Group #20256488 — subgroups

Subgroups — C₂×#15689541 #20256488

Order 6016.

368 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 213 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×#13094030	C₂	1	1	yes	maximal, characteristic
2	#15689541	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13093963	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₆×D₄₇	C₂	1	2	yes	maximal, complemented
6	C₂×M₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#10625350	C₄	1	1	yes	characteristic
8	#13094030	C₄	1	4	yes	—
9	#13094030	C₂²	1	4	yes	—
10	C₂×#10625329	C₄	1	1	yes	characteristic
11	C₂×#10625324	C₄	1	2	yes	—
12	C₂×C₈×D₄₇	C₂²	1	2	yes	—
13	#13093963	C₂²	1	8	yes	complemented
14	C₂×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₄₇⋊C₁₆	C₂²	1	4	yes	complemented
17	C₁₆×D₄₇	C₂²	1	8	yes	complemented
18	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	M₃₂×C₄₇	C₂²	1	4	yes	complemented
20	C₂×C₁₆×C₄₇	C₂²	1	2	yes	—
21	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₂²⋊D₄₇	C₈	1	2	yes	—
23	#10625350	C₈	1	4	yes	—
24	#10625350	C₂×C₄	1	4	yes	—
25	C₂×D₁₈₈	C₈	1	1	yes	characteristic
26	C₂×C₄×D₄₇	C₂×C₄	1	2	yes	—
27	C₂×Dic₉₄	C₈	1	1	yes	characteristic
28	C₂×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₄₇	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₄₇	C₂³	1	2	yes	—
31	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
32	#10625329	C₂×C₄	1	4	yes	—
33	C₂×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
34	#10625324	C₂×C₄	1	8	yes	—
35	C₈×D₄₇	C₂×C₄	1	4	yes	—
36	C₈×D₄₇	C₂³	1	4	yes	—
37	C₁₆×C₄₇	C₂³	1	4	yes	—
38	C₄₇⋊C₁₆	C₂³	1	2	yes	—
39	C₄₇⋊C₁₆	C₂³	1	2	yes	—
40	C₂³×C₄₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×D₄₇	C₂×C₈	1	2	yes	—
42	C₂×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×C₄₇	C₂×C₈	1	2	yes	—
44	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
46	C₂²⋊D₄₇	C₂×C₈	1	8	yes	—
47	C₂×Dic₄₇	C₂×C₈	1	2	yes	—
48	D₁₈₈	C₂×C₈	1	4	yes	—
49	C₄×D₄₇	C₂×C₈	1	4	yes	—
50	C₄×D₄₇	C₂²×C₄	1	4	yes	—
51	Dic₉₄	C₂×C₈	1	4	yes	—
52	C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₈×C₄₇	C₂²×C₄	1	2	yes	—
54	C₈×C₄₇	C₂⁴	1	1	yes	characteristic
55	C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
56	C₂²×C₄₇	—	2	2	no	—
57	C₂²×C₄₇	C₂²×C₈	1	2	yes	—
58	C₂²×C₄₇	C₂²×C₈	1	1	yes	characteristic
59	C₂×D₄₇	—	2	4	no	—
60	C₂×D₄₇	C₂²×C₈	1	4	yes	—
61	C₄×C₄₇	C₂²×C₈	1	2	yes	—
62	C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
63	C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
64	Dic₄₇	C₂²×C₈	1	4	yes	—
65	C₂×#504	—	47	1	no	maximal, Sylow
66	C₂×C₄₇	—	2	2	no	—
67	C₂×C₄₇	#504	1	2	yes	complemented
68	C₂×C₄₇	C₂³×C₈	1	1	yes	characteristic
69	D₄₇	—	2	4	no	—
70	C₂×#131	—	47	1	no	—
71	#504	—	47	8	no	—
72	C₂×M₃₂	—	47	2	no	—
73	C₂×M₃₂	D₄₇	1	1	yes	characteristic, complemented
74	C₂²×C₁₆	—	47	1	no	—
75	C₂²×C₁₆	—	47	2	no	—
76	C₄₇	C₂×#504	1	1	yes	characteristic, complemented, Sylow
77	C₂×#41	—	47	1	no	—
78	#131	—	47	4	no	—
79	#131	—	47	4	no	—
80	C₂×M₁₆	—	47	1	no	—
81	C₂×M₁₆	—	47	2	no	—
82	C₂²×C₈	—	47	2	no	—
83	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
84	M₃₂	—	47	8	no	—
85	M₃₂	C₂×D₄₇	1	4	yes	complemented
86	C₂×C₁₆	—	47	4	no	—
87	C₂×C₁₆	—	47	1	no	—
88	C₂×C₁₆	—	47	1	no	—
89	C₂×C₁₆	—	47	8	no	—
90	C₂×C₁₆	C₂×D₄₇	1	2	yes	—
91	#41	—	47	4	no	—
92	#41	—	47	4	no	—
93	C₂×Q₈	—	47	1	no	—
94	C₂×D₄	—	47	2	no	—
95	C₂×D₄	—	47	1	no	—
96	C₂²×C₄	—	47	2	no	—
97	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
98	M₁₆	—	47	4	no	—
99	M₁₆	—	47	8	no	—
100	C₂×C₈	—	47	2	no	—
101	C₂×C₈	—	47	4	no	—
102	C₂×C₈	—	47	4	no	—
103	C₂×C₈	C₄×D₄₇	1	1	yes	characteristic
104	C₂×C₈	C₄×D₄₇	1	2	yes	—
105	C₂×C₈	C₂²×D₄₇	1	2	yes	—
106	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
107	C₁₆	—	47	2	no	—
108	C₁₆	—	47	2	no	—
109	C₁₆	C₂²×D₄₇	1	4	yes	—
110	C₂³	—	47	2	no	—
111	C₂³	C₈×D₄₇	1	1	yes	characteristic
112	Q₈	—	47	4	no	—
113	D₄	—	47	8	no	—
114	D₄	—	47	4	no	—
115	C₂×C₄	—	47	2	no	—
116	C₂×C₄	—	47	4	no	—
117	C₂×C₄	—	47	4	no	—
118	C₂×C₄	C₈×D₄₇	1	1	yes	characteristic
119	C₂×C₄	C₈×D₄₇	1	2	yes	—
120	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
121	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
122	C₈	—	47	4	no	—
123	C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
124	C₈	C₂×C₄×D₄₇	1	2	yes	—
125	C₈	C₂³×D₄₇	1	1	yes	characteristic
126	C₂²	—	2	2	no	—
127	C₂²	—	47	4	no	—
128	C₂²	—	94	4	no	—
129	C₂²	C₂×C₈×D₄₇	1	2	yes	—
130	C₂²	C₂×C₈×D₄₇	1	1	yes	characteristic
131	C₄	—	47	4	no	—
132	C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
133	C₄	C₂×C₈×D₄₇	1	2	yes	—
134	C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
135	C₂	—	2	2	no	—
136	C₂	—	94	4	no	—
137	C₂	#15689541	1	2	yes	direct product factor
138	C₂	C₂²×C₈×D₄₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094030

C₂

yes

maximal, characteristic

#15689541

C₂

yes

maximal, direct product factor

C₂×#13093963

C₂

yes

maximal, complemented

C₂²×C₄₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₄₇

C₂

yes

maximal, complemented

C₂×M₃₂×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×#10625350

C₄

yes

characteristic

#13094030

C₄

yes

—

#13094030

C₂²

yes

—

C₂×#10625329

C₄

yes

characteristic

C₂×#10625324

C₄

yes

—

C₂×C₈×D₄₇

C₂²

yes

—

#13093963

C₂²

yes

complemented

C₂×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₂×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₂×C₄₇⋊C₁₆

C₂²

yes

complemented

C₁₆×D₄₇

C₂²

yes

complemented

C₂²×C₈×C₄₇

C₂²

yes

characteristic

M₃₂×C₄₇

C₂²

yes

complemented

C₂×C₁₆×C₄₇

C₂²

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₄₇

C₈

yes

—

#10625350

C₈

yes

—

#10625350

C₂×C₄

yes

—

C₂×D₁₈₈

C₈

yes

characteristic

C₂×C₄×D₄₇

C₂×C₄

yes

—

C₂×Dic₉₄

C₈

yes

characteristic

C₂×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₈×C₄₇

C₂³

yes

—

C₂×C₈×C₄₇

C₂³

yes

characteristic

#10625329

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

#10625324

C₂×C₄

yes

—

C₈×D₄₇

C₂×C₄

yes

—

C₈×D₄₇

C₂³

yes

—

C₁₆×C₄₇

C₂³

yes

—

C₄₇⋊C₁₆

C₂³

yes

—

C₄₇⋊C₁₆

C₂³

yes

—

C₂³×C₄₇

C₂×C₈

yes

characteristic

C₂²×D₄₇

C₂×C₈

yes

—

C₂×C₄×C₄₇

C₂×C₈

yes

characteristic

C₂×C₄×C₄₇

C₂×C₈

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂²⋊D₄₇

C₂×C₈

yes

—

C₂×Dic₄₇

C₂×C₈

yes

—

D₁₈₈

C₂×C₈

yes

—

C₄×D₄₇

C₂×C₈

yes

—

C₄×D₄₇

C₂²×C₄

yes

—

Dic₉₄

C₂×C₈

yes

—

C₈×C₄₇

C₂²×C₄

yes

characteristic

C₈×C₄₇

C₂²×C₄

yes

—

C₈×C₄₇

C₂⁴

yes

characteristic

C₄₇⋊C₈

C₂²×C₄

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×C₈

yes

—

C₂²×C₄₇

C₂²×C₈

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

C₂²×C₈

yes

—

C₄×C₄₇

C₂²×C₈

yes

—

C₄×C₄₇

C₂²×C₈

yes

characteristic

C₄×C₄₇

C₂³×C₄

yes

characteristic

Dic₄₇

C₂²×C₈

yes

—

C₂×#504

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#504

yes

complemented

C₂×C₄₇

C₂³×C₈

yes

characteristic

D₄₇

—

C₂×#131

—

#504

—

C₂×M₃₂

—

C₂×M₃₂

D₄₇

yes

characteristic, complemented

C₂²×C₁₆

—

C₂²×C₁₆

—

C₄₇

C₂×#504

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄₇

yes

characteristic

M₃₂

—

M₃₂

C₂×D₄₇

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄₇

yes

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄₇

yes

characteristic

M₁₆

—

M₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

C₄×D₄₇

yes

characteristic

104

C₂×C₈

C₄×D₄₇

yes

—

105

C₂×C₈

C₂²×D₄₇

yes

—

106

C₂×C₈

C₂²×D₄₇

yes

characteristic

107

C₁₆

—

108

C₁₆

—

109

C₁₆

C₂²×D₄₇

yes

—

110

C₂³

—

111

C₂³

C₈×D₄₇

yes

characteristic

112

Q₈

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₈×D₄₇

yes

characteristic

119

C₂×C₄

C₈×D₄₇

yes

—

120

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

121

C₂×C₄

C₂×C₄×D₄₇

yes

—

122

C₈

—

123

C₈

C₂×C₄×D₄₇

yes

characteristic

124

C₈

C₂×C₄×D₄₇

yes

—

125

C₈

C₂³×D₄₇

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×C₈×D₄₇

yes

—

130

C₂²

C₂×C₈×D₄₇

yes

characteristic

131

C₄

—

132

C₄

C₂×C₈×D₄₇

yes

characteristic

133

C₄

C₂×C₈×D₄₇

yes

—

134

C₄

C₂²×C₄×D₄₇

yes

characteristic

135

C₂

—

136

C₂

—

137

C₂

#15689541

yes

direct product factor

138

C₂

C₂²×C₈×D₄₇

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₄₇
commutator	C₂×C₄₇	C₂³×C₈
Frattini	C₈	C₂³×D₄₇
Fitting	C₂×M₃₂×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×C₈
hypercenter	C₂×M₃₂	D₄₇
hypocenter	C₄₇	C₂×#504
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂×C₈

C₂²×D₄₇

commutator

C₂×C₄₇

C₂³×C₈

Frattini

C₈

C₂³×D₄₇

Fitting

C₂×M₃₂×C₄₇

C₂

socle

C₂²×C₄₇

C₂²×C₈

hypercenter

C₂×M₃₂

D₄₇

C₄₇

G ▷ C₂×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
2. #15689541	C₂	direct (N × H)
3. C₂×#13093963	C₂	semidirect
4. C₂²×C₄₇⋊C₁₆	C₂	semidirect
5. C₂×C₁₆×D₄₇	C₂	semidirect
6. C₂×M₃₂×C₄₇	C₂	semidirect
13. #13093963	C₂²	semidirect
16. C₂×C₄₇⋊C₁₆	C₂²	semidirect
17. C₁₆×D₄₇	C₂²	semidirect
19. M₃₂×C₄₇	C₂²	semidirect
67. C₂×C₄₇	#504	semidirect
73. C₂×M₃₂	D₄₇	semidirect
76. C₄₇	C₂×#504	semidirect
85. M₃₂	C₂×D₄₇	semidirect
137. C₂	#15689541	direct (N × H)

H ≅ G/N

type

2. #15689541

C₂

direct (N × H)

3. C₂×#13093963

C₂

semidirect

4. C₂²×C₄₇⋊C₁₆

C₂

semidirect

5. C₂×C₁₆×D₄₇

C₂

semidirect

6. C₂×M₃₂×C₄₇

C₂

semidirect

13. #13093963

C₂²

semidirect

16. C₂×C₄₇⋊C₁₆

C₂²

semidirect

17. C₁₆×D₄₇

C₂²

semidirect

19. M₃₂×C₄₇

C₂²

semidirect

67. C₂×C₄₇

#504

semidirect

73. C₂×M₃₂

D₄₇

semidirect

76. C₄₇

C₂×#504

semidirect

85. M₃₂

C₂×D₄₇

semidirect

137. C₂

#15689541

direct (N × H)

Subgroups — C₂×#15689541 #20256488

Conjugacy classes of subgroups

Complements

Subgroups — C2×#15689541 #20256488

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689541 #20256488