Group #31807899 — subgroups

Subgroups — #31807899

Order 17792.

224 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 89 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×#17984776	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22580852	C₂	1	2	yes	maximal
4	#22580715	C₂	1	2	yes	maximal
5	#340×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₃₉	—	2	2	no	—
8	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
9	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
10	#17984776	—	2	2	no	—
11	Dic₁₃₉⋊C₈	—	2	2	no	—
12	C₁₃₉⋊#95	—	2	2	no	—
13	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
15	C₂²×Dic₁₃₉	—	2	2	no	—
16	C₂²×Dic₁₃₉	D₄	1	1	yes	characteristic
17	C₂²×Dic₁₃₉	D₄	1	1	yes	characteristic
18	C₂×C₁₃₉⋊C₈	—	2	1	no	—
19	C₂×C₁₃₉⋊C₈	—	2	2	no	—
20	C₄⋊Dic₁₃₉	—	2	2	no	—
21	C₄⋊Dic₁₃₉	—	2	2	no	—
22	C₄⋊Dic₁₃₉	—	4	4	no	—
23	C₄×Dic₁₃₉	—	2	1	no	—
24	C₄×Dic₁₃₉	—	2	1	no	—
25	C₂²×C₄×C₁₃₉	—	2	2	no	—
26	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₃₉	—	2	2	no	—
28	C₂×C₈×C₁₃₉	D₄	1	2	yes	—
29	C₄⋊C₄×C₁₃₉	—	2	2	no	—
30	C₄⋊C₄×C₁₃₉	Q₈	1	4	yes	—
31	C₂×Dic₁₃₉	—	2	1	no	—
32	C₂×Dic₁₃₉	—	2	1	no	—
33	C₂×Dic₁₃₉	—	2	1	no	—
34	C₂×Dic₁₃₉	—	2	2	no	—
35	C₂×Dic₁₃₉	—	4	2	no	—
36	C₂×Dic₁₃₉	—	4	2	no	—
37	C₂×Dic₁₃₉	—	4	2	no	—
38	C₂×Dic₁₃₉	Q₁₆	1	2	yes	—
39	C₁₃₉⋊C₈	—	4	1	no	—
40	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃₉	—	2	4	no	—
42	C₂×C₄×C₁₃₉	—	4	2	no	—
43	C₂×C₄×C₁₃₉	—	4	2	no	—
44	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₁₃₉	#41	1	2	yes	—
47	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
48	C₈×C₁₃₉	—	2	2	no	—
49	C₂²×C₁₃₉	#121	1	1	yes	characteristic
50	C₂²×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
51	C₂²×C₁₃₉	C₂×Q₁₆	1	1	yes	characteristic
52	C₂²×C₁₃₉	#135	1	1	yes	characteristic
53	C₂²×C₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
54	C₄×C₁₃₉	—	4	4	no	—
55	C₄×C₁₃₉	#123	1	1	yes	characteristic
56	C₄×C₁₃₉	#125	1	2	yes	—
57	C₄×C₁₃₉	C₄⋊Q₈	1	1	yes	characteristic
58	Dic₁₃₉	—	2	2	no	—
59	Dic₁₃₉	—	4	1	no	—
60	Dic₁₃₉	—	8	2	no	—
61	C₂×C₁₃₉	#398	1	1	yes	characteristic
62	C₂×C₁₃₉	#467	1	1	yes	characteristic
63	C₂×C₁₃₉	#469	1	1	yes	characteristic
64	C₂×C₁₃₉	#477	1	2	yes	—
65	C₂×C₁₃₉	#478	1	2	yes	—
66	C₁₃₉	#2087	1	1	yes	characteristic, complemented, Sylow
67	#2087	—	139	1	no	maximal, Sylow
68	C₂×#107	—	139	1	no	—
69	C₂×C₄⋊C₈	—	139	1	no	—
70	#389	—	139	2	no	—
71	#340	—	139	2	no	—
72	#340	D₁₃₉	1	1	yes	characteristic
73	C₂²×C₈	—	139	1	no	—
74	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	278	2	no	—
76	C₂×C₄⋊C₄	—	139	2	no	—
77	C₂×C₄⋊C₄	C₂×D₁₃₉	1	2	yes	—
78	C₂×C₄²	—	139	1	no	—
79	#107	—	278	2	no	—
80	C₄⋊C₈	—	278	2	no	—
81	#95	—	278	2	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	278	2	no	—
84	C₂²×C₄	—	139	1	no	—
85	C₂²×C₄	—	139	1	no	—
86	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	278	1	no	—
89	C₂×C₈	—	278	2	no	—
90	C₂×C₈	D₅₅₆	1	2	yes	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	278	2	no	—
93	C₄⋊C₄	—	278	2	no	—
94	C₄⋊C₄	—	556	4	no	—
95	C₄⋊C₄	Dic₂₇₈	1	4	yes	—
96	C₄²	—	278	1	no	—
97	C₄²	—	278	1	no	—
98	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	278	1	no	—
103	C₂×C₄	—	278	1	no	—
104	C₂×C₄	—	278	1	no	—
105	C₂×C₄	—	278	2	no	—
106	C₂×C₄	—	556	2	no	—
107	C₂×C₄	—	556	2	no	—
108	C₂×C₄	—	556	2	no	—
109	C₂×C₄	—	139	2	no	—
110	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
111	C₂×C₄	C₂×Dic₂₇₈	1	2	yes	—
112	C₂×C₄	#14555784	1	2	yes	—
113	C₂×C₄	#14555786	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	556	1	no	—
116	C₂²	#17984817	1	1	yes	characteristic
117	C₂²	#17984819	1	2	yes	—
118	C₂²	Q₁₆×D₁₃₉	1	1	yes	characteristic
119	C₂²	#17984861	1	1	yes	characteristic
120	C₂²	#17984869	1	2	yes	—
121	C₄	—	4	4	no	—
122	C₄	—	278	2	no	—
123	C₄	—	556	1	no	—
124	C₄	—	1112	2	no	—
125	C₄	#17984826	1	1	yes	characteristic
126	C₄	#17984833	1	2	yes	—
127	C₄	#17984842	1	1	yes	characteristic
128	C₂	#22580885	1	1	yes	characteristic
129	C₂	#22581195	1	1	yes	characteristic
130	C₂	#22581207	1	1	yes	characteristic
131	C₂	#22581246	1	2	yes	—
132	C₂	#22581251	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984776

C₂

yes

maximal, characteristic

C₂×Dic₁₃₉⋊C₈

C₂

yes

maximal, characteristic

#22580852

C₂

yes

maximal

#22580715

C₂

yes

maximal

#340×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃₉

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

C₂×C₄×Dic₁₃₉

C₂²

yes

characteristic

#17984776

—

Dic₁₃₉⋊C₈

—

C₁₃₉⋊#95

—

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₃₉

C₂²

yes

—

C₂²×Dic₁₃₉

—

C₂²×Dic₁₃₉

D₄

yes

characteristic

C₂²×Dic₁₃₉

D₄

yes

characteristic

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

D₄

yes

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

Q₈

yes

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

Q₁₆

yes

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

—

C₂×C₄×C₁₃₉

#41

yes

—

C₂×C₄×C₁₃₉

#41

yes

characteristic

C₈×C₁₃₉

—

C₂²×C₁₃₉

#121

yes

characteristic

C₂²×C₁₃₉

C₂²⋊Q₈

yes

—

C₂²×C₁₃₉

C₂×Q₁₆

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#123

yes

characteristic

C₄×C₁₃₉

#125

yes

—

C₄×C₁₃₉

C₄⋊Q₈

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

—

C₂×C₁₃₉

#478

yes

—

C₁₃₉

#2087

yes

characteristic, complemented, Sylow

#2087

—

139

maximal, Sylow

C₂×#107

—

139

—

C₂×C₄⋊C₈

—

139

—

#389

—

139

—

#340

—

139

—

#340

D₁₃₉

yes

characteristic

C₂²×C₈

—

139

—

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

C₂×C₄⋊C₄

—

278

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄⋊C₄

C₂×D₁₃₉

yes

—

C₂×C₄²

—

139

—

#107

—

278

—

C₄⋊C₈

—

278

—

#95

—

278

—

C₂²×C₄

—

C₂²×C₄

—

278

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

278

—

C₂×C₈

—

278

—

C₂×C₈

D₅₅₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

556

—

C₄⋊C₄

Dic₂₇₈

yes

—

C₄²

—

278

—

C₄²

—

278

—

C₂³

D₄×D₁₃₉

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

278

—

103

C₂×C₄

—

278

—

104

C₂×C₄

—

278

—

105

C₂×C₄

—

278

—

106

C₂×C₄

—

556

—

107

C₂×C₄

—

556

—

108

C₂×C₄

—

556

—

109

C₂×C₄

—

139

—

110

C₂×C₄

C₂×D₅₅₆

yes

characteristic

111

C₂×C₄

C₂×Dic₂₇₈

yes

—

112

C₂×C₄

#14555784

yes

—

113

C₂×C₄

#14555786

yes

characteristic

114

C₈

—

115

C₈

—

556

—

116

C₂²

#17984817

yes

characteristic

117

C₂²

#17984819

yes

—

118

C₂²

Q₁₆×D₁₃₉

yes

characteristic

119

C₂²

#17984861

yes

characteristic

120

C₂²

#17984869

yes

—

121

C₄

—

122

C₄

—

278

—

123

C₄

—

556

—

124

C₄

—

1112

—

125

C₄

#17984826

yes

characteristic

126

C₄

#17984833

yes

—

127

C₄

#17984842

yes

characteristic

128

C₂

#22580885

yes

characteristic

129

C₂

#22581195

yes

characteristic

130

C₂

#22581207

yes

characteristic

131

C₂

#22581246

yes

—

132

C₂

#22581251

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₃₉
commutator, cosocle	C₂²×C₄×C₁₃₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#340×C₁₃₉	C₂
socle, prime radical	C₂³×C₁₃₉	C₂×D₄
hypercenter	#340	D₁₃₉
hypocenter	C₁₃₉	#2087
LCS	G ▷ C₂²×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₁₃₉

commutator, cosocle

C₂²×C₄×C₁₃₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#340×C₁₃₉

C₂

socle, prime radical

C₂³×C₁₃₉

C₂×D₄

D₁₃₉

C₁₃₉

G ▷ C₂²×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
66. C₁₃₉	#2087	semidirect

H ≅ G/N

type

66. C₁₃₉

#2087

semidirect

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

subgroup	index
C₂×#17984776	2
C₂×Dic₁₃₉⋊C₈	2
#22580852	2
#22580715	2
#340×C₁₃₉	2
#2087	139

subgroup

index

C₂×Dic₁₃₉⋊C₈

#340×C₁₃₉

139

Subgroups — #31807899

Conjugacy classes of subgroups

Complements

Maximal subgroups