Group #31805461 — subgroups

Subgroups — #31805461

Order 17792.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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	#22581689	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581150	C₂	1	2	yes	maximal, complemented
3	#22580697	C₂	1	1	yes	maximal, characteristic, complemented
4	#22580685	C₂	1	2	yes	maximal, complemented
5	#493×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555773	—	2	2	no	—
7	#17984829	—	2	2	no	—
8	#17984823	—	2	2	no	—
9	C₄×Dic₂₇₈	C₂²	1	2	yes	—
10	D₄×Dic₁₃₉	—	2	2	no	—
11	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	2	yes	—
12	#17984763	—	2	2	no	—
13	C₂×D₈×C₁₃₉	—	2	2	no	—
14	#127×C₁₃₉	C₂²	1	1	yes	characteristic
15	#127×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₇₈	D₄	1	2	yes	—
18	C₂²×Dic₁₃₉	—	2	2	no	—
19	#14555773	—	4	4	no	—
20	C₂×C₁₃₉⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₃₉	D₄	1	2	yes	—
22	Dic₁₃₉⋊C₄	—	2	2	no	—
23	C₂²⋊Dic₁₃₉	—	2	4	no	—
24	C₂²⋊Dic₁₃₉	—	2	2	no	—
25	C₄×Dic₁₃₉	—	2	2	no	—
26	C₂×D₄×C₁₃₉	—	2	2	no	—
27	C₂×D₄×C₁₃₉	—	2	2	no	—
28	C₂×D₄×C₁₃₉	—	2	1	no	—
29	C₂×D₄×C₁₃₉	D₄	1	2	yes	—
30	D₈×C₁₃₉	—	4	4	no	—
31	C₂×C₈×C₁₃₉	—	2	1	no	—
32	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
33	Dic₂₇₈	—	2	2	no	—
34	Dic₂₇₈	—	2	2	no	—
35	C₂×Dic₁₃₉	—	2	2	no	—
36	C₂×Dic₁₃₉	—	2	2	no	—
37	C₂×Dic₁₃₉	—	8	2	no	—
38	C₁₃₉⋊C₈	—	4	2	no	—
39	C₂³×C₁₃₉	—	2	2	no	—
40	C₂³×C₁₃₉	—	4	1	no	—
41	D₄×C₁₃₉	—	4	2	no	—
42	D₄×C₁₃₉	—	4	2	no	—
43	D₄×C₁₃₉	—	4	2	no	—
44	D₄×C₁₃₉	—	4	2	no	—
45	D₄×C₁₃₉	—	4	2	no	—
46	D₄×C₁₃₉	—	4	2	no	—
47	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
49	C₈×C₁₃₉	—	2	2	no	—
50	C₂²×C₁₃₉	—	4	4	no	—
51	C₂²×C₁₃₉	—	4	2	no	—
52	C₂²×C₁₃₉	—	8	2	no	—
53	C₂²×C₁₃₉	—	8	1	no	—
54	C₂²×C₁₃₉	#120	1	1	yes	characteristic
55	C₄×C₁₃₉	—	2	1	no	—
56	C₄×C₁₃₉	#135	1	2	yes	—
57	C₄×C₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
58	Dic₁₃₉	—	4	2	no	—
59	Dic₁₃₉	—	4	2	no	—
60	C₂×C₁₃₉	—	8	2	no	—
61	C₂×C₁₃₉	—	16	1	no	—
62	C₂×C₁₃₉	#449	1	2	yes	—
63	C₂×C₁₃₉	#453	1	1	yes	characteristic
64	C₁₃₉	#1620	1	1	yes	characteristic, complemented, Sylow
65	#1620	—	139	1	no	maximal, Sylow
66	#559	—	139	1	no	—
67	#493	D₁₃₉	1	1	yes	characteristic
68	#460	—	139	2	no	—
69	#331	—	139	1	no	—
70	Q₈⋊C₈	—	139	2	no	—
71	C₂×SD₁₆	—	278	2	no	—
72	C₂×D₈	—	2	2	no	—
73	#127	C₂×D₁₃₉	1	1	yes	characteristic
74	#127	C₂×D₁₃₉	1	1	yes	characteristic
75	#124	—	278	2	no	—
76	#123	—	278	2	no	—
77	C₄×Q₈	—	139	2	no	—
78	C₄×D₄	—	278	2	no	—
79	C₄⋊C₈	—	139	2	no	—
80	#102	—	278	2	no	—
81	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
82	C₂×Q₈	—	139	2	no	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	C₂²⋊D₁₃₉	1	2	yes	—
87	C₂²×C₄	—	278	2	no	—
88	SD₁₆	—	556	4	no	—
89	D₈	—	4	4	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	278	2	no	—
92	C₄⋊C₄	—	278	2	no	—
93	C₄⋊C₄	—	139	2	no	—
94	C₂²⋊C₄	—	278	4	no	—
95	C₂²⋊C₄	—	278	2	no	—
96	C₄²	—	278	2	no	—
97	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	—	4	1	no	—
100	Q₈	—	278	2	no	—
101	Q₈	—	278	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	2	no	—
106	D₄	—	4	2	no	—
107	D₄	—	4	2	no	—
108	C₂×C₄	—	278	2	no	—
109	C₂×C₄	—	278	2	no	—
110	C₂×C₄	—	1112	2	no	—
111	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	556	2	no	—
115	C₂²	—	4	4	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	8	2	no	—
118	C₂²	—	8	1	no	—
119	C₂²	#17984810	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	556	2	no	—
122	C₄	—	556	2	no	—
123	C₄	#17984861	1	2	yes	—
124	C₄	#17984866	1	2	yes	—
125	C₂	—	8	2	no	—
126	C₂	—	16	1	no	—
127	C₂	#22581095	1	2	yes	—
128	C₂	#22581120	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581689

C₂

yes

maximal, characteristic, complemented

#22581150

C₂

yes

maximal, complemented

#22580697

C₂

yes

maximal, characteristic, complemented

#22580685

C₂

yes

maximal, complemented

#493×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555773

—

#17984829

—

#17984823

—

C₄×Dic₂₇₈

C₂²

yes

—

D₄×Dic₁₃₉

—

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

—

#17984763

—

C₂×D₈×C₁₃₉

—

#127×C₁₃₉

C₂²

yes

characteristic

#127×C₁₃₉

C₂²

yes

characteristic

C₄×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

D₄

yes

—

C₂²×Dic₁₃₉

—

#14555773

—

C₂×C₁₃₉⋊C₈

—

C₄⋊Dic₁₃₉

D₄

yes

—

Dic₁₃₉⋊C₄

—

C₂²⋊Dic₁₃₉

—

C₂²⋊Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

D₄

yes

—

D₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₄²×C₁₃₉

C₂³

yes

characteristic

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

#120

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

#135

yes

—

C₄×C₁₃₉

AGL(1,ℤ/8)

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#449

yes

—

C₂×C₁₃₉

#453

yes

characteristic

C₁₃₉

#1620

yes

characteristic, complemented, Sylow

#1620

—

139

maximal, Sylow

#559

—

139

—

#493

D₁₃₉

yes

characteristic

#460

—

139

—

#331

—

139

—

Q₈⋊C₈

—

139

—

C₂×SD₁₆

—

278

—

C₂×D₈

—

#127

C₂×D₁₃₉

yes

characteristic

#127

C₂×D₁₃₉

yes

characteristic

#124

—

278

—

#123

—

278

—

C₄×Q₈

—

139

—

C₄×D₄

—

278

—

C₄⋊C₈

—

139

—

#102

—

278

—

C₄×C₈

C₂×D₁₃₉

yes

characteristic

C₂×Q₈

—

139

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₃₉

yes

—

C₂²×C₄

—

278

—

SD₁₆

—

556

—

D₈

—

C₂×C₈

—

C₂×C₈

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

139

—

C₂²⋊C₄

—

278

—

C₂²⋊C₄

—

278

—

C₄²

—

278

—

C₄²

C₂²×D₁₃₉

yes

characteristic

C₂³

—

C₂³

—

100

Q₈

—

278

—

101

Q₈

—

278

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

106

D₄

—

107

D₄

—

108

C₂×C₄

—

278

—

109

C₂×C₄

—

278

—

110

C₂×C₄

—

1112

—

111

C₂×C₄

D₄×D₁₃₉

yes

—

112

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

113

C₈

—

114

C₈

—

556

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

#17984810

yes

characteristic

120

C₄

—

121

C₄

—

556

—

122

C₄

—

556

—

123

C₄

#17984861

yes

—

124

C₄

#17984866

yes

—

125

C₂

—

126

C₂

—

127

C₂

#22581095

yes

—

128

C₂

#22581120

yes

characteristic

	subgroup	G/H
center	C₂²	#17984810
commutator, cosocle	C₄²×C₁₃₉	C₂³
Frattini	C₄²	C₂²×D₁₃₉
Fitting, prime radical	#493×C₁₃₉	C₂
socle	C₂²×C₁₃₉	#120
hypercenter	#493	D₁₃₉
hypocenter	C₁₃₉	#1620
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#17984810

commutator, cosocle

C₄²×C₁₃₉

C₂³

Frattini

C₄²

C₂²×D₁₃₉

Fitting, prime radical

#493×C₁₃₉

C₂

C₂²×C₁₃₉

D₁₃₉

C₁₃₉

G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #22581689	C₂	semidirect
2. #22581150	C₂	semidirect
3. #22580697	C₂	semidirect
4. #22580685	C₂	semidirect
64. C₁₃₉	#1620	semidirect

H ≅ G/N

type

1. #22581689

C₂

semidirect

2. #22581150

C₂

semidirect

3. #22580697

C₂

semidirect

4. #22580685

C₂

semidirect

64. C₁₃₉

#1620

semidirect

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

subgroup	index
#22581689	2
#22581150	2
#22580697	2
#22580685	2
#493×C₁₃₉	2
#1620	139

subgroup

index

#493×C₁₃₉

139

Subgroups — #31805461

Conjugacy classes of subgroups

Complements

Maximal subgroups