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	2	yes	maximal
2	#15688876	C₂	1	2	yes	maximal, complemented
3	#15688874	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
6	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
7	C₄×C₄₇⋊C₈	—	2	2	no	—
8	C₂²×C₈×C₄₇	C₄	1	1	yes	characteristic
9	C₂²×C₈×C₄₇	C₄	1	1	yes	characteristic
10	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
12	C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
13	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₄₇	Q₈	1	1	yes	characteristic
18	C₂×C₈×C₄₇	—	2	2	no	—
19	C₂×C₈×C₄₇	C₈	1	4	yes	complemented
20	C₂×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₄₇	D₄	1	1	yes	characteristic
26	C₄²×C₄₇	Q₈	1	1	yes	characteristic
27	C₄²×C₄₇	C₂³	1	1	yes	characteristic
28	C₂×C₄₇⋊C₈	—	2	4	no	—
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	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₄₇	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₄₇	M₁₆	1	2	yes	—
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	—	2	1	no	—
41	C₈×C₄₇	—	2	1	no	—
42	C₈×C₄₇	C₂×C₈	1	2	yes	—
43	C₈×C₄₇	C₂×C₈	1	2	yes	—
44	C₄₇⋊C₈	—	4	4	no	—
45	C₂²×C₄₇	—	2	1	no	—
46	C₂²×C₄₇	—	2	1	no	—
47	C₂²×C₄₇	C₄⋊C₈	1	2	yes	—
48	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₄×C₄₇	—	2	1	no	—
50	C₄×C₄₇	—	2	1	no	—
51	C₄×C₄₇	C₄⋊C₈	1	2	yes	—
52	C₄×C₄₇	#108	1	2	yes	—
53	C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
54	C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
55	C₄×C₄₇	#135	1	2	yes	—
56	#1564	—	47	1	no	maximal, Sylow
57	C₂×C₄₇	—	2	2	no	—
58	C₂×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
59	C₂×C₄₇	#427	1	1	yes	characteristic
60	C₂×C₄₇	C₂×#108	1	1	yes	characteristic
61	#431	—	47	2	no	—
62	C₂×C₄×C₈	D₄₇	1	1	yes	characteristic
63	#335	—	47	2	no	—
64	#334	—	47	2	no	—
65	C₄₇	#1564	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₈	Dic₄₇	1	1	yes	characteristic
67	C₂²×C₈	Dic₄₇	1	1	yes	characteristic
68	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
69	C₄⋊C₈	—	47	4	no	—
70	C₂²⋊C₈	—	94	2	no	—
71	C₄×C₈	—	94	2	no	—
72	C₄×C₈	Dic₄₇	1	1	yes	characteristic
73	C₄×C₈	Dic₄₇	1	1	yes	characteristic
74	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
75	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
76	C₂²×C₄	Dic₉₄	1	1	yes	characteristic
77	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
78	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	94	4	no	—
81	C₂×C₈	C₄₇⋊C₈	1	4	yes	complemented
82	C₂×C₈	C₂×Dic₄₇	1	1	yes	characteristic
83	C₂×C₈	C₂×Dic₄₇	1	1	yes	characteristic
84	C₂×C₈	C₂×Dic₄₇	1	1	yes	characteristic
85	C₂×C₈	C₂×Dic₄₇	1	1	yes	characteristic
86	C₄²	Dic₉₄	1	1	yes	characteristic
87	C₄²	D₁₈₈	1	1	yes	characteristic
88	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
89	C₄²	C₂²×D₄₇	1	1	yes	characteristic
90	C₂³	C₄⋊Dic₄₇	1	1	yes	characteristic
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
94	C₂×C₄	#10625329	1	2	yes	—
95	C₂×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
96	C₂×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
97	C₂×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
98	C₂×C₄	C₂×Dic₉₄	1	1	yes	characteristic
99	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
100	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	2	1	no	—
103	C₈	—	188	4	no	—
104	C₈	C₂×C₄₇⋊C₈	1	2	yes	—
105	C₈	C₂×C₄₇⋊C₈	1	2	yes	—
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
109	C₂²	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
113	C₄	#13093960	1	2	yes	—
114	C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
115	C₄	C₂×#10625329	1	1	yes	characteristic
116	C₄	#13094046	1	2	yes	—
117	C₂	—	2	2	no	—
118	C₂	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
119	C₂	#15689161	1	1	yes	characteristic
120	C₂	C₂×#13093960	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689176

C₂

yes

maximal

#15688876

C₂

yes

maximal, complemented

#15688874

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₄₇

C₂

yes

maximal, characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

C₄

yes

characteristic

C₂²×C₈×C₄₇

C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

C₄

yes

characteristic

C₄×C₈×C₄₇

C₄

yes

characteristic

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

C₂²

yes

characteristic

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

C₈

yes

complemented

C₂×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

Q₈

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₄⋊C₄

yes

characteristic

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

—

C₈×C₄₇

C₂×C₈

yes

—

C₈×C₄₇

C₂×C₈

yes

—

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⋊C₈

yes

—

C₄×C₄₇

#108

yes

—

C₄×C₄₇

C₂²×C₈

yes

characteristic

C₄×C₄₇

C₂×M₁₆

yes

characteristic

C₄×C₄₇

#135

yes

—

#1564

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×C₄⋊C₈

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄×C₈

D₄₇

yes

characteristic

#335

—

#334

—

C₄₇

#1564

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₄₇

yes

characteristic

C₂²×C₈

Dic₄₇

yes

characteristic

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

Dic₄₇

yes

characteristic

C₄×C₈

Dic₄₇

yes

characteristic

C₄×C₈

C₂×D₄₇

yes

characteristic

C₄×C₈

C₂×D₄₇

yes

characteristic

C₂²×C₄

Dic₉₄

yes

characteristic

C₂²×C₄

D₁₈₈

yes

characteristic

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄₇⋊C₈

yes

complemented

C₂×C₈

C₂×Dic₄₇

yes

characteristic

C₂×C₈

C₂×Dic₄₇

yes

characteristic

C₂×C₈

C₂×Dic₄₇

yes

characteristic

C₂×C₈

C₂×Dic₄₇

yes

characteristic

C₄²

Dic₉₄

yes

characteristic

C₄²

D₁₈₈

yes

characteristic

C₄²

C₂×Dic₄₇

yes

characteristic

C₄²

C₂²×D₄₇

yes

characteristic

C₂³

C₄⋊Dic₄₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄₇⋊C₈

yes

—

C₂×C₄

#10625329

yes

—

C₂×C₄

C₄⋊Dic₄₇

yes

characteristic

C₂×C₄

C₄⋊Dic₄₇

yes

characteristic

C₂×C₄

C₄⋊Dic₄₇

yes

characteristic

C₂×C₄

C₂×Dic₉₄

yes

characteristic

C₂×C₄

C₂×D₁₈₈

yes

characteristic

100

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

188

—

104

C₈

C₂×C₄₇⋊C₈

yes

—

105

C₈

C₂×C₄₇⋊C₈

yes

—

106

C₂²

—

107

C₂²

—

108

C₂²

C₄⋊(C₄₇⋊C₈)

yes

—

109

C₂²

C₂×C₄⋊Dic₄₇

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

C₄⋊(C₄₇⋊C₈)

yes

—

113

C₄

#13093960

yes

—

114

C₄

C₂²×C₄₇⋊C₈

yes

characteristic

115

C₄

C₂×#10625329

yes

characteristic

116

C₄

#13094046

yes

—

117

C₂

—

118

C₂

C₂×C₄⋊(C₄₇⋊C₈)

yes

characteristic

119

C₂

#15689161

yes

characteristic

120

C₂

C₂×#13093960

yes

characteristic

	subgroup	G/H
center	C₄²	D₁₈₈
commutator	C₄×C₄₇	C₂²×C₈
Frattini	C₄²	C₂²×D₄₇
Fitting	C₂×C₄×C₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₄⋊C₄
hypercenter	C₂×C₄×C₈	D₄₇
hypocenter	C₄₇	#1564
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₄²

D₁₈₈

commutator

C₄×C₄₇

C₂²×C₈

Frattini

C₄²

C₂²×D₄₇

Fitting

C₂×C₄×C₈×C₄₇

C₂

socle

C₂²×C₄₇

C₂×C₄⋊C₄

hypercenter

C₂×C₄×C₈

D₄₇

hypocenter

C₄₇

#1564

LCS

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #15688876	C₂	semidirect
3. #15688874	C₂	semidirect
19. C₂×C₈×C₄₇	C₈	semidirect
65. C₄₇	#1564	semidirect
81. C₂×C₈	C₄₇⋊C₈	semidirect

H ≅ G/N

type

2. #15688876

C₂

semidirect

3. #15688874

C₂

semidirect

19. C₂×C₈×C₄₇

C₈

semidirect

65. C₄₇

#1564

semidirect

81. C₂×C₈

C₄₇⋊C₈

semidirect

Subgroups — #20242136

Conjugacy classes of subgroups

Complements