Group #20242072 — subgroups

Subgroups — #20242072

Order 6016.

240 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂×#13094024	C₂	1	1	yes	maximal, characteristic
2	#15689147	C₂	1	1	yes	maximal, characteristic
3	#15688870	C₂	1	4	yes	maximal, complemented
4	C₄₇×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₉₄	—	2	1	no	—
6	C₂×#10625329	—	2	1	no	—
7	#13094024	—	2	1	no	—
8	#13094024	—	2	1	no	—
9	#13094024	C₄	1	2	yes	—
10	#13094024	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₄₇	—	2	1	no	—
12	C₂×C₄⋊Dic₄₇	C₄	1	2	yes	—
13	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
14	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
17	C₂²⋊C₈×C₄₇	—	2	1	no	—
18	C₄×C₈×C₄₇	—	2	1	no	—
19	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
22	C₂²×Dic₄₇	—	2	2	no	—
23	C₂×Dic₉₄	—	2	2	no	—
24	C₂×Dic₉₄	—	2	2	no	—
25	C₂×Dic₉₄	—	4	4	no	—
26	C₂×C₈×C₄₇	—	2	2	no	—
27	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₄₇	D₄	1	1	yes	characteristic
29	C₄²×C₄₇	D₄	1	1	yes	characteristic
30	C₄²×C₄₇	C₂³	1	1	yes	characteristic
31	C₄⋊Dic₄₇	—	2	2	no	—
32	C₄⋊Dic₄₇	—	2	2	no	—
33	C₄⋊Dic₄₇	—	2	2	no	—
34	C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
35	#10625329	—	4	2	no	—
36	C₂×C₄₇⋊C₈	—	2	2	no	—
37	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₄₇	—	2	2	no	—
39	C₂×C₄×C₄₇	—	2	2	no	—
40	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₄₇	SD₁₆	1	2	yes	—
44	C₂×C₄×C₄₇	Q₁₆	1	2	yes	—
45	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₄₇	—	2	4	no	—
49	C₂×Dic₄₇	—	4	4	no	—
50	Dic₉₄	—	4	4	no	—
51	Dic₉₄	—	4	4	no	—
52	C₈×C₄₇	—	4	2	no	—
53	C₄₇⋊C₈	—	4	2	no	—
54	C₂²×C₄₇	—	2	1	no	—
55	C₂²×C₄₇	—	2	1	no	—
56	C₂²×C₄₇	#103	1	2	yes	—
57	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₄₇	—	2	1	no	—
59	C₄×C₄₇	—	2	1	no	—
60	C₄×C₄₇	#101	1	2	yes	—
61	C₄×C₄₇	#103	1	2	yes	—
62	C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
64	C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
65	Dic₄₇	—	4	4	no	—
66	#1549	—	47	1	no	maximal, Sylow
67	C₂×C₄₇	—	2	2	no	—
68	C₂×C₄₇	C₂×#101	1	1	yes	characteristic
69	C₂×C₄₇	C₂×#103	1	1	yes	characteristic
70	C₂×C₄₇	#418	1	1	yes	characteristic
71	C₂×C₄⋊Q₈	—	47	1	no	—
72	#431	D₄₇	1	1	yes	characteristic
73	#423	—	47	1	no	—
74	#332	—	47	4	no	—
75	C₄₇	#1549	1	1	yes	characteristic, complemented, Sylow
76	C₂²×Q₈	—	94	1	no	—
77	C₂×M₁₆	—	94	1	no	—
78	C₄⋊Q₈	—	47	2	no	—
79	C₄⋊Q₈	—	47	2	no	—
80	C₄⋊Q₈	—	94	1	no	—
81	C₄⋊Q₈	—	94	1	no	—
82	C₂×C₄⋊C₄	—	47	2	no	—
83	C₂×C₄⋊C₄	—	94	1	no	—
84	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
85	C₄⋊C₈	—	47	2	no	—
86	C₄⋊C₈	—	94	1	no	—
87	C₄⋊C₈	C₂×D₄₇	1	2	yes	—
88	C₂²⋊C₈	—	2	1	no	—
89	C₄×C₈	—	2	1	no	—
90	C₂×Q₈	—	94	2	no	—
91	C₂×Q₈	—	94	2	no	—
92	C₂×Q₈	—	188	4	no	—
93	C₂²×C₄	—	94	2	no	—
94	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
95	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
97	M₁₆	—	188	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	94	2	no	—
100	C₄⋊C₄	—	47	4	no	—
101	C₄⋊C₄	—	94	2	no	—
102	C₄⋊C₄	—	94	2	no	—
103	C₄⋊C₄	—	94	2	no	—
104	C₄²	C₄×D₄₇	1	1	yes	characteristic
105	C₄²	D₁₈₈	1	1	yes	characteristic
106	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
107	C₄²	C₂²×D₄₇	1	1	yes	characteristic
108	C₂³	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
109	Q₈	—	188	4	no	—
110	Q₈	—	188	4	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	94	4	no	—
114	C₂×C₄	—	188	4	no	—
115	C₂×C₄	#10625325	1	2	yes	—
116	C₂×C₄	Dic₁₈₈	1	2	yes	—
117	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
118	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
119	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
120	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
121	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
122	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
123	C₈	—	4	2	no	—
124	C₈	—	188	2	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	#13093949	1	2	yes	—
128	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	188	4	no	—
132	C₄	#13093942	1	2	yes	—
133	C₄	#13093949	1	2	yes	—
134	C₄	C₂×#10625325	1	1	yes	characteristic
135	C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
136	C₄	#13094049	1	2	yes	—
137	C₂	—	2	2	no	—
138	C₂	C₂×#13093942	1	1	yes	characteristic
139	C₂	C₂×#13093949	1	1	yes	characteristic
140	C₂	#15689127	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094024

C₂

yes

maximal, characteristic

#15689147

C₂

yes

maximal, characteristic

#15688870

C₂

yes

maximal, complemented

C₄₇×#431

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

—

—

—

—

C₄

yes

—

#13094024

C₂²

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₄

yes

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

Q₁₆

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#103

yes

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#101

yes

—

C₄×C₄₇

#103

yes

—

C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₄×C₄₇

C₂×Q₁₆

yes

characteristic

C₄×C₄₇

AGL(1,ℤ/8)

yes

—

Dic₄₇

—

#1549

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄⋊Q₈

—

#431

D₄₇

yes

characteristic

#423

—

#332

—

C₄₇

#1549

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

188

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄₇

yes

characteristic

C₂²×C₄

D₁₈₈

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

M₁₆

—

188

—

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄²

C₄×D₄₇

yes

characteristic

105

C₄²

D₁₈₈

yes

characteristic

106

C₄²

C₂²⋊D₄₇

yes

characteristic

107

C₄²

C₂²×D₄₇

yes

characteristic

108

C₂³

(C₂×D₄₇)⋊C₄

yes

characteristic

109

Q₈

—

188

—

110

Q₈

—

188

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

188

—

115

C₂×C₄

#10625325

yes

—

116

C₂×C₄

Dic₁₈₈

yes

—

117

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

118

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

119

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

120

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

121

C₂×C₄

C₂×D₁₈₈

yes

characteristic

122

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

123

C₈

—

124

C₈

—

188

—

125

C₂²

—

126

C₂²

—

127

C₂²

#13093949

yes

—

128

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

188

—

132

C₄

#13093942

yes

—

133

C₄

#13093949

yes

—

134

C₄

C₂×#10625325

yes

characteristic

135

C₄

C₂×Dic₁₈₈

yes

characteristic

136

C₄

#13094049

yes

—

137

C₂

—

138

C₂

C₂×#13093942

yes

characteristic

139

C₂

C₂×#13093949

yes

characteristic

140

C₂

#15689127

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#431	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#431	D₄₇
hypocenter	C₄₇	#1549
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂²

C₂×(C₂×D₄₇)⋊C₄

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#431

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #15688870	C₂	semidirect
75. C₄₇	#1549	semidirect

H ≅ G/N

type

3. #15688870

C₂

semidirect

75. C₄₇

#1549

semidirect

Subgroups — #20242072

Conjugacy classes of subgroups

Complements