Group #20254782 — subgroups

Subgroups — #20254782

Order 6016.

342 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 125 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	#15689680	C₂	1	2	yes	maximal
2	#15689409	C₂	1	4	yes	maximal, complemented
3	#15689381	C₂	1	4	yes	maximal
4	#15689334	C₂	1	4	yes	maximal, complemented
5	C₄₇×#423	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₈₈	C₂²	1	4	yes	—
7	#13094024	C₂²	1	2	yes	—
8	#13094024	C₂²	1	2	yes	—
9	#13094016	—	2	2	no	—
10	#13094003	—	2	2	no	—
11	#13094003	C₂²	1	4	yes	—
12	C₄×Dic₉₄	—	2	2	no	—
13	C₄×Dic₉₄	C₂²	1	4	yes	—
14	#13093983	—	2	2	no	—
15	#13093956	C₂²	1	4	yes	—
16	#13093949	C₂²	1	8	yes	—
17	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
18	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
20	C₂²×C₄×C₄₇	D₄	1	2	yes	—
21	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
22	C₂×Dic₉₄	—	2	2	no	—
23	C₂×Dic₉₄	C₂³	1	4	yes	—
24	M₁₆×C₄₇	—	4	2	no	—
25	C₂×C₈×C₄₇	C₂³	1	4	yes	—
26	C₄²×C₄₇	D₄	1	1	yes	characteristic
27	C₄²×C₄₇	D₄	1	1	yes	characteristic
28	C₄²×C₄₇	C₂³	1	2	yes	—
29	C₂²⋊Dic₄₇	—	2	4	no	—
30	C₄⋊Dic₄₇	—	2	2	no	—
31	C₄⋊Dic₄₇	—	2	2	no	—
32	C₄⋊Dic₄₇	—	2	2	no	—
33	C₄⋊Dic₄₇	C₂³	1	4	yes	—
34	Dic₄₇⋊C₄	—	2	4	no	—
35	Dic₄₇⋊C₄	—	2	4	no	—
36	C₄×Dic₄₇	—	2	4	no	—
37	Dic₁₈₈	—	4	4	no	—
38	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₄₇	—	2	4	no	—
40	C₂×C₄×C₄₇	—	2	1	no	—
41	C₂×C₄×C₄₇	—	2	1	no	—
42	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₄₇	—	2	4	no	—
47	C₂×Dic₄₇	—	2	4	no	—
48	Dic₉₄	—	2	4	no	—
49	Dic₉₄	—	2	4	no	—
50	Dic₉₄	—	4	4	no	—
51	C₈×C₄₇	—	2	4	no	—
52	C₂²×C₄₇	—	2	1	no	—
53	C₂²×C₄₇	—	2	2	no	—
54	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
55	C₄×C₄₇	—	2	1	no	—
56	C₄×C₄₇	—	2	2	no	—
57	C₄×C₄₇	#137	1	4	yes	—
58	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
59	C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
60	Dic₄₇	—	4	4	no	—
61	Dic₄₇	—	4	4	no	—
62	#3255	—	47	1	no	maximal, Sylow
63	C₂×C₄₇	—	4	1	no	—
64	C₂×C₄₇	#537	1	1	yes	characteristic
65	C₂×C₄₇	C₂×#137	1	2	yes	—
66	#533	—	47	2	no	—
67	#475	—	47	4	no	—
68	#470	—	47	4	no	—
69	#462	—	47	4	no	—
70	#423	D₄₇	1	1	yes	characteristic
71	C₄₇	#3255	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₁₆	—	47	4	no	—
73	C₂×M₁₆	C₂×D₄₇	1	2	yes	—
74	C₄⋊Q₈	—	47	2	no	—
75	C₄⋊Q₈	—	47	2	no	—
76	#125	—	94	2	no	—
77	C₂²⋊Q₈	—	47	4	no	—
78	C₂²⋊Q₈	—	94	2	no	—
79	C₄×Q₈	—	47	4	no	—
80	C₄×Q₈	—	94	2	no	—
81	#117	—	94	2	no	—
82	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
83	#106	—	47	4	no	—
84	C₄⋊C₈	C₂×D₄₇	1	4	yes	—
85	#103	—	47	8	no	—
86	C₂×Q₈	—	47	4	no	—
87	C₂×Q₈	—	94	2	no	—
88	C₂²×C₄	D₁₈₈	1	2	yes	—
89	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
90	Q₁₆	—	188	4	no	—
91	M₁₆	—	4	2	no	—
92	C₂×C₈	C₂²×D₄₇	1	4	yes	—
93	C₄⋊C₄	—	47	4	no	—
94	C₄⋊C₄	—	94	4	no	—
95	C₄⋊C₄	—	94	4	no	—
96	C₄⋊C₄	—	94	2	no	—
97	C₄⋊C₄	—	94	2	no	—
98	C₄⋊C₄	—	94	2	no	—
99	C₂²⋊C₄	—	94	4	no	—
100	C₄²	—	94	4	no	—
101	C₄²	D₁₈₈	1	1	yes	characteristic
102	C₄²	D₁₈₈	1	1	yes	characteristic
103	C₄²	C₂²×D₄₇	1	2	yes	—
104	C₂³	C₂×D₁₈₈	1	1	yes	characteristic
105	Q₈	—	94	4	no	—
106	Q₈	—	94	4	no	—
107	Q₈	—	188	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	94	4	no	—
112	C₂×C₄	—	94	4	no	—
113	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
114	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
115	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
116	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
117	C₈	—	2	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	1	no	—
123	C₄	—	188	4	no	—
124	C₄	—	188	4	no	—
125	C₄	#13094057	1	4	yes	—
126	C₄	#13094069	1	1	yes	characteristic
127	C₄	#13094072	1	1	yes	characteristic
128	C₂	—	4	1	no	—
129	C₂	#15689704	1	1	yes	characteristic
130	C₂	C₂×#13094057	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689680

C₂

yes

maximal

#15689409

C₂

yes

maximal, complemented

#15689381

C₂

yes

maximal

#15689334

C₂

yes

maximal, complemented

C₄₇×#423

C₂

yes

maximal, characteristic

C₂×Dic₁₈₈

C₂²

yes

—

#13094024

C₂²

yes

—

#13094024

C₂²

yes

—

#13094016

—

#13094003

—

#13094003

C₂²

yes

—

C₄×Dic₉₄

—

C₄×Dic₉₄

C₂²

yes

—

#13093983

—

#13093956

C₂²

yes

—

#13093949

C₂²

yes

—

C₂×M₁₆×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂³

yes

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

—

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

Dic₁₈₈

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#137

yes

—

C₄×C₄₇

H(5,𝔽₂)

yes

characteristic

C₄×C₄₇

2₋¹⁺⁴

yes

characteristic

Dic₄₇

—

Dic₄₇

—

#3255

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#537

yes

characteristic

C₂×C₄₇

C₂×#137

yes

—

—

—

—

—

D₄₇

yes

characteristic

C₄₇

#3255

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₄₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

#106

—

C₄⋊C₈

C₂×D₄₇

yes

—

#103

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

D₁₈₈

yes

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

Q₁₆

—

188

—

M₁₆

—

C₂×C₈

C₂²×D₄₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

100

C₄²

—

101

C₄²

D₁₈₈

yes

characteristic

102

C₄²

D₁₈₈

yes

characteristic

103

C₄²

C₂²×D₄₇

yes

—

104

C₂³

C₂×D₁₈₈

yes

characteristic

105

Q₈

—

106

Q₈

—

107

Q₈

—

188

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×D₁₈₈

yes

characteristic

114

C₂×C₄

C₂×D₁₈₈

yes

—

115

C₂×C₄

C₂×D₁₈₈

yes

—

116

C₂×C₄

C₂³×D₄₇

yes

characteristic

117

C₈

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂²×D₁₈₈

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

188

—

124

C₄

—

188

—

125

C₄

#13094057

yes

—

126

C₄

#13094069

yes

characteristic

127

C₄

#13094072

yes

characteristic

128

C₂

—

129

C₂

#15689704

yes

characteristic

130

C₂

C₂×#13094057

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#423	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#423	D₄₇
hypocenter	C₄₇	#3255
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂²×D₁₈₈

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#423

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. #15689409	C₂	semidirect
4. #15689334	C₂	semidirect
71. C₄₇	#3255	semidirect

H ≅ G/N

type

2. #15689409

C₂

semidirect

4. #15689334

C₂

semidirect

71. C₄₇

#3255

semidirect

Subgroups — #20254782

Conjugacy classes of subgroups

Complements