Group #31818203 — subgroups

Subgroups — #31818203

Order 17792.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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	#22581754	C₂	1	4	yes	maximal, complemented
2	#22581220	C₂	1	4	yes	maximal, complemented
3	#22581120	C₂	1	4	yes	maximal, complemented
4	#22581071	C₂	1	2	yes	maximal, complemented
5	#493×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984887	C₂²	1	4	yes	complemented
7	#17984868	C₂²	1	8	yes	complemented
8	#17984866	—	2	8	no	—
9	#17984864	C₂²	1	2	yes	—
10	#17984861	—	2	4	no	—
11	D₈×D₁₃₉	—	2	4	no	—
12	#17984848	C₂²	1	4	yes	complemented
13	#17984778	C₂²	1	2	yes	complemented
14	#17984770	C₂²	1	4	yes	complemented
15	#17984759	C₂²	1	4	yes	complemented
16	C₂×D₈×C₁₃₉	C₂²	1	4	yes	complemented
17	#127×C₁₃₉	C₂²	1	2	yes	complemented
18	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
19	#14555785	C₂³	1	4	yes	—
20	#14555784	—	2	8	no	—
21	C₂×C₂²⋊D₁₃₉	—	4	4	no	—
22	D₄×D₁₃₉	—	2	8	no	—
23	Dic₅₅₆	D₄	1	2	yes	complemented
24	#14555774	D₄	1	4	yes	complemented
25	#14555773	—	2	8	no	—
26	D₁₁₁₂	D₄	1	2	yes	complemented
27	#14555770	—	2	8	no	—
28	#14555769	C₂³	1	4	yes	—
29	#14555768	—	2	4	no	—
30	C₈×D₁₃₉	—	2	4	no	—
31	C₂×D₄×C₁₃₉	—	4	2	no	—
32	C₂×D₄×C₁₃₉	C₂³	1	4	yes	—
33	D₈×C₁₃₉	—	2	4	no	—
34	D₈×C₁₃₉	—	2	4	no	—
35	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
36	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
37	C₂²×D₁₃₉	—	4	4	no	—
38	Dic₂₇₈	C₂×D₄	1	4	yes	—
39	D₅₅₆	C₂×D₄	1	4	yes	—
40	C₂²⋊D₁₃₉	—	2	4	no	—
41	C₂²⋊D₁₃₉	—	4	8	no	—
42	C₂×Dic₁₃₉	—	4	4	no	—
43	C₄×D₁₃₉	—	2	4	no	—
44	C₁₃₉⋊C₈	—	2	4	no	—
45	C₂³×C₁₃₉	—	2	4	no	—
46	D₄×C₁₃₉	—	2	8	no	—
47	D₄×C₁₃₉	—	8	4	no	—
48	C₂×C₄×C₁₃₉	—	2	1	no	—
49	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
50	C₈×C₁₃₉	C₂×D₄	1	2	yes	—
51	C₈×C₁₃₉	C₂×D₄	1	2	yes	—
52	C₂²×C₁₃₉	—	4	4	no	—
53	C₂²×C₁₃₉	—	8	4	no	—
54	C₂²×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
55	C₂×D₁₃₉	—	2	4	no	—
56	C₂×D₁₃₉	—	16	4	no	—
57	C₄×C₁₃₉	—	2	2	no	—
58	C₄×C₁₃₉	C₂²×D₄	1	2	yes	—
59	Dic₁₃₉	—	2	4	no	—
60	C₂×C₁₃₉	—	2	1	no	—
61	C₂×C₁₃₉	—	8	4	no	—
62	C₂×C₁₃₉	D₄²	1	1	yes	characteristic
63	D₁₃₉	—	4	4	no	—
64	C₁₃₉	#3292	1	1	yes	characteristic, complemented, Sylow
65	#3292	—	139	1	no	maximal, Sylow
66	#576	—	139	4	no	—
67	#493	D₁₃₉	1	1	yes	characteristic, complemented
68	#472	—	139	4	no	—
69	#453	—	139	4	no	—
70	#443	—	139	2	no	—
71	H(5,𝔽₂)	—	139	4	no	—
72	AGL(1,ℤ/8)	—	278	8	no	—
73	AGL(1,ℤ/8)	—	139	8	no	—
74	#135	—	278	4	no	—
75	#135	—	139	2	no	—
76	C₂×D₈	—	278	4	no	—
77	C₂×D₈	C₂×D₁₃₉	1	4	yes	complemented
78	#131	—	139	4	no	—
79	#127	C₂×D₁₃₉	1	2	yes	complemented
80	#108	—	139	2	no	—
81	#104	—	139	4	no	—
82	#100	—	139	4	no	—
83	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic, complemented
84	#41	—	278	8	no	—
85	#41	—	139	4	no	—
86	C₂×D₄	—	4	2	no	—
87	C₂×D₄	—	278	8	no	—
88	C₂×D₄	—	556	4	no	—
89	C₂×D₄	C₂²×D₁₃₉	1	4	yes	—
90	Q₁₆	—	139	2	no	—
91	SD₁₆	—	278	8	no	—
92	SD₁₆	—	139	4	no	—
93	D₈	—	2	4	no	—
94	D₈	—	2	4	no	—
95	D₈	—	278	8	no	—
96	D₈	—	139	2	no	—
97	M₁₆	—	278	4	no	—
98	M₁₆	—	139	4	no	—
99	C₂×C₈	—	278	4	no	—
100	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
101	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
102	C₂³	—	2	4	no	—
103	C₂³	—	556	4	no	—
104	Q₈	—	139	4	no	—
105	D₄	—	2	8	no	—
106	D₄	—	8	4	no	—
107	D₄	—	278	4	no	—
108	D₄	—	556	8	no	—
109	D₄	—	139	4	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	278	4	no	—
112	C₂×C₄	—	556	4	no	—
113	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
114	C₈	—	278	4	no	—
115	C₈	D₄×D₁₃₉	1	2	yes	—
116	C₈	D₄×D₁₃₉	1	2	yes	—
117	C₂²	—	4	4	no	—
118	C₂²	—	8	4	no	—
119	C₂²	—	278	4	no	—
120	C₂²	—	2224	4	no	—
121	C₂²	#17984887	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	278	4	no	—
124	C₄	C₂×D₄×D₁₃₉	1	2	yes	—
125	C₂	—	2	1	no	—
126	C₂	—	8	4	no	—
127	C₂	—	556	4	no	—
128	C₂	#22581588	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581754

C₂

yes

maximal, complemented

#22581220

C₂

yes

maximal, complemented

#22581120

C₂

yes

maximal, complemented

#22581071

C₂

yes

maximal, complemented

#493×C₁₃₉

C₂

yes

maximal, characteristic, complemented

#17984887

C₂²

yes

complemented

#17984868

C₂²

yes

complemented

#17984866

—

#17984864

C₂²

yes

—

#17984861

—

D₈×D₁₃₉

—

#17984848

C₂²

yes

complemented

#17984778

C₂²

yes

complemented

#17984770

C₂²

yes

complemented

#17984759

C₂²

yes

complemented

C₂×D₈×C₁₃₉

C₂²

yes

complemented

#127×C₁₃₉

C₂²

yes

complemented

C₄×C₈×C₁₃₉

C₂²

yes

characteristic, complemented

#14555785

C₂³

yes

—

#14555784

—

C₂×C₂²⋊D₁₃₉

—

D₄×D₁₃₉

—

Dic₅₅₆

D₄

yes

complemented

#14555774

D₄

yes

complemented

#14555773

—

D₁₁₁₂

D₄

yes

complemented

#14555770

—

#14555769

C₂³

yes

—

#14555768

—

C₈×D₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

C₂³

yes

—

D₈×C₁₃₉

—

D₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂³

yes

—

C₄²×C₁₃₉

C₂³

yes

characteristic

C₂²×D₁₃₉

—

Dic₂₇₈

C₂×D₄

yes

—

D₅₅₆

C₂×D₄

yes

—

C₂²⋊D₁₃₉

—

C₂²⋊D₁₃₉

—

C₂×Dic₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂⁴

yes

characteristic

C₈×C₁₃₉

C₂×D₄

yes

—

C₈×C₁₃₉

C₂×D₄

yes

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

H(5,𝔽₂)

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂²×D₄

yes

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

D₄²

yes

characteristic

D₁₃₉

—

C₁₃₉

#3292

yes

characteristic, complemented, Sylow

#3292

—

139

maximal, Sylow

#576

—

139

—

#493

D₁₃₉

yes

characteristic, complemented

#472

—

139

—

#453

—

139

—

#443

—

139

—

H(5,𝔽₂)

—

139

—

AGL(1,ℤ/8)

—

278

—

AGL(1,ℤ/8)

—

139

—

#135

—

278

—

#135

—

139

—

C₂×D₈

—

278

—

C₂×D₈

C₂×D₁₃₉

yes

complemented

#131

—

139

—

#127

C₂×D₁₃₉

yes

complemented

#108

—

139

—

#104

—

139

—

#100

—

139

—

C₄×C₈

C₂×D₁₃₉

yes

characteristic, complemented

#41

—

278

—

#41

—

139

—

C₂×D₄

—

C₂×D₄

—

278

—

C₂×D₄

—

556

—

C₂×D₄

C₂²×D₁₃₉

yes

—

Q₁₆

—

139

—

SD₁₆

—

278

—

SD₁₆

—

139

—

D₈

—

D₈

—

D₈

—

278

—

D₈

—

139

—

M₁₆

—

278

—

M₁₆

—

139

—

C₂×C₈

—

278

—

100

C₂×C₈

C₂²×D₁₃₉

yes

—

101

C₄²

C₂²×D₁₃₉

yes

characteristic

102

C₂³

—

103

C₂³

—

556

—

104

Q₈

—

139

—

105

D₄

—

106

D₄

—

107

D₄

—

278

—

108

D₄

—

556

—

109

D₄

—

139

—

110

C₂×C₄

—

111

C₂×C₄

—

278

—

112

C₂×C₄

—

556

—

113

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

114

C₈

—

278

—

115

C₈

D₄×D₁₃₉

yes

—

116

C₈

D₄×D₁₃₉

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

278

—

120

C₂²

—

2224

—

121

C₂²

#17984887

yes

characteristic

122

C₄

—

123

C₄

—

278

—

124

C₄

C₂×D₄×D₁₃₉

yes

—

125

C₂

—

126

C₂

—

127

C₂

—

556

—

128

C₂

#22581588

yes

characteristic

	subgroup	G/H
center	C₂	#22581588
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#493×C₁₃₉	C₂
socle	C₂×C₁₃₉	D₄²
hypercenter	#493	D₁₃₉
hypocenter	C₁₃₉	#3292
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

subgroup

G/H

center

C₂

#22581588

commutator, cosocle

C₂×C₄×C₁₃₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₉

Fitting

#493×C₁₃₉

C₂

socle

C₂×C₁₃₉

D₄²

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

N	H ≅ G/N	type
1. #22581754	C₂	semidirect
2. #22581220	C₂	semidirect
3. #22581120	C₂	semidirect
4. #22581071	C₂	semidirect
5. #493×C₁₃₉	C₂	semidirect
6. #17984887	C₂²	semidirect
7. #17984868	C₂²	semidirect
12. #17984848	C₂²	semidirect
13. #17984778	C₂²	semidirect
14. #17984770	C₂²	semidirect
15. #17984759	C₂²	semidirect
16. C₂×D₈×C₁₃₉	C₂²	semidirect
17. #127×C₁₃₉	C₂²	semidirect
18. C₄×C₈×C₁₃₉	C₂²	semidirect
23. Dic₅₅₆	D₄	semidirect
24. #14555774	D₄	semidirect
26. D₁₁₁₂	D₄	semidirect
64. C₁₃₉	#3292	semidirect
67. #493	D₁₃₉	semidirect
77. C₂×D₈	C₂×D₁₃₉	semidirect
79. #127	C₂×D₁₃₉	semidirect
83. C₄×C₈	C₂×D₁₃₉	semidirect

H ≅ G/N

type

1. #22581754

C₂

semidirect

2. #22581220

C₂

semidirect

3. #22581120

C₂

semidirect

4. #22581071

C₂

semidirect

5. #493×C₁₃₉

C₂

semidirect

6. #17984887

C₂²

semidirect

7. #17984868

C₂²

semidirect

12. #17984848

C₂²

semidirect

13. #17984778

C₂²

semidirect

14. #17984770

C₂²

semidirect

15. #17984759

C₂²

semidirect

16. C₂×D₈×C₁₃₉

C₂²

semidirect

17. #127×C₁₃₉

C₂²

semidirect

18. C₄×C₈×C₁₃₉

C₂²

semidirect

23. Dic₅₅₆

D₄

semidirect

24. #14555774

D₄

semidirect

26. D₁₁₁₂

D₄

semidirect

64. C₁₃₉

#3292

semidirect

67. #493

D₁₃₉

semidirect

77. C₂×D₈

C₂×D₁₃₉

semidirect

79. #127

C₂×D₁₃₉

semidirect

83. C₄×C₈

C₂×D₁₃₉

semidirect

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

subgroup	index
#22581754	2
#22581220	2
#22581120	2
#22581071	2
#493×C₁₃₉	2
#3292	139

subgroup

index

#493×C₁₃₉

139

Subgroups — #31818203

Conjugacy classes of subgroups

Complements

Maximal subgroups