Group #31808310 — subgroups

Subgroups — #31808310

Order 17792.

126 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 71 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	C₈×Dic₂₇₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₁₃₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	Dic₁₃₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	Dic₁₃₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#22580728	C₂	1	1	yes	maximal, characteristic
6	C₁₆×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
7	#346×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₂₇₈	C₄	1	1	yes	characteristic
9	C₂×C₁₃₉⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₃₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₄⋊(C₁₃₉⋊C₈)	C₄	1	1	yes	characteristic
12	Dic₁₃₉⋊C₈	C₄	1	1	yes	characteristic
13	Dic₁₃₉⋊C₈	C₄	1	1	yes	characteristic
14	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
15	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
18	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₇₈	C₈	1	1	yes	characteristic
20	C₂×C₁₃₉⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₃₉⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₄⋊Dic₁₃₉	C₈	1	1	yes	characteristic
23	Dic₁₃₉⋊C₄	C₈	1	1	yes	characteristic
24	Dic₁₃₉⋊C₄	C₈	1	1	yes	characteristic
25	C₄×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
27	C₁₃₉⋊C₁₆	—	2	1	no	—
28	C₁₃₉⋊C₁₆	—	2	1	no	—
29	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
31	C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
32	C₁₆×C₁₃₉	—	2	1	no	—
33	C₁₆×C₁₃₉	Q₈	1	2	yes	complemented
34	Dic₂₇₈	—	2	2	no	—
35	C₂×Dic₁₃₉	C₂×C₈	1	1	yes	characteristic
36	C₂×Dic₁₃₉	C₂×C₈	1	1	yes	characteristic
37	C₂×Dic₁₃₉	C₂×C₈	1	1	yes	characteristic
38	C₂×Dic₁₃₉	C₂×C₈	1	1	yes	characteristic
39	C₁₃₉⋊C₈	—	2	1	no	—
40	C₁₃₉⋊C₈	—	2	1	no	—
41	C₂×C₄×C₁₃₉	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄×C₁₃₉	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
44	C₈×C₁₃₉	—	2	1	no	—
45	C₈×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
46	C₈×C₁₃₉	#41	1	1	yes	characteristic
47	C₂²×C₁₃₉	C₂²×C₈	1	1	yes	characteristic
48	C₄×C₁₃₉	—	2	1	no	—
49	C₄×C₁₃₉	—	2	1	no	—
50	C₄×C₁₃₉	C₄×Q₈	1	1	yes	characteristic
51	C₄×C₁₃₉	#131	1	1	yes	characteristic
52	Dic₁₃₉	—	2	1	no	—
53	Dic₁₃₉	—	2	1	no	—
54	Dic₁₃₉	—	2	1	no	—
55	Dic₁₃₉	M₃₂	1	2	yes	—
56	C₂×C₁₃₉	C₈×Q₈	1	1	yes	characteristic
57	C₂×C₁₃₉	C₂×M₃₂	1	1	yes	characteristic
58	C₂×C₁₃₉	#504	1	1	yes	characteristic
59	C₁₃₉	#2183	1	1	yes	characteristic, complemented, Sylow
60	#2183	—	139	1	no	maximal, Sylow
61	C₈×Q₈	—	139	1	no	—
62	C₄⋊C₁₆	—	139	1	no	—
63	C₄⋊C₁₆	—	139	1	no	—
64	C₄⋊C₁₆	—	139	1	no	—
65	#346	—	139	1	no	—
66	#346	D₁₃₉	1	1	yes	characteristic
67	C₄×C₁₆	—	139	1	no	—
68	C₄×Q₈	—	139	1	no	—
69	C₂×C₁₆	—	139	1	no	—
70	C₂×C₁₆	—	139	1	no	—
71	C₂×C₁₆	C₂×D₁₃₉	1	1	yes	characteristic
72	C₂×C₁₆	C₂×D₁₃₉	1	1	yes	characteristic
73	C₄⋊C₈	—	139	1	no	—
74	C₄⋊C₈	—	139	1	no	—
75	C₄⋊C₈	—	139	1	no	—
76	C₄×C₈	—	139	1	no	—
77	C₄×C₈	—	139	1	no	—
78	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
79	C₂×Q₈	—	139	1	no	—
80	C₂×C₈	—	139	1	no	—
81	C₂×C₈	—	139	1	no	—
82	C₂×C₈	C₄×D₁₃₉	1	1	yes	characteristic
83	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
84	C₄⋊C₄	—	139	1	no	—
85	C₄⋊C₄	—	139	1	no	—
86	C₄⋊C₄	—	139	1	no	—
87	C₄²	—	139	1	no	—
88	C₄²	—	139	1	no	—
89	C₄²	C₄×D₁₃₉	1	1	yes	characteristic
90	C₁₆	—	2	1	no	—
91	C₁₆	—	278	1	no	—
92	C₁₆	—	278	1	no	—
93	C₁₆	Dic₂₇₈	1	2	yes	complemented
94	Q₈	—	278	2	no	—
95	C₂×C₄	—	139	1	no	—
96	C₂×C₄	—	139	1	no	—
97	C₂×C₄	—	139	1	no	—
98	C₂×C₄	—	139	1	no	—
99	C₂×C₄	C₈×D₁₃₉	1	1	yes	characteristic
100	C₂×C₄	C₈×D₁₃₉	1	1	yes	characteristic
101	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	278	1	no	—
104	C₈	—	278	1	no	—
105	C₈	C₂×Dic₂₇₈	1	1	yes	characteristic
106	C₈	#14555785	1	1	yes	characteristic
107	C₂²	C₂×C₈×D₁₃₉	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	278	1	no	—
111	C₄	—	278	1	no	—
112	C₄	—	278	1	no	—
113	C₄	—	139	2	no	—
114	C₄	C₄×Dic₂₇₈	1	1	yes	characteristic
115	C₄	#17984848	1	1	yes	characteristic
116	C₂	C₈×Dic₂₇₈	1	1	yes	characteristic
117	C₂	M₃₂×D₁₃₉	1	1	yes	characteristic
118	C₂	#22581371	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Dic₂₇₈

C₂

yes

maximal, characteristic

C₄⋊(C₁₃₉⋊C₁₆)

C₂

yes

maximal, characteristic

Dic₁₃₉⋊C₁₆

C₂

yes

maximal, characteristic

Dic₁₃₉⋊C₁₆

C₂

yes

maximal, characteristic

#22580728

C₂

yes

maximal, characteristic

C₁₆×Dic₁₃₉

C₂

yes

maximal, characteristic

#346×C₁₃₉

C₂

yes

maximal, characteristic

C₄×Dic₂₇₈

C₄

yes

characteristic

C₂×C₁₃₉⋊C₁₆

C₂²

yes

characteristic

C₂×C₁₃₉⋊C₁₆

C₂²

yes

characteristic

C₄⋊(C₁₃₉⋊C₈)

C₄

yes

characteristic

Dic₁₃₉⋊C₈

C₄

yes

characteristic

Dic₁₃₉⋊C₈

C₄

yes

characteristic

C₈×Dic₁₃₉

C₂²

yes

characteristic

C₈×Dic₁₃₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₃₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₃₉

C₂²

yes

characteristic

C₄×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

C₈

yes

characteristic

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

characteristic

C₄⋊Dic₁₃₉

C₈

yes

characteristic

Dic₁₃₉⋊C₄

C₈

yes

characteristic

Dic₁₃₉⋊C₄

C₈

yes

characteristic

C₄×Dic₁₃₉

C₂×C₄

yes

characteristic

C₄×Dic₁₃₉

C₂×C₄

yes

characteristic

C₁₃₉⋊C₁₆

—

C₁₃₉⋊C₁₆

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₉

C₂³

yes

characteristic

C₄²×C₁₃₉

C₂×C₄

yes

characteristic

C₁₆×C₁₃₉

—

C₁₆×C₁₃₉

Q₈

yes

complemented

Dic₂₇₈

—

C₂×Dic₁₃₉

C₂×C₈

yes

characteristic

C₂×Dic₁₃₉

C₂×C₈

yes

characteristic

C₂×Dic₁₃₉

C₂×C₈

yes

characteristic

C₂×Dic₁₃₉

C₂×C₈

yes

characteristic

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₂×C₄×C₁₃₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₃₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₈×C₁₃₉

—

C₈×C₁₃₉

C₂×Q₈

yes

characteristic

C₈×C₁₃₉

#41

yes

characteristic

C₂²×C₁₃₉

C₂²×C₈

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₄×Q₈

yes

characteristic

C₄×C₁₃₉

#131

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

M₃₂

yes

—

C₂×C₁₃₉

C₈×Q₈

yes

characteristic

C₂×C₁₃₉

C₂×M₃₂

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#2183

—

139

maximal, Sylow

C₈×Q₈

—

139

—

C₄⋊C₁₆

—

139

—

C₄⋊C₁₆

—

139

—

C₄⋊C₁₆

—

139

—

#346

—

139

—

#346

D₁₃₉

yes

characteristic

C₄×C₁₆

—

139

—

C₄×Q₈

—

139

—

C₂×C₁₆

—

139

—

C₂×C₁₆

—

139

—

C₂×C₁₆

C₂×D₁₃₉

yes

characteristic

C₂×C₁₆

C₂×D₁₃₉

yes

characteristic

C₄⋊C₈

—

139

—

C₄⋊C₈

—

139

—

C₄⋊C₈

—

139

—

C₄×C₈

—

139

—

C₄×C₈

—

139

—

C₄×C₈

C₂×D₁₃₉

yes

characteristic

C₂×Q₈

—

139

—

C₂×C₈

—

139

—

C₂×C₈

—

139

—

C₂×C₈

C₄×D₁₃₉

yes

characteristic

C₂×C₈

C₂²×D₁₃₉

yes

characteristic

C₄⋊C₄

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

—

139

—

C₄²

—

139

—

C₄²

—

139

—

C₄²

C₄×D₁₃₉

yes

characteristic

C₁₆

—

C₁₆

—

278

—

C₁₆

—

278

—

C₁₆

Dic₂₇₈

yes

complemented

Q₈

—

278

—

C₂×C₄

—

139

—

C₂×C₄

—

139

—

C₂×C₄

—

139

—

C₂×C₄

—

139

—

C₂×C₄

C₈×D₁₃₉

yes

characteristic

100

C₂×C₄

C₈×D₁₃₉

yes

characteristic

101

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

102

C₈

—

103

C₈

—

278

—

104

C₈

—

278

—

105

C₈

C₂×Dic₂₇₈

yes

characteristic

106

C₈

#14555785

yes

characteristic

107

C₂²

C₂×C₈×D₁₃₉

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

278

—

111

C₄

—

278

—

112

C₄

—

278

—

113

C₄

—

139

—

114

C₄

C₄×Dic₂₇₈

yes

characteristic

115

C₄

#17984848

yes

characteristic

116

C₂

C₈×Dic₂₇₈

yes

characteristic

117

C₂

M₃₂×D₁₃₉

yes

characteristic

118

C₂

#22581371

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₃₉
commutator, socle, prime radical	C₂²×C₁₃₉	C₂²×C₈
Fitting	#346×C₁₃₉	C₂
cosocle	C₂×C₈×C₁₃₉	C₂³
hypercenter	#346	D₁₃₉
hypocenter	C₁₃₉	#2183
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₈ ◁ #346

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₃₉

commutator, socle, prime radical

C₂²×C₁₃₉

C₂²×C₈

Fitting

#346×C₁₃₉

C₂

cosocle

C₂×C₈×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂²×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂×C₈ ◁ #346

N	H ≅ G/N	type
33. C₁₆×C₁₃₉	Q₈	semidirect
59. C₁₃₉	#2183	semidirect
93. C₁₆	Dic₂₇₈	semidirect

H ≅ G/N

type

33. C₁₆×C₁₃₉

Q₈

semidirect

59. C₁₃₉

#2183

semidirect

93. C₁₆

Dic₂₇₈

semidirect

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

subgroup	index
C₈×Dic₂₇₈	2
C₄⋊(C₁₃₉⋊C₁₆)	2
Dic₁₃₉⋊C₁₆	2
#22580728	2
C₁₆×Dic₁₃₉	2
#346×C₁₃₉	2
#2183	139

subgroup

index

C₈×Dic₂₇₈

C₄⋊(C₁₃₉⋊C₁₆)

Dic₁₃₉⋊C₁₆

#22580728

C₁₆×Dic₁₃₉

#346×C₁₃₉

#2183

139

Subgroups — #31808310

Conjugacy classes of subgroups

Complements

Maximal subgroups