Group #31808573 — subgroups

Subgroups — #31808573

Order 17792.

126 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 45 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	#22581350	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581070	C₂	1	1	yes	maximal, characteristic
3	#22580776	C₂	1	1	yes	maximal, characteristic, complemented
4	#22580775	C₂	1	1	yes	maximal, characteristic, complemented
5	#22580773	C₂	1	1	yes	maximal, characteristic, complemented
6	#22580763	C₂	1	1	yes	maximal, characteristic
7	#360×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	#17984849	—	2	1	no	—
9	#17984841	—	2	1	no	—
10	#17984782	C₂²	1	1	yes	characteristic
11	#17984778	C₂²	1	1	yes	characteristic
12	#17984774	C₂²	1	1	yes	characteristic
13	#17984773	—	2	1	no	—
14	#17984769	—	2	1	no	—
15	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
16	#135×C₁₃₉	C₂²	1	1	yes	characteristic
17	M₃₂×C₁₃₉	C₂²	1	1	yes	characteristic
18	#106×C₁₃₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₇₈	—	4	1	no	—
20	#14555769	—	2	1	no	—
21	#14555769	—	4	1	no	—
22	C₂×C₁₃₉⋊C₈	—	4	1	no	—
23	C₂×C₁₃₉⋊C₈	D₄	1	1	yes	characteristic
24	C₄⋊Dic₁₃₉	—	2	1	no	—
25	Dic₁₃₉⋊C₄	—	4	1	no	—
26	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
27	C₁₃₉⋊C₁₆	—	2	1	no	—
28	#41×C₁₃₉	—	2	1	no	—
29	Q₁₆×C₁₃₉	Q₈	1	1	yes	characteristic, complemented
30	SD₁₆×C₁₃₉	—	2	1	no	—
31	D₈×C₁₃₉	Q₈	1	1	yes	characteristic, complemented
32	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₃₉	—	2	1	no	—
34	C₁₆×C₁₃₉	—	2	1	no	—
35	Dic₂₇₈	—	8	2	no	—
36	C₂×Dic₁₃₉	—	2	1	no	—
37	C₂×Dic₁₃₉	—	4	1	no	—
38	C₁₃₉⋊C₈	—	4	1	no	—
39	C₁₃₉⋊C₈	SD₁₆	1	1	yes	characteristic
40	C₁₃₉⋊C₈	SD₁₆	1	1	yes	characteristic
41	Q₈×C₁₃₉	—	2	1	no	—
42	D₄×C₁₃₉	—	2	1	no	—
43	D₄×C₁₃₉	—	4	1	no	—
44	C₂×C₄×C₁₃₉	—	4	1	no	—
45	C₂×C₄×C₁₃₉	—	4	1	no	—
46	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
47	C₈×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
48	C₈×C₁₃₉	#41	1	1	yes	characteristic
49	C₂²×C₁₃₉	—	4	1	no	—
50	C₂²×C₁₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
51	C₄×C₁₃₉	—	4	1	no	—
52	C₄×C₁₃₉	—	8	1	no	—
53	C₄×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
54	C₄×C₁₃₉	C₂×SD₁₆	1	1	yes	characteristic
55	Dic₁₃₉	—	2	2	no	—
56	Dic₁₃₉	—	8	1	no	—
57	C₂×C₁₃₉	—	2	1	no	—
58	C₂×C₁₃₉	—	8	1	no	—
59	C₂×C₁₃₉	#476	1	1	yes	characteristic
60	C₁₃₉	#2230	1	1	yes	characteristic, complemented, Sylow
61	#2230	—	139	1	no	maximal, Sylow
62	#498	—	139	1	no	—
63	#443	—	139	1	no	—
64	#365	—	139	1	no	—
65	#365	—	139	1	no	—
66	#364	—	139	1	no	—
67	#360	—	139	1	no	—
68	#360	D₁₃₉	1	1	yes	characteristic
69	#135	C₂×D₁₃₉	1	1	yes	characteristic
70	#131	—	278	1	no	—
71	C₄⋊Q₈	—	278	1	no	—
72	M₃₂	—	139	1	no	—
73	M₃₂	C₂×D₁₃₉	1	1	yes	characteristic
74	#108	—	139	1	no	—
75	#106	—	278	1	no	—
76	#106	—	139	1	no	—
77	#106	C₂×D₁₃₉	1	1	yes	characteristic
78	#104	—	278	1	no	—
79	C₄×C₈	—	139	1	no	—
80	#41	—	2	1	no	—
81	C₂×Q₈	—	556	1	no	—
82	Q₁₆	Dic₂₇₈	1	1	yes	characteristic, complemented
83	SD₁₆	—	2	1	no	—
84	D₈	Dic₂₇₈	1	1	yes	characteristic, complemented
85	M₁₆	—	278	1	no	—
86	M₁₆	—	556	1	no	—
87	C₂×C₈	—	556	1	no	—
88	C₂×C₈	—	139	1	no	—
89	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	278	1	no	—
92	C₄⋊C₄	—	556	1	no	—
93	C₄²	—	139	1	no	—
94	C₁₆	—	2	1	no	—
95	C₁₆	—	278	1	no	—
96	Q₈	—	2	1	no	—
97	Q₈	—	1112	2	no	—
98	D₄	—	2	1	no	—
99	D₄	—	4	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	278	1	no	—
103	C₂×C₄	—	556	1	no	—
104	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
105	C₈	—	556	1	no	—
106	C₈	—	139	1	no	—
107	C₈	—	139	1	no	—
108	C₈	C₂×Dic₂₇₈	1	1	yes	characteristic
109	C₈	#14555784	1	1	yes	characteristic
110	C₂²	—	4	1	no	—
111	C₂²	#17984866	1	1	yes	characteristic
112	C₄	—	4	1	no	—
113	C₄	—	8	1	no	—
114	C₄	—	278	2	no	—
115	C₄	—	1112	1	no	—
116	C₄	#17984819	1	1	yes	characteristic
117	C₄	SD₁₆×D₁₃₉	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	8	1	no	—
120	C₂	#22581241	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581350

C₂

yes

maximal, characteristic, complemented

#22581070

C₂

yes

maximal, characteristic

#22580776

C₂

yes

maximal, characteristic, complemented

#22580775

C₂

yes

maximal, characteristic, complemented

#22580773

C₂

yes

maximal, characteristic, complemented

#22580763

C₂

yes

maximal, characteristic

#360×C₁₃₉

C₂

yes

maximal, characteristic

#17984849

—

#17984841

—

#17984782

C₂²

yes

characteristic

#17984778

C₂²

yes

characteristic

#17984774

C₂²

yes

characteristic

#17984773

—

#17984769

—

C₈×Dic₁₃₉

C₂²

yes

characteristic

#135×C₁₃₉

C₂²

yes

characteristic

M₃₂×C₁₃₉

C₂²

yes

characteristic

#106×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

#14555769

—

#14555769

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

D₄

yes

characteristic

C₄⋊Dic₁₃₉

—

Dic₁₃₉⋊C₄

—

C₄×Dic₁₃₉

D₄

yes

characteristic

C₁₃₉⋊C₁₆

—

#41×C₁₃₉

—

Q₁₆×C₁₃₉

Q₈

yes

characteristic, complemented

SD₁₆×C₁₃₉

—

D₈×C₁₃₉

Q₈

yes

characteristic, complemented

C₂×C₈×C₁₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₉

—

C₁₆×C₁₃₉

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

SD₁₆

yes

characteristic

C₁₃₉⋊C₈

SD₁₆

yes

characteristic

Q₈×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₈×C₁₃₉

C₂×Q₈

yes

characteristic

C₈×C₁₃₉

#41

yes

characteristic

C₂²×C₁₃₉

—

C₂²×C₁₃₉

AGL(1,ℤ/8)

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂²⋊Q₈

yes

characteristic

C₄×C₁₃₉

C₂×SD₁₆

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#476

yes

characteristic

C₁₃₉

#2230

yes

characteristic, complemented, Sylow

#2230

—

139

maximal, Sylow

#498

—

139

—

#443

—

139

—

#365

—

139

—

#365

—

139

—

#364

—

139

—

#360

—

139

—

#360

D₁₃₉

yes

characteristic

#135

C₂×D₁₃₉

yes

characteristic

#131

—

278

—

C₄⋊Q₈

—

278

—

M₃₂

—

139

—

M₃₂

C₂×D₁₃₉

yes

characteristic

#108

—

139

—

#106

—

278

—

#106

—

139

—

#106

C₂×D₁₃₉

yes

characteristic

#104

—

278

—

C₄×C₈

—

139

—

#41

—

C₂×Q₈

—

556

—

Q₁₆

Dic₂₇₈

yes

characteristic, complemented

SD₁₆

—

D₈

Dic₂₇₈

yes

characteristic, complemented

M₁₆

—

278

—

M₁₆

—

556

—

C₂×C₈

—

556

—

C₂×C₈

—

139

—

C₂×C₈

C₂²×D₁₃₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

556

—

C₄²

—

139

—

C₁₆

—

C₁₆

—

278

—

Q₈

—

Q₈

—

1112

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

278

—

103

C₂×C₄

—

556

—

104

C₂×C₄

D₄×D₁₃₉

yes

characteristic

105

C₈

—

556

—

106

C₈

—

139

—

107

C₈

—

139

—

108

C₈

C₂×Dic₂₇₈

yes

characteristic

109

C₈

#14555784

yes

characteristic

110

C₂²

—

111

C₂²

#17984866

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

278

—

115

C₄

—

1112

—

116

C₄

#17984819

yes

characteristic

117

C₄

SD₁₆×D₁₃₉

yes

characteristic

118

C₂

—

119

C₂

—

120

C₂

#22581241

yes

characteristic

	subgroup	G/H
center	C₂	#22581241
commutator, cosocle	C₂×C₈×C₁₃₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₃₉
Fitting	#360×C₁₃₉	C₂
socle	C₂×C₁₃₉	#476
prime radical	#135×C₁₃₉	C₂²
hypercenter	#360	D₁₃₉
hypocenter	C₁₃₉	#2230
LCS	G ▷ C₂×C₈×C₁₃₉ ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #360

subgroup

G/H

center

C₂

#22581241

commutator, cosocle

C₂×C₈×C₁₃₉

C₂³

Frattini

C₂×C₈

C₂²×D₁₃₉

Fitting

#360×C₁₃₉

C₂

socle

C₂×C₁₃₉

#476

prime radical

#135×C₁₃₉

C₂²

D₁₃₉

C₁₃₉

G ▷ C₂×C₈×C₁₃₉ ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #360

N	H ≅ G/N	type
1. #22581350	C₂	semidirect
3. #22580776	C₂	semidirect
4. #22580775	C₂	semidirect
5. #22580773	C₂	semidirect
29. Q₁₆×C₁₃₉	Q₈	semidirect
31. D₈×C₁₃₉	Q₈	semidirect
60. C₁₃₉	#2230	semidirect
82. Q₁₆	Dic₂₇₈	semidirect
84. D₈	Dic₂₇₈	semidirect

H ≅ G/N

type

1. #22581350

C₂

semidirect

3. #22580776

C₂

semidirect

4. #22580775

C₂

semidirect

5. #22580773

C₂

semidirect

29. Q₁₆×C₁₃₉

Q₈

semidirect

31. D₈×C₁₃₉

Q₈

semidirect

60. C₁₃₉

#2230

semidirect

82. Q₁₆

Dic₂₇₈

semidirect

84. D₈

Dic₂₇₈

semidirect

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

subgroup	index
#22581350	2
#22581070	2
#22580776	2
#22580775	2
#22580773	2
#22580763	2
#360×C₁₃₉	2
#2230	139

subgroup

index

#360×C₁₃₉

139

Subgroups — #31808573

Conjugacy classes of subgroups

Complements

Maximal subgroups