Group #31805154 — subgroups

Subgroups — #31805154

Order 17792.

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

normal subgroups only

This group has 180 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	#22581006	C₂	1	1	yes	maximal, characteristic
2	C₄×#14555769	C₂	1	1	yes	maximal, characteristic
3	#22580705	C₂	1	2	yes	maximal, complemented
4	#22580703	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊C₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555769	C₄	1	1	yes	characteristic
7	C₂×#14555769	C₄	1	1	yes	characteristic
8	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
9	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂²⋊(C₁₃₉⋊C₈)	—	2	1	no	—
11	#17984753	C₄	1	2	yes	—
12	C₄×C₁₃₉⋊C₈	—	2	1	no	—
13	C₄×C₁₃₉⋊C₈	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₃₉	—	2	1	no	—
15	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₃₉	—	2	1	no	—
17	C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
18	#14555769	—	2	2	no	—
19	#14555769	C₈	1	4	yes	complemented
20	C₂×C₁₃₉⋊C₈	—	2	1	no	—
21	C₂×C₁₃₉⋊C₈	—	2	1	no	—
22	C₂×C₁₃₉⋊C₈	C₂×C₄	1	2	yes	—
23	C₂×C₁₃₉⋊C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₃₉	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₃₉	Q₈	1	1	yes	characteristic
27	C₂×C₈×C₁₃₉	—	2	2	no	—
28	C₂×C₈×C₁₃₉	—	4	2	no	—
29	C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₃₉	D₄	1	1	yes	characteristic
31	C₄²×C₁₃₉	Q₈	1	1	yes	characteristic
32	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
33	C₁₃₉⋊C₈	—	2	2	no	—
34	C₁₃₉⋊C₈	—	4	1	no	—
35	C₁₃₉⋊C₈	—	4	1	no	—
36	C₁₃₉⋊C₈	C₂×C₈	1	4	yes	complemented
37	C₂³×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₃₉	—	2	1	no	—
39	C₂×C₄×C₁₃₉	—	2	1	no	—
40	C₂×C₄×C₁₃₉	—	4	1	no	—
41	C₂×C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₃₉	C₂×C₈	1	2	yes	—
45	C₂×C₄×C₁₃₉	M₁₆	1	2	yes	—
46	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
49	C₈×C₁₃₉	—	4	2	no	—
50	C₂²×C₁₃₉	—	2	1	no	—
51	C₂²×C₁₃₉	—	2	1	no	—
52	C₂²×C₁₃₉	C₄⋊C₈	1	2	yes	—
53	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₁₃₉	—	2	1	no	—
55	C₄×C₁₃₉	—	2	1	no	—
56	C₄×C₁₃₉	—	2	1	no	—
57	C₄×C₁₃₉	C₄⋊C₈	1	2	yes	—
58	C₄×C₁₃₉	C₂²×C₈	1	1	yes	characteristic
59	C₄×C₁₃₉	C₂×M₁₆	1	1	yes	characteristic
60	C₄×C₁₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₄×C₁₃₉	#137	1	1	yes	characteristic
62	C₂×C₁₃₉	—	2	2	no	—
63	C₂×C₁₃₉	C₂×C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₁₃₉	#428	1	1	yes	characteristic
65	C₂×C₁₃₉	#430	1	1	yes	characteristic
66	C₁₃₉	#1565	1	1	yes	characteristic, complemented, Sylow
67	#1565	—	139	1	no	maximal, Sylow
68	#431	—	139	1	no	—
69	C₂×C₄⋊C₈	D₁₃₉	1	1	yes	characteristic
70	C₄×M₁₆	—	139	1	no	—
71	#335	—	139	2	no	—
72	#334	—	139	2	no	—
73	C₂×M₁₆	—	139	1	no	—
74	C₂×M₁₆	—	139	1	no	—
75	C₂²×C₈	—	2	1	no	—
76	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
77	C₄⋊C₈	—	2	1	no	—
78	C₄⋊C₈	—	139	1	no	—
79	C₄⋊C₈	—	139	1	no	—
80	C₄⋊C₈	C₂×D₁₃₉	1	2	yes	—
81	C₂²⋊C₈	—	278	1	no	—
82	#97	—	139	2	no	—
83	C₄×C₈	—	278	1	no	—
84	C₄×C₈	—	139	2	no	—
85	C₂²×C₄	C₄×D₁₃₉	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
87	C₂²×C₄	Dic₂₇₈	1	1	yes	characteristic
88	M₁₆	—	278	2	no	—
89	M₁₆	—	139	4	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	4	2	no	—
92	C₂×C₈	—	278	1	no	—
93	C₂×C₈	—	278	1	no	—
94	C₂×C₈	—	139	2	no	—
95	C₂×C₈	—	139	2	no	—
96	C₄²	C₄×D₁₃₉	1	1	yes	characteristic
97	C₄²	C₂²⋊D₁₃₉	1	1	yes	characteristic
98	C₄²	Dic₂₇₈	1	1	yes	characteristic
99	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
100	C₂³	Dic₁₃₉⋊C₄	1	1	yes	characteristic
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	Dic₁₃₉⋊C₄	1	1	yes	characteristic
105	C₂×C₄	Dic₁₃₉⋊C₄	1	1	yes	characteristic
106	C₂×C₄	Dic₁₃₉⋊C₄	1	1	yes	characteristic
107	C₂×C₄	C₈×D₁₃₉	1	2	yes	—
108	C₂×C₄	#14555768	1	2	yes	—
109	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
110	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
111	C₂×C₄	C₂×Dic₂₇₈	1	1	yes	characteristic
112	C₈	—	4	2	no	—
113	C₈	—	278	2	no	—
114	C₈	—	556	1	no	—
115	C₈	—	556	1	no	—
116	C₈	—	139	4	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	Dic₁₃₉⋊C₈	1	2	yes	—
120	C₂²	C₂×Dic₁₃₉⋊C₄	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	Dic₁₃₉⋊C₈	1	2	yes	—
125	C₄	C₂×C₈×D₁₃₉	1	1	yes	characteristic
126	C₄	C₂×#14555768	1	1	yes	characteristic
127	C₄	#17984868	1	1	yes	characteristic
128	C₄	#17984873	1	1	yes	characteristic
129	C₂	—	2	2	no	—
130	C₂	C₂×Dic₁₃₉⋊C₈	1	1	yes	characteristic
131	C₂	#22580993	1	1	yes	characteristic
132	C₂	#22581001	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581006

C₂

yes

maximal, characteristic

C₄×#14555769

C₂

yes

maximal, characteristic

#22580705

C₂

yes

maximal, complemented

#22580703

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555769

C₄

yes

characteristic

C₂×#14555769

C₄

yes

characteristic

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

characteristic

C₂²⋊(C₁₃₉⋊C₈)

—

#17984753

C₄

yes

—

C₄×C₁₃₉⋊C₈

—

C₄×C₁₃₉⋊C₈

C₂²

yes

—

C₂²×C₈×C₁₃₉

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₉

—

C₄⋊C₈×C₁₃₉

C₂²

yes

—

#14555769

—

#14555769

C₈

yes

complemented

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

C₂×C₈×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

—

C₁₃₉⋊C₈

C₂×C₈

yes

complemented

C₂³×C₁₃₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₉

—

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

C₂×C₈

yes

—

C₂×C₄×C₁₃₉

M₁₆

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

C₄⋊C₈

yes

—

C₂²×C₁₃₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₄⋊C₈

yes

—

C₄×C₁₃₉

C₂²×C₈

yes

characteristic

C₄×C₁₃₉

C₂×M₁₆

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

C₂×C₁₃₉

—

C₂×C₁₃₉

C₂×C₄⋊C₈

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#1565

—

139

maximal, Sylow

#431

—

139

—

C₂×C₄⋊C₈

D₁₃₉

yes

characteristic

C₄×M₁₆

—

139

—

#335

—

139

—

#334

—

139

—

C₂×M₁₆

—

139

—

C₂×M₁₆

—

139

—

C₂²×C₈

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

139

—

C₄⋊C₈

—

139

—

C₄⋊C₈

C₂×D₁₃₉

yes

—

C₂²⋊C₈

—

278

—

#97

—

139

—

C₄×C₈

—

278

—

C₄×C₈

—

139

—

C₂²×C₄

C₄×D₁₃₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₃₉

yes

characteristic

C₂²×C₄

Dic₂₇₈

yes

characteristic

M₁₆

—

278

—

M₁₆

—

139

—

C₂×C₈

—

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

100

C₂³

Dic₁₃₉⋊C₄

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

Dic₁₃₉⋊C₄

yes

characteristic

105

C₂×C₄

Dic₁₃₉⋊C₄

yes

characteristic

106

C₂×C₄

Dic₁₃₉⋊C₄

yes

characteristic

107

C₂×C₄

C₈×D₁₃₉

yes

—

108

C₂×C₄

#14555768

yes

—

109

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

110

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

111

C₂×C₄

C₂×Dic₂₇₈

yes

characteristic

112

C₈

—

113

C₈

—

278

—

114

C₈

—

556

—

115

C₈

—

556

—

116

C₈

—

139

—

117

C₂²

—

118

C₂²

—

119

C₂²

Dic₁₃₉⋊C₈

yes

—

120

C₂²

C₂×Dic₁₃₉⋊C₄

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

Dic₁₃₉⋊C₈

yes

—

125

C₄

C₂×C₈×D₁₃₉

yes

characteristic

126

C₄

C₂×#14555768

yes

characteristic

127

C₄

#17984868

yes

characteristic

128

C₄

#17984873

yes

characteristic

129

C₂

—

130

C₂

C₂×Dic₁₃₉⋊C₈

yes

characteristic

131

C₂

#22580993

yes

characteristic

132

C₂

#22581001

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃₉
commutator	C₄×C₁₃₉	C₂²×C₈
Frattini	C₄²	C₂²×D₁₃₉
Fitting	C₂×C₄⋊C₈×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×C₄⋊C₄
prime radical	C₂³×C₁₃₉	C₄⋊C₄
cosocle	C₄²×C₁₃₉	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₃₉
hypocenter	C₁₃₉	#1565
LCS	G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₃₉

commutator

C₄×C₁₃₉

C₂²×C₈

Frattini

C₄²

C₂²×D₁₃₉

Fitting

C₂×C₄⋊C₈×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂×C₄⋊C₄

prime radical

C₂³×C₁₃₉

C₄⋊C₄

cosocle

C₄²×C₁₃₉

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₃₉

hypocenter

C₁₃₉

#1565

LCS

G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #22580705	C₂	semidirect
4. #22580703	C₂	semidirect
19. #14555769	C₈	semidirect
36. C₁₃₉⋊C₈	C₂×C₈	semidirect
66. C₁₃₉	#1565	semidirect

H ≅ G/N

type

3. #22580705

C₂

semidirect

4. #22580703

C₂

semidirect

19. #14555769

C₈

semidirect

36. C₁₃₉⋊C₈

C₂×C₈

semidirect

66. C₁₃₉

#1565

semidirect

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

subgroup	index
#22581006	2
C₄×#14555769	2
#22580705	2
#22580703	2
C₂×C₄⋊C₈×C₁₃₉	2
#1565	139

subgroup

index

C₂×C₄⋊C₈×C₁₃₉

139

Subgroups — #31805154

Conjugacy classes of subgroups

Complements

Maximal subgroups