Group #20256507 — subgroups

Subgroups — C₂×#15689556 #20256507

Order 6016.

368 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 117 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×#13094047	C₂	1	2	yes	maximal, complemented
2	#15689556	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13093967	C₂	1	2	yes	maximal, complemented
4	C₂×D₇₅₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₆×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#10625354	—	2	2	no	—
8	#13094047	—	2	4	no	—
9	#13094047	C₂²	1	8	yes	complemented
10	C₂×#10625336	—	2	2	no	—
11	C₂×D₃₇₆	C₂²	1	2	yes	—
12	C₂×C₈×D₄₇	C₂²	1	1	yes	characteristic
13	#13093967	C₂²	1	8	yes	complemented
14	D₇₅₂	C₂²	1	4	yes	complemented
15	C₂×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₁₆×D₄₇	C₂²	1	4	yes	complemented
17	C₂×Q₁₆×C₄₇	C₂²	1	2	yes	—
18	Q₃₂×C₄₇	C₂²	1	4	yes	complemented
19	C₂×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×Q₈×C₄₇	—	2	2	no	—
21	#10625354	—	2	8	no	—
22	#10625354	—	4	4	no	—
23	C₂×D₁₈₈	—	2	2	no	—
24	C₂×D₁₈₈	—	4	2	no	—
25	C₂×C₄×D₄₇	—	4	2	no	—
26	C₂×C₄×D₄₇	D₄	1	1	yes	characteristic
27	Q₁₆×C₄₇	—	2	2	no	—
28	Q₁₆×C₄₇	C₂³	1	4	yes	—
29	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
30	#10625336	—	2	8	no	—
31	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
32	D₃₇₆	—	2	2	no	—
33	D₃₇₆	C₂³	1	4	yes	—
34	C₈×D₄₇	D₄	1	2	yes	—
35	C₈×D₄₇	C₂³	1	2	yes	—
36	C₁₆×C₄₇	C₂³	1	2	yes	—
37	C₄₇⋊C₁₆	C₂³	1	2	yes	—
38	C₂²×D₄₇	—	4	2	no	—
39	C₂²×D₄₇	D₈	1	1	yes	characteristic
40	Q₈×C₄₇	—	2	4	no	—
41	Q₈×C₄₇	—	4	2	no	—
42	C₂×C₄×C₄₇	—	4	2	no	—
43	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₄₇	D₈	1	1	yes	characteristic
45	D₁₈₈	—	2	4	no	—
46	D₁₈₈	—	4	8	no	—
47	D₁₈₈	—	4	2	no	—
48	C₄×D₄₇	—	4	8	no	—
49	C₄×D₄₇	D₈	1	2	yes	—
50	C₄×D₄₇	C₂×D₄	1	2	yes	—
51	C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₈×C₄₇	C₂⁴	1	1	yes	characteristic
53	C₄₇⋊C₈	C₂×D₄	1	2	yes	—
54	C₂²×C₄₇	C₂×D₈	1	1	yes	characteristic
55	C₂×D₄₇	—	2	2	no	—
56	C₂×D₄₇	—	4	4	no	—
57	C₂×D₄₇	—	8	4	no	—
58	C₂×D₄₇	C₂×D₈	1	2	yes	—
59	C₄×C₄₇	—	4	4	no	—
60	C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
61	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
62	Dic₄₇	C₂×D₈	1	2	yes	—
63	C₂×#508	—	47	1	no	maximal, Sylow
64	C₂×C₄₇	#508	1	2	yes	complemented
65	C₂×C₄₇	C₂²×D₈	1	1	yes	characteristic
66	D₄₇	—	2	2	no	—
67	D₄₇	—	8	4	no	—
68	C₂×#135	—	47	2	no	—
69	#508	—	47	8	no	—
70	C₂×Q₃₂	D₄₇	1	1	yes	characteristic, complemented
71	C₂×SD₃₂	—	47	2	no	—
72	C₂×D₁₆	—	47	1	no	—
73	C₂²×C₁₆	—	47	1	no	—
74	C₄₇	C₂×#508	1	1	yes	characteristic, complemented, Sylow
75	C₂×#41	—	94	2	no	—
76	#135	—	47	8	no	—
77	#135	—	94	4	no	—
78	C₂×Q₁₆	C₂×D₄₇	1	2	yes	—
79	C₂×SD₁₆	—	94	2	no	—
80	C₂×D₈	—	47	2	no	—
81	C₂²×C₈	—	47	1	no	—
82	Q₃₂	C₂×D₄₇	1	4	yes	complemented
83	SD₃₂	—	47	8	no	—
84	D₁₆	—	47	4	no	—
85	C₂×C₁₆	—	47	1	no	—
86	C₂×C₁₆	—	47	4	no	—
87	C₂×C₁₆	C₂×D₄₇	1	1	yes	characteristic
88	#41	—	94	8	no	—
89	#41	—	188	4	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×D₄	—	94	2	no	—
92	C₂×D₄	—	188	2	no	—
93	C₂²×C₄	—	47	1	no	—
94	C₂²×C₄	—	188	2	no	—
95	Q₁₆	—	2	2	no	—
96	Q₁₆	C₂²×D₄₇	1	4	yes	—
97	SD₁₆	—	94	8	no	—
98	D₈	—	47	4	no	—
99	D₈	—	94	2	no	—
100	C₂×C₈	—	47	1	no	—
101	C₂×C₈	—	47	2	no	—
102	C₂×C₈	—	47	2	no	—
103	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
104	C₁₆	—	47	2	no	—
105	C₁₆	C₂²×D₄₇	1	2	yes	—
106	C₂³	—	47	1	no	—
107	C₂³	—	188	2	no	—
108	Q₈	—	2	4	no	—
109	Q₈	—	4	2	no	—
110	D₄	—	94	4	no	—
111	D₄	—	188	8	no	—
112	D₄	—	188	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	47	1	no	—
115	C₂×C₄	—	47	2	no	—
116	C₂×C₄	—	47	2	no	—
117	C₂×C₄	—	188	8	no	—
118	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
119	C₈	—	47	2	no	—
120	C₈	D₄×D₄₇	1	1	yes	characteristic
121	C₈	C₂³×D₄₇	1	1	yes	characteristic
122	C₂²	—	47	2	no	—
123	C₂²	—	94	2	no	—
124	C₂²	—	376	4	no	—
125	C₂²	—	188	4	no	—
126	C₂²	D₈×D₄₇	1	1	yes	characteristic
127	C₄	—	4	4	no	—
128	C₄	—	47	2	no	—
129	C₄	D₈×D₄₇	1	1	yes	characteristic
130	C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
131	C₂	—	94	2	no	—
132	C₂	—	376	4	no	—
133	C₂	#15689556	1	2	yes	direct product factor
134	C₂	C₂×D₈×D₄₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094047

C₂

yes

maximal, complemented

#15689556

C₂

yes

maximal, direct product factor

C₂×#13093967

C₂

yes

maximal, complemented

C₂×D₇₅₂

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₄₇

C₂

yes

maximal, characteristic, complemented

C₂×Q₃₂×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×#10625354

—

#13094047

—

#13094047

C₂²

yes

complemented

C₂×#10625336

—

C₂×D₃₇₆

C₂²

yes

—

C₂×C₈×D₄₇

C₂²

yes

characteristic

#13093967

C₂²

yes

complemented

D₇₅₂

C₂²

yes

complemented

C₂×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₁₆×D₄₇

C₂²

yes

complemented

C₂×Q₁₆×C₄₇

C₂²

yes

—

Q₃₂×C₄₇

C₂²

yes

complemented

C₂×C₁₆×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

#10625354

—

#10625354

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

D₄

yes

characteristic

Q₁₆×C₄₇

—

Q₁₆×C₄₇

C₂³

yes

—

C₂×C₈×C₄₇

C₂³

yes

characteristic

#10625336

—

C₂×C₄₇⋊C₈

D₄

yes

characteristic

D₃₇₆

—

D₃₇₆

C₂³

yes

—

C₈×D₄₇

D₄

yes

—

C₈×D₄₇

C₂³

yes

—

C₁₆×C₄₇

C₂³

yes

—

C₄₇⋊C₁₆

C₂³

yes

—

C₂²×D₄₇

—

C₂²×D₄₇

D₈

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×Dic₄₇

D₈

yes

characteristic

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

D₈

yes

—

C₄×D₄₇

C₂×D₄

yes

—

C₈×C₄₇

C₂×D₄

yes

characteristic

C₈×C₄₇

C₂⁴

yes

characteristic

C₄₇⋊C₈

C₂×D₄

yes

—

C₂²×C₄₇

C₂×D₈

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

C₂×D₈

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₈

yes

characteristic

C₄×C₄₇

C₂²×D₄

yes

characteristic

Dic₄₇

C₂×D₈

yes

—

—

C₂×C₄₇

yes

C₂×C₄₇

C₂²×D₈

yes

characteristic

D₄₇

—

D₄₇

—

C₂×#135

—

#508

—

C₂×Q₃₂

D₄₇

yes

characteristic, complemented

C₂×SD₃₂

—

C₂×D₁₆

—

C₂²×C₁₆

—

C₄₇

C₂×#508

yes

characteristic, complemented, Sylow

C₂×#41

—

#135

—

#135

—

C₂×Q₁₆

C₂×D₄₇

yes

—

C₂×SD₁₆

—

C₂×D₈

—

C₂²×C₈

—

Q₃₂

C₂×D₄₇

yes

complemented

SD₃₂

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄₇

yes

characteristic

#41

—

#41

—

188

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

188

—

C₂²×C₄

—

C₂²×C₄

—

188

—

Q₁₆

—

Q₁₆

C₂²×D₄₇

yes

—

SD₁₆

—

D₈

—

D₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

C₂²×D₄₇

yes

characteristic

104

C₁₆

—

105

C₁₆

C₂²×D₄₇

yes

—

106

C₂³

—

107

C₂³

—

188

—

108

Q₈

—

109

Q₈

—

110

D₄

—

111

D₄

—

188

—

112

D₄

—

188

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

188

—

118

C₂×C₄

D₄×D₄₇

yes

characteristic

119

C₈

—

120

C₈

D₄×D₄₇

yes

characteristic

121

C₈

C₂³×D₄₇

yes

characteristic

122

C₂²

—

123

C₂²

—

124

C₂²

—

376

—

125

C₂²

—

188

—

126

C₂²

D₈×D₄₇

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

D₈×D₄₇

yes

characteristic

130

C₄

C₂×D₄×D₄₇

yes

characteristic

131

C₂

—

132

C₂

—

376

—

133

C₂

#15689556

yes

direct product factor

134

C₂

C₂×D₈×D₄₇

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₄₇
commutator	C₈×C₄₇	C₂⁴
Frattini	C₈	C₂³×D₄₇
Fitting	C₂×Q₃₂×C₄₇	C₂
socle	C₂²×C₄₇	C₂×D₈
hypercenter	C₂×Q₃₂	D₄₇
hypocenter	C₄₇	C₂×#508
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×D₄₇

commutator

C₈×C₄₇

C₂⁴

Frattini

C₈

C₂³×D₄₇

Fitting

C₂×Q₃₂×C₄₇

C₂

socle

C₂²×C₄₇

C₂×D₈

hypercenter

C₂×Q₃₂

D₄₇

hypocenter

C₄₇

C₂×#508

LCS

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
1. C₂×#13094047	C₂	semidirect
2. #15689556	C₂	direct (N × H)
3. C₂×#13093967	C₂	semidirect
4. C₂×D₇₅₂	C₂	semidirect
5. C₂×C₁₆×D₄₇	C₂	semidirect
6. C₂×Q₃₂×C₄₇	C₂	semidirect
9. #13094047	C₂²	semidirect
13. #13093967	C₂²	semidirect
14. D₇₅₂	C₂²	semidirect
16. C₁₆×D₄₇	C₂²	semidirect
18. Q₃₂×C₄₇	C₂²	semidirect
64. C₂×C₄₇	#508	semidirect
70. C₂×Q₃₂	D₄₇	semidirect
74. C₄₇	C₂×#508	semidirect
82. Q₃₂	C₂×D₄₇	semidirect
133. C₂	#15689556	direct (N × H)

H ≅ G/N

type

1. C₂×#13094047

C₂

semidirect

2. #15689556

C₂

direct (N × H)

3. C₂×#13093967

C₂

semidirect

4. C₂×D₇₅₂

C₂

semidirect

5. C₂×C₁₆×D₄₇

C₂

semidirect

6. C₂×Q₃₂×C₄₇

C₂

semidirect

9. #13094047

C₂²

semidirect

13. #13093967

C₂²

semidirect

14. D₇₅₂

C₂²

semidirect

16. C₁₆×D₄₇

C₂²

semidirect

18. Q₃₂×C₄₇

C₂²

semidirect

64. C₂×C₄₇

#508

semidirect

70. C₂×Q₃₂

D₄₇

semidirect

74. C₄₇

C₂×#508

semidirect

82. Q₃₂

C₂×D₄₇

semidirect

133. C₂

#15689556

direct (N × H)

Subgroups — C₂×#15689556 #20256507

Conjugacy classes of subgroups

Complements

Subgroups — C2×#15689556 #20256507

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689556 #20256507