Group #31806450 — subgroups

Subgroups — #31806450

Order 17792.

260 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 105 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	#22580991	C₂	1	2	yes	maximal, complemented
2	#22580927	C₂	1	2	yes	maximal
3	#22580710	C₂	1	2	yes	maximal, complemented
4	C₂×#131×C₁₃₉	C₂	1	1	yes	maximal, characteristic
5	#17984801	C₂²	1	4	yes	—
6	#17984776	—	2	2	no	—
7	#17984774	—	2	2	no	—
8	#17984757	—	2	4	no	—
9	C₂×#41×C₁₃₉	C₂²	1	1	yes	characteristic
10	#131×C₁₃₉	—	2	2	no	—
11	#131×C₁₃₉	—	2	2	no	—
12	C₂×M₁₆×C₁₃₉	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₃₉	C₄	1	2	yes	complemented
15	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₃₉	—	2	4	no	—
17	C₂²⋊Dic₁₃₉	—	2	4	no	—
18	C₄×Dic₁₃₉	—	2	4	no	—
19	#41×C₁₃₉	—	2	2	no	—
20	#41×C₁₃₉	—	2	2	no	—
21	C₂×Q₈×C₁₃₉	Q₈	1	1	yes	characteristic
22	C₂×D₄×C₁₃₉	D₄	1	2	yes	—
23	C₂×D₄×C₁₃₉	Q₈	1	1	yes	characteristic
24	C₂²×C₄×C₁₃₉	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₃₉	—	2	2	no	—
27	M₁₆×C₁₃₉	—	2	4	no	—
28	C₂×C₈×C₁₃₉	—	2	2	no	—
29	C₂×C₈×C₁₃₉	—	2	2	no	—
30	C₂×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₃₉	D₄	1	2	yes	—
34	C₂×C₈×C₁₃₉	D₄	1	2	yes	—
35	C₂×Dic₁₃₉	—	4	4	no	—
36	C₂³×C₁₃₉	C₄⋊C₄	1	2	yes	—
37	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₁₃₉	—	2	2	no	—
39	D₄×C₁₃₉	—	2	4	no	—
40	D₄×C₁₃₉	—	2	2	no	—
41	C₂×C₄×C₁₃₉	—	2	2	no	—
42	C₂×C₄×C₁₃₉	—	2	2	no	—
43	C₂×C₄×C₁₃₉	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₃₉	#41	1	2	yes	—
48	C₈×C₁₃₉	—	2	2	no	—
49	C₈×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₈×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₈×C₁₃₉	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₁₃₉	—	2	2	no	—
53	C₂²×C₁₃₉	—	4	1	no	—
54	C₂²×C₁₃₉	—	4	2	no	—
55	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₃₉	#121	1	2	yes	—
57	C₄×C₁₃₉	—	2	2	no	—
58	C₄×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₄×C₁₃₉	#117	1	1	yes	characteristic
60	C₄×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
61	Dic₁₃₉	—	8	4	no	—
62	C₂×C₁₃₉	—	2	2	no	—
63	C₂×C₁₃₉	—	2	1	no	—
64	C₂×C₁₃₉	—	4	2	no	—
65	C₂×C₁₃₉	#380	1	1	yes	characteristic
66	C₁₃₉	#1812	1	1	yes	characteristic, complemented, Sylow
67	#1812	—	139	1	no	maximal, Sylow
68	C₂×#131	D₁₃₉	1	1	yes	characteristic
69	#427	—	139	2	no	—
70	#410	—	139	2	no	—
71	H(3,ℤ/4)	—	139	2	no	—
72	C₂×#41	C₂×D₁₃₉	1	1	yes	characteristic
73	#131	—	2	2	no	—
74	#131	—	2	2	no	—
75	C₂×M₁₆	Dic₁₃₉	1	2	yes	complemented
76	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
77	C₂²×C₈	Dic₁₃₉	1	2	yes	complemented
78	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
79	#117	—	139	4	no	—
80	#107	—	278	2	no	—
81	#106	—	278	2	no	—
82	#99	—	278	4	no	—
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	Dic₂₇₈	1	1	yes	characteristic
86	C₂×D₄	D₅₅₆	1	2	yes	—
87	C₂×D₄	Dic₂₇₈	1	1	yes	characteristic
88	C₂²×C₄	C₂×Dic₁₃₉	1	2	yes	—
89	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	C₂×Dic₁₃₉	1	2	yes	—
95	C₂×C₈	C₂×Dic₁₃₉	1	1	yes	characteristic
96	C₂×C₈	C₂×Dic₁₃₉	1	1	yes	characteristic
97	C₂×C₈	C₂²⋊D₁₃₉	1	2	yes	—
98	C₂×C₈	C₂²⋊D₁₃₉	1	2	yes	—
99	C₄⋊C₄	—	278	4	no	—
100	C₂²⋊C₄	—	278	4	no	—
101	C₄²	—	278	4	no	—
102	C₂³	C₄⋊Dic₁₃₉	1	2	yes	—
103	C₂³	C₂×D₅₅₆	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	556	4	no	—
110	C₂×C₄	C₄⋊Dic₁₃₉	1	2	yes	—
111	C₂×C₄	C₂²×Dic₁₃₉	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	2	yes	—
113	C₂×C₄	C₂×Dic₂₇₈	1	1	yes	characteristic
114	C₂×C₄	#14555785	1	2	yes	—
115	C₈	—	2	2	no	—
116	C₈	C₂²⋊Dic₁₃₉	1	1	yes	characteristic
117	C₈	C₂²⋊Dic₁₃₉	1	1	yes	characteristic
118	C₈	C₂²⋊Dic₁₃₉	1	2	yes	—
119	C₂²	—	2	2	no	—
120	C₂²	—	4	1	no	—
121	C₂²	—	4	2	no	—
122	C₂²	C₂×C₄⋊Dic₁₃₉	1	1	yes	characteristic
123	C₂²	#17984816	1	2	yes	—
124	C₄	—	2	2	no	—
125	C₄	—	1112	4	no	—
126	C₄	C₂×C₂²⋊Dic₁₃₉	1	1	yes	characteristic
127	C₄	#17984801	1	1	yes	characteristic
128	C₄	#17984821	1	2	yes	—
129	C₂	—	2	2	no	—
130	C₂	—	2	1	no	—
131	C₂	—	4	2	no	—
132	C₂	#22580813	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22580991

C₂

yes

maximal, complemented

#22580927

C₂

yes

maximal

#22580710

C₂

yes

maximal, complemented

C₂×#131×C₁₃₉

C₂

yes

maximal, characteristic

#17984801

C₂²

yes

—

#17984776

—

#17984774

—

#17984757

—

C₂×#41×C₁₃₉

C₂²

yes

characteristic

#131×C₁₃₉

—

#131×C₁₃₉

—

C₂×M₁₆×C₁₃₉

C₄

yes

complemented

C₂×M₁₆×C₁₃₉

C₂²

yes

characteristic

C₂²×C₈×C₁₃₉

C₄

yes

complemented

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

C₄⋊Dic₁₃₉

—

C₂²⋊Dic₁₃₉

—

C₄×Dic₁₃₉

—

#41×C₁₃₉

—

#41×C₁₃₉

—

C₂×Q₈×C₁₃₉

Q₈

yes

characteristic

C₂×D₄×C₁₃₉

D₄

yes

—

C₂×D₄×C₁₃₉

Q₈

yes

characteristic

C₂²×C₄×C₁₃₉

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₉

D₄

yes

—

C₂×C₈×C₁₃₉

D₄

yes

—

C₂×Dic₁₃₉

—

C₂³×C₁₃₉

C₄⋊C₄

yes

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

Q₈×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₈×C₁₃₉

—

C₈×C₁₃₉

C₂²⋊C₄

yes

characteristic

C₈×C₁₃₉

C₂²⋊C₄

yes

characteristic

C₈×C₁₃₉

C₂²⋊C₄

yes

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃₉

#121

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₃₉

#117

yes

characteristic

C₄×C₁₃₉

C₂²⋊Q₈

yes

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#380

yes

characteristic

C₁₃₉

#1812

yes

characteristic, complemented, Sylow

#1812

—

139

maximal, Sylow

C₂×#131

D₁₃₉

yes

characteristic

#427

—

139

—

#410

—

139

—

H(3,ℤ/4)

—

139

—

C₂×#41

C₂×D₁₃₉

yes

characteristic

#131

—

#131

—

C₂×M₁₆

Dic₁₃₉

yes

complemented

C₂×M₁₆

C₂×D₁₃₉

yes

characteristic

C₂²×C₈

Dic₁₃₉

yes

complemented

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

#117

—

139

—

#107

—

278

—

#106

—

278

—

#99

—

278

—

#41

—

#41

—

C₂×Q₈

Dic₂₇₈

yes

characteristic

C₂×D₄

D₅₅₆

yes

—

C₂×D₄

Dic₂₇₈

yes

characteristic

C₂²×C₄

C₂×Dic₁₃₉

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₁₃₉

yes

—

C₂×C₈

C₂×Dic₁₃₉

yes

characteristic

C₂×C₈

C₂×Dic₁₃₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₃₉

yes

—

C₂×C₈

C₂²⋊D₁₃₉

yes

—

C₄⋊C₄

—

278

—

100

C₂²⋊C₄

—

278

—

101

C₄²

—

278

—

102

C₂³

C₄⋊Dic₁₃₉

yes

—

103

C₂³

C₂×D₅₅₆

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

556

—

110

C₂×C₄

C₄⋊Dic₁₃₉

yes

—

111

C₂×C₄

C₂²×Dic₁₃₉

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

113

C₂×C₄

C₂×Dic₂₇₈

yes

characteristic

114

C₂×C₄

#14555785

yes

—

115

C₈

—

116

C₈

C₂²⋊Dic₁₃₉

yes

characteristic

117

C₈

C₂²⋊Dic₁₃₉

yes

characteristic

118

C₈

C₂²⋊Dic₁₃₉

yes

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂×C₄⋊Dic₁₃₉

yes

characteristic

123

C₂²

#17984816

yes

—

124

C₄

—

125

C₄

—

1112

—

126

C₄

C₂×C₂²⋊Dic₁₃₉

yes

characteristic

127

C₄

#17984801

yes

characteristic

128

C₄

#17984821

yes

—

129

C₂

—

130

C₂

—

131

C₂

—

132

C₂

#22580813

yes

characteristic

	subgroup	G/H
center	C₄	#17984801
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	C₂×#131×C₁₃₉	C₂
socle	C₂×C₁₃₉	#380
prime radical	C₂×#41×C₁₃₉	C₂²
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	C₂×#131	D₁₃₉
hypocenter	C₁₃₉	#1812
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

subgroup

G/H

center

C₄

#17984801

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

C₂×#131×C₁₃₉

C₂

socle

C₂×C₁₃₉

#380

prime radical

C₂×#41×C₁₃₉

C₂²

cosocle

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

N	H ≅ G/N	type
1. #22580991	C₂	semidirect
3. #22580710	C₂	semidirect
12. C₂×M₁₆×C₁₃₉	C₄	semidirect
14. C₂²×C₈×C₁₃₉	C₄	semidirect
66. C₁₃₉	#1812	semidirect
75. C₂×M₁₆	Dic₁₃₉	semidirect
77. C₂²×C₈	Dic₁₃₉	semidirect

H ≅ G/N

type

1. #22580991

C₂

semidirect

3. #22580710

C₂

semidirect

12. C₂×M₁₆×C₁₃₉

C₄

semidirect

14. C₂²×C₈×C₁₃₉

C₄

semidirect

66. C₁₃₉

#1812

semidirect

75. C₂×M₁₆

Dic₁₃₉

semidirect

77. C₂²×C₈

Dic₁₃₉

semidirect

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

subgroup	index
#22580991	2
#22580927	2
#22580710	2
C₂×#131×C₁₃₉	2
#1812	139

subgroup

index

C₂×#131×C₁₃₉

139

Subgroups — #31806450

Conjugacy classes of subgroups

Complements

Maximal subgroups