Group #31806441 — subgroups

Subgroups — #31806441

Order 17792.

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

normal subgroups only

This group has 292 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	C₂²×C₈×D₁₃₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984777	C₂	1	2	yes	maximal, complemented
3	#22580916	C₂	1	2	yes	maximal
4	#22580718	C₂	1	1	yes	maximal, characteristic, complemented
5	#341×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
7	C₂×#14555769	C₂²	1	2	yes	—
8	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₈×D₁₃₉	—	2	4	no	—
10	C₂×C₈×D₁₃₉	C₄	1	4	yes	—
11	#17984777	—	2	4	no	—
12	(C₂×D₁₃₉)⋊C₈	—	2	4	no	—
13	C₂×M₁₆×C₁₃₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂³×D₁₃₉	Q₈	1	1	yes	characteristic
16	C₂²×Dic₁₃₉	Q₈	1	1	yes	characteristic
17	C₂×C₄×D₁₃₉	—	2	2	no	—
18	C₂×C₄×D₁₃₉	—	2	2	no	—
19	C₂×C₄×D₁₃₉	C₂×C₄	1	2	yes	—
20	C₂×C₄×D₁₃₉	D₄	1	2	yes	—
21	#14555769	—	2	2	no	—
22	#14555769	—	2	2	no	—
23	C₂×C₁₃₉⋊C₈	—	2	2	no	—
24	C₂×C₁₃₉⋊C₈	C₂×C₄	1	2	yes	—
25	C₂×C₁₃₉⋊C₈	D₄	1	4	yes	—
26	C₈×D₁₃₉	—	2	8	no	—
27	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
28	M₁₆×C₁₃₉	—	2	2	no	—
29	M₁₆×C₁₃₉	—	2	2	no	—
30	C₂×C₈×C₁₃₉	—	2	2	no	—
31	C₂×C₈×C₁₃₉	—	2	2	no	—
32	C₂×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
33	C₂²×D₁₃₉	—	2	2	no	—
34	C₂²×D₁₃₉	—	2	4	no	—
35	C₂²×D₁₃₉	C₄⋊C₄	1	2	yes	—
36	C₂×Dic₁₃₉	—	2	2	no	—
37	C₂×Dic₁₃₉	C₄⋊C₄	1	2	yes	—
38	C₄×D₁₃₉	—	2	2	no	—
39	C₄×D₁₃₉	—	2	2	no	—
40	C₄×D₁₃₉	—	2	4	no	—
41	C₁₃₉⋊C₈	—	4	2	no	—
42	C₁₃₉⋊C₈	C₂²⋊C₄	1	4	yes	—
43	C₂³×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₃₉	#41	1	2	yes	—
48	C₈×C₁₃₉	—	2	2	no	—
49	C₈×C₁₃₉	—	4	2	no	—
50	C₂²×C₁₃₉	—	2	2	no	—
51	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
53	C₂×D₁₃₉	—	2	8	no	—
54	C₂×D₁₃₉	—	2	2	no	—
55	C₂×D₁₃₉	—	2	2	no	—
56	C₂×D₁₃₉	#108	1	4	yes	complemented
57	C₄×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₁₃₉	#117	1	1	yes	characteristic
59	C₄×C₁₃₉	#121	1	2	yes	—
60	Dic₁₃₉	—	2	2	no	—
61	C₂×C₁₃₉	—	2	2	no	—
62	C₂×C₁₃₉	#380	1	1	yes	characteristic
63	C₂×C₁₃₉	C₂×#108	1	2	yes	—
64	D₁₃₉	—	2	4	no	—
65	C₁₃₉	#1810	1	1	yes	characteristic, complemented, Sylow
66	#1810	—	139	1	no	maximal, Sylow
67	C₂³×C₈	—	139	1	no	—
68	C₂×#108	—	139	2	no	—
69	#407	—	139	2	no	—
70	#341	—	139	1	no	—
71	#341	D₁₃₉	1	1	yes	characteristic, complemented
72	C₂³×C₄	—	139	1	no	—
73	C₂×M₁₆	—	139	2	no	—
74	C₂×M₁₆	C₂×D₁₃₉	1	2	yes	—
75	C₂²×C₈	—	278	4	no	—
76	C₂²×C₈	—	139	4	no	—
77	C₂²×C₈	—	139	1	no	—
78	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
79	#108	—	278	4	no	—
80	C₂²⋊C₈	—	278	4	no	—
81	C₂⁴	—	139	1	no	—
82	C₂²×C₄	—	278	2	no	—
83	C₂²×C₄	—	278	2	no	—
84	C₂²×C₄	—	139	1	no	—
85	C₂²×C₄	—	139	2	no	—
86	C₂²×C₄	—	139	2	no	—
87	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
88	M₁₆	—	2	2	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	—	278	2	no	—
91	M₁₆	—	278	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	278	8	no	—
95	C₂×C₈	—	278	2	no	—
96	C₂×C₈	—	139	2	no	—
97	C₂×C₈	—	139	4	no	—
98	C₂×C₈	C₄×D₁₃₉	1	2	yes	—
99	C₂³	—	278	4	no	—
100	C₂³	—	278	2	no	—
101	C₂³	—	139	2	no	—
102	C₂³	Q₈×D₁₃₉	1	1	yes	characteristic
103	C₂×C₄	—	278	2	no	—
104	C₂×C₄	—	278	2	no	—
105	C₂×C₄	—	278	2	no	—
106	C₂×C₄	—	278	4	no	—
107	C₂×C₄	—	139	2	no	—
108	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
109	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
110	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
111	C₂×C₄	#14555785	1	2	yes	—
112	C₈	—	2	2	no	—
113	C₈	—	4	2	no	—
114	C₈	—	556	2	no	—
115	C₈	—	139	4	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	278	8	no	—
118	C₂²	—	278	2	no	—
119	C₂²	—	278	2	no	—
120	C₂²	—	139	4	no	—
121	C₂²	C₄⋊C₄×D₁₃₉	1	1	yes	characteristic
122	C₂²	#17984820	1	2	yes	—
123	C₄	—	278	2	no	—
124	C₄	C₂²⋊C₄×D₁₃₉	1	1	yes	characteristic
125	C₄	#17984799	1	1	yes	characteristic
126	C₄	#17984815	1	2	yes	—
127	C₂	—	2	2	no	—
128	C₂	—	278	4	no	—
129	C₂	#22580812	1	1	yes	characteristic
130	C₂	#108×D₁₃₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₁₃₉

C₂

yes

maximal, characteristic

C₂×#17984777

C₂

yes

maximal, complemented

#22580916

C₂

yes

maximal

#22580718

C₂

yes

maximal, characteristic, complemented

#341×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₃₉

C₂²

yes

characteristic

C₂×#14555769

C₂²

yes

—

C₂²×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₁₃₉

—

C₂×C₈×D₁₃₉

C₄

yes

—

#17984777

—

(C₂×D₁₃₉)⋊C₈

—

C₂×M₁₆×C₁₃₉

C₂²

yes

—

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

C₂³×D₁₃₉

Q₈

yes

characteristic

C₂²×Dic₁₃₉

Q₈

yes

characteristic

C₂×C₄×D₁₃₉

—

C₂×C₄×D₁₃₉

—

C₂×C₄×D₁₃₉

C₂×C₄

yes

—

C₂×C₄×D₁₃₉

D₄

yes

—

#14555769

—

#14555769

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

—

C₂×C₁₃₉⋊C₈

D₄

yes

—

C₈×D₁₃₉

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

C₄⋊C₄

yes

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₄⋊C₄

yes

—

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₁₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃₉

C₂²⋊Q₈

yes

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

#108

yes

complemented

C₄×C₁₃₉

C₂×C₂²⋊C₄

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#380

yes

characteristic

C₂×C₁₃₉

C₂×#108

yes

—

D₁₃₉

—

C₁₃₉

#1810

yes

characteristic, complemented, Sylow

#1810

—

139

maximal, Sylow

C₂³×C₈

—

139

—

C₂×#108

—

139

—

#407

—

139

—

#341

—

139

—

#341

D₁₃₉

yes

characteristic, complemented

C₂³×C₄

—

139

—

C₂×M₁₆

—

139

—

C₂×M₁₆

C₂×D₁₃₉

yes

—

C₂²×C₈

—

278

—

C₂²×C₈

—

139

—

C₂²×C₈

—

139

—

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

#108

—

278

—

C₂²⋊C₈

—

278

—

C₂⁴

—

139

—

C₂²×C₄

—

278

—

C₂²×C₄

—

278

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

278

—

M₁₆

—

278

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

278

—

C₂×C₈

—

278

—

C₂×C₈

—

139

—

C₂×C₈

—

139

—

C₂×C₈

C₄×D₁₃₉

yes

—

C₂³

—

278

—

100

C₂³

—

278

—

101

C₂³

—

139

—

102

C₂³

Q₈×D₁₃₉

yes

characteristic

103

C₂×C₄

—

278

—

104

C₂×C₄

—

278

—

105

C₂×C₄

—

278

—

106

C₂×C₄

—

278

—

107

C₂×C₄

—

139

—

108

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

109

C₂×C₄

D₄×D₁₃₉

yes

—

110

C₂×C₄

D₄×D₁₃₉

yes

characteristic

111

C₂×C₄

#14555785

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

556

—

115

C₈

—

139

—

116

C₂²

—

117

C₂²

—

278

—

118

C₂²

—

278

—

119

C₂²

—

278

—

120

C₂²

—

139

—

121

C₂²

C₄⋊C₄×D₁₃₉

yes

characteristic

122

C₂²

#17984820

yes

—

123

C₄

—

278

—

124

C₄

C₂²⋊C₄×D₁₃₉

yes

characteristic

125

C₄

#17984799

yes

characteristic

126

C₄

#17984815

yes

—

127

C₂

—

128

C₂

—

278

—

129

C₂

#22580812

yes

characteristic

130

C₂

#108×D₁₃₉

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#341×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×C₄⋊C₄
prime radical	C₂³×D₁₃₉	Q₈
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	#341	D₁₃₉
hypocenter	C₁₃₉	#1810
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₁₃₉

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#341×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂×C₄⋊C₄

prime radical

C₂³×D₁₃₉

Q₈

cosocle

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
2. C₂×#17984777	C₂	semidirect
4. #22580718	C₂	semidirect
5. #341×C₁₃₉	C₂	semidirect
56. C₂×D₁₃₉	#108	semidirect
65. C₁₃₉	#1810	semidirect
71. #341	D₁₃₉	semidirect

H ≅ G/N

type

2. C₂×#17984777

C₂

semidirect

4. #22580718

C₂

semidirect

5. #341×C₁₃₉

C₂

semidirect

56. C₂×D₁₃₉

#108

semidirect

65. C₁₃₉

#1810

semidirect

71. #341

D₁₃₉

semidirect

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

subgroup	index
C₂²×C₈×D₁₃₉	2
C₂×#17984777	2
#22580916	2
#22580718	2
#341×C₁₃₉	2
#1810	139

subgroup

index

C₂²×C₈×D₁₃₉

#341×C₁₃₉

139

Subgroups — #31806441

Conjugacy classes of subgroups

Complements

Maximal subgroups