Group #31807958 — subgroups

Subgroups — #31807958

Order 17792.

210 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 67 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	#22581002	C₂	1	1	yes	maximal, characteristic, complemented
2	#22580980	C₂	1	2	yes	maximal, complemented
3	#22580971	C₂	1	2	yes	maximal
4	#22580921	C₂	1	1	yes	maximal, characteristic
5	H(3,ℤ/4)×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14555785	C₂²	1	1	yes	characteristic
7	C₂×#14555769	C₂²	1	2	yes	—
8	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
9	#17984778	—	2	2	no	—
10	C₄⋊(C₁₃₉⋊C₈)	—	2	4	no	—
11	#17984770	—	2	4	no	—
12	(C₂×D₁₃₉)⋊C₈	—	2	2	no	—
13	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
14	#117×C₁₃₉	C₂²	1	2	yes	—
15	#14555785	—	2	2	no	—
16	#14555785	—	2	2	no	—
17	C₂×Dic₂₇₈	Q₈	1	1	yes	characteristic
18	C₂×D₅₅₆	Q₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
20	C₂×C₄×D₁₃₉	—	2	1	no	—
21	#14555769	—	2	2	no	—
22	#14555769	—	2	2	no	—
23	C₂×C₁₃₉⋊C₈	—	2	2	no	—
24	C₂×C₁₃₉⋊C₈	—	2	2	no	—
25	C₂×C₁₃₉⋊C₈	—	2	1	no	—
26	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₃₉	—	2	2	no	—
28	C₂×C₈×C₁₃₉	D₄	1	2	yes	—
29	C₄⋊C₄×C₁₃₉	—	2	2	no	—
30	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
31	C₄²×C₁₃₉	D₄	1	4	yes	—
32	C₂²×D₁₃₉	—	2	1	no	—
33	Dic₂₇₈	—	2	2	no	—
34	D₅₅₆	—	2	2	no	—
35	C₂²⋊D₁₃₉	—	4	2	no	—
36	C₂×Dic₁₃₉	—	2	1	no	—
37	C₄×D₁₃₉	—	4	1	no	—
38	C₄×D₁₃₉	—	4	1	no	—
39	C₁₃₉⋊C₈	—	4	1	no	—
40	C₁₃₉⋊C₈	—	4	2	no	—
41	C₂³×C₁₃₉	#41	1	1	yes	characteristic
42	C₂×C₄×C₁₃₉	—	2	4	no	—
43	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₃₉	#41	1	2	yes	—
47	C₈×C₁₃₉	—	2	2	no	—
48	C₂²×C₁₃₉	—	4	1	no	—
49	C₂²×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₃₉	#123	1	2	yes	—
51	C₂×D₁₃₉	—	4	2	no	—
52	C₂×D₁₃₉	—	4	1	no	—
53	C₄×C₁₃₉	—	4	4	no	—
54	C₄×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₁₃₉	#123	1	1	yes	characteristic
56	C₄×C₁₃₉	#127	1	1	yes	characteristic
57	Dic₁₃₉	—	4	1	no	—
58	C₂×C₁₃₉	—	2	2	no	—
59	C₂×C₁₃₉	—	2	1	no	—
60	C₂×C₁₃₉	#399	1	1	yes	characteristic
61	D₁₃₉	—	8	1	no	—
62	C₁₃₉	#2098	1	1	yes	characteristic, complemented, Sylow
63	#2098	—	139	1	no	maximal, Sylow
64	#430	—	139	1	no	—
65	#424	—	139	2	no	—
66	#421	—	139	2	no	—
67	#408	—	139	1	no	—
68	H(3,ℤ/4)	D₁₃₉	1	1	yes	characteristic, complemented
69	C₂×#41	—	139	1	no	—
70	C₂×M₁₆	—	139	2	no	—
71	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
72	C₂²×C₈	—	139	1	no	—
73	#117	C₂×D₁₃₉	1	2	yes	—
74	#108	—	278	2	no	—
75	C₄⋊C₈	—	278	4	no	—
76	#104	—	278	4	no	—
77	C₂²⋊C₈	—	278	2	no	—
78	#41	—	278	2	no	—
79	#41	—	278	2	no	—
80	C₂×Q₈	—	139	1	no	—
81	C₂×D₄	—	278	1	no	—
82	C₂×D₄	—	139	1	no	—
83	C₂²×C₄	—	278	1	no	—
84	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	278	2	no	—
87	M₁₆	—	278	2	no	—
88	C₂×C₈	—	278	1	no	—
89	C₂×C₈	—	278	2	no	—
90	C₂×C₈	—	278	2	no	—
91	C₂×C₈	D₅₅₆	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₅₅₆	1	4	yes	—
95	C₂³	—	278	1	no	—
96	C₂³	#14555784	1	1	yes	characteristic
97	Q₈	—	278	2	no	—
98	D₄	—	278	2	no	—
99	D₄	—	556	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	278	1	no	—
102	C₂×C₄	—	556	1	no	—
103	C₂×C₄	—	556	1	no	—
104	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
105	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₃₉	1	1	yes	characteristic
107	C₂×C₄	#14555784	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	556	1	no	—
110	C₈	—	556	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	556	2	no	—
113	C₂²	—	556	1	no	—
114	C₂²	#17984822	1	1	yes	characteristic
115	C₂²	#17984826	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	556	1	no	—
118	C₄	#17984822	1	2	yes	—
119	C₄	#17984826	1	1	yes	characteristic
120	C₄	#17984839	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	1112	1	no	—
124	C₂	#22580890	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581002

C₂

yes

maximal, characteristic, complemented

#22580980

C₂

yes

maximal, complemented

#22580971

C₂

yes

maximal

#22580921

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×#14555785

C₂²

yes

characteristic

C₂×#14555769

C₂²

yes

—

C₂²×C₁₃₉⋊C₈

C₂²

yes

characteristic

#17984778

—

C₄⋊(C₁₃₉⋊C₈)

—

#17984770

—

(C₂×D₁₃₉)⋊C₈

—

C₂×M₁₆×C₁₃₉

C₂²

yes

characteristic

#117×C₁₃₉

C₂²

yes

—

#14555785

—

#14555785

—

C₂×Dic₂₇₈

Q₈

yes

characteristic

C₂×D₅₅₆

Q₈

yes

characteristic

C₂×C₂²⋊D₁₃₉

—

C₂×C₄×D₁₃₉

—

#14555769

—

#14555769

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

D₄

yes

—

C₄⋊C₄×C₁₃₉

—

C₂²⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

D₄

yes

—

C₂²×D₁₃₉

—

Dic₂₇₈

—

D₅₅₆

—

C₂²⋊D₁₃₉

—

C₂×Dic₁₃₉

—

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

#41

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₃₉

#123

yes

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂²⋊Q₈

yes

—

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#399

yes

characteristic

D₁₃₉

—

C₁₃₉

#2098

yes

characteristic, complemented, Sylow

#2098

—

139

maximal, Sylow

#430

—

139

—

#424

—

139

—

#421

—

139

—

#408

—

139

—

H(3,ℤ/4)

D₁₃₉

yes

characteristic, complemented

C₂×#41

—

139

—

C₂×M₁₆

—

139

—

C₂×M₁₆

C₂×D₁₃₉

yes

characteristic

C₂²×C₈

—

139

—

#117

C₂×D₁₃₉

yes

—

#108

—

278

—

C₄⋊C₈

—

278

—

#104

—

278

—

C₂²⋊C₈

—

278

—

#41

—

278

—

#41

—

278

—

C₂×Q₈

—

139

—

C₂×D₄

—

278

—

C₂×D₄

—

139

—

C₂²×C₄

—

278

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

M₁₆

—

M₁₆

—

278

—

M₁₆

—

278

—

C₂×C₈

—

278

—

C₂×C₈

—

278

—

C₂×C₈

—

278

—

C₂×C₈

D₅₅₆

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₅₅₆

yes

—

C₂³

—

278

—

C₂³

#14555784

yes

characteristic

Q₈

—

278

—

D₄

—

278

—

D₄

—

556

—

100

C₂×C₄

—

101

C₂×C₄

—

278

—

102

C₂×C₄

—

556

—

103

C₂×C₄

—

556

—

104

C₂×C₄

C₂×D₅₅₆

yes

—

105

C₂×C₄

C₂×D₅₅₆

yes

characteristic

106

C₂×C₄

Q₈×D₁₃₉

yes

characteristic

107

C₂×C₄

#14555784

yes

—

108

C₈

—

109

C₈

—

556

—

110

C₈

—

556

—

111

C₂²

—

112

C₂²

—

556

—

113

C₂²

—

556

—

114

C₂²

#17984822

yes

characteristic

115

C₂²

#17984826

yes

—

116

C₄

—

117

C₄

—

556

—

118

C₄

#17984822

yes

—

119

C₄

#17984826

yes

characteristic

120

C₄

#17984839

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

1112

—

124

C₂

#22580890

yes

characteristic

	subgroup	G/H
center	C₄	#17984839
commutator, cosocle	C₂²×C₄×C₁₃₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	H(3,ℤ/4)×C₁₃₉	C₂
socle	C₂×C₁₃₉	#399
prime radical	C₂×#14555785	C₂²
hypercenter	H(3,ℤ/4)	D₁₃₉
hypocenter	C₁₃₉	#2098
LCS	G ▷ C₂²×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#17984839

commutator, cosocle

C₂²×C₄×C₁₃₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

H(3,ℤ/4)×C₁₃₉

C₂

C₂×C₁₃₉

C₂²

D₁₃₉

C₁₃₉

G ▷ C₂²×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #22581002	C₂	semidirect
2. #22580980	C₂	semidirect
5. H(3,ℤ/4)×C₁₃₉	C₂	semidirect
62. C₁₃₉	#2098	semidirect
68. H(3,ℤ/4)	D₁₃₉	semidirect

H ≅ G/N

type

1. #22581002

C₂

semidirect

2. #22580980

C₂

semidirect

5. H(3,ℤ/4)×C₁₃₉

C₂

semidirect

62. C₁₃₉

#2098

semidirect

68. H(3,ℤ/4)

D₁₃₉

semidirect

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

subgroup	index
#22581002	2
#22580980	2
#22580971	2
#22580921	2
H(3,ℤ/4)×C₁₃₉	2
#2098	139

subgroup

index

H(3,ℤ/4)×C₁₃₉

139

Subgroups — #31807958

Conjugacy classes of subgroups

Complements

Maximal subgroups