Group #31808574 — subgroups

Subgroups — #31808574

Order 17792.

126 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 43 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	#22581069	C₂	1	1	yes	maximal, characteristic
3	#22580777	C₂	1	1	yes	maximal, characteristic, complemented
4	#22580773	C₂	1	1	yes	maximal, characteristic, complemented
5	#22580764	C₂	1	1	yes	maximal, characteristic
6	#22580762	C₂	1	1	yes	maximal, characteristic
7	#365×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#17984864	C₂²	1	1	yes	characteristic
9	#17984850	—	2	1	no	—
10	#17984841	—	2	1	no	—
11	#17984782	C₂²	1	1	yes	characteristic
12	#17984774	C₂²	1	1	yes	characteristic
13	#17984773	—	2	1	no	—
14	#17984768	—	2	1	no	—
15	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
16	M₃₂×C₁₃₉	C₂²	1	1	yes	characteristic
17	#108×C₁₃₉	C₂²	1	1	yes	characteristic
18	#106×C₁₃₉	C₂²	1	1	yes	characteristic
19	#14555785	—	2	1	no	—
20	C₂×Dic₂₇₈	—	4	1	no	—
21	Dic₅₅₆	Q₈	1	1	yes	characteristic, complemented
22	#14555774	—	2	1	no	—
23	D₁₁₁₂	Q₈	1	1	yes	characteristic, complemented
24	#14555768	—	4	1	no	—
25	C₂×C₁₃₉⋊C₈	D₄	1	1	yes	characteristic
26	C₈×D₁₃₉	—	4	1	no	—
27	C₄⋊Dic₁₃₉	—	2	1	no	—
28	Dic₁₃₉⋊C₄	—	4	1	no	—
29	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
30	C₁₃₉⋊C₁₆	—	2	1	no	—
31	M₁₆×C₁₃₉	—	2	1	no	—
32	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₃₉	—	2	1	no	—
34	C₁₆×C₁₃₉	—	2	1	no	—
35	Dic₂₇₈	—	2	1	no	—
36	Dic₂₇₈	—	8	2	no	—
37	D₅₅₆	—	2	1	no	—
38	C₂²⋊D₁₃₉	—	4	1	no	—
39	C₂×Dic₁₃₉	—	2	1	no	—
40	C₂×Dic₁₃₉	—	4	1	no	—
41	C₄×D₁₃₉	—	4	1	no	—
42	C₁₃₉⋊C₈	SD₁₆	1	1	yes	characteristic
43	C₁₃₉⋊C₈	SD₁₆	1	1	yes	characteristic
44	C₂×C₄×C₁₃₉	—	4	1	no	—
45	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
46	C₈×C₁₃₉	—	4	1	no	—
47	C₈×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
48	C₈×C₁₃₉	#41	1	1	yes	characteristic
49	C₂²×C₁₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₁₃₉	—	4	1	no	—
51	C₄×C₁₃₉	—	8	1	no	—
52	C₄×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
53	C₄×C₁₃₉	C₂×SD₁₆	1	1	yes	characteristic
54	Dic₁₃₉	—	2	2	no	—
55	Dic₁₃₉	—	4	1	no	—
56	Dic₁₃₉	—	8	1	no	—
57	C₂×C₁₃₉	—	2	1	no	—
58	C₂×C₁₃₉	#476	1	1	yes	characteristic
59	D₁₃₉	—	8	1	no	—
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	D₁₃₉	1	1	yes	characteristic, complemented
66	#364	—	139	1	no	—
67	#360	—	139	1	no	—
68	#360	—	139	1	no	—
69	#135	—	139	1	no	—
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	C₂×D₁₃₉	1	1	yes	characteristic
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	—	278	1	no	—
81	C₂×Q₈	—	556	1	no	—
82	Q₁₆	—	139	1	no	—
83	SD₁₆	—	278	1	no	—
84	D₈	—	139	1	no	—
85	M₁₆	—	2	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	—	278	1	no	—
97	Q₈	—	1112	2	no	—
98	D₄	—	278	1	no	—
99	D₄	—	556	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	278	1	no	—
102	C₂×C₄	—	556	1	no	—
103	C₂×C₄	—	556	1	no	—
104	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
105	C₈	—	4	1	no	—
106	C₈	—	139	1	no	—
107	C₈	—	139	1	no	—
108	C₈	Q₈×D₁₃₉	1	1	yes	characteristic
109	C₈	#14555785	1	1	yes	characteristic
110	C₂²	—	556	1	no	—
111	C₂²	#17984869	1	1	yes	characteristic
112	C₄	—	8	1	no	—
113	C₄	—	278	2	no	—
114	C₄	—	556	1	no	—
115	C₄	—	1112	1	no	—
116	C₄	#17984820	1	1	yes	characteristic
117	C₄	SD₁₆×D₁₃₉	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	1112	1	no	—
120	C₂	#22581242	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581350

C₂

yes

maximal, characteristic, complemented

#22581069

C₂

yes

maximal, characteristic

#22580777

C₂

yes

maximal, characteristic, complemented

#22580773

C₂

yes

maximal, characteristic, complemented

#22580764

C₂

yes

maximal, characteristic

#22580762

C₂

yes

maximal, characteristic

#365×C₁₃₉

C₂

yes

maximal, characteristic, complemented

#17984864

C₂²

yes

—

—

C₂²

yes

characteristic

#17984774

C₂²

yes

characteristic

#17984773

—

#17984768

—

C₈×Dic₁₃₉

C₂²

yes

characteristic

M₃₂×C₁₃₉

C₂²

yes

characteristic

#108×C₁₃₉

C₂²

yes

characteristic

#106×C₁₃₉

C₂²

yes

characteristic

#14555785

—

C₂×Dic₂₇₈

—

Dic₅₅₆

Q₈

yes

characteristic, complemented

#14555774

—

D₁₁₁₂

Q₈

yes

characteristic, complemented

#14555768

—

C₂×C₁₃₉⋊C₈

D₄

yes

characteristic

C₈×D₁₃₉

—

C₄⋊Dic₁₃₉

—

Dic₁₃₉⋊C₄

—

C₄×Dic₁₃₉

D₄

yes

characteristic

C₁₃₉⋊C₁₆

—

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₉

—

C₁₆×C₁₃₉

—

Dic₂₇₈

—

Dic₂₇₈

—

D₅₅₆

—

C₂²⋊D₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

SD₁₆

yes

characteristic

C₁₃₉⋊C₈

SD₁₆

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₈×C₁₃₉

—

C₈×C₁₃₉

C₂×Q₈

yes

C₈×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂²⋊Q₈

yes

characteristic

C₄×C₁₃₉

C₂×SD₁₆

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#476

yes

characteristic

D₁₃₉

—

C₁₃₉

#2230

yes

characteristic, complemented, Sylow

#2230

—

139

maximal, Sylow

#498

—

139

—

#443

—

139

—

#365

—

139

—

#365

D₁₃₉

yes

characteristic, complemented

#364

—

139

—

#360

—

139

—

#360

—

139

—

#135

—

139

—

#131

—

278

—

C₄⋊Q₈

—

278

—

M₃₂

—

139

—

M₃₂

C₂×D₁₃₉

yes

characteristic

#108

C₂×D₁₃₉

yes

characteristic

#106

—

278

—

#106

—

139

—

#106

C₂×D₁₃₉

yes

characteristic

#104

—

278

—

C₄×C₈

—

139

—

#41

—

278

—

C₂×Q₈

—

556

—

Q₁₆

—

139

—

SD₁₆

—

278

—

D₈

—

139

—

M₁₆

—

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

—

278

—

Q₈

—

1112

—

D₄

—

278

—

D₄

—

556

—

100

C₂×C₄

—

101

C₂×C₄

—

278

—

102

C₂×C₄

—

556

—

103

C₂×C₄

—

556

—

104

C₂×C₄

D₄×D₁₃₉

yes

characteristic

105

C₈

—

106

C₈

—

139

—

107

C₈

—

139

—

108

C₈

Q₈×D₁₃₉

yes

characteristic

109

C₈

#14555785

yes

characteristic

110

C₂²

—

556

—

111

C₂²

#17984869

yes

characteristic

112

C₄

—

113

C₄

—

278

—

114

C₄

—

556

—

115

C₄

—

1112

—

116

C₄

#17984820

yes

characteristic

117

C₄

SD₁₆×D₁₃₉

yes

characteristic

118

C₂

—

119

C₂

—

1112

—

120

C₂

#22581242

yes

characteristic

	subgroup	G/H
center	C₂	#22581242
commutator, cosocle	C₂×C₈×C₁₃₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₃₉
Fitting	#365×C₁₃₉	C₂
socle	C₂×C₁₃₉	#476
prime radical	#17984864	C₂²
hypercenter	#365	D₁₃₉
hypocenter	C₁₃₉	#2230
LCS	G ▷ C₂×C₈×C₁₃₉ ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

subgroup

G/H

center

C₂

#22581242

commutator, cosocle

C₂×C₈×C₁₃₉

C₂³

Frattini

C₂×C₈

C₂²×D₁₃₉

Fitting

#365×C₁₃₉

C₂

C₂×C₁₃₉

C₂²

D₁₃₉

C₁₃₉

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

N	H ≅ G/N	type
1. #22581350	C₂	semidirect
3. #22580777	C₂	semidirect
4. #22580773	C₂	semidirect
7. #365×C₁₃₉	C₂	semidirect
21. Dic₅₅₆	Q₈	semidirect
23. D₁₁₁₂	Q₈	semidirect
60. C₁₃₉	#2230	semidirect
65. #365	D₁₃₉	semidirect

H ≅ G/N

type

1. #22581350

C₂

semidirect

3. #22580777

C₂

semidirect

4. #22580773

C₂

semidirect

7. #365×C₁₃₉

C₂

semidirect

21. Dic₅₅₆

Q₈

semidirect

23. D₁₁₁₂

Q₈

semidirect

60. C₁₃₉

#2230

semidirect

65. #365

D₁₃₉

semidirect

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

subgroup	index
#22581350	2
#22581069	2
#22580777	2
#22580773	2
#22580764	2
#22580762	2
#365×C₁₃₉	2
#2230	139

subgroup

index

#365×C₁₃₉

139

Subgroups — #31808574

Conjugacy classes of subgroups

Complements

Maximal subgroups