Group #48028572 — subgroups

Subgroups — C₂²×SD₆₄×C₁₃₉ #48028572

Order 35584.

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×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₁₆×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	C₂×SD₆₄×C₁₃₉	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₃₂×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×Q₁₆×C₁₃₉	—	2	1	no	—
6	C₂²×D₈×C₁₃₉	—	2	1	no	—
7	C₂×Q₃₂×C₁₃₉	—	2	3	no	—
8	C₂×Q₃₂×C₁₃₉	C₂²	1	6	yes	complemented
9	C₂×D₁₆×C₁₃₉	—	2	3	no	—
10	C₂×D₁₆×C₁₃₉	C₂²	1	6	yes	—
11	C₂²×C₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
12	SD₆₄×C₁₃₉	C₂²	1	16	yes	direct product factor
13	C₂×C₃₂×C₁₃₉	C₂²	1	6	yes	complemented
14	C₂²×Q₈×C₁₃₉	—	4	1	no	—
15	C₂²×D₄×C₁₃₉	—	4	1	no	—
16	C₂×Q₁₆×C₁₃₉	—	2	6	no	—
17	C₂×Q₁₆×C₁₃₉	—	4	3	no	—
18	C₂×D₈×C₁₃₉	—	2	6	no	—
19	C₂×D₈×C₁₃₉	—	4	3	no	—
20	C₂²×C₈×C₁₃₉	D₄	1	1	yes	characteristic
21	Q₃₂×C₁₃₉	—	2	6	no	—
22	Q₃₂×C₁₃₉	C₂³	1	4	yes	complemented
23	D₁₆×C₁₃₉	—	2	6	no	—
24	D₁₆×C₁₃₉	C₂³	1	4	yes	—
25	C₂×C₁₆×C₁₃₉	D₄	1	3	yes	—
26	C₂×C₁₆×C₁₃₉	C₂³	1	3	yes	—
27	C₃₂×C₁₃₉	C₂³	1	4	yes	complemented
28	C₂⁴×C₁₃₉	—	8	1	no	—
29	C₂×Q₈×C₁₃₉	—	4	6	no	—
30	C₂×Q₈×C₁₃₉	—	8	3	no	—
31	C₂×D₄×C₁₃₉	—	4	6	no	—
32	C₂×D₄×C₁₃₉	—	8	3	no	—
33	C₂²×C₄×C₁₃₉	—	8	1	no	—
34	C₂²×C₄×C₁₃₉	D₈	1	1	yes	characteristic
35	Q₁₆×C₁₃₉	—	2	4	no	—
36	Q₁₆×C₁₃₉	—	4	6	no	—
37	D₈×C₁₃₉	—	2	4	no	—
38	D₈×C₁₃₉	—	4	6	no	—
39	C₂×C₈×C₁₃₉	D₈	1	3	yes	—
40	C₂×C₈×C₁₃₉	C₂×D₄	1	3	yes	—
41	C₁₆×C₁₃₉	C₂×D₄	1	3	yes	—
42	C₁₆×C₁₃₉	C₂⁴	1	1	yes	characteristic
43	C₂³×C₁₃₉	—	8	6	no	—
44	C₂³×C₁₃₉	—	16	4	no	—
45	C₂³×C₁₃₉	D₁₆	1	1	yes	characteristic
46	Q₈×C₁₃₉	—	4	4	no	—
47	Q₈×C₁₃₉	—	8	6	no	—
48	D₄×C₁₃₉	—	4	4	no	—
49	D₄×C₁₃₉	—	8	6	no	—
50	C₂×C₄×C₁₃₉	—	8	6	no	—
51	C₂×C₄×C₁₃₉	D₁₆	1	3	yes	—
52	C₂×C₄×C₁₃₉	C₂×D₈	1	3	yes	—
53	C₈×C₁₃₉	C₂×D₈	1	3	yes	—
54	C₈×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
55	C₂²×C₁₃₉	—	8	4	no	—
56	C₂²×C₁₃₉	—	16	12	no	—
57	C₂²×C₁₃₉	SD₆₄	1	4	yes	direct product factor
58	C₂²×C₁₃₉	C₂×D₁₆	1	3	yes	—
59	C₄×C₁₃₉	—	8	4	no	—
60	C₄×C₁₃₉	C₂×D₁₆	1	3	yes	—
61	C₄×C₁₃₉	C₂²×D₈	1	1	yes	characteristic
62	C₂×C₁₃₉	—	16	4	no	—
63	C₂×C₁₃₉	C₂×SD₆₄	1	6	yes	direct product factor
64	C₂×C₁₃₉	C₂²×D₁₆	1	1	yes	characteristic
65	C₂²×SD₆₄	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₁₃₉	C₂²×SD₆₄	1	1	yes	characteristic, direct product factor, Sylow
67	C₂²×Q₃₂	C₂×C₁₃₉	1	1	yes	characteristic, complemented
68	C₂²×D₁₆	C₂×C₁₃₉	1	1	yes	characteristic
69	C₂×SD₆₄	C₂×C₁₃₉	1	12	yes	direct product factor
70	C₂²×C₃₂	C₂×C₁₃₉	1	1	yes	characteristic, complemented
71	C₂²×Q₁₆	—	2	1	no	—
72	C₂²×D₈	—	2	1	no	—
73	C₂×Q₃₂	—	2	3	no	—
74	C₂×Q₃₂	C₂²×C₁₃₉	1	6	yes	complemented
75	C₂×D₁₆	—	2	3	no	—
76	C₂×D₁₆	C₂²×C₁₃₉	1	6	yes	—
77	C₂²×C₁₆	C₂²×C₁₃₉	1	1	yes	characteristic
78	SD₆₄	C₂²×C₁₃₉	1	16	yes	direct product factor
79	C₂×C₃₂	C₂²×C₁₃₉	1	6	yes	complemented
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×C₁₃₉	1	1	yes	characteristic
87	Q₃₂	—	2	6	no	—
88	Q₃₂	C₂³×C₁₃₉	1	4	yes	complemented
89	D₁₆	—	2	6	no	—
90	D₁₆	C₂³×C₁₃₉	1	4	yes	—
91	C₂×C₁₆	D₄×C₁₃₉	1	3	yes	—
92	C₂×C₁₆	C₂³×C₁₃₉	1	3	yes	—
93	C₃₂	C₂³×C₁₃₉	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×C₁₃₉	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×C₁₃₉	1	3	yes	—
106	C₂×C₈	C₂×D₄×C₁₃₉	1	3	yes	—
107	C₁₆	C₂×D₄×C₁₃₉	1	3	yes	—
108	C₁₆	C₂⁴×C₁₃₉	1	1	yes	characteristic
109	C₂³	—	8	6	no	—
110	C₂³	—	16	4	no	—
111	C₂³	D₁₆×C₁₃₉	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×C₁₃₉	1	3	yes	—
118	C₂×C₄	C₂×D₈×C₁₃₉	1	3	yes	—
119	C₈	C₂×D₈×C₁₃₉	1	3	yes	—
120	C₈	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
121	C₂²	—	8	4	no	—
122	C₂²	—	16	12	no	—
123	C₂²	SD₆₄×C₁₃₉	1	4	yes	direct product factor
124	C₂²	C₂×D₁₆×C₁₃₉	1	3	yes	—
125	C₄	—	8	4	no	—
126	C₄	C₂×D₁₆×C₁₃₉	1	3	yes	—
127	C₄	C₂²×D₈×C₁₃₉	1	1	yes	characteristic
128	C₂	—	16	4	no	—
129	C₂	C₂×SD₆₄×C₁₃₉	1	6	yes	direct product factor
130	C₂	C₂²×D₁₆×C₁₃₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₃₂×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₆×C₁₃₉

C₂

yes

maximal, characteristic

C₂×SD₆₄×C₁₃₉

C₂

yes

maximal, direct product factor

C₂²×C₃₂×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂²×Q₁₆×C₁₃₉

—

C₂²×D₈×C₁₃₉

—

C₂×Q₃₂×C₁₃₉

—

C₂×Q₃₂×C₁₃₉

C₂²

yes

complemented

C₂×D₁₆×C₁₃₉

—

C₂×D₁₆×C₁₃₉

C₂²

yes

—

C₂²×C₁₆×C₁₃₉

C₂²

yes

characteristic

SD₆₄×C₁₃₉

C₂²

yes

direct product factor

C₂×C₃₂×C₁₃₉

C₂²

yes

complemented

C₂²×Q₈×C₁₃₉

—

C₂²×D₄×C₁₃₉

—

C₂×Q₁₆×C₁₃₉

—

C₂×Q₁₆×C₁₃₉

—

C₂×D₈×C₁₃₉

—

C₂×D₈×C₁₃₉

—

C₂²×C₈×C₁₃₉

D₄

yes

characteristic

Q₃₂×C₁₃₉

—

Q₃₂×C₁₃₉

C₂³

yes

complemented

D₁₆×C₁₃₉

—

D₁₆×C₁₃₉

C₂³

yes

—

C₂×C₁₆×C₁₃₉

D₄

yes

—

C₂×C₁₆×C₁₃₉

C₂³

yes

—

C₃₂×C₁₃₉

C₂³

yes

complemented

C₂⁴×C₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

D₈

yes

characteristic

Q₁₆×C₁₃₉

—

Q₁₆×C₁₃₉

—

D₈×C₁₃₉

—

D₈×C₁₃₉

—

C₂×C₈×C₁₃₉

D₈

yes

—

C₂×C₈×C₁₃₉

C₂×D₄

yes

—

C₁₆×C₁₃₉

C₂×D₄

yes

—

C₁₆×C₁₃₉

C₂⁴

yes

characteristic

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

D₁₆

yes

characteristic

Q₈×C₁₃₉

—

Q₈×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

D₁₆

yes

—

C₂×C₄×C₁₃₉

C₂×D₈

yes

—

C₈×C₁₃₉

C₂×D₈

yes

—

C₈×C₁₃₉

C₂²×D₄

yes

characteristic

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

SD₆₄

yes

direct product factor

C₂²×C₁₃₉

C₂×D₁₆

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×D₁₆

yes

—

C₄×C₁₃₉

C₂²×D₈

yes

characteristic

C₂×C₁₃₉

—

C₂×C₁₃₉

C₂×SD₆₄

yes

direct product factor

C₂×C₁₃₉

C₂²×D₁₆

yes

characteristic

C₂²×SD₆₄

C₁₃₉

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₉

C₂²×SD₆₄

yes

characteristic, direct product factor, Sylow

C₂²×Q₃₂

C₂×C₁₃₉

yes

characteristic, complemented

C₂²×D₁₆

C₂×C₁₃₉

yes

characteristic

C₂×SD₆₄

C₂×C₁₃₉

yes

direct product factor

C₂²×C₃₂

C₂×C₁₃₉

yes

characteristic, complemented

C₂²×Q₁₆

—

C₂²×D₈

—

C₂×Q₃₂

—

C₂×Q₃₂

C₂²×C₁₃₉

yes

complemented

C₂×D₁₆

—

C₂×D₁₆

C₂²×C₁₃₉

yes

—

C₂²×C₁₆

C₂²×C₁₃₉

yes

characteristic

SD₆₄

C₂²×C₁₃₉

yes

direct product factor

C₂×C₃₂

C₂²×C₁₃₉

yes

complemented

C₂²×Q₈

—

C₂²×D₄

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

D₄×C₁₃₉

yes

characteristic

Q₃₂

—

Q₃₂

C₂³×C₁₃₉

yes

complemented

D₁₆

—

D₁₆

C₂³×C₁₃₉

yes

—

C₂×C₁₆

D₄×C₁₃₉

yes

—

C₂×C₁₆

C₂³×C₁₃₉

yes

—

C₃₂

C₂³×C₁₃₉

yes

complemented

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

100

C₂²×C₄

D₈×C₁₃₉

yes

characteristic

101

Q₁₆

—

102

Q₁₆

—

103

D₈

—

104

D₈

—

105

C₂×C₈

D₈×C₁₃₉

yes

—

106

C₂×C₈

C₂×D₄×C₁₃₉

yes

—

107

C₁₆

C₂×D₄×C₁₃₉

yes

—

108

C₁₆

C₂⁴×C₁₃₉

yes

characteristic

109

C₂³

—

110

C₂³

—

111

C₂³

D₁₆×C₁₃₉

yes

characteristic

112

Q₈

—

113

Q₈

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

D₁₆×C₁₃₉

yes

—

118

C₂×C₄

C₂×D₈×C₁₃₉

yes

—

119

C₈

C₂×D₈×C₁₃₉

yes

—

120

C₈

C₂²×D₄×C₁₃₉

yes

characteristic

121

C₂²

—

122

C₂²

—

123

C₂²

SD₆₄×C₁₃₉

yes

direct product factor

124

C₂²

C₂×D₁₆×C₁₃₉

yes

—

125

C₄

—

126

C₄

C₂×D₁₆×C₁₃₉

yes

—

127

C₄

C₂²×D₈×C₁₃₉

yes

characteristic

128

C₂

—

129

C₂

C₂×SD₆₄×C₁₃₉

yes

direct product factor

130

C₂

C₂²×D₁₆×C₁₃₉

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₉	D₁₆
commutator, Frattini, cosocle	C₁₆	C₂⁴×C₁₃₉
prime radical	C₂²×D₁₆×C₁₃₉	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₉ ◁ C₂²×C₄×C₁₃₉ ◁ C₂²×C₈×C₁₃₉ ◁ C₂²×C₁₆×C₁₃₉ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₃₉

D₁₆

commutator, Frattini, cosocle

C₁₆

C₂⁴×C₁₃₉

prime radical

C₂²×D₁₆×C₁₃₉

C₂

LCS

G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₃₉ ◁ C₂²×C₄×C₁₃₉ ◁ C₂²×C₈×C₁₃₉ ◁ C₂²×C₁₆×C₁₃₉ ◁ G

N	H ≅ G/N	type
1. C₂²×Q₃₂×C₁₃₉	C₂	semidirect
3. C₂×SD₆₄×C₁₃₉	C₂	direct (N × H)
4. C₂²×C₃₂×C₁₃₉	C₂	semidirect
8. C₂×Q₃₂×C₁₃₉	C₂²	semidirect
12. SD₆₄×C₁₃₉	C₂²	direct (N × H)
13. C₂×C₃₂×C₁₃₉	C₂²	semidirect
22. Q₃₂×C₁₃₉	C₂³	semidirect
27. C₃₂×C₁₃₉	C₂³	semidirect
57. C₂²×C₁₃₉	SD₆₄	direct (N × H)
63. C₂×C₁₃₉	C₂×SD₆₄	direct (N × H)
65. C₂²×SD₆₄	C₁₃₉	direct (N × H)
66. C₁₃₉	C₂²×SD₆₄	direct (N × H)
67. C₂²×Q₃₂	C₂×C₁₃₉	semidirect
69. C₂×SD₆₄	C₂×C₁₃₉	direct (N × H)
70. C₂²×C₃₂	C₂×C₁₃₉	semidirect
74. C₂×Q₃₂	C₂²×C₁₃₉	semidirect
78. SD₆₄	C₂²×C₁₃₉	direct (N × H)
79. C₂×C₃₂	C₂²×C₁₃₉	semidirect
88. Q₃₂	C₂³×C₁₃₉	semidirect
93. C₃₂	C₂³×C₁₃₉	semidirect
123. C₂²	SD₆₄×C₁₃₉	direct (N × H)
129. C₂	C₂×SD₆₄×C₁₃₉	direct (N × H)

H ≅ G/N

type

1. C₂²×Q₃₂×C₁₃₉

C₂

semidirect

3. C₂×SD₆₄×C₁₃₉

C₂

direct (N × H)

4. C₂²×C₃₂×C₁₃₉

C₂

semidirect

8. C₂×Q₃₂×C₁₃₉

C₂²

semidirect

12. SD₆₄×C₁₃₉

C₂²

direct (N × H)

13. C₂×C₃₂×C₁₃₉

C₂²

semidirect

22. Q₃₂×C₁₃₉

C₂³

semidirect

27. C₃₂×C₁₃₉

C₂³

semidirect

57. C₂²×C₁₃₉

SD₆₄

direct (N × H)

63. C₂×C₁₃₉

C₂×SD₆₄

direct (N × H)

65. C₂²×SD₆₄

C₁₃₉

direct (N × H)

66. C₁₃₉

C₂²×SD₆₄

direct (N × H)

67. C₂²×Q₃₂

C₂×C₁₃₉

semidirect

69. C₂×SD₆₄

C₂×C₁₃₉

direct (N × H)

70. C₂²×C₃₂

C₂×C₁₃₉

semidirect

74. C₂×Q₃₂

C₂²×C₁₃₉

semidirect

78. SD₆₄

C₂²×C₁₃₉

direct (N × H)

79. C₂×C₃₂

C₂²×C₁₃₉

semidirect

88. Q₃₂

C₂³×C₁₃₉

semidirect

93. C₃₂

C₂³×C₁₃₉

semidirect

123. C₂²

SD₆₄×C₁₃₉

direct (N × H)

129. C₂

C₂×SD₆₄×C₁₃₉

direct (N × H)

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

subgroup	index
C₂²×Q₃₂×C₁₃₉	2
C₂²×D₁₆×C₁₃₉	2
C₂×SD₆₄×C₁₃₉	2
C₂²×C₃₂×C₁₃₉	2
C₂²×SD₆₄	139

subgroup

index

C₂²×Q₃₂×C₁₃₉

C₂²×D₁₆×C₁₃₉

C₂×SD₆₄×C₁₃₉

C₂²×C₃₂×C₁₃₉

C₂²×SD₆₄

139

Subgroups — C₂²×SD₆₄×C₁₃₉ #48028572

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×SD64×C139 #48028572

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×SD₆₄×C₁₃₉ #48028572