Group #31807150 — subgroups

Subgroups — #31807150

Order 17792.

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

normal subgroups only

This group has 208 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	#22580991	C₂	1	1	yes	maximal, characteristic
2	#22580981	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
4	#22580714	C₂	1	2	yes	maximal
5	C₂×#108×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
7	#17984801	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
9	#17984776	C₄	1	2	yes	complemented
10	#17984774	C₄	1	2	yes	complemented
11	Dic₁₃₉⋊C₈	—	2	4	no	—
12	C₈×Dic₁₃₉	—	2	2	no	—
13	C₂×M₁₆×C₁₃₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	#108×C₁₃₉	—	2	2	no	—
16	C₂²×Dic₁₃₉	—	2	1	no	—
17	C₂×C₁₃₉⋊C₈	—	2	1	no	—
18	C₂×C₁₃₉⋊C₈	—	2	1	no	—
19	C₂×C₁₃₉⋊C₈	D₄	1	2	yes	—
20	C₄⋊Dic₁₃₉	C₂×C₄	1	4	yes	—
21	C₂²⋊Dic₁₃₉	—	2	2	no	—
22	C₄×Dic₁₃₉	—	2	2	no	—
23	C₄×Dic₁₃₉	—	2	1	no	—
24	C₄×Dic₁₃₉	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₃₉	—	2	2	no	—
27	M₁₆×C₁₃₉	—	2	2	no	—
28	C₂×C₈×C₁₃₉	—	2	2	no	—
29	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₃₉	D₄	1	2	yes	—
32	C₂×C₈×C₁₃₉	Q₈	1	2	yes	—
33	C₂×Dic₁₃₉	—	2	4	no	—
34	C₂×Dic₁₃₉	—	2	2	no	—
35	C₂×Dic₁₃₉	—	2	4	no	—
36	C₁₃₉⋊C₈	—	2	2	no	—
37	C₂³×C₁₃₉	#41	1	1	yes	characteristic
38	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
44	C₈×C₁₃₉	—	4	2	no	—
45	C₈×C₁₃₉	C₄⋊C₄	1	2	yes	—
46	C₈×C₁₃₉	C₄⋊C₄	1	2	yes	—
47	C₂²×C₁₃₉	—	2	2	no	—
48	C₂²×C₁₃₉	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₃₉	#125	1	1	yes	characteristic
51	C₄×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₁₃₉	C₄×Q₈	1	1	yes	characteristic
53	C₄×C₁₃₉	#121	1	1	yes	characteristic
54	C₄×C₁₃₉	C₄⋊Q₈	1	1	yes	characteristic
55	Dic₁₃₉	—	2	4	no	—
56	Dic₁₃₉	—	4	4	no	—
57	C₂×C₁₃₉	—	2	1	no	—
58	C₂×C₁₃₉	—	2	1	no	—
59	C₂×C₁₃₉	#389	1	1	yes	characteristic
60	C₂×C₁₃₉	#443	1	2	yes	—
61	C₁₃₉	#1945	1	1	yes	characteristic, complemented, Sylow
62	#1945	—	139	1	no	maximal, Sylow
63	C₂×#108	D₁₃₉	1	1	yes	characteristic
64	#427	—	139	1	no	—
65	#424	—	139	2	no	—
66	C₂×C₄×C₈	—	139	1	no	—
67	#339	—	139	2	no	—
68	C₂×M₁₆	C₂×D₁₃₉	1	2	yes	—
69	C₂²×C₈	—	139	1	no	—
70	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
71	#117	—	139	2	no	—
72	C₂×C₄²	—	139	1	no	—
73	#108	—	2	2	no	—
74	#107	—	139	2	no	—
75	#106	—	139	2	no	—
76	C₄⋊C₈	—	278	4	no	—
77	C₄×C₈	—	278	2	no	—
78	C₂²×C₄	—	278	1	no	—
79	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
80	M₁₆	—	2	2	no	—
81	M₁₆	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	278	1	no	—
84	C₂×C₈	—	278	1	no	—
85	C₂×C₈	—	139	2	no	—
86	C₂×C₈	C₄×D₁₃₉	1	1	yes	characteristic
87	C₂×C₈	C₄×D₁₃₉	1	1	yes	characteristic
88	C₂×C₈	C₂²⋊D₁₃₉	1	2	yes	—
89	C₂×C₈	Dic₂₇₈	1	2	yes	—
90	C₄⋊C₄	—	139	4	no	—
91	C₂²⋊C₄	—	278	2	no	—
92	C₄²	—	278	2	no	—
93	C₄²	—	278	1	no	—
94	C₄²	—	139	2	no	—
95	C₂³	#14555786	1	1	yes	characteristic
96	C₂×C₄	—	278	4	no	—
97	C₂×C₄	—	278	2	no	—
98	C₂×C₄	—	278	4	no	—
99	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
101	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₁₃₉	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₂₇₈	1	1	yes	characteristic
104	C₂×C₄	#14555784	1	1	yes	characteristic
105	C₈	—	4	2	no	—
106	C₈	—	278	2	no	—
107	C₈	Dic₁₃₉⋊C₄	1	2	yes	—
108	C₈	Dic₁₃₉⋊C₄	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	#17984804	1	1	yes	characteristic
111	C₂²	#17984818	1	1	yes	characteristic
112	C₂²	#17984833	1	1	yes	characteristic
113	C₄	—	278	4	no	—
114	C₄	—	556	4	no	—
115	C₄	C₂×Dic₁₃₉⋊C₄	1	1	yes	characteristic
116	C₄	#17984807	1	1	yes	characteristic
117	C₄	#17984813	1	1	yes	characteristic
118	C₄	#17984841	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#22580849	1	1	yes	characteristic
122	C₂	#22581069	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22580991

C₂

yes

maximal, characteristic

#22580981

C₂

yes

maximal

C₂×C₈×Dic₁₃₉

C₂

yes

maximal, characteristic

#22580714

C₂

yes

maximal

C₂×#108×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×C₁₃₉⋊C₈

C₂²

yes

characteristic

#17984801

C₂²

yes

—

C₂×C₄×Dic₁₃₉

C₂²

yes

characteristic

#17984776

C₄

yes

complemented

#17984774

C₄

yes

complemented

Dic₁₃₉⋊C₈

—

C₈×Dic₁₃₉

—

C₂×M₁₆×C₁₃₉

C₂²

yes

—

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

#108×C₁₃₉

—

C₂²×Dic₁₃₉

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

D₄

yes

—

C₄⋊Dic₁₃₉

C₂×C₄

yes

—

C₂²⋊Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

Q₈

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

—

C₂×C₈×C₁₃₉

Q₈

yes

—

C₂×Dic₁₃₉

—

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×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

characteristic

C₈×C₁₃₉

—

C₈×C₁₃₉

C₄⋊C₄

yes

—

C₈×C₁₃₉

C₄⋊C₄

yes

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₄×D₄

yes

characteristic

C₂²×C₁₃₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₃₉

#125

yes

characteristic

C₄×C₁₃₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₃₉

C₄×Q₈

yes

characteristic

C₄×C₁₃₉

#121

yes

characteristic

C₄×C₁₃₉

C₄⋊Q₈

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#389

yes

characteristic

C₂×C₁₃₉

#443

yes

—

C₁₃₉

#1945

yes

characteristic, complemented, Sylow

#1945

—

139

maximal, Sylow

C₂×#108

D₁₃₉

yes

characteristic

#427

—

139

—

#424

—

139

—

C₂×C₄×C₈

—

139

—

#339

—

139

—

C₂×M₁₆

C₂×D₁₃₉

yes

—

C₂²×C₈

—

139

—

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

#117

—

139

—

C₂×C₄²

—

139

—

#108

—

#107

—

139

—

#106

—

139

—

C₄⋊C₈

—

278

—

C₄×C₈

—

278

—

C₂²×C₄

—

278

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

278

—

C₂×C₈

—

278

—

C₂×C₈

—

139

—

C₂×C₈

C₄×D₁₃₉

yes

characteristic

C₂×C₈

C₄×D₁₃₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₃₉

yes

—

C₂×C₈

Dic₂₇₈

yes

—

C₄⋊C₄

—

139

—

C₂²⋊C₄

—

278

—

C₄²

—

278

—

C₄²

—

278

—

C₄²

—

139

—

C₂³

#14555786

yes

characteristic

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

100

C₂×C₄

D₄×D₁₃₉

yes

characteristic

101

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

102

C₂×C₄

Q₈×D₁₃₉

yes

characteristic

103

C₂×C₄

C₂×Dic₂₇₈

yes

characteristic

104

C₂×C₄

#14555784

yes

characteristic

105

C₈

—

106

C₈

—

278

—

107

C₈

Dic₁₃₉⋊C₄

yes

—

108

C₈

Dic₁₃₉⋊C₄

yes

—

109

C₂²

—

110

C₂²

#17984804

yes

characteristic

111

C₂²

#17984818

yes

characteristic

112

C₂²

#17984833

yes

characteristic

113

C₄

—

278

—

114

C₄

—

556

—

115

C₄

C₂×Dic₁₃₉⋊C₄

yes

characteristic

116

C₄

#17984807

yes

characteristic

117

C₄

#17984813

yes

characteristic

118

C₄

#17984841

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#22580849

yes

characteristic

122

C₂

#22581069

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	C₂×#108×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₄×D₄
prime radical	C₂³×C₁₃₉	#41
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	C₂×#108	D₁₃₉
hypocenter	C₁₃₉	#1945
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₁₃₉

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

C₂×#108×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₄×D₄

prime radical

C₂³×C₁₃₉

#41

cosocle

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
9. #17984776	C₄	semidirect
10. #17984774	C₄	semidirect
61. C₁₃₉	#1945	semidirect

H ≅ G/N

type

9. #17984776

C₄

semidirect

10. #17984774

C₄

semidirect

61. C₁₃₉

#1945

semidirect

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

subgroup	index
#22580991	2
#22580981	2
C₂×C₈×Dic₁₃₉	2
#22580714	2
C₂×#108×C₁₃₉	2
#1945	139

subgroup

index

C₂×C₈×Dic₁₃₉

C₂×#108×C₁₃₉

139

Subgroups — #31807150

Conjugacy classes of subgroups

Complements

Maximal subgroups