Group #20256961 — subgroups

Subgroups — C₂×#15689852 #20256961

Order 6016.

1048 conjugacy classes of subgroups; only the 581 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 581 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	#15689852	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13094024	C₂	1	3	yes	maximal
3	C₂×#13094015	C₂	1	6	yes	maximal
4	C₂×Q₈×Dic₄₇	C₂	1	2	yes	maximal
5	C₂×C₄×Dic₉₄	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₉₄	C₂²	1	3	yes	—
8	#13094024	C₂²	1	24	yes	—
9	#13094015	C₂²	1	48	yes	—
10	Q₈×Dic₄₇	C₂²	1	16	yes	—
11	C₄×Dic₉₄	C₂²	1	24	yes	—
12	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₄₇	C₂²	1	6	yes	—
14	C₂×C₄⋊Dic₄₇	C₂²	1	6	yes	—
15	C₂×Dic₄₇⋊C₄	C₂²	1	6	yes	—
16	C₂×C₄×Dic₄₇	C₂²	1	6	yes	—
17	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₄₇	C₂²	1	8	yes	—
19	C₂×C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
20	C₂×C₄²×C₄₇	C₂²	1	3	yes	—
21	C₂×Q₈×C₄₇	Q₈	1	8	yes	complemented
22	C₂×Q₈×C₄₇	C₂³	1	4	yes	—
23	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
24	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
25	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₄₇	C₂³	1	2	yes	—
27	C₂²×Dic₄₇	C₂³	1	6	yes	—
28	C₂×Dic₉₄	—	2	12	no	—
29	C₂×Dic₉₄	C₂³	1	12	yes	—
30	C₄⋊C₄×C₄₇	C₂³	1	12	yes	—
31	C₄²×C₄₇	C₂³	1	12	yes	—
32	C₄⋊Dic₄₇	C₂³	1	4	yes	—
33	C₄⋊Dic₄₇	C₂³	1	24	yes	—
34	C₄⋊Dic₄₇	C₂³	1	24	yes	—
35	Dic₄₇⋊C₄	C₂³	1	24	yes	—
36	C₄×Dic₄₇	C₂³	1	24	yes	—
37	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
38	Q₈×C₄₇	C₂×Q₈	1	16	yes	complemented
39	C₂×C₄×C₄₇	—	2	6	no	—
40	C₂×C₄×C₄₇	C₂×Q₈	1	12	yes	—
41	C₂×C₄×C₄₇	#41	1	4	yes	—
42	C₂×C₄×C₄₇	C₂⁴	1	6	yes	—
43	C₂×C₄×C₄₇	C₂⁴	1	6	yes	—
44	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
45	C₂×Dic₄₇	—	2	12	no	—
46	C₂×Dic₄₇	—	2	4	no	—
47	C₂×Dic₄₇	C₂⁴	1	4	yes	—
48	C₂×Dic₄₇	C₂⁴	1	12	yes	—
49	Dic₉₄	—	2	24	no	—
50	C₂²×C₄₇	C₂²×Q₈	1	2	yes	—
51	C₂²×C₄₇	C₂×#41	1	2	yes	—
52	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
53	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
54	C₄×C₄₇	—	2	6	no	—
55	C₄×C₄₇	C₂²×Q₈	1	12	yes	—
56	C₄×C₄₇	C₂×#41	1	4	yes	—
57	Dic₄₇	—	2	12	no	—
58	Dic₄₇	—	2	4	no	—
59	C₂×#557	—	47	1	no	maximal, Sylow
60	C₂×C₄₇	#557	1	4	yes	complemented
61	C₂×C₄₇	C₂³×Q₈	1	1	yes	characteristic
62	C₂×C₄₇	C₂²×#41	1	1	yes	characteristic
63	C₂×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
64	#557	—	47	16	no	—
65	C₂×C₄⋊Q₈	—	47	3	no	—
66	C₂×#125	—	47	6	no	—
67	C₂×C₄×Q₈	—	47	2	no	—
68	C₂×C₄×Q₈	—	47	3	no	—
69	C₂×C₄×Q₈	D₄₇	1	1	yes	characteristic
70	C₄₇	C₂×#557	1	1	yes	characteristic, complemented, Sylow
71	C₂²×Q₈	—	47	3	no	—
72	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
73	C₄⋊Q₈	—	47	24	no	—
74	#125	—	47	48	no	—
75	C₄×Q₈	—	47	16	no	—
76	C₄×Q₈	—	47	24	no	—
77	C₄×Q₈	C₂×D₄₇	1	8	yes	—
78	C₂×C₄⋊C₄	—	47	1	no	—
79	C₂×C₄⋊C₄	—	47	6	no	—
80	C₂×C₄⋊C₄	—	47	6	no	—
81	C₂×C₄⋊C₄	—	47	6	no	—
82	C₂×C₄⋊C₄	C₂×D₄₇	1	3	yes	—
83	C₂×C₄²	—	47	6	no	—
84	C₂×C₄²	C₂×D₄₇	1	3	yes	—
85	C₂×Q₈	—	47	12	no	—
86	C₂×Q₈	—	94	12	no	—
87	C₂×Q₈	Dic₉₄	1	8	yes	complemented
88	C₂×Q₈	C₂²×D₄₇	1	4	yes	—
89	C₂²×C₄	—	47	2	no	—
90	C₂²×C₄	—	47	6	no	—
91	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
92	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
93	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
94	C₄⋊C₄	—	47	24	no	—
95	C₄⋊C₄	—	47	4	no	—
96	C₄⋊C₄	—	47	24	no	—
97	C₄⋊C₄	—	47	24	no	—
98	C₄⋊C₄	C₂²×D₄₇	1	12	yes	—
99	C₄²	—	47	24	no	—
100	C₄²	C₂²×D₄₇	1	12	yes	—
101	C₂³	C₂³×D₄₇	1	1	yes	characteristic
102	Q₈	—	94	24	no	—
103	Q₈	C₂×Dic₉₄	1	16	yes	complemented
104	C₂×C₄	—	2	6	no	—
105	C₂×C₄	—	47	4	no	—
106	C₂×C₄	—	47	12	no	—
107	C₂×C₄	—	94	12	no	—
108	C₂×C₄	—	94	4	no	—
109	C₂×C₄	C₂×Dic₉₄	1	12	yes	—
110	C₂×C₄	#10625354	1	4	yes	—
111	C₂×C₄	C₂³×D₄₇	1	6	yes	—
112	C₂×C₄	C₂³×D₄₇	1	6	yes	—
113	C₂×C₄	C₂³×D₄₇	1	2	yes	—
114	C₂²	C₂²×Dic₉₄	1	2	yes	—
115	C₂²	C₂×#10625354	1	2	yes	—
116	C₂²	#13094071	1	2	yes	—
117	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
118	C₄	—	2	6	no	—
119	C₄	—	94	12	no	—
120	C₄	—	94	4	no	—
121	C₄	C₂²×Dic₉₄	1	12	yes	—
122	C₄	C₂×#10625354	1	4	yes	—
123	C₂	#15689852	1	4	yes	direct product factor
124	C₂	C₂³×Dic₉₄	1	1	yes	characteristic
125	C₂	C₂²×#10625354	1	1	yes	characteristic
126	C₂	C₂×#13094071	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689852

C₂

yes

maximal, direct product factor

C₂×#13094024

C₂

yes

maximal

C₂×#13094015

C₂

yes

maximal

C₂×Q₈×Dic₄₇

C₂

yes

maximal

C₂×C₄×Dic₉₄

C₂

yes

maximal

C₂×C₄×Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

—

#13094024

C₂²

yes

—

#13094015

C₂²

yes

—

Q₈×Dic₄₇

C₂²

yes

—

C₄×Dic₉₄

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₄×Q₈×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

—

C₂×Q₈×C₄₇

Q₈

yes

complemented

C₂×Q₈×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂³

yes

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₄²×C₄₇

C₂³

yes

—

C₄⋊Dic₄₇

C₂³

yes

—

C₄⋊Dic₄₇

C₂³

yes

—

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

C₂³

yes

—

C₄×Dic₄₇

C₂³

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

Q₈×C₄₇

C₂×Q₈

yes

complemented

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂⁴

yes

—

C₂×Dic₄₇

C₂⁴

yes

—

Dic₉₄

—

C₂²×C₄₇

C₂²×Q₈

yes

—

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₂²×C₄₇

C₂⁵

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

C₂²×Q₈

yes

—

C₄×C₄₇

C₂×#41

yes

—

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

C₂³×Q₈

yes

characteristic

C₂×C₄₇

C₂²×#41

yes

characteristic

C₂×C₄₇

C₂×2₋¹⁺⁴

yes

characteristic

#557

—

C₂×C₄⋊Q₈

—

C₂×#125

—

C₂×C₄×Q₈

—

C₂×C₄×Q₈

—

C₂×C₄×Q₈

D₄₇

yes

characteristic

C₄₇

C₂×#557

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₄₇

yes

characteristic

C₄⋊Q₈

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₄₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₉₄

yes

complemented

C₂×Q₈

C₂²×D₄₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₄₇

yes

—

C₄²

—

100

C₄²

C₂²×D₄₇

yes

—

101

C₂³

C₂³×D₄₇

yes

characteristic

102

Q₈

—

103

Q₈

C₂×Dic₉₄

yes

complemented

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂×Dic₉₄

yes

—

110

C₂×C₄

#10625354

yes

—

111

C₂×C₄

C₂³×D₄₇

yes

—

112

C₂×C₄

C₂³×D₄₇

yes

—

113

C₂×C₄

C₂³×D₄₇

yes

—

114

C₂²

C₂²×Dic₉₄

yes

—

115

C₂²

C₂×#10625354

yes

—

116

C₂²

#13094071

yes

—

117

C₂²

C₂⁴×D₄₇

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₂²×Dic₉₄

yes

—

122

C₄

C₂×#10625354

yes

—

123

C₂

#15689852

yes

direct product factor

124

C₂

C₂³×Dic₉₄

yes

characteristic

125

C₂

C₂²×#10625354

yes

characteristic

126

C₂

C₂×#13094071

yes

characteristic

	subgroup	G/H
center	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁵
Frattini	C₂²	C₂⁴×D₄₇
Fitting	C₂×C₄×Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂×C₄×Q₈	D₄₇
hypocenter	C₄₇	C₂×#557
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂³

C₂³×D₄₇

commutator

C₂²×C₄₇

C₂⁵

Frattini

C₂²

C₂⁴×D₄₇

Fitting

C₂×C₄×Q₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂⁴

hypercenter

C₂×C₄×Q₈

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
1. #15689852	C₂	direct (N × H)
21. C₂×Q₈×C₄₇	Q₈	semidirect
38. Q₈×C₄₇	C₂×Q₈	semidirect
60. C₂×C₄₇	#557	semidirect
70. C₄₇	C₂×#557	semidirect
87. C₂×Q₈	Dic₉₄	semidirect
103. Q₈	C₂×Dic₉₄	semidirect
123. C₂	#15689852	direct (N × H)

H ≅ G/N

type

1. #15689852

C₂

direct (N × H)

21. C₂×Q₈×C₄₇

Q₈

semidirect

38. Q₈×C₄₇

C₂×Q₈

semidirect

60. C₂×C₄₇

#557

semidirect

70. C₄₇

C₂×#557

semidirect

87. C₂×Q₈

Dic₉₄

semidirect

103. Q₈

C₂×Dic₉₄

semidirect

123. C₂

#15689852

direct (N × H)

Subgroups — C₂×#15689852 #20256961

Conjugacy classes of subgroups

Complements

Subgroups — C2×#15689852 #20256961

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689852 #20256961