Group #26073674 — subgroups

Subgroups — C₂²×C₄₇×SD₆₄ #26073674

Order 12032.

502 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 222 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	C₂²×Q₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₇×SD₆₄	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×Q₁₆×C₄₇	—	2	1	no	—
6	C₂²×D₈×C₄₇	—	2	1	no	—
7	C₂×Q₃₂×C₄₇	—	2	3	no	—
8	C₂×Q₃₂×C₄₇	C₂²	1	6	yes	complemented
9	C₂×D₁₆×C₄₇	—	2	3	no	—
10	C₂×D₁₆×C₄₇	C₂²	1	6	yes	—
11	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×SD₆₄	C₂²	1	16	yes	direct product factor
13	C₂×C₃₂×C₄₇	C₂²	1	6	yes	complemented
14	C₂²×Q₈×C₄₇	—	4	1	no	—
15	C₂²×D₄×C₄₇	—	4	1	no	—
16	C₂×Q₁₆×C₄₇	—	2	6	no	—
17	C₂×Q₁₆×C₄₇	—	4	3	no	—
18	C₂×D₈×C₄₇	—	2	6	no	—
19	C₂×D₈×C₄₇	—	4	3	no	—
20	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
21	Q₃₂×C₄₇	—	2	6	no	—
22	Q₃₂×C₄₇	C₂³	1	4	yes	complemented
23	D₁₆×C₄₇	—	2	6	no	—
24	D₁₆×C₄₇	C₂³	1	4	yes	—
25	C₂×C₁₆×C₄₇	D₄	1	3	yes	—
26	C₂×C₁₆×C₄₇	C₂³	1	3	yes	—
27	C₃₂×C₄₇	C₂³	1	4	yes	complemented
28	C₂⁴×C₄₇	—	8	1	no	—
29	C₂×Q₈×C₄₇	—	4	6	no	—
30	C₂×Q₈×C₄₇	—	8	3	no	—
31	C₂×D₄×C₄₇	—	4	6	no	—
32	C₂×D₄×C₄₇	—	8	3	no	—
33	C₂²×C₄×C₄₇	—	8	1	no	—
34	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
35	Q₁₆×C₄₇	—	2	4	no	—
36	Q₁₆×C₄₇	—	4	6	no	—
37	D₈×C₄₇	—	2	4	no	—
38	D₈×C₄₇	—	4	6	no	—
39	C₂×C₈×C₄₇	D₈	1	3	yes	—
40	C₂×C₈×C₄₇	C₂×D₄	1	3	yes	—
41	C₁₆×C₄₇	C₂×D₄	1	3	yes	—
42	C₁₆×C₄₇	C₂⁴	1	1	yes	characteristic
43	C₂³×C₄₇	—	8	6	no	—
44	C₂³×C₄₇	—	16	4	no	—
45	C₂³×C₄₇	D₁₆	1	1	yes	characteristic
46	Q₈×C₄₇	—	4	4	no	—
47	Q₈×C₄₇	—	8	6	no	—
48	D₄×C₄₇	—	4	4	no	—
49	D₄×C₄₇	—	8	6	no	—
50	C₂×C₄×C₄₇	—	8	6	no	—
51	C₂×C₄×C₄₇	D₁₆	1	3	yes	—
52	C₂×C₄×C₄₇	C₂×D₈	1	3	yes	—
53	C₈×C₄₇	C₂×D₈	1	3	yes	—
54	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
55	C₂²×SD₆₄	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂²×C₄₇	—	8	4	no	—
57	C₂²×C₄₇	—	16	12	no	—
58	C₂²×C₄₇	SD₆₄	1	4	yes	direct product factor
59	C₂²×C₄₇	C₂×D₁₆	1	3	yes	—
60	C₄×C₄₇	—	8	4	no	—
61	C₄×C₄₇	C₂×D₁₆	1	3	yes	—
62	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
63	C₂²×Q₃₂	C₂×C₄₇	1	1	yes	characteristic, complemented
64	C₂²×D₁₆	C₂×C₄₇	1	1	yes	characteristic
65	C₂×SD₆₄	C₂×C₄₇	1	12	yes	direct product factor
66	C₂²×C₃₂	C₂×C₄₇	1	1	yes	characteristic, complemented
67	C₂×C₄₇	—	16	4	no	—
68	C₂×C₄₇	C₂×SD₆₄	1	6	yes	direct product factor
69	C₂×C₄₇	C₂²×D₁₆	1	1	yes	characteristic
70	C₂²×Q₁₆	—	2	1	no	—
71	C₂²×D₈	—	2	1	no	—
72	C₂×Q₃₂	—	2	3	no	—
73	C₂×Q₃₂	C₂²×C₄₇	1	6	yes	complemented
74	C₂×D₁₆	—	2	3	no	—
75	C₂×D₁₆	C₂²×C₄₇	1	6	yes	—
76	C₂²×C₁₆	C₂²×C₄₇	1	1	yes	characteristic
77	SD₆₄	C₂²×C₄₇	1	16	yes	direct product factor
78	C₂×C₃₂	C₂²×C₄₇	1	6	yes	complemented
79	C₄₇	C₂²×SD₆₄	1	1	yes	characteristic, direct product factor, Sylow
80	C₂²×Q₈	—	4	1	no	—
81	C₂²×D₄	—	4	1	no	—
82	C₂×Q₁₆	—	2	6	no	—
83	C₂×Q₁₆	—	4	3	no	—
84	C₂×D₈	—	2	6	no	—
85	C₂×D₈	—	4	3	no	—
86	C₂²×C₈	D₄×C₄₇	1	1	yes	characteristic
87	Q₃₂	—	2	6	no	—
88	Q₃₂	C₂³×C₄₇	1	4	yes	complemented
89	D₁₆	—	2	6	no	—
90	D₁₆	C₂³×C₄₇	1	4	yes	—
91	C₂×C₁₆	D₄×C₄₇	1	3	yes	—
92	C₂×C₁₆	C₂³×C₄₇	1	3	yes	—
93	C₃₂	C₂³×C₄₇	1	4	yes	complemented
94	C₂⁴	—	8	1	no	—
95	C₂×Q₈	—	4	6	no	—
96	C₂×Q₈	—	8	3	no	—
97	C₂×D₄	—	4	6	no	—
98	C₂×D₄	—	8	3	no	—
99	C₂²×C₄	—	8	1	no	—
100	C₂²×C₄	D₈×C₄₇	1	1	yes	characteristic
101	Q₁₆	—	2	4	no	—
102	Q₁₆	—	4	6	no	—
103	D₈	—	2	4	no	—
104	D₈	—	4	6	no	—
105	C₂×C₈	D₈×C₄₇	1	3	yes	—
106	C₂×C₈	C₂×D₄×C₄₇	1	3	yes	—
107	C₁₆	C₂×D₄×C₄₇	1	3	yes	—
108	C₁₆	C₂⁴×C₄₇	1	1	yes	characteristic
109	C₂³	—	8	6	no	—
110	C₂³	—	16	4	no	—
111	C₂³	D₁₆×C₄₇	1	1	yes	characteristic
112	Q₈	—	4	4	no	—
113	Q₈	—	8	6	no	—
114	D₄	—	4	4	no	—
115	D₄	—	8	6	no	—
116	C₂×C₄	—	8	6	no	—
117	C₂×C₄	D₁₆×C₄₇	1	3	yes	—
118	C₂×C₄	C₂×D₈×C₄₇	1	3	yes	—
119	C₈	C₂×D₈×C₄₇	1	3	yes	—
120	C₈	C₂²×D₄×C₄₇	1	1	yes	characteristic
121	C₂²	—	8	4	no	—
122	C₂²	—	16	12	no	—
123	C₂²	C₄₇×SD₆₄	1	4	yes	direct product factor
124	C₂²	C₂×D₁₆×C₄₇	1	3	yes	—
125	C₄	—	8	4	no	—
126	C₄	C₂×D₁₆×C₄₇	1	3	yes	—
127	C₄	C₂²×D₈×C₄₇	1	1	yes	characteristic
128	C₂	—	16	4	no	—
129	C₂	C₂×C₄₇×SD₆₄	1	6	yes	direct product factor
130	C₂	C₂²×D₁₆×C₄₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₃₂×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₆×C₄₇

C₂

yes

maximal, characteristic

C₂×C₄₇×SD₆₄

C₂

yes

maximal, direct product factor

C₂²×C₃₂×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×Q₁₆×C₄₇

—

C₂²×D₈×C₄₇

—

C₂×Q₃₂×C₄₇

—

C₂×Q₃₂×C₄₇

C₂²

yes

complemented

C₂×D₁₆×C₄₇

—

C₂×D₁₆×C₄₇

C₂²

yes

—

C₂²×C₁₆×C₄₇

C₂²

yes

characteristic

C₄₇×SD₆₄

C₂²

yes

direct product factor

C₂×C₃₂×C₄₇

C₂²

yes

complemented

C₂²×Q₈×C₄₇

—

C₂²×D₄×C₄₇

—

C₂×Q₁₆×C₄₇

—

C₂×Q₁₆×C₄₇

—

C₂×D₈×C₄₇

—

C₂×D₈×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

characteristic

Q₃₂×C₄₇

—

Q₃₂×C₄₇

C₂³

yes

complemented

D₁₆×C₄₇

—

D₁₆×C₄₇

C₂³

yes

—

C₂×C₁₆×C₄₇

D₄

yes

—

C₂×C₁₆×C₄₇

C₂³

yes

—

C₃₂×C₄₇

C₂³

yes

complemented

C₂⁴×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₈

yes

characteristic

Q₁₆×C₄₇

—

Q₁₆×C₄₇

—

D₈×C₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₁₆×C₄₇

C₂×D₄

yes

—

C₁₆×C₄₇

C₂⁴

yes

characteristic

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

D₁₆

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

D₁₆

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₈×C₄₇

C₂×D₈

yes

—

C₈×C₄₇

C₂²×D₄

yes

characteristic

C₂²×SD₆₄

C₄₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

SD₆₄

yes

direct product factor

C₂²×C₄₇

C₂×D₁₆

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₁₆

yes

—

C₄×C₄₇

C₂²×D₈

yes

characteristic

C₂²×Q₃₂

C₂×C₄₇

yes

characteristic, complemented

C₂²×D₁₆

C₂×C₄₇

yes

characteristic

C₂×SD₆₄

C₂×C₄₇

yes

direct product factor

C₂²×C₃₂

C₂×C₄₇

yes

characteristic, complemented

C₂×C₄₇

—

C₂×C₄₇

C₂×SD₆₄

yes

direct product factor

C₂×C₄₇

C₂²×D₁₆

yes

characteristic

C₂²×Q₁₆

—

C₂²×D₈

—

C₂×Q₃₂

—

C₂×Q₃₂

C₂²×C₄₇

yes

complemented

C₂×D₁₆

—

C₂×D₁₆

C₂²×C₄₇

yes

—

C₂²×C₁₆

C₂²×C₄₇

yes

characteristic

SD₆₄

C₂²×C₄₇

yes

direct product factor

C₂×C₃₂

C₂²×C₄₇

yes

complemented

C₄₇

C₂²×SD₆₄

yes

characteristic, direct product factor, Sylow

C₂²×Q₈

—

C₂²×D₄

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

D₄×C₄₇

yes

characteristic

Q₃₂

—

Q₃₂

C₂³×C₄₇

yes

complemented

D₁₆

—

D₁₆

C₂³×C₄₇

yes

—

C₂×C₁₆

D₄×C₄₇

yes

—

C₂×C₁₆

C₂³×C₄₇

yes

—

C₃₂

C₂³×C₄₇

yes

complemented

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

100

C₂²×C₄

D₈×C₄₇

yes

characteristic

101

Q₁₆

—

102

Q₁₆

—

103

D₈

—

104

D₈

—

105

C₂×C₈

D₈×C₄₇

yes

—

106

C₂×C₈

C₂×D₄×C₄₇

yes

—

107

C₁₆

C₂×D₄×C₄₇

yes

—

108

C₁₆

C₂⁴×C₄₇

yes

characteristic

109

C₂³

—

110

C₂³

—

111

C₂³

D₁₆×C₄₇

yes

characteristic

112

Q₈

—

113

Q₈

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

D₁₆×C₄₇

yes

—

118

C₂×C₄

C₂×D₈×C₄₇

yes

—

119

C₈

C₂×D₈×C₄₇

yes

—

120

C₈

C₂²×D₄×C₄₇

yes

characteristic

121

C₂²

—

122

C₂²

—

123

C₂²

C₄₇×SD₆₄

yes

direct product factor

124

C₂²

C₂×D₁₆×C₄₇

yes

—

125

C₄

—

126

C₄

C₂×D₁₆×C₄₇

yes

—

127

C₄

C₂²×D₈×C₄₇

yes

characteristic

128

C₂

—

129

C₂

C₂×C₄₇×SD₆₄

yes

direct product factor

130

C₂

C₂²×D₁₆×C₄₇

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₄₇	D₁₆
commutator, Frattini, cosocle	C₁₆	C₂⁴×C₄₇
prime radical	C₂²×D₁₆×C₄₇	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ C₂²×C₄×C₄₇ ◁ C₂²×C₈×C₄₇ ◁ C₂²×C₁₆×C₄₇ ◁ G

subgroup

G/H

center, socle

C₂³×C₄₇

D₁₆

commutator, Frattini, cosocle

C₁₆

C₂⁴×C₄₇

prime radical

C₂²×D₁₆×C₄₇

C₂

LCS

G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₄₇ ◁ C₂²×C₄×C₄₇ ◁ C₂²×C₈×C₄₇ ◁ C₂²×C₁₆×C₄₇ ◁ G

N	H ≅ G/N	type
1. C₂²×Q₃₂×C₄₇	C₂	semidirect
3. C₂×C₄₇×SD₆₄	C₂	direct (N × H)
4. C₂²×C₃₂×C₄₇	C₂	semidirect
8. C₂×Q₃₂×C₄₇	C₂²	semidirect
12. C₄₇×SD₆₄	C₂²	direct (N × H)
13. C₂×C₃₂×C₄₇	C₂²	semidirect
22. Q₃₂×C₄₇	C₂³	semidirect
27. C₃₂×C₄₇	C₂³	semidirect
55. C₂²×SD₆₄	C₄₇	direct (N × H)
58. C₂²×C₄₇	SD₆₄	direct (N × H)
63. C₂²×Q₃₂	C₂×C₄₇	semidirect
65. C₂×SD₆₄	C₂×C₄₇	direct (N × H)
66. C₂²×C₃₂	C₂×C₄₇	semidirect
68. C₂×C₄₇	C₂×SD₆₄	direct (N × H)
73. C₂×Q₃₂	C₂²×C₄₇	semidirect
77. SD₆₄	C₂²×C₄₇	direct (N × H)
78. C₂×C₃₂	C₂²×C₄₇	semidirect
79. C₄₇	C₂²×SD₆₄	direct (N × H)
88. Q₃₂	C₂³×C₄₇	semidirect
93. C₃₂	C₂³×C₄₇	semidirect
123. C₂²	C₄₇×SD₆₄	direct (N × H)
129. C₂	C₂×C₄₇×SD₆₄	direct (N × H)

H ≅ G/N

type

1. C₂²×Q₃₂×C₄₇

C₂

semidirect

3. C₂×C₄₇×SD₆₄

C₂

direct (N × H)

4. C₂²×C₃₂×C₄₇

C₂

semidirect

8. C₂×Q₃₂×C₄₇

C₂²

semidirect

12. C₄₇×SD₆₄

C₂²

direct (N × H)

13. C₂×C₃₂×C₄₇

C₂²

semidirect

22. Q₃₂×C₄₇

C₂³

semidirect

27. C₃₂×C₄₇

C₂³

semidirect

55. C₂²×SD₆₄

C₄₇

direct (N × H)

58. C₂²×C₄₇

SD₆₄

direct (N × H)

63. C₂²×Q₃₂

C₂×C₄₇

semidirect

65. C₂×SD₆₄

C₂×C₄₇

direct (N × H)

66. C₂²×C₃₂

C₂×C₄₇

semidirect

68. C₂×C₄₇

C₂×SD₆₄

direct (N × H)

73. C₂×Q₃₂

C₂²×C₄₇

semidirect

77. SD₆₄

C₂²×C₄₇

direct (N × H)

78. C₂×C₃₂

C₂²×C₄₇

semidirect

79. C₄₇

C₂²×SD₆₄

direct (N × H)

88. Q₃₂

C₂³×C₄₇

semidirect

93. C₃₂

C₂³×C₄₇

semidirect

123. C₂²

C₄₇×SD₆₄

direct (N × H)

129. C₂

C₂×C₄₇×SD₆₄

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×Q₃₂×C₄₇	2
C₂²×D₁₆×C₄₇	2
C₂×C₄₇×SD₆₄	2
C₂²×C₃₂×C₄₇	2
C₂²×SD₆₄	47

subgroup

index

C₂²×Q₃₂×C₄₇

C₂²×D₁₆×C₄₇

C₂×C₄₇×SD₆₄

C₂²×C₃₂×C₄₇

C₂²×SD₆₄

Subgroups — C₂²×C₄₇×SD₆₄ #26073674

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×C47×SD64 #26073674

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄₇×SD₆₄ #26073674