Group #31805158 — subgroups

Subgroups — #31805158

Order 17792.

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

normal subgroups only

This group has 184 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	#22580975	C₂	1	2	yes	maximal
2	C₄×C₈×D₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#22580704	C₂	1	1	yes	maximal, characteristic, complemented
4	#22580703	C₂	1	2	yes	maximal, complemented
5	#334×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14555768	—	2	2	no	—
7	C₂×C₈×D₁₃₉	C₄	1	1	yes	characteristic
8	C₂×C₈×D₁₃₉	C₄	1	1	yes	characteristic
9	C₄²×D₁₃₉	C₂²	1	1	yes	characteristic
10	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	2	yes	—
11	Dic₁₃₉⋊C₈	—	2	2	no	—
12	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₈×Dic₁₃₉	C₄	1	1	yes	characteristic
14	C₈×Dic₁₃₉	C₄	1	1	yes	characteristic
15	C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
16	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₁₃₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×D₁₃₉	D₄	1	1	yes	characteristic
19	C₂×C₄×D₁₃₉	Q₈	1	1	yes	characteristic
20	#14555768	—	4	4	no	—
21	C₂×C₁₃₉⋊C₈	—	2	2	no	—
22	C₂×C₁₃₉⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₃₉⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₈×D₁₃₉	—	2	2	no	—
25	C₈×D₁₃₉	—	2	2	no	—
26	C₄×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
28	C₄×Dic₁₃₉	Q₈	1	1	yes	characteristic
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	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
33	C₂²×D₁₃₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂×Dic₁₃₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×Dic₁₃₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×Dic₁₃₉	C₄⋊C₄	1	1	yes	characteristic
37	C₄×D₁₃₉	—	2	1	no	—
38	C₄×D₁₃₉	—	2	1	no	—
39	C₄×D₁₃₉	—	2	2	no	—
40	C₄×D₁₃₉	SD₁₆	1	4	yes	—
41	C₁₃₉⋊C₈	—	2	1	no	—
42	C₁₃₉⋊C₈	—	4	2	no	—
43	C₁₃₉⋊C₈	M₁₆	1	2	yes	complemented
44	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
47	C₈×C₁₃₉	—	2	1	no	—
48	C₈×C₁₃₉	—	4	2	no	—
49	C₈×C₁₃₉	M₁₆	1	2	yes	complemented
50	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×D₁₃₉	—	2	1	no	—
52	C₂×D₁₃₉	—	2	1	no	—
53	C₂×D₁₃₉	#106	1	2	yes	—
54	C₄×C₁₃₉	—	2	1	no	—
55	C₄×C₁₃₉	C₂×M₁₆	1	1	yes	characteristic
56	C₄×C₁₃₉	C₂×M₁₆	1	1	yes	characteristic
57	C₄×C₁₃₉	C₂×SD₁₆	1	2	yes	—
58	Dic₁₃₉	—	2	1	no	—
59	Dic₁₃₉	#106	1	2	yes	—
60	Dic₁₃₉	#108	1	2	yes	—
61	C₂×C₁₃₉	#423	1	1	yes	characteristic
62	C₂×C₁₃₉	C₂×#106	1	1	yes	characteristic
63	C₂×C₁₃₉	C₂×#108	1	1	yes	characteristic
64	D₁₃₉	—	2	2	no	—
65	C₁₃₉	#1566	1	1	yes	characteristic, complemented, Sylow
66	#1566	—	139	1	no	maximal, Sylow
67	#423	—	139	2	no	—
68	C₂×C₄×C₈	—	139	1	no	—
69	#334	—	139	2	no	—
70	#334	—	139	1	no	—
71	#334	D₁₃₉	1	1	yes	characteristic, complemented
72	C₂×M₁₆	—	278	2	no	—
73	C₂²×C₈	—	139	1	no	—
74	C₂²×C₈	—	139	1	no	—
75	C₂×C₄²	—	139	1	no	—
76	C₄⋊C₈	—	278	2	no	—
77	C₄⋊C₈	—	139	2	no	—
78	C₄⋊C₈	C₂×D₁₃₉	1	2	yes	—
79	C₄×C₈	—	139	1	no	—
80	C₄×C₈	—	139	1	no	—
81	C₄×C₈	—	139	1	no	—
82	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
83	C₂²×C₄	—	139	1	no	—
84	C₂²×C₄	—	139	1	no	—
85	C₂²×C₄	—	139	1	no	—
86	M₁₆	—	556	4	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	278	2	no	—
89	C₂×C₈	—	278	2	no	—
90	C₂×C₈	—	278	2	no	—
91	C₂×C₈	—	139	1	no	—
92	C₂×C₈	—	139	1	no	—
93	C₂×C₈	C₄×D₁₃₉	1	1	yes	characteristic
94	C₂×C₈	C₄×D₁₃₉	1	1	yes	characteristic
95	C₄²	—	139	1	no	—
96	C₄²	—	139	1	no	—
97	C₄²	—	139	1	no	—
98	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
99	C₂³	—	139	1	no	—
100	C₂×C₄	—	278	2	no	—
101	C₂×C₄	—	278	1	no	—
102	C₂×C₄	—	278	1	no	—
103	C₂×C₄	—	139	1	no	—
104	C₂×C₄	—	139	1	no	—
105	C₂×C₄	—	139	1	no	—
106	C₂×C₄	—	139	4	no	—
107	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
108	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
109	C₂×C₄	Q₈×D₁₃₉	1	1	yes	characteristic
110	C₈	—	2	1	no	—
111	C₈	—	4	2	no	—
112	C₈	—	278	1	no	—
113	C₈	—	556	2	no	—
114	C₈	—	139	2	no	—
115	C₈	#14555768	1	2	yes	complemented
116	C₂²	—	278	1	no	—
117	C₂²	—	278	1	no	—
118	C₂²	—	139	2	no	—
119	C₂²	C₄⋊C₄×D₁₃₉	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	278	1	no	—
122	C₄	—	139	2	no	—
123	C₄	—	139	2	no	—
124	C₄	M₁₆×D₁₃₉	1	1	yes	characteristic
125	C₄	C₂×#14555768	1	1	yes	characteristic
126	C₄	SD₁₆×D₁₃₉	1	2	yes	—
127	C₂	—	278	2	no	—
128	C₂	#22580975	1	1	yes	characteristic
129	C₂	#106×D₁₃₉	1	1	yes	characteristic
130	C₂	#108×D₁₃₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22580975

C₂

yes

maximal

C₄×C₈×D₁₃₉

C₂

yes

maximal, characteristic

#22580704

C₂

yes

maximal, characteristic, complemented

#22580703

C₂

yes

maximal, complemented

#334×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×#14555768

—

C₂×C₈×D₁₃₉

C₄

yes

characteristic

C₂×C₈×D₁₃₉

C₄

yes

characteristic

C₄²×D₁₃₉

C₂²

yes

characteristic

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

—

Dic₁₃₉⋊C₈

—

C₄×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₈×Dic₁₃₉

C₄

yes

characteristic

C₈×Dic₁₃₉

C₄

yes

characteristic

C₄⋊C₈×C₁₃₉

C₂²

yes

—

C₄×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×C₄×D₁₃₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₃₉

D₄

yes

characteristic

C₂×C₄×D₁₃₉

Q₈

yes

characteristic

#14555768

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

characteristic

C₈×D₁₃₉

—

C₈×D₁₃₉

—

C₄×Dic₁₃₉

C₂×C₄

yes

characteristic

C₄×Dic₁₃₉

D₄

yes

characteristic

C₄×Dic₁₃₉

Q₈

yes

characteristic

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₉

C₂×C₄

yes

characteristic

C₄²×C₁₃₉

C₂³

yes

characteristic

C₂²×D₁₃₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₃₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₃₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₃₉

C₄⋊C₄

yes

characteristic

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₄×D₁₃₉

SD₁₆

yes

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

M₁₆

yes

complemented

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

characteristic

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₈×C₁₃₉

M₁₆

yes

complemented

C₂²×C₁₃₉

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

#106

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×M₁₆

yes

characteristic

C₄×C₁₃₉

C₂×M₁₆

yes

characteristic

C₄×C₁₃₉

C₂×SD₁₆

yes

—

Dic₁₃₉

—

Dic₁₃₉

#106

yes

—

Dic₁₃₉

#108

yes

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

D₁₃₉

—

C₁₃₉

#1566

yes

characteristic, complemented, Sylow

#1566

—

139

maximal, Sylow

#423

—

139

—

C₂×C₄×C₈

—

139

—

#334

—

139

—

#334

—

139

—

#334

D₁₃₉

yes

characteristic, complemented

C₂×M₁₆

—

278

—

C₂²×C₈

—

139

—

C₂²×C₈

—

139

—

C₂×C₄²

—

139

—

C₄⋊C₈

—

278

—

C₄⋊C₈

—

139

—

C₄⋊C₈

C₂×D₁₃₉

yes

—

C₄×C₈

—

139

—

C₄×C₈

—

139

—

C₄×C₈

—

139

—

C₄×C₈

C₂×D₁₃₉

yes

characteristic

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

M₁₆

—

556

—

C₂×C₈

—

C₂×C₈

—

278

—

C₂×C₈

—

278

—

C₂×C₈

—

278

—

C₂×C₈

—

139

—

C₂×C₈

—

139

—

C₂×C₈

C₄×D₁₃₉

yes

characteristic

C₂×C₈

C₄×D₁₃₉

yes

characteristic

C₄²

—

139

—

C₄²

—

139

—

C₄²

—

139

—

C₄²

C₂²×D₁₃₉

yes

characteristic

C₂³

—

139

—

100

C₂×C₄

—

278

—

101

C₂×C₄

—

278

—

102

C₂×C₄

—

278

—

103

C₂×C₄

—

139

—

104

C₂×C₄

—

139

—

105

C₂×C₄

—

139

—

106

C₂×C₄

—

139

—

107

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

108

C₂×C₄

D₄×D₁₃₉

yes

characteristic

109

C₂×C₄

Q₈×D₁₃₉

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

278

—

113

C₈

—

556

—

114

C₈

—

139

—

115

C₈

#14555768

yes

complemented

116

C₂²

—

278

—

117

C₂²

—

278

—

118

C₂²

—

139

—

119

C₂²

C₄⋊C₄×D₁₃₉

yes

characteristic

120

C₄

—

121

C₄

—

278

—

122

C₄

—

139

—

123

C₄

—

139

—

124

C₄

M₁₆×D₁₃₉

yes

characteristic

125

C₄

C₂×#14555768

yes

characteristic

126

C₄

SD₁₆×D₁₃₉

yes

—

127

C₂

—

278

—

128

C₂

#22580975

yes

characteristic

129

C₂

#106×D₁₃₉

yes

characteristic

130

C₂

#108×D₁₃₉

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₃₉
Fitting	#334×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×C₄⋊C₄
prime radical	C₂²×D₁₃₉	C₄⋊C₄
cosocle	C₄²×C₁₃₉	C₂³
hypercenter	#334	D₁₃₉
hypocenter	C₁₃₉	#1566
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×D₁₃₉

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₄²

C₂²×D₁₃₉

Fitting

#334×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂×C₄⋊C₄

prime radical

C₂²×D₁₃₉

C₄⋊C₄

cosocle

C₄²×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
3. #22580704	C₂	semidirect
4. #22580703	C₂	semidirect
5. #334×C₁₃₉	C₂	semidirect
43. C₁₃₉⋊C₈	M₁₆	semidirect
49. C₈×C₁₃₉	M₁₆	semidirect
65. C₁₃₉	#1566	semidirect
71. #334	D₁₃₉	semidirect
115. C₈	#14555768	semidirect

H ≅ G/N

type

3. #22580704

C₂

semidirect

4. #22580703

C₂

semidirect

5. #334×C₁₃₉

C₂

semidirect

43. C₁₃₉⋊C₈

M₁₆

semidirect

49. C₈×C₁₃₉

M₁₆

semidirect

65. C₁₃₉

#1566

semidirect

71. #334

D₁₃₉

semidirect

115. C₈

#14555768

semidirect

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

subgroup	index
#22580975	2
C₄×C₈×D₁₃₉	2
#22580704	2
#22580703	2
#334×C₁₃₉	2
#1566	139

subgroup

index

C₄×C₈×D₁₃₉

#334×C₁₃₉

139

Subgroups — #31805158

Conjugacy classes of subgroups

Complements

Maximal subgroups