Group #48028558 — subgroups

Subgroups — C₁₃₉×#33962 #48028558

Order 35584.

496 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 218 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	#3395×C₁₃₉	C₂	1	2	yes	maximal
2	#2247×C₁₃₉	C₂	1	3	yes	maximal
3	#2209×C₁₃₉	C₂	1	6	yes	maximal
4	Q₈×C₁₆×C₁₃₉	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₃₂×C₁₃₉	C₂	1	3	yes	maximal
6	#557×C₁₃₉	—	2	2	no	—
7	C₂×Q₃₂×C₁₃₉	C₂²	1	3	yes	—
8	C₂×Q₃₂×C₁₃₉	C₂²	1	6	yes	—
9	#494×C₁₃₉	C₂²	1	6	yes	—
10	#462×C₁₃₉	—	2	6	no	—
11	C₈×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
12	C₄×Q₁₆×C₁₃₉	C₂²	1	6	yes	—
13	#366×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₁₃₉	C₂²	1	3	yes	—
15	#358×C₁₃₉	C₂²	1	6	yes	—
16	C₄×C₁₆×C₁₃₉	C₂²	1	3	yes	—
17	C₂×Q₁₆×C₁₃₉	—	2	6	no	—
18	C₂×Q₁₆×C₁₃₉	C₂³	1	6	yes	—
19	C₄⋊Q₈×C₁₃₉	—	2	6	no	—
20	#125×C₁₃₉	—	4	6	no	—
21	C₄×Q₈×C₁₃₉	—	2	6	no	—
22	C₄×Q₈×C₁₃₉	—	4	2	no	—
23	C₄×Q₈×C₁₃₉	D₄	1	1	yes	characteristic
24	Q₃₂×C₁₃₉	—	2	6	no	—
25	Q₃₂×C₁₃₉	—	4	6	no	—
26	C₂×C₁₆×C₁₃₉	C₂³	1	3	yes	—
27	C₂×C₁₆×C₁₃₉	C₂³	1	1	yes	characteristic
28	#107×C₁₃₉	C₂³	1	2	yes	—
29	C₄⋊C₈×C₁₃₉	D₄	1	3	yes	—
30	#103×C₁₃₉	—	2	6	no	—
31	C₄×C₈×C₁₃₉	C₂³	1	3	yes	—
32	C₂×Q₈×C₁₃₉	—	2	6	no	—
33	C₂×Q₈×C₁₃₉	D₈	1	1	yes	characteristic
34	Q₁₆×C₁₃₉	—	2	6	no	—
35	Q₁₆×C₁₃₉	—	2	6	no	—
36	Q₁₆×C₁₃₉	—	8	6	no	—
37	C₂×C₈×C₁₃₉	C₂×D₄	1	3	yes	—
38	C₂×C₈×C₁₃₉	C₂⁴	1	1	yes	characteristic
39	C₄⋊C₄×C₁₃₉	—	2	2	no	—
40	C₄⋊C₄×C₁₃₉	—	4	6	no	—
41	C₄⋊C₄×C₁₃₉	—	4	6	no	—
42	C₄⋊C₄×C₁₃₉	—	4	6	no	—
43	C₄⋊C₄×C₁₃₉	D₈	1	3	yes	—
44	C₄²×C₁₃₉	—	4	6	no	—
45	C₄²×C₁₃₉	C₂×D₄	1	3	yes	—
46	C₁₆×C₁₃₉	—	2	3	no	—
47	C₁₆×C₁₃₉	#41	1	2	yes	—
48	Q₈×C₁₃₉	—	4	6	no	—
49	Q₈×C₁₃₉	—	4	6	no	—
50	Q₈×C₁₃₉	Q₃₂	1	4	yes	complemented
51	C₂×C₄×C₁₃₉	—	4	2	no	—
52	C₂×C₄×C₁₃₉	—	4	6	no	—
53	C₂×C₄×C₁₃₉	C₂×D₈	1	3	yes	—
54	C₂×C₄×C₁₃₉	C₂×D₈	1	3	yes	—
55	C₂×C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
56	C₈×C₁₃₉	—	2	3	no	—
57	C₈×C₁₃₉	C₂×#41	1	1	yes	characteristic
58	C₈×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
59	C₂²×C₁₃₉	C₂²×D₈	1	1	yes	characteristic
60	C₄×C₁₃₉	—	2	3	no	—
61	C₄×C₁₃₉	—	8	6	no	—
62	C₄×C₁₃₉	—	8	2	no	—
63	C₄×C₁₃₉	C₂×Q₃₂	1	6	yes	—
64	C₄×C₁₃₉	#550	1	1	yes	characteristic
65	C₄×C₁₃₉	#576	1	1	yes	characteristic
66	C₂×C₁₃₉	#3391	1	1	yes	characteristic
67	C₂×C₁₃₉	C₂²×Q₃₂	1	1	yes	characteristic
68	C₂×C₁₃₉	#3417	1	1	yes	characteristic
69	#33962	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₁₃₉	#33962	1	1	yes	characteristic, direct product factor, Sylow
71	#3395	C₂×C₁₃₉	1	2	yes	—
72	#2247	C₂×C₁₃₉	1	3	yes	—
73	#2209	C₂×C₁₃₉	1	6	yes	—
74	Q₈×C₁₆	C₂×C₁₃₉	1	1	yes	characteristic
75	C₄×Q₃₂	C₂×C₁₃₉	1	3	yes	—
76	#557	—	2	2	no	—
77	C₂×Q₃₂	C₂²×C₁₃₉	1	3	yes	—
78	C₂×Q₃₂	C₂²×C₁₃₉	1	6	yes	—
79	#494	C₂²×C₁₃₉	1	6	yes	—
80	#462	—	2	6	no	—
81	C₈×Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
82	C₄×Q₁₆	C₂²×C₁₃₉	1	6	yes	—
83	#366	C₂²×C₁₃₉	1	1	yes	characteristic
84	C₄⋊C₁₆	C₂²×C₁₃₉	1	3	yes	—
85	#358	C₂²×C₁₃₉	1	6	yes	—
86	C₄×C₁₆	C₂²×C₁₃₉	1	3	yes	—
87	C₂×Q₁₆	—	2	6	no	—
88	C₂×Q₁₆	C₂³×C₁₃₉	1	6	yes	—
89	C₄⋊Q₈	—	2	6	no	—
90	#125	—	4	6	no	—
91	C₄×Q₈	—	2	6	no	—
92	C₄×Q₈	—	4	2	no	—
93	C₄×Q₈	D₄×C₁₃₉	1	1	yes	characteristic
94	Q₃₂	—	2	6	no	—
95	Q₃₂	—	4	6	no	—
96	C₂×C₁₆	C₂³×C₁₃₉	1	3	yes	—
97	C₂×C₁₆	C₂³×C₁₃₉	1	1	yes	characteristic
98	#107	C₂³×C₁₃₉	1	2	yes	—
99	C₄⋊C₈	D₄×C₁₃₉	1	3	yes	—
100	#103	—	2	6	no	—
101	C₄×C₈	C₂³×C₁₃₉	1	3	yes	—
102	C₂×Q₈	—	2	6	no	—
103	C₂×Q₈	D₈×C₁₃₉	1	1	yes	characteristic
104	Q₁₆	—	2	6	no	—
105	Q₁₆	—	2	6	no	—
106	Q₁₆	—	8	6	no	—
107	C₂×C₈	C₂×D₄×C₁₃₉	1	3	yes	—
108	C₂×C₈	C₂⁴×C₁₃₉	1	1	yes	characteristic
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	4	6	no	—
111	C₄⋊C₄	—	4	6	no	—
112	C₄⋊C₄	—	4	6	no	—
113	C₄⋊C₄	D₈×C₁₃₉	1	3	yes	—
114	C₄²	—	4	6	no	—
115	C₄²	C₂×D₄×C₁₃₉	1	3	yes	—
116	C₁₆	—	2	3	no	—
117	C₁₆	#41×C₁₃₉	1	2	yes	—
118	Q₈	—	4	6	no	—
119	Q₈	—	4	6	no	—
120	Q₈	Q₃₂×C₁₃₉	1	4	yes	complemented
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	6	no	—
123	C₂×C₄	C₂×D₈×C₁₃₉	1	3	yes	—
124	C₂×C₄	C₂×D₈×C₁₃₉	1	3	yes	—
125	C₂×C₄	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
126	C₈	—	2	3	no	—
127	C₈	C₂×#41×C₁₃₉	1	1	yes	characteristic
128	C₈	H(5,𝔽₂)×C₁₃₉	1	1	yes	characteristic
129	C₂²	C₂²×D₈×C₁₃₉	1	1	yes	characteristic
130	C₄	—	2	3	no	—
131	C₄	—	8	6	no	—
132	C₄	—	8	2	no	—
133	C₄	C₂×Q₃₂×C₁₃₉	1	6	yes	—
134	C₄	#550×C₁₃₉	1	1	yes	characteristic
135	C₄	#576×C₁₃₉	1	1	yes	characteristic
136	C₂	#3391×C₁₃₉	1	1	yes	characteristic
137	C₂	C₂²×Q₃₂×C₁₃₉	1	1	yes	characteristic
138	C₂	#3417×C₁₃₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3395×C₁₃₉

C₂

yes

maximal

#2247×C₁₃₉

C₂

yes

maximal

#2209×C₁₃₉

C₂

yes

maximal

Q₈×C₁₆×C₁₃₉

C₂

yes

maximal, characteristic

C₄×Q₃₂×C₁₃₉

C₂

yes

maximal

#557×C₁₃₉

—

C₂×Q₃₂×C₁₃₉

C₂²

yes

—

C₂×Q₃₂×C₁₃₉

C₂²

yes

—

#494×C₁₃₉

C₂²

yes

—

#462×C₁₃₉

—

C₈×Q₈×C₁₃₉

C₂²

yes

characteristic

C₄×Q₁₆×C₁₃₉

C₂²

yes

—

#366×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₁₃₉

C₂²

yes

—

#358×C₁₃₉

C₂²

yes

—

C₄×C₁₆×C₁₃₉

C₂²

yes

—

C₂×Q₁₆×C₁₃₉

—

C₂×Q₁₆×C₁₃₉

C₂³

yes

—

C₄⋊Q₈×C₁₃₉

—

#125×C₁₃₉

—

C₄×Q₈×C₁₃₉

—

C₄×Q₈×C₁₃₉

—

C₄×Q₈×C₁₃₉

D₄

yes

characteristic

Q₃₂×C₁₃₉

—

Q₃₂×C₁₃₉

—

C₂×C₁₆×C₁₃₉

C₂³

yes

—

C₂×C₁₆×C₁₃₉

C₂³

yes

characteristic

#107×C₁₃₉

C₂³

yes

—

C₄⋊C₈×C₁₃₉

D₄

yes

—

#103×C₁₃₉

—

C₄×C₈×C₁₃₉

C₂³

yes

—

C₂×Q₈×C₁₃₉

—

C₂×Q₈×C₁₃₉

D₈

yes

characteristic

Q₁₆×C₁₃₉

—

Q₁₆×C₁₃₉

—

Q₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂×D₄

yes

—

C₂×C₈×C₁₃₉

C₂⁴

yes

characteristic

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

D₈

yes

—

C₄²×C₁₃₉

—

C₄²×C₁₃₉

C₂×D₄

yes

—

C₁₆×C₁₃₉

—

C₁₆×C₁₃₉

#41

yes

—

Q₈×C₁₃₉

—

Q₈×C₁₃₉

—

Q₈×C₁₃₉

Q₃₂

yes

complemented

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₈

yes

—

C₂×C₄×C₁₃₉

C₂×D₈

yes

—

C₂×C₄×C₁₃₉

C₂²×D₄

yes

characteristic

C₈×C₁₃₉

—

C₈×C₁₃₉

C₂×#41

yes

characteristic

C₈×C₁₃₉

H(5,𝔽₂)

yes

characteristic

C₂²×C₁₃₉

C₂²×D₈

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×Q₃₂

yes

—

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

C₂²×Q₃₂

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₉

#33962

yes

characteristic, direct product factor, Sylow

#3395

C₂×C₁₃₉

yes

—

#2247

C₂×C₁₃₉

yes

—

#2209

C₂×C₁₃₉

yes

—

Q₈×C₁₆

C₂×C₁₃₉

yes

characteristic

C₄×Q₃₂

C₂×C₁₃₉

yes

—

#557

—

C₂×Q₃₂

C₂²×C₁₃₉

yes

—

C₂×Q₃₂

C₂²×C₁₃₉

yes

—

#494

C₂²×C₁₃₉

yes

—

#462

—

C₈×Q₈

C₂²×C₁₃₉

yes

characteristic

C₄×Q₁₆

C₂²×C₁₃₉

yes

—

#366

C₂²×C₁₃₉

yes

characteristic

C₄⋊C₁₆

C₂²×C₁₃₉

yes

—

#358

C₂²×C₁₃₉

yes

—

C₄×C₁₆

C₂²×C₁₃₉

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂³×C₁₃₉

yes

—

C₄⋊Q₈

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

D₄×C₁₃₉

yes

characteristic

Q₃₂

—

Q₃₂

—

C₂×C₁₆

C₂³×C₁₃₉

yes

—

C₂×C₁₆

C₂³×C₁₃₉

yes

characteristic

#107

C₂³×C₁₃₉

yes

—

C₄⋊C₈

D₄×C₁₃₉

yes

—

100

#103

—

101

C₄×C₈

C₂³×C₁₃₉

yes

—

102

C₂×Q₈

—

103

C₂×Q₈

D₈×C₁₃₉

yes

characteristic

104

Q₁₆

—

105

Q₁₆

—

106

Q₁₆

—

107

C₂×C₈

C₂×D₄×C₁₃₉

yes

—

108

C₂×C₈

C₂⁴×C₁₃₉

yes

characteristic

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

D₈×C₁₃₉

yes

—

114

C₄²

—

115

C₄²

C₂×D₄×C₁₃₉

yes

—

116

C₁₆

—

117

C₁₆

#41×C₁₃₉

yes

—

118

Q₈

—

119

Q₈

—

120

Q₈

Q₃₂×C₁₃₉

yes

complemented

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₂×D₈×C₁₃₉

yes

—

124

C₂×C₄

C₂×D₈×C₁₃₉

yes

—

125

C₂×C₄

C₂²×D₄×C₁₃₉

yes

characteristic

126

C₈

—

127

C₈

C₂×#41×C₁₃₉

yes

characteristic

128

C₈

H(5,𝔽₂)×C₁₃₉

yes

characteristic

129

C₂²

C₂²×D₈×C₁₃₉

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

C₂×Q₃₂×C₁₃₉

yes

—

134

C₄

#550×C₁₃₉

yes

characteristic

135

C₄

#576×C₁₃₉

yes

characteristic

136

C₂

#3391×C₁₃₉

yes

characteristic

137

C₂

C₂²×Q₃₂×C₁₃₉

yes

characteristic

138

C₂

#3417×C₁₃₉

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₃₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₉
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₉ ◁ C₄×Q₈×C₁₃₉ ◁ C₈×Q₈×C₁₃₉ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₁₃₉

C₂²×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂⁴×C₁₃₉

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₃₉ ◁ C₄×Q₈×C₁₃₉ ◁ C₈×Q₈×C₁₃₉ ◁ G

N	H ≅ G/N	type
50. Q₈×C₁₃₉	Q₃₂	semidirect
69. #33962	C₁₃₉	direct (N × H)
70. C₁₃₉	#33962	direct (N × H)
120. Q₈	Q₃₂×C₁₃₉	semidirect

H ≅ G/N

type

50. Q₈×C₁₃₉

Q₃₂

semidirect

69. #33962

C₁₃₉

direct (N × H)

70. C₁₃₉

#33962

direct (N × H)

120. Q₈

Q₃₂×C₁₃₉

semidirect

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

subgroup	index
#3395×C₁₃₉	2
#2247×C₁₃₉	2
#2209×C₁₃₉	2
Q₈×C₁₆×C₁₃₉	2
C₄×Q₃₂×C₁₃₉	2
#33962	139

subgroup

index

#3395×C₁₃₉

#2247×C₁₃₉

#2209×C₁₃₉

Q₈×C₁₆×C₁₃₉

C₄×Q₃₂×C₁₃₉

#33962

139

Subgroups — C₁₃₉×#33962 #48028558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C139×#33962 #48028558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#33962 #48028558