Group #31806529 — subgroups

Subgroups — #31806529

Order 17792.

172 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 81 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	#22580991	C₂	1	1	yes	maximal, characteristic
2	#22580908	C₂	1	1	yes	maximal, characteristic
3	#22580725	C₂	1	2	yes	maximal
4	#22580710	C₂	1	2	yes	maximal
5	C₂×#108×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555769	C₂²	1	1	yes	characteristic
7	#17984801	C₂²	1	2	yes	—
8	#17984801	C₂²	1	1	yes	characteristic
9	#17984776	—	2	1	no	—
10	#17984774	—	2	1	no	—
11	#17984752	C₄	1	2	yes	complemented
12	C₈×Dic₁₃₉	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₁₃₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	#108×C₁₃₉	—	2	1	no	—
16	#108×C₁₃₉	—	2	1	no	—
17	#14555769	—	2	1	no	—
18	#14555769	—	2	1	no	—
19	C₂×C₁₃₉⋊C₈	C₂×C₄	1	2	yes	—
20	C₄⋊Dic₁₃₉	—	2	2	no	—
21	C₄⋊Dic₁₃₉	—	2	1	no	—
22	C₂²⋊Dic₁₃₉	—	2	1	no	—
23	C₂²⋊Dic₁₃₉	—	2	2	no	—
24	C₄×Dic₁₃₉	—	2	2	no	—
25	C₄×Dic₁₃₉	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₃₉	—	2	2	no	—
28	M₁₆×C₁₃₉	—	2	2	no	—
29	C₂×C₈×C₁₃₉	—	2	2	no	—
30	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₃₉	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₃₉	D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₃₉	Q₈	1	1	yes	characteristic
35	C₂×C₈×C₁₃₉	Q₈	1	1	yes	characteristic
36	C₂×Dic₁₃₉	—	2	2	no	—
37	C₂×Dic₁₃₉	—	4	2	no	—
38	C₁₃₉⋊C₈	—	2	2	no	—
39	C₂³×C₁₃₉	#41	1	1	yes	characteristic
40	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
44	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
45	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
46	C₈×C₁₃₉	—	4	2	no	—
47	C₈×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
48	C₈×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
49	C₈×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
50	C₈×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₃₉	—	4	1	no	—
52	C₂²×C₁₃₉	#117	1	1	yes	characteristic
53	C₂²×C₁₃₉	#124	1	1	yes	characteristic
54	C₂²×C₁₃₉	#125	1	1	yes	characteristic
55	C₄×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₁₃₉	#117	1	1	yes	characteristic
57	C₄×C₁₃₉	#124	1	1	yes	characteristic
58	C₄×C₁₃₉	#125	1	1	yes	characteristic
59	Dic₁₃₉	—	4	2	no	—
60	Dic₁₃₉	—	8	2	no	—
61	C₂×C₁₃₉	—	2	1	no	—
62	C₂×C₁₃₉	—	2	1	no	—
63	C₂×C₁₃₉	—	2	1	no	—
64	C₂×C₁₃₉	#382	1	1	yes	characteristic
65	C₁₃₉	#1833	1	1	yes	characteristic, complemented, Sylow
66	#1833	—	139	1	no	maximal, Sylow
67	C₂×#108	D₁₃₉	1	1	yes	characteristic
68	#427	—	139	1	no	—
69	#405	—	139	1	no	—
70	#344	—	139	2	no	—
71	H(3,ℤ/4)	—	139	2	no	—
72	C₂×M₁₆	—	139	1	no	—
73	C₂×M₁₆	C₂×D₁₃₉	1	2	yes	—
74	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
75	#117	—	139	2	no	—
76	#117	—	139	1	no	—
77	#108	—	2	1	no	—
78	#108	—	2	1	no	—
79	#107	—	278	1	no	—
80	#106	—	278	1	no	—
81	#97	—	139	2	no	—
82	C₄×C₈	—	139	2	no	—
83	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	2	2	no	—
86	M₁₆	—	278	1	no	—
87	M₁₆	—	278	1	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	139	2	no	—
90	C₂×C₈	C₄×D₁₃₉	1	1	yes	characteristic
91	C₂×C₈	C₄×D₁₃₉	1	1	yes	characteristic
92	C₂×C₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
93	C₂×C₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
94	C₂×C₈	Dic₂₇₈	1	1	yes	characteristic
95	C₂×C₈	Dic₂₇₈	1	1	yes	characteristic
96	C₄⋊C₄	—	278	1	no	—
97	C₄⋊C₄	—	278	2	no	—
98	C₂²⋊C₄	—	278	1	no	—
99	C₂²⋊C₄	—	278	2	no	—
100	C₄²	—	278	2	no	—
101	C₄²	—	139	2	no	—
102	C₂³	#14555786	1	1	yes	characteristic
103	C₂×C₄	—	278	2	no	—
104	C₂×C₄	—	556	2	no	—
105	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₂₇₈	1	1	yes	characteristic
108	C₂×C₄	#14555784	1	1	yes	characteristic
109	C₂×C₄	#14555784	1	1	yes	characteristic
110	C₂×C₄	#14555786	1	1	yes	characteristic
111	C₈	—	4	2	no	—
112	C₈	—	278	2	no	—
113	C₈	Dic₁₃₉⋊C₄	1	1	yes	characteristic
114	C₈	Dic₁₃₉⋊C₄	1	1	yes	characteristic
115	C₈	Dic₁₃₉⋊C₄	1	1	yes	characteristic
116	C₈	Dic₁₃₉⋊C₄	1	1	yes	characteristic
117	C₂²	—	4	1	no	—
118	C₂²	#17984800	1	1	yes	characteristic
119	C₂²	#17984831	1	1	yes	characteristic
120	C₂²	#17984833	1	1	yes	characteristic
121	C₄	—	556	2	no	—
122	C₄	—	1112	2	no	—
123	C₄	C₂×Dic₁₃₉⋊C₄	1	1	yes	characteristic
124	C₄	#17984800	1	1	yes	characteristic
125	C₄	#17984829	1	1	yes	characteristic
126	C₄	#17984833	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#22580818	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22580991

C₂

yes

maximal, characteristic

#22580908

C₂

yes

maximal, characteristic

#22580725

C₂

yes

maximal

#22580710

C₂

yes

maximal

C₂×#108×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555769

C₂²

yes

characteristic

#17984801

C₂²

yes

—

#17984801

C₂²

yes

—

—

C₄

yes

complemented

C₈×Dic₁₃₉

C₄

yes

complemented

C₂×M₁₆×C₁₃₉

C₂²

yes

—

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

#108×C₁₃₉

—

#108×C₁₃₉

—

#14555769

—

#14555769

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₂²⋊Dic₁₃₉

—

C₂²⋊Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₉

D₄

yes

characteristic

C₂×C₈×C₁₃₉

D₄

yes

characteristic

C₂×C₈×C₁₃₉

Q₈

yes

characteristic

C₂×C₈×C₁₃₉

Q₈

yes

characteristic

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

#41

yes

characteristic

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

C₂×C₄×C₁₃₉

yes

C₂×C₄×C₁₃₉

yes

C₂×C₄×C₁₃₉

yes

C₈×C₁₃₉

—

C₈×C₁₃₉

C₄⋊C₄

yes

characteristic

C₈×C₁₃₉

C₄⋊C₄

yes

characteristic

C₈×C₁₃₉

C₄⋊C₄

yes

characteristic

C₈×C₁₃₉

C₄⋊C₄

yes

characteristic

C₂²×C₁₃₉

—

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₄×C₁₃₉

C₂×C₄⋊C₄

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#382

yes

characteristic

C₁₃₉

#1833

yes

characteristic, complemented, Sylow

#1833

—

139

maximal, Sylow

C₂×#108

D₁₃₉

yes

characteristic

#427

—

139

—

#405

—

139

—

#344

—

139

—

H(3,ℤ/4)

—

139

—

C₂×M₁₆

—

139

—

C₂×M₁₆

C₂×D₁₃₉

yes

—

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

#117

—

139

—

#117

—

139

—

#108

—

#108

—

#107

—

278

—

#106

—

278

—

#97

—

139

—

C₄×C₈

—

139

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

278

—

M₁₆

—

278

—

C₂×C₈

—

C₂×C₈

—

139

—

C₂×C₈

C₄×D₁₃₉

yes

characteristic

C₂×C₈

C₄×D₁₃₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₃₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₃₉

yes

characteristic

C₂×C₈

Dic₂₇₈

yes

characteristic

C₂×C₈

Dic₂₇₈

yes

characteristic

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₂²⋊C₄

—

278

—

C₂²⋊C₄

—

278

—

100

C₄²

—

278

—

101

C₄²

—

139

—

102

C₂³

#14555786

yes

characteristic

103

C₂×C₄

—

278

—

104

C₂×C₄

—

556

—

105

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

107

C₂×C₄

C₂×Dic₂₇₈

yes

characteristic

108

C₂×C₄

#14555784

yes

characteristic

109

C₂×C₄

#14555784

yes

characteristic

110

C₂×C₄

#14555786

yes

characteristic

111

C₈

—

112

C₈

—

278

—

113

C₈

Dic₁₃₉⋊C₄

yes

characteristic

114

C₈

Dic₁₃₉⋊C₄

yes

characteristic

115

C₈

Dic₁₃₉⋊C₄

yes

characteristic

116

C₈

Dic₁₃₉⋊C₄

yes

characteristic

117

C₂²

—

118

C₂²

#17984800

yes

characteristic

119

C₂²

#17984831

yes

characteristic

120

C₂²

#17984833

yes

characteristic

121

C₄

—

556

—

122

C₄

—

1112

—

123

C₄

C₂×Dic₁₃₉⋊C₄

yes

characteristic

124

C₄

#17984800

yes

characteristic

125

C₄

#17984829

yes

characteristic

126

C₄

#17984833

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#22580818

yes

characteristic

	subgroup	G/H
center	C₄	#17984800
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	C₂×#108×C₁₃₉	C₂
socle	C₂×C₁₃₉	#382
prime radical	C₂³×C₁₃₉	#41
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	C₂×#108	D₁₃₉
hypocenter	C₁₃₉	#1833
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#17984800

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

C₂×#108×C₁₃₉

C₂

C₂×C₁₃₉

C₂³×C₁₃₉

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
11. #17984752	C₄	semidirect
12. C₈×Dic₁₃₉	C₄	semidirect
65. C₁₃₉	#1833	semidirect

H ≅ G/N

type

11. #17984752

C₄

semidirect

12. C₈×Dic₁₃₉

C₄

semidirect

65. C₁₃₉

#1833

semidirect

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

subgroup	index
#22580991	2
#22580908	2
#22580725	2
#22580710	2
C₂×#108×C₁₃₉	2
#1833	139

subgroup

index

C₂×#108×C₁₃₉

139

Subgroups — #31806529

Conjugacy classes of subgroups

Complements

Maximal subgroups