Group #20242818 — subgroups

Subgroups — #20242818

Order 6016.

150 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 57 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	#15689474	C₂	1	1	yes	maximal, characteristic
2	#15689408	C₂	1	1	yes	maximal, characteristic
3	#15689333	C₂	1	1	yes	maximal, characteristic
4	#15688874	C₂	1	1	yes	maximal, characteristic
5	#15688870	C₂	1	1	yes	maximal, characteristic
6	#15688868	C₂	1	1	yes	maximal, characteristic
7	C₈×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#10625337	—	2	1	no	—
9	#13094024	C₂²	1	1	yes	characteristic
10	#13094024	C₂²	1	1	yes	characteristic
11	#13093955	—	2	1	no	—
12	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
14	#13093949	—	2	1	no	—
15	#13093949	—	2	1	no	—
16	#13093948	—	2	1	no	—
17	#13093947	—	2	1	no	—
18	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₄₇	—	2	1	no	—
20	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₄₇	—	2	1	no	—
22	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
24	C₂×Dic₉₄	—	2	1	no	—
25	C₂×Dic₉₄	—	2	1	no	—
26	C₂×C₈×C₄₇	—	2	1	no	—
27	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
28	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
29	C₄⋊C₄×C₄₇	—	2	1	no	—
30	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
31	C₄²×C₄₇	—	2	1	no	—
32	C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	#10625337	—	4	2	no	—
34	C₄⋊Dic₄₇	—	2	1	no	—
35	C₄⋊Dic₄₇	—	2	1	no	—
36	C₄⋊Dic₄₇	—	2	1	no	—
37	C₄⋊Dic₄₇	—	2	1	no	—
38	C₂×C₄₇⋊C₈	—	2	1	no	—
39	C₂×C₄₇⋊C₈	—	2	1	no	—
40	Q₈×C₄₇	—	2	1	no	—
41	Q₈×C₄₇	SD₁₆	1	2	yes	—
42	C₂×C₄×C₄₇	—	2	1	no	—
43	C₂×C₄×C₄₇	—	2	1	no	—
44	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
47	C₂×Dic₄₇	—	4	1	no	—
48	C₂×Dic₄₇	—	4	1	no	—
49	Dic₉₄	—	4	1	no	—
50	Dic₉₄	—	4	1	no	—
51	Dic₉₄	—	4	1	no	—
52	Dic₉₄	—	4	1	no	—
53	C₈×C₄₇	—	2	1	no	—
54	C₈×C₄₇	—	2	1	no	—
55	C₈×C₄₇	—	4	1	no	—
56	C₄₇⋊C₈	—	4	1	no	—
57	C₄₇⋊C₈	—	4	1	no	—
58	C₂²×C₄₇	#121	1	1	yes	characteristic
59	C₄×C₄₇	—	2	2	no	—
60	C₄×C₄₇	—	2	1	no	—
61	C₄×C₄₇	—	4	1	no	—
62	C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₄×C₄₇	#135	1	1	yes	characteristic
64	C₄×C₄₇	#135	1	1	yes	characteristic
65	C₄×C₄₇	#137	1	1	yes	characteristic
66	Dic₄₇	—	8	1	no	—
67	Dic₄₇	—	8	1	no	—
68	#1680	—	47	1	no	maximal, Sylow
69	C₂×C₄₇	#464	1	1	yes	characteristic
70	C₂×C₄₇	#465	1	1	yes	characteristic
71	C₂×C₄₇	#473	1	1	yes	characteristic
72	#487	—	47	1	no	—
73	#475	—	47	1	no	—
74	#462	—	47	1	no	—
75	C₈×Q₈	D₄₇	1	1	yes	characteristic
76	#334	—	47	1	no	—
77	#332	—	47	1	no	—
78	#332	—	47	1	no	—
79	C₄₇	#1680	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₁₆	—	94	1	no	—
81	C₄⋊Q₈	—	47	1	no	—
82	C₄⋊Q₈	—	47	1	no	—
83	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
84	#106	—	94	1	no	—
85	C₄⋊C₈	—	2	1	no	—
86	C₄⋊C₈	—	47	1	no	—
87	C₄⋊C₈	—	47	1	no	—
88	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
89	#103	—	94	1	no	—
90	#103	—	94	1	no	—
91	#103	—	94	1	no	—
92	#103	—	94	1	no	—
93	C₄×C₈	—	2	1	no	—
94	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
95	C₂×Q₈	—	94	1	no	—
96	C₂×Q₈	—	94	1	no	—
97	C₂×Q₈	D₁₈₈	1	1	yes	characteristic
98	Q₁₆	—	188	2	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	94	1	no	—
101	C₂×C₈	—	94	1	no	—
102	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
103	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	94	1	no	—
106	C₄⋊C₄	—	94	1	no	—
107	C₄⋊C₄	—	94	1	no	—
108	C₄⋊C₄	—	94	1	no	—
109	C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
110	C₄²	—	2	1	no	—
111	C₄²	C₂²×D₄₇	1	1	yes	characteristic
112	Q₈	—	2	1	no	—
113	Q₈	—	188	1	no	—
114	Q₈	—	188	1	no	—
115	Q₈	—	188	1	no	—
116	Q₈	—	188	1	no	—
117	Q₈	#10625325	1	2	yes	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	188	1	no	—
121	C₂×C₄	—	188	1	no	—
122	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
123	C₂×C₄	#10625350	1	1	yes	characteristic
124	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	2	1	no	—
127	C₈	—	4	1	no	—
128	C₈	—	188	1	no	—
129	C₈	—	188	1	no	—
130	C₂²	#13093998	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	1	no	—
133	C₄	—	4	1	no	—
134	C₄	—	376	1	no	—
135	C₄	—	376	1	no	—
136	C₄	C₂×#10625325	1	1	yes	characteristic
137	C₄	#13094044	1	1	yes	characteristic
138	C₄	#13094046	1	1	yes	characteristic
139	C₄	#13094055	1	1	yes	characteristic
140	C₂	#15689345	1	1	yes	characteristic
141	C₂	#15689352	1	1	yes	characteristic
142	C₂	#15689398	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689474

C₂

yes

maximal, characteristic

#15689408

C₂

yes

maximal, characteristic

#15689333

C₂

yes

maximal, characteristic

#15688874

C₂

yes

maximal, characteristic

#15688870

C₂

yes

maximal, characteristic

#15688868

C₂

yes

maximal, characteristic

C₈×Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂×#10625337

—

#13094024

C₂²

yes

characteristic

#13094024

C₂²

yes

characteristic

#13093955

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

—

—

—

C₄×Q₈×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

D₄

yes

characteristic

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₄

yes

characteristic

C₂×C₈×C₄₇

D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625337

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

Q₈×C₄₇

—

Q₈×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

#121

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×SD₁₆

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

—

C₈×Q₈

D₄₇

yes

—

—

—

C₄₇

#1680

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

C₂×D₄₇

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

—

—

—

—

C₄×C₈

—

C₄×C₈

C₂×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₁₈₈

yes

characteristic

Q₁₆

—

188

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

103

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

D₁₈₈

yes

characteristic

110

C₄²

—

111

C₄²

C₂²×D₄₇

yes

characteristic

112

Q₈

—

113

Q₈

—

188

—

114

Q₈

—

188

—

115

Q₈

—

188

—

116

Q₈

—

188

—

117

Q₈

#10625325

yes

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

188

—

121

C₂×C₄

—

188

—

122

C₂×C₄

C₂×D₁₈₈

yes

characteristic

123

C₂×C₄

#10625350

yes

characteristic

124

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

188

—

129

C₈

—

188

—

130

C₂²

#13093998

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

376

—

135

C₄

—

376

—

136

C₄

C₂×#10625325

yes

characteristic

137

C₄

#13094044

yes

characteristic

138

C₄

#13094046

yes

characteristic

139

C₄

#13094055

yes

characteristic

140

C₂

#15689345

yes

characteristic

141

C₂

#15689352

yes

characteristic

142

C₂

#15689398

yes

characteristic

	subgroup	G/H
center	C₂²	#13093998
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₈×Q₈×C₄₇	C₂
socle	C₂²×C₄₇	#121
hypercenter	C₈×Q₈	D₄₇
hypocenter	C₄₇	#1680
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#13093998

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₈×Q₈×C₄₇

C₂

socle

C₂²×C₄₇

#121

hypercenter

C₈×Q₈

D₄₇

hypocenter

C₄₇

#1680

LCS

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
79. C₄₇	#1680	semidirect

H ≅ G/N

type

79. C₄₇

#1680

semidirect

Subgroups — #20242818

Conjugacy classes of subgroups

Complements