Group #31805923 — subgroups

Subgroups — #31805923

Order 17792.

152 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 55 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	#22581358	C₂	1	1	yes	maximal, characteristic
2	#22581164	C₂	1	2	yes	maximal
3	#22581083	C₂	1	1	yes	maximal, characteristic
4	#22580700	C₂	1	2	yes	maximal
5	#334×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₅₆	—	2	2	no	—
7	#17984842	C₂²	1	2	yes	—
8	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
9	#17984776	—	2	1	no	—
10	#17984776	—	2	1	no	—
11	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
12	Dic₁₃₉⋊C₈	—	2	1	no	—
13	#17984767	—	2	2	no	—
14	#17984752	—	2	1	no	—
15	C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
16	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₇₈	—	2	2	no	—
18	C₂×Dic₂₇₈	D₄	1	1	yes	characteristic
19	Dic₅₅₆	—	4	4	no	—
20	C₂×C₁₃₉⋊C₈	—	2	1	no	—
21	C₄⋊Dic₁₃₉	—	2	2	no	—
22	C₄⋊Dic₁₃₉	—	2	2	no	—
23	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
24	Dic₁₃₉⋊C₄	—	2	1	no	—
25	C₄×Dic₁₃₉	—	2	1	no	—
26	C₂×C₈×C₁₃₉	—	2	2	no	—
27	C₂×C₈×C₁₃₉	D₄	1	1	yes	characteristic
28	C₂×C₈×C₁₃₉	D₄	1	1	yes	characteristic
29	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
30	Dic₂₇₈	—	2	2	no	—
31	Dic₂₇₈	—	4	2	no	—
32	Dic₂₇₈	—	4	2	no	—
33	C₂×Dic₁₃₉	—	2	1	no	—
34	C₂×Dic₁₃₉	—	2	1	no	—
35	C₂×Dic₁₃₉	—	4	2	no	—
36	C₁₃₉⋊C₈	—	4	1	no	—
37	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
40	C₈×C₁₃₉	—	2	1	no	—
41	C₈×C₁₃₉	—	4	2	no	—
42	C₈×C₁₃₉	Q₁₆	1	2	yes	complemented
43	C₂²×C₁₃₉	#121	1	1	yes	characteristic
44	C₄×C₁₃₉	—	2	1	no	—
45	C₄×C₁₃₉	C₂×Q₁₆	1	1	yes	characteristic
46	C₄×C₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
47	C₄×C₁₃₉	#137	1	1	yes	characteristic
48	Dic₁₃₉	—	4	1	no	—
49	Dic₁₃₉	—	4	1	no	—
50	Dic₁₃₉	—	8	2	no	—
51	C₂×C₁₃₉	#462	1	1	yes	characteristic
52	C₂×C₁₃₉	#469	1	1	yes	characteristic
53	C₂×C₁₃₉	#473	1	1	yes	characteristic
54	C₁₃₉	#1694	1	1	yes	characteristic, complemented, Sylow
55	#1694	—	139	1	no	maximal, Sylow
56	#500	—	139	1	no	—
57	#462	—	139	2	no	—
58	#446	—	139	1	no	—
59	#334	D₁₃₉	1	1	yes	characteristic
60	#332	—	139	2	no	—
61	C₂×Q₁₆	—	278	2	no	—
62	C₄⋊Q₈	—	139	2	no	—
63	C₄×Q₈	—	139	1	no	—
64	#107	—	278	1	no	—
65	#107	—	278	1	no	—
66	C₄⋊C₈	—	278	1	no	—
67	C₄⋊C₈	—	139	1	no	—
68	C₄⋊C₈	C₂×D₁₃₉	1	2	yes	—
69	#103	—	278	2	no	—
70	#97	—	278	1	no	—
71	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
72	C₂×Q₈	—	278	2	no	—
73	C₂×Q₈	—	139	1	no	—
74	Q₁₆	—	556	4	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	278	1	no	—
77	C₂×C₈	D₅₅₆	1	1	yes	characteristic
78	C₂×C₈	D₅₅₆	1	1	yes	characteristic
79	C₄⋊C₄	—	278	1	no	—
80	C₄⋊C₄	—	278	2	no	—
81	C₄⋊C₄	—	278	2	no	—
82	C₄⋊C₄	—	139	1	no	—
83	C₄²	—	278	1	no	—
84	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
85	Q₈	—	278	2	no	—
86	Q₈	—	556	2	no	—
87	Q₈	—	556	2	no	—
88	C₂×C₄	—	278	1	no	—
89	C₂×C₄	—	278	1	no	—
90	C₂×C₄	—	556	2	no	—
91	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
92	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
93	C₂×C₄	#14555786	1	1	yes	characteristic
94	C₈	—	2	1	no	—
95	C₈	—	4	2	no	—
96	C₈	—	556	1	no	—
97	C₈	Dic₅₅₆	1	2	yes	complemented
98	C₂²	#17984817	1	1	yes	characteristic
99	C₄	—	2	1	no	—
100	C₄	—	556	1	no	—
101	C₄	—	556	1	no	—
102	C₄	—	1112	2	no	—
103	C₄	C₂×Dic₅₅₆	1	1	yes	characteristic
104	C₄	#17984869	1	2	yes	—
105	C₄	#17984875	1	1	yes	characteristic
106	C₂	#22581164	1	1	yes	characteristic
107	C₂	#22581207	1	1	yes	characteristic
108	C₂	#22581229	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581358

C₂

yes

maximal, characteristic

#22581164

C₂

yes

maximal

#22581083

C₂

yes

maximal, characteristic

#22580700

C₂

yes

maximal

#334×C₁₃₉

C₂

yes

maximal, characteristic

C₂×Dic₅₅₆

—

#17984842

C₂²

yes

—

C₄×Dic₂₇₈

C₂²

yes

characteristic

#17984776

—

#17984776

—

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

characteristic

Dic₁₃₉⋊C₈

—

#17984767

—

#17984752

—

C₄⋊C₈×C₁₃₉

C₂²

yes

—

C₄×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

D₄

yes

characteristic

Dic₅₅₆

—

C₂×C₁₃₉⋊C₈

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

D₄

yes

characteristic

Dic₁₃₉⋊C₄

—

C₄×Dic₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

D₄

yes

characteristic

C₂×C₈×C₁₃₉

D₄

yes

characteristic

C₄²×C₁₃₉

C₂³

yes

characteristic

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

characteristic

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₈×C₁₃₉

Q₁₆

yes

complemented

C₂²×C₁₃₉

#121

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×Q₁₆

yes

characteristic

C₄×C₁₃₉

AGL(1,ℤ/8)

yes

—

C₄×C₁₃₉

#137

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#1694

—

139

maximal, Sylow

#500

—

139

—

#462

—

139

—

#446

—

139

—

#334

D₁₃₉

yes

characteristic

#332

—

139

—

C₂×Q₁₆

—

278

—

C₄⋊Q₈

—

139

—

C₄×Q₈

—

139

—

#107

—

278

—

#107

—

278

—

C₄⋊C₈

—

278

—

C₄⋊C₈

—

139

—

C₄⋊C₈

C₂×D₁₃₉

yes

—

#103

—

278

—

#97

—

278

—

C₄×C₈

C₂×D₁₃₉

yes

characteristic

C₂×Q₈

—

278

—

C₂×Q₈

—

139

—

Q₁₆

—

556

—

C₂×C₈

—

C₂×C₈

—

278

—

C₂×C₈

D₅₅₆

yes

characteristic

C₂×C₈

D₅₅₆

yes

characteristic

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

139

—

C₄²

—

278

—

C₄²

C₂²×D₁₃₉

yes

characteristic

Q₈

—

278

—

Q₈

—

556

—

Q₈

—

556

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

—

556

—

C₂×C₄

D₄×D₁₃₉

yes

characteristic

C₂×C₄

C₂×D₅₅₆

yes

characteristic

C₂×C₄

#14555786

yes

characteristic

C₈

—

C₈

—

C₈

—

556

—

C₈

Dic₅₅₆

yes

complemented

C₂²

#17984817

yes

characteristic

C₄

—

100

C₄

—

556

—

101

C₄

—

556

—

102

C₄

—

1112

—

103

C₄

C₂×Dic₅₅₆

yes

characteristic

104

C₄

#17984869

yes

—

105

C₄

#17984875

yes

characteristic

106

C₂

#22581164

yes

characteristic

107

C₂

#22581207

yes

characteristic

108

C₂

#22581229

yes

characteristic

	subgroup	G/H
center	C₂²	#17984817
commutator, cosocle	C₄²×C₁₃₉	C₂³
Frattini	C₄²	C₂²×D₁₃₉
Fitting	#334×C₁₃₉	C₂
socle, prime radical	C₂²×C₁₃₉	#121
hypercenter	#334	D₁₃₉
hypocenter	C₁₃₉	#1694
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#17984817

commutator, cosocle

C₄²×C₁₃₉

C₂³

Frattini

C₄²

C₂²×D₁₃₉

Fitting

#334×C₁₃₉

C₂

C₂²×C₁₃₉

D₁₃₉

C₁₃₉

G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
42. C₈×C₁₃₉	Q₁₆	semidirect
54. C₁₃₉	#1694	semidirect
97. C₈	Dic₅₅₆	semidirect

H ≅ G/N

type

42. C₈×C₁₃₉

Q₁₆

semidirect

54. C₁₃₉

#1694

semidirect

97. C₈

Dic₅₅₆

semidirect

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

subgroup	index
#22581358	2
#22581164	2
#22581083	2
#22580700	2
#334×C₁₃₉	2
#1694	139

subgroup

index

#334×C₁₃₉

139

Subgroups — #31805923

Conjugacy classes of subgroups

Complements

Maximal subgroups