Group #31805159 — subgroups

Subgroups — #31805159

Order 17792.

184 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 97 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	#22580977	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22580703	C₂	1	4	yes	maximal, complemented
4	#423×C₁₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14555769	—	2	1	no	—
6	C₂²×C₁₃₉⋊C₈	C₄	1	2	yes	—
7	C₄⋊(C₁₃₉⋊C₈)	—	2	1	no	—
8	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	2	yes	—
9	C₄×C₁₃₉⋊C₈	C₄	1	1	yes	characteristic
10	C₄×C₁₃₉⋊C₈	C₄	1	1	yes	characteristic
11	C₄×C₁₃₉⋊C₈	C₂²	1	2	yes	—
12	C₂×M₁₆×C₁₃₉	—	2	1	no	—
13	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₃₉	—	2	1	no	—
15	C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
16	#14555769	—	4	2	no	—
17	C₂×C₁₃₉⋊C₈	—	2	4	no	—
18	C₂×C₁₃₉⋊C₈	—	2	2	no	—
19	C₂×C₁₃₉⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₁₃₉⋊C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₃₉	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₃₉	Q₈	1	1	yes	characteristic
24	M₁₆×C₁₃₉	—	4	2	no	—
25	C₂×C₈×C₁₃₉	—	2	2	no	—
26	C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₁₃₉	D₄	1	1	yes	characteristic
28	C₄²×C₁₃₉	Q₈	1	1	yes	characteristic
29	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
30	C₁₃₉⋊C₈	—	2	2	no	—
31	C₁₃₉⋊C₈	—	4	2	no	—
32	C₁₃₉⋊C₈	M₁₆	1	4	yes	complemented
33	C₂³×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₃₉	—	2	2	no	—
35	C₂×C₄×C₁₃₉	—	2	2	no	—
36	C₂×C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃₉	SD₁₆	1	2	yes	—
40	C₂×C₄×C₁₃₉	SD₁₆	1	2	yes	—
41	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
44	C₈×C₁₃₉	—	4	2	no	—
45	C₂²×C₁₃₉	—	2	1	no	—
46	C₂²×C₁₃₉	—	2	1	no	—
47	C₂²×C₁₃₉	#106	1	2	yes	—
48	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₄×C₁₃₉	—	2	1	no	—
50	C₄×C₁₃₉	—	2	1	no	—
51	C₄×C₁₃₉	#106	1	2	yes	—
52	C₄×C₁₃₉	#108	1	2	yes	—
53	C₄×C₁₃₉	C₂×M₁₆	1	2	yes	—
54	C₄×C₁₃₉	C₂×SD₁₆	1	1	yes	characteristic
55	C₄×C₁₃₉	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×C₁₃₉	—	2	2	no	—
57	C₂×C₁₃₉	#423	1	1	yes	characteristic
58	C₂×C₁₃₉	C₂×#106	1	1	yes	characteristic
59	C₂×C₁₃₉	C₂×#108	1	1	yes	characteristic
60	C₁₃₉	#1566	1	1	yes	characteristic, complemented, Sylow
61	#1566	—	139	1	no	maximal, Sylow
62	#423	—	139	1	no	—
63	#423	D₁₃₉	1	1	yes	characteristic
64	C₂×C₄×C₈	—	139	1	no	—
65	#334	—	139	4	no	—
66	C₂×M₁₆	—	2	1	no	—
67	C₂×M₁₆	—	278	1	no	—
68	C₂²×C₈	—	139	2	no	—
69	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	—	278	1	no	—
72	C₄⋊C₈	—	139	2	no	—
73	C₄⋊C₈	C₂×D₁₃₉	1	2	yes	—
74	C₄×C₈	—	139	1	no	—
75	C₄×C₈	—	139	1	no	—
76	C₄×C₈	—	139	2	no	—
77	C₂²×C₄	C₄×D₁₃₉	1	1	yes	characteristic
78	C₂²×C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
79	C₂²×C₄	Dic₂₇₈	1	1	yes	characteristic
80	M₁₆	—	4	2	no	—
81	M₁₆	—	556	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	278	4	no	—
84	C₂×C₈	—	278	2	no	—
85	C₂×C₈	—	139	2	no	—
86	C₂×C₈	—	139	2	no	—
87	C₄²	C₄×D₁₃₉	1	1	yes	characteristic
88	C₄²	C₂²⋊D₁₃₉	1	1	yes	characteristic
89	C₄²	Dic₂₇₈	1	1	yes	characteristic
90	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
91	C₂³	Dic₁₃₉⋊C₄	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	Dic₁₃₉⋊C₄	1	1	yes	characteristic
95	C₂×C₄	Dic₁₃₉⋊C₄	1	1	yes	characteristic
96	C₂×C₄	Dic₁₃₉⋊C₄	1	1	yes	characteristic
97	C₂×C₄	#14555772	1	2	yes	—
98	C₂×C₄	#14555773	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
101	C₂×C₄	C₂×Dic₂₇₈	1	1	yes	characteristic
102	C₈	—	4	2	no	—
103	C₈	—	278	2	no	—
104	C₈	—	556	2	no	—
105	C₈	—	139	4	no	—
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	#17984773	1	2	yes	—
109	C₂²	C₂×Dic₁₃₉⋊C₄	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	#17984773	1	2	yes	—
113	C₄	#17984777	1	2	yes	—
114	C₄	M₁₆×D₁₃₉	1	2	yes	—
115	C₄	C₂×#14555772	1	1	yes	characteristic
116	C₄	C₂×#14555773	1	1	yes	characteristic
117	C₂	—	2	2	no	—
118	C₂	#22580976	1	1	yes	characteristic
119	C₂	C₂×#17984773	1	1	yes	characteristic
120	C₂	C₂×#17984777	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22580977

C₂

yes

maximal, characteristic

C₂×C₄×C₁₃₉⋊C₈

C₂

yes

maximal, characteristic

#22580703

C₂

yes

maximal, complemented

#423×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555769

—

C₂²×C₁₃₉⋊C₈

C₄

yes

—

C₄⋊(C₁₃₉⋊C₈)

—

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

—

C₄×C₁₃₉⋊C₈

C₄

yes

characteristic

C₄×C₁₃₉⋊C₈

C₄

yes

characteristic

C₄×C₁₃₉⋊C₈

C₂²

yes

—

C₂×M₁₆×C₁₃₉

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₉

—

C₄⋊C₈×C₁₃₉

C₂²

yes

—

#14555769

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₉

D₄

yes

characteristic

C₂²×C₄×C₁₃₉

Q₈

yes

characteristic

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₄²×C₁₃₉

C₂×C₄

yes

characteristic

C₄²×C₁₃₉

D₄

yes

characteristic

C₄²×C₁₃₉

Q₈

yes

characteristic

C₄²×C₁₃₉

C₂³

yes

characteristic

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

M₁₆

yes

complemented

C₂³×C₁₃₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₉

SD₁₆

yes

—

C₂×C₄×C₁₃₉

SD₁₆

yes

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

characteristic

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

#106

yes

—

C₂²×C₁₃₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#106

yes

—

C₄×C₁₃₉

#108

yes

—

C₄×C₁₃₉

C₂×M₁₆

yes

—

C₄×C₁₃₉

C₂×SD₁₆

yes

characteristic

C₄×C₁₃₉

C₂×SD₁₆

yes

characteristic

C₂×C₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#1566

—

139

maximal, Sylow

#423

—

139

—

#423

D₁₃₉

yes

characteristic

C₂×C₄×C₈

—

139

—

#334

—

139

—

C₂×M₁₆

—

C₂×M₁₆

—

278

—

C₂²×C₈

—

139

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

278

—

C₄⋊C₈

—

139

—

C₄⋊C₈

C₂×D₁₃₉

yes

—

C₄×C₈

—

139

—

C₄×C₈

—

139

—

C₄×C₈

—

139

—

C₂²×C₄

C₄×D₁₃₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₃₉

yes

characteristic

C₂²×C₄

Dic₂₇₈

yes

characteristic

M₁₆

—

M₁₆

—

556

—

C₂×C₈

—

C₂×C₈

—

278

—

C₂×C₈

—

278

—

C₂×C₈

—

139

—

C₂×C₈

—

139

—

C₄²

C₄×D₁₃₉

yes

characteristic

C₄²

C₂²⋊D₁₃₉

yes

characteristic

C₄²

Dic₂₇₈

yes

characteristic

C₄²

C₂²×D₁₃₉

yes

characteristic

C₂³

Dic₁₃₉⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

Dic₁₃₉⋊C₄

yes

characteristic

C₂×C₄

Dic₁₃₉⋊C₄

yes

characteristic

C₂×C₄

Dic₁₃₉⋊C₄

yes

characteristic

C₂×C₄

#14555772

yes

—

C₂×C₄

#14555773

yes

—

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

101

C₂×C₄

C₂×Dic₂₇₈

yes

characteristic

102

C₈

—

103

C₈

—

278

—

104

C₈

—

556

—

105

C₈

—

139

—

106

C₂²

—

107

C₂²

—

108

C₂²

#17984773

yes

—

109

C₂²

C₂×Dic₁₃₉⋊C₄

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

#17984773

yes

—

113

C₄

#17984777

yes

—

114

C₄

M₁₆×D₁₃₉

yes

—

115

C₄

C₂×#14555772

yes

characteristic

116

C₄

C₂×#14555773

yes

characteristic

117

C₂

—

118

C₂

#22580976

yes

characteristic

119

C₂

C₂×#17984773

yes

characteristic

120

C₂

C₂×#17984777

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₃₉
Fitting	#423×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×C₄⋊C₄
prime radical	C₂³×C₁₃₉	C₄⋊C₄
cosocle	C₄²×C₁₃₉	C₂³
hypercenter	#423	D₁₃₉
hypocenter	C₁₃₉	#1566
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₃₉

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₄²

C₂²×D₁₃₉

Fitting

#423×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂×C₄⋊C₄

prime radical

C₂³×C₁₃₉

C₄⋊C₄

cosocle

C₄²×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
3. #22580703	C₂	semidirect
32. C₁₃₉⋊C₈	M₁₆	semidirect
60. C₁₃₉	#1566	semidirect

H ≅ G/N

type

3. #22580703

C₂

semidirect

32. C₁₃₉⋊C₈

M₁₆

semidirect

60. C₁₃₉

#1566

semidirect

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

subgroup	index
#22580977	2
C₂×C₄×C₁₃₉⋊C₈	2
#22580703	2
#423×C₁₃₉	2
#1566	139

subgroup

index

#22580977

C₂×C₄×C₁₃₉⋊C₈

#22580703

#423×C₁₃₉

#1566

139

Subgroups — #31805159

Conjugacy classes of subgroups

Complements

Maximal subgroups