Group #20253904 — subgroups

Subgroups — C₂×#15689529 #20253904

Order 6016.

384 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 165 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×#13094016	C₂	1	1	yes	maximal, characteristic
2	#15689529	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13093957	C₂	1	2	yes	maximal
4	C₂×#13093955	C₂	1	2	yes	maximal
5	C₂×#13093935	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
8	#13094016	—	2	2	no	—
9	#13094016	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₄₇⋊C₄	—	2	1	no	—
13	C₂×Dic₄₇⋊C₄	—	2	1	no	—
14	#13093957	C₂²	1	8	yes	—
15	#13093955	C₂²	1	8	yes	—
16	#13093935	C₂²	1	4	yes	—
17	C₂²×Q₈×C₄₇	—	2	1	no	—
18	C₄⋊Q₈×C₄₇	—	2	2	no	—
19	C₄⋊Q₈×C₄₇	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
21	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₄₇	—	2	4	no	—
24	C₂×Q₈×C₄₇	—	4	4	no	—
25	C₂²×C₄×C₄₇	—	2	2	no	—
26	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
29	C₂²×Dic₄₇	—	2	1	no	—
30	C₂²×Dic₄₇	—	2	1	no	—
31	C₄⋊C₄×C₄₇	—	2	4	no	—
32	C₄⋊C₄×C₄₇	—	2	2	no	—
33	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
34	C₄²×C₄₇	D₄	1	2	yes	—
35	C₄²×C₄₇	C₂³	1	2	yes	—
36	C₄⋊Dic₄₇	—	2	1	no	—
37	C₄⋊Dic₄₇	—	2	1	no	—
38	C₄⋊Dic₄₇	C₂³	1	2	yes	—
39	C₄⋊Dic₄₇	C₂³	1	2	yes	—
40	Dic₄₇⋊C₄	—	2	4	no	—
41	Dic₄₇⋊C₄	—	2	4	no	—
42	C₂×C₄₇⋊C₈	Q₈	1	8	yes	complemented
43	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
44	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₄₇	—	4	8	no	—
46	C₂×C₄×C₄₇	—	2	2	no	—
47	C₂×C₄×C₄₇	—	2	2	no	—
48	C₂×C₄×C₄₇	—	2	4	no	—
49	C₂×C₄×C₄₇	—	4	4	no	—
50	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	—
54	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₄₇	—	2	2	no	—
56	C₂×Dic₄₇	—	2	2	no	—
57	C₂×Dic₄₇	—	4	2	no	—
58	C₂×Dic₄₇	—	4	2	no	—
59	C₄₇⋊C₈	C₂×Q₈	1	8	yes	complemented
60	C₂²×C₄₇	C₄⋊Q₈	1	2	yes	—
61	C₂²×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
62	C₂²×C₄₇	#137	1	2	yes	—
63	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₄₇	—	2	2	no	—
65	C₄×C₄₇	—	2	2	no	—
66	C₄×C₄₇	—	4	4	no	—
67	C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
68	C₄×C₄₇	C₂²×Q₈	1	2	yes	—
69	Dic₄₇	—	4	2	no	—
70	Dic₄₇	—	4	2	no	—
71	C₂×#501	—	47	1	no	maximal, Sylow
72	C₂×C₄₇	#501	1	4	yes	complemented
73	C₂×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₂×C₄₇	C₂×#137	1	1	yes	characteristic
76	C₂×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
77	C₂×#125	—	47	1	no	—
78	#501	—	47	8	no	—
79	C₂×#107	—	47	2	no	—
80	C₂×#106	—	47	2	no	—
81	C₂×#97	—	47	1	no	—
82	C₄₇	C₂×#501	1	1	yes	characteristic, complemented, Sylow
83	C₂²×Q₈	—	2	1	no	—
84	C₂²×C₈	—	47	2	no	—
85	C₄⋊Q₈	—	2	2	no	—
86	C₄⋊Q₈	C₂×D₄₇	1	4	yes	—
87	#125	—	47	4	no	—
88	#125	—	94	2	no	—
89	C₂×C₄⋊C₄	—	2	1	no	—
90	C₂×C₄⋊C₄	—	47	1	no	—
91	C₂×C₄⋊C₄	—	47	1	no	—
92	C₂×C₄⋊C₄	—	94	1	no	—
93	C₂×C₄⋊C₄	—	94	1	no	—
94	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
95	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
96	#107	—	47	8	no	—
97	#106	—	47	8	no	—
98	#97	—	47	4	no	—
99	C₂×Q₈	—	2	4	no	—
100	C₂×Q₈	—	4	4	no	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	—	94	1	no	—
103	C₂²×C₄	—	94	1	no	—
104	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
107	C₂×C₈	—	47	8	no	—
108	C₂×C₈	—	47	4	no	—
109	C₄⋊C₄	—	2	4	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	47	2	no	—
112	C₄⋊C₄	—	47	2	no	—
113	C₄⋊C₄	—	94	4	no	—
114	C₄⋊C₄	—	94	4	no	—
115	C₄⋊C₄	—	94	1	no	—
116	C₄⋊C₄	—	94	1	no	—
117	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
118	C₄²	C₂²⋊D₄₇	1	2	yes	—
119	C₄²	C₂²×D₄₇	1	2	yes	—
120	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
121	Q₈	—	4	8	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	4	4	no	—
126	C₂×C₄	—	94	2	no	—
127	C₂×C₄	—	94	2	no	—
128	C₂×C₄	—	188	2	no	—
129	C₂×C₄	—	188	2	no	—
130	C₂×C₄	Q₈×D₄₇	1	4	yes	—
131	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
132	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
133	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
134	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
135	C₈	—	47	8	no	—
136	C₂²	#13094022	1	2	yes	—
137	C₂²	#13094050	1	2	yes	—
138	C₂²	#13094055	1	2	yes	—
139	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	—	4	4	no	—
143	C₄	—	188	2	no	—
144	C₄	—	188	2	no	—
145	C₄	#13094022	1	2	yes	—
146	C₄	C₂×Q₈×D₄₇	1	2	yes	—
147	C₂	#15689529	1	4	yes	direct product factor
148	C₂	C₂×#13094022	1	1	yes	characteristic
149	C₂	C₂×#13094050	1	1	yes	characteristic
150	C₂	C₂×#13094055	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094016

C₂

yes

maximal, characteristic

#15689529

C₂

yes

maximal, direct product factor

C₂×#13093957

C₂

yes

maximal

C₂×#13093955

C₂

yes

maximal

C₂×#13093935

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

#13094016

—

#13094016

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×Dic₄₇⋊C₄

—

C₂×Dic₄₇⋊C₄

—

#13093957

C₂²

yes

—

#13093955

C₂²

yes

—

#13093935

C₂²

yes

—

C₂²×Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₄²×C₄₇

D₄

yes

—

C₄²×C₄₇

C₂³

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

—

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₂×C₄₇⋊C₈

Q₈

yes

complemented

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

C₂×Q₈

yes

complemented

C₂²×C₄₇

C₄⋊Q₈

yes

—

C₂²×C₄₇

AGL(1,ℤ/8)

yes

—

C₂²×C₄₇

#137

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄⋊Q₈

yes

—

C₄×C₄₇

C₂²×Q₈

yes

—

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

C₂×C₄⋊Q₈

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄⋊Q₈

D₄₇

yes

—

—

—

—

C₂×#97

—

C₄₇

C₂×#501

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₄₇

yes

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

—

—

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

105

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

106

C₂²×C₄

C₂²×D₄₇

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

C₂²×D₄₇

yes

—

118

C₄²

C₂²⋊D₄₇

yes

—

119

C₄²

C₂²×D₄₇

yes

—

120

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

188

—

129

C₂×C₄

—

188

—

130

C₂×C₄

Q₈×D₄₇

yes

—

131

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

132

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

133

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

134

C₂×C₄

C₂³×D₄₇

yes

characteristic

135

C₈

—

136

C₂²

#13094022

yes

—

137

C₂²

#13094050

yes

—

138

C₂²

#13094055

yes

—

139

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

188

—

144

C₄

—

188

—

145

C₄

#13094022

yes

—

146

C₄

C₂×Q₈×D₄₇

yes

—

147

C₂

#15689529

yes

direct product factor

148

C₂

C₂×#13094022

yes

characteristic

149

C₂

C₂×#13094050

yes

characteristic

150

C₂

C₂×#13094055

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₂×C₄⋊Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	C₂×#501
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₂×C₄⋊Q₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₄₇

hypocenter

C₄₇

C₂×#501

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #15689529	C₂	direct (N × H)
42. C₂×C₄₇⋊C₈	Q₈	semidirect
59. C₄₇⋊C₈	C₂×Q₈	semidirect
72. C₂×C₄₇	#501	semidirect
82. C₄₇	C₂×#501	semidirect
147. C₂	#15689529	direct (N × H)

H ≅ G/N

type

2. #15689529

C₂

direct (N × H)

42. C₂×C₄₇⋊C₈

Q₈

semidirect

59. C₄₇⋊C₈

C₂×Q₈

semidirect

72. C₂×C₄₇

#501

semidirect

82. C₄₇

C₂×#501

semidirect

147. C₂

#15689529

direct (N × H)

Subgroups — C₂×#15689529 #20253904

Conjugacy classes of subgroups

Complements

Subgroups — C2×#15689529 #20253904

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689529 #20253904