Group #31807192 — subgroups

Subgroups — #31807192

Order 17792.

184 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 85 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	#22581001	C₂	1	2	yes	maximal
2	C₂×#17984778	C₂	1	1	yes	maximal, characteristic
3	#22580908	C₂	1	1	yes	maximal, characteristic
4	#22580724	C₂	1	2	yes	maximal
5	C₂×#108×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555769	C₂²	1	2	yes	—
7	C₂×#14555769	C₂²	1	1	yes	characteristic
8	#17984801	C₂²	1	1	yes	characteristic
9	#17984778	C₄	1	4	yes	complemented
10	#17984777	—	2	4	no	—
11	#17984752	—	2	1	no	—
12	C₈×Dic₁₃₉	—	2	1	no	—
13	C₂×M₁₆×C₁₃₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	#108×C₁₃₉	—	2	1	no	—
16	#108×C₁₃₉	—	2	1	no	—
17	#14555769	—	2	1	no	—
18	#14555769	—	2	2	no	—
19	#14555769	C₂×C₄	1	4	yes	—
20	#14555769	D₄	1	2	yes	—
21	C₂×C₁₃₉⋊C₈	—	2	2	no	—
22	C₂×C₁₃₉⋊C₈	—	2	1	no	—
23	C₄⋊Dic₁₃₉	—	2	1	no	—
24	C₂²⋊Dic₁₃₉	Q₈	1	2	yes	—
25	C₄×Dic₁₃₉	—	2	1	no	—
26	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₃₉	—	2	2	no	—
28	M₁₆×C₁₃₉	—	2	2	no	—
29	C₂×C₈×C₁₃₉	—	2	2	no	—
30	C₂×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₁₃₉	D₄	1	1	yes	characteristic
32	C₂×C₈×C₁₃₉	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₃₉	Q₈	1	2	yes	—
34	C₂×Dic₁₃₉	—	2	2	no	—
35	C₁₃₉⋊C₈	—	2	4	no	—
36	C₁₃₉⋊C₈	—	2	2	no	—
37	C₂³×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
43	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
44	C₈×C₁₃₉	—	4	2	no	—
45	C₈×C₁₃₉	C₄⋊C₄	1	2	yes	—
46	C₈×C₁₃₉	C₄⋊C₄	1	2	yes	—
47	C₂²×C₁₃₉	—	4	1	no	—
48	C₂²×C₁₃₉	C₄×Q₈	1	1	yes	characteristic
49	C₂²×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₃₉	C₄⋊Q₈	1	1	yes	characteristic
51	C₄×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
53	C₄×C₁₃₉	#121	1	1	yes	characteristic
54	C₄×C₁₃₉	#125	1	1	yes	characteristic
55	Dic₁₃₉	—	4	2	no	—
56	C₂×C₁₃₉	—	2	1	no	—
57	C₂×C₁₃₉	—	2	1	no	—
58	C₂×C₁₃₉	—	2	1	no	—
59	C₂×C₁₃₉	#389	1	1	yes	characteristic
60	C₁₃₉	#1953	1	1	yes	characteristic, complemented, Sylow
61	#1953	—	139	1	no	maximal, Sylow
62	#430	—	139	2	no	—
63	C₂×#108	—	139	1	no	—
64	C₂×#108	D₁₃₉	1	1	yes	characteristic
65	#405	—	139	1	no	—
66	#344	—	139	2	no	—
67	C₂×M₁₆	—	139	2	no	—
68	C₂×M₁₆	—	139	1	no	—
69	C₂×M₁₆	C₂×D₁₃₉	1	2	yes	—
70	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
71	#117	—	139	1	no	—
72	#108	—	2	1	no	—
73	#108	—	2	1	no	—
74	#108	—	278	4	no	—
75	#108	—	139	4	no	—
76	#97	—	278	1	no	—
77	C₄×C₈	—	278	1	no	—
78	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
79	M₁₆	—	2	2	no	—
80	M₁₆	—	2	2	no	—
81	M₁₆	—	278	1	no	—
82	M₁₆	—	278	2	no	—
83	M₁₆	—	139	4	no	—
84	M₁₆	—	139	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	278	2	no	—
87	C₂×C₈	—	278	1	no	—
88	C₂×C₈	C₄×D₁₃₉	1	2	yes	—
89	C₂×C₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
91	C₂×C₈	Dic₂₇₈	1	2	yes	—
92	C₄⋊C₄	—	278	1	no	—
93	C₂²⋊C₄	—	139	2	no	—
94	C₄²	—	278	1	no	—
95	C₂³	Q₈×D₁₃₉	1	1	yes	characteristic
96	C₂×C₄	—	278	2	no	—
97	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
98	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
99	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
100	C₂×C₄	C₂×Dic₂₇₈	1	1	yes	characteristic
101	C₂×C₄	#14555784	1	1	yes	characteristic
102	C₂×C₄	#14555786	1	1	yes	characteristic
103	C₈	—	4	2	no	—
104	C₈	—	278	4	no	—
105	C₈	—	278	2	no	—
106	C₈	Dic₁₃₉⋊C₄	1	2	yes	—
107	C₈	Dic₁₃₉⋊C₄	1	2	yes	—
108	C₂²	—	4	1	no	—
109	C₂²	#17984807	1	1	yes	characteristic
110	C₂²	#17984818	1	1	yes	characteristic
111	C₂²	#17984841	1	1	yes	characteristic
112	C₄	—	556	2	no	—
113	C₄	C₂×Dic₁₃₉⋊C₄	1	1	yes	characteristic
114	C₄	#17984804	1	1	yes	characteristic
115	C₄	#17984813	1	1	yes	characteristic
116	C₄	#17984833	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	1	no	—
119	C₂	—	2	1	no	—
120	C₂	#22580849	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581001

C₂

yes

maximal

C₂×#17984778

C₂

yes

maximal, characteristic

#22580908

C₂

yes

maximal, characteristic

#22580724

C₂

yes

maximal

C₂×#108×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555769

C₂²

yes

—

C₂×#14555769

C₂²

yes

characteristic

#17984801

C₂²

yes

characteristic

#17984778

C₄

yes

complemented

#17984777

—

#17984752

—

C₈×Dic₁₃₉

—

C₂×M₁₆×C₁₃₉

C₂²

yes

—

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

#108×C₁₃₉

—

#108×C₁₃₉

—

#14555769

—

#14555769

—

#14555769

C₂×C₄

yes

—

#14555769

D₄

yes

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₄⋊Dic₁₃₉

—

C₂²⋊Dic₁₃₉

Q₈

yes

—

C₄×Dic₁₃₉

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

—

C₂×C₈×C₁₃₉

D₄

yes

characteristic

C₂×C₈×C₁₃₉

D₄

yes

characteristic

C₂×C₈×C₁₃₉

Q₈

yes

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

C₂×C₄×C₁₃₉

yes

C₂×C₄×C₁₃₉

yes

C₈×C₁₃₉

—

C₈×C₁₃₉

C₄⋊C₄

yes

—

C₈×C₁₃₉

C₄⋊C₄

yes

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₄×Q₈

yes

characteristic

C₂²×C₁₃₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₃₉

C₄⋊Q₈

yes

characteristic

C₄×C₁₃₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₃₉

C₄×D₄

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#389

yes

characteristic

C₁₃₉

#1953

yes

characteristic, complemented, Sylow

#1953

—

139

maximal, Sylow

#430

—

139

—

C₂×#108

—

139

—

C₂×#108

D₁₃₉

yes

characteristic

#405

—

139

—

#344

—

139

—

C₂×M₁₆

—

139

—

C₂×M₁₆

—

139

—

C₂×M₁₆

C₂×D₁₃₉

yes

—

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

#117

—

139

—

#108

—

#108

—

#108

—

278

—

#108

—

139

—

#97

—

278

—

C₄×C₈

—

278

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

278

—

M₁₆

—

278

—

M₁₆

—

139

—

M₁₆

—

139

—

C₂×C₈

—

C₂×C₈

—

278

—

C₂×C₈

—

278

—

C₂×C₈

C₄×D₁₃₉

yes

—

C₂×C₈

C₂²⋊D₁₃₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₃₉

yes

characteristic

C₂×C₈

Dic₂₇₈

yes

—

C₄⋊C₄

—

278

—

C₂²⋊C₄

—

139

—

C₄²

—

278

—

C₂³

Q₈×D₁₃₉

yes

characteristic

C₂×C₄

—

278

—

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

C₂×C₄

D₄×D₁₃₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

100

C₂×C₄

C₂×Dic₂₇₈

yes

characteristic

101

C₂×C₄

#14555784

yes

characteristic

102

C₂×C₄

#14555786

yes

characteristic

103

C₈

—

104

C₈

—

278

—

105

C₈

—

278

—

106

C₈

Dic₁₃₉⋊C₄

yes

—

107

C₈

Dic₁₃₉⋊C₄

yes

—

108

C₂²

—

109

C₂²

#17984807

yes

characteristic

110

C₂²

#17984818

yes

characteristic

111

C₂²

#17984841

yes

characteristic

112

C₄

—

556

—

113

C₄

C₂×Dic₁₃₉⋊C₄

yes

characteristic

114

C₄

#17984804

yes

characteristic

115

C₄

#17984813

yes

characteristic

116

C₄

#17984833

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

#22580849

yes

characteristic

	subgroup	G/H
center	C₄	#17984813
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	C₂×#108×C₁₃₉	C₂
socle	C₂×C₁₃₉	#389
prime radical	C₂³×C₁₃₉	C₂×Q₈
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	C₂×#108	D₁₃₉
hypocenter	C₁₃₉	#1953
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#17984813

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

C₂×#108×C₁₃₉

C₂

socle

C₂×C₁₃₉

#389

prime radical

C₂³×C₁₃₉

C₂×Q₈

cosocle

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
9. #17984778	C₄	semidirect
60. C₁₃₉	#1953	semidirect

H ≅ G/N

type

9. #17984778

C₄

semidirect

60. C₁₃₉

#1953

semidirect

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

subgroup	index
#22581001	2
C₂×#17984778	2
#22580908	2
#22580724	2
C₂×#108×C₁₃₉	2
#1953	139

subgroup

index

C₂×#108×C₁₃₉

139

Subgroups — #31807192

Conjugacy classes of subgroups

Complements

Maximal subgroups