Group #20253894 — subgroups

Subgroups — C₂×#15689526 #20253894

Order 6016.

424 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 197 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	#15689526	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13093957	C₂	1	4	yes	maximal
4	C₂×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₉₄	—	2	1	no	—
7	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
8	#13094024	—	2	2	no	—
9	#13094024	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₄₇	—	2	1	no	—
11	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	#13093957	C₂²	1	16	yes	—
13	C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
14	C₂²×Q₈×C₄₇	—	2	1	no	—
15	C₄⋊Q₈×C₄₇	—	2	2	no	—
16	C₄⋊Q₈×C₄₇	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
18	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
19	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×Q₈×C₄₇	—	2	4	no	—
21	C₂×Q₈×C₄₇	—	4	4	no	—
22	C₂²×C₄×C₄₇	—	2	2	no	—
23	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₄₇	—	2	2	no	—
27	C₂×Dic₉₄	—	2	4	no	—
28	C₂×Dic₉₄	—	4	4	no	—
29	C₄⋊C₄×C₄₇	—	2	4	no	—
30	C₄⋊C₄×C₄₇	—	2	2	no	—
31	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
32	C₄²×C₄₇	D₄	1	2	yes	—
33	C₄²×C₄₇	C₂³	1	2	yes	—
34	C₄⋊Dic₄₇	—	2	4	no	—
35	C₄⋊Dic₄₇	—	2	2	no	—
36	C₄⋊Dic₄₇	C₂³	1	4	yes	—
37	C₂×C₄₇⋊C₈	Q₈	1	8	yes	complemented
38	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
39	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
40	Q₈×C₄₇	—	4	8	no	—
41	C₂×C₄×C₄₇	—	2	4	no	—
42	C₂×C₄×C₄₇	—	4	4	no	—
43	C₂×C₄×C₄₇	D₈	1	4	yes	—
44	C₂×C₄×C₄₇	Q₁₆	1	4	yes	—
45	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	—
49	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₄₇	—	2	4	no	—
51	C₂×Dic₄₇	—	4	4	no	—
52	Dic₉₄	—	4	8	no	—
53	C₄₇⋊C₈	C₂×Q₈	1	8	yes	complemented
54	C₂²×C₄₇	C₄⋊Q₈	1	2	yes	—
55	C₂²×C₄₇	C₂×D₈	1	2	yes	—
56	C₂²×C₄₇	C₂×Q₁₆	1	2	yes	—
57	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₄₇	—	4	4	no	—
59	C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
60	C₄×C₄₇	C₂×D₈	1	4	yes	—
61	C₄×C₄₇	C₂×Q₁₆	1	4	yes	—
62	C₄×C₄₇	C₂²×Q₈	1	2	yes	—
63	Dic₄₇	—	4	4	no	—
64	C₂×#500	—	47	1	no	maximal, Sylow
65	C₂×C₄₇	#500	1	4	yes	complemented
66	C₂×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄₇	C₂²×D₈	1	1	yes	characteristic
68	C₂×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
69	C₂×C₄⋊Q₈	—	47	1	no	—
70	C₂×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
71	#500	—	47	8	no	—
72	C₂×#107	—	47	4	no	—
73	C₂×C₄×C₈	—	47	1	no	—
74	C₄₇	C₂×#500	1	1	yes	characteristic, complemented, Sylow
75	C₂²×Q₈	—	2	1	no	—
76	C₂²×Q₈	—	94	1	no	—
77	C₂²×C₈	—	47	2	no	—
78	C₄⋊Q₈	—	2	2	no	—
79	C₄⋊Q₈	—	47	4	no	—
80	C₄⋊Q₈	—	94	2	no	—
81	C₄⋊Q₈	C₂×D₄₇	1	4	yes	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	47	2	no	—
84	C₂×C₄⋊C₄	—	94	1	no	—
85	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
86	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
87	#107	—	47	16	no	—
88	C₄×C₈	—	47	4	no	—
89	C₂×Q₈	—	2	4	no	—
90	C₂×Q₈	—	4	4	no	—
91	C₂×Q₈	—	94	4	no	—
92	C₂×Q₈	—	188	4	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	94	2	no	—
95	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
98	C₂×C₈	—	47	8	no	—
99	C₂×C₈	—	47	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	47	4	no	—
103	C₄⋊C₄	—	94	2	no	—
104	C₄⋊C₄	—	94	4	no	—
105	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
106	C₄²	C₂²⋊D₄₇	1	2	yes	—
107	C₄²	C₂²×D₄₇	1	2	yes	—
108	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
109	Q₈	—	4	8	no	—
110	Q₈	—	188	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	4	4	no	—
113	C₂×C₄	—	94	4	no	—
114	C₂×C₄	—	188	4	no	—
115	C₂×C₄	#10625334	1	4	yes	—
116	C₂×C₄	#10625337	1	4	yes	—
117	C₂×C₄	Q₈×D₄₇	1	4	yes	—
118	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
120	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
121	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
122	C₈	—	47	8	no	—
123	C₂²	#13094022	1	2	yes	—
124	C₂²	C₂×#10625334	1	2	yes	—
125	C₂²	C₂×#10625337	1	2	yes	—
126	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
127	C₄	—	4	4	no	—
128	C₄	—	188	4	no	—
129	C₄	#13094022	1	2	yes	—
130	C₄	C₂×#10625334	1	4	yes	—
131	C₄	C₂×#10625337	1	4	yes	—
132	C₄	C₂×Q₈×D₄₇	1	2	yes	—
133	C₂	#15689526	1	4	yes	direct product factor
134	C₂	C₂×#13094022	1	1	yes	characteristic
135	C₂	C₂²×#10625334	1	1	yes	characteristic
136	C₂	C₂²×#10625337	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094024

C₂

yes

maximal, characteristic

#15689526

C₂

yes

maximal, direct product factor

C₂×#13093957

C₂

yes

maximal

C₂×C₄×C₄₇⋊C₈

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

—

C₂²×C₄₇⋊C₈

C₂²

yes

—

#13094024

—

#13094024

C₂²

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

#13093957

C₂²

yes

—

C₄×C₄₇⋊C₈

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

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

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

—

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

D₈

yes

—

C₂×C₄×C₄₇

Q₁₆

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

C₄₇⋊C₈

C₂×Q₈

yes

complemented

C₂²×C₄₇

C₄⋊Q₈

yes

—

C₂²×C₄₇

C₂×D₈

yes

—

C₂²×C₄₇

C₂×Q₁₆

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

C₄⋊Q₈

yes

—

C₄×C₄₇

C₂×D₈

yes

—

C₄×C₄₇

C₂×Q₁₆

yes

—

C₄×C₄₇

C₂²×Q₈

yes

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₄₇

C₂²×D₈

yes

characteristic

C₂×C₄₇

C₂²×Q₁₆

yes

characteristic

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₄₇

yes

characteristic

#500

—

C₂×#107

—

C₂×C₄×C₈

—

C₄₇

C₂×#500

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₄₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

#107

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

188

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

C₂²×C₄

C₂²×D₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

C₂²×D₄₇

yes

—

106

C₄²

C₂²⋊D₄₇

yes

—

107

C₄²

C₂²×D₄₇

yes

—

108

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

109

Q₈

—

110

Q₈

—

188

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

188

—

115

C₂×C₄

#10625334

yes

—

116

C₂×C₄

#10625337

yes

—

117

C₂×C₄

Q₈×D₄₇

yes

—

118

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

120

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

121

C₂×C₄

C₂³×D₄₇

yes

characteristic

122

C₈

—

123

C₂²

#13094022

yes

—

124

C₂²

C₂×#10625334

yes

—

125

C₂²

C₂×#10625337

yes

—

126

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

127

C₄

—

128

C₄

—

188

—

129

C₄

#13094022

yes

—

130

C₄

C₂×#10625334

yes

—

131

C₄

C₂×#10625337

yes

—

132

C₄

C₂×Q₈×D₄₇

yes

—

133

C₂

#15689526

yes

direct product factor

134

C₂

C₂×#13094022

yes

characteristic

135

C₂

C₂²×#10625334

yes

characteristic

136

C₂

C₂²×#10625337

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×#500
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×#500

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #15689526	C₂	direct (N × H)
37. C₂×C₄₇⋊C₈	Q₈	semidirect
53. C₄₇⋊C₈	C₂×Q₈	semidirect
65. C₂×C₄₇	#500	semidirect
74. C₄₇	C₂×#500	semidirect
133. C₂	#15689526	direct (N × H)

H ≅ G/N

type

2. #15689526

C₂

direct (N × H)

37. C₂×C₄₇⋊C₈

Q₈

semidirect

53. C₄₇⋊C₈

C₂×Q₈

semidirect

65. C₂×C₄₇

#500

semidirect

74. C₄₇

C₂×#500

semidirect

133. C₂

#15689526

direct (N × H)

Subgroups — C₂×#15689526 #20253894

Conjugacy classes of subgroups

Complements

Subgroups — C2×#15689526 #20253894

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689526 #20253894