Group #31818208 — subgroups

Subgroups — #31818208

Order 17792.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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	#22581766	C₂	1	4	yes	maximal, complemented
2	#22581225	C₂	1	4	yes	maximal, complemented
3	#22581135	C₂	1	4	yes	maximal, complemented
4	#22581071	C₂	1	2	yes	maximal, complemented
5	#494×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984890	C₂²	1	4	yes	—
7	#17984874	—	2	8	no	—
8	#17984873	C₂²	1	8	yes	—
9	#17984865	—	2	4	no	—
10	#17984864	C₂²	1	2	yes	—
11	Q₁₆×D₁₃₉	—	2	4	no	—
12	#17984848	C₂²	1	4	yes	—
13	#17984778	C₂²	1	2	yes	—
14	#17984770	C₂²	1	4	yes	—
15	#17984761	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₁₃₉	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₁₃₉	C₂²	1	2	yes	—
18	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
19	#14555786	—	2	8	no	—
20	#14555785	—	4	4	no	—
21	#14555785	C₂³	1	4	yes	—
22	Q₈×D₁₃₉	—	2	8	no	—
23	Dic₅₅₆	D₄	1	2	yes	—
24	#14555775	—	2	8	no	—
25	#14555774	D₄	1	4	yes	—
26	#14555772	—	2	8	no	—
27	D₁₁₁₂	D₄	1	2	yes	—
28	#14555769	C₂³	1	4	yes	—
29	#14555768	—	2	4	no	—
30	C₈×D₁₃₉	—	2	4	no	—
31	C₂×Q₈×C₁₃₉	C₂³	1	4	yes	—
32	Q₁₆×C₁₃₉	—	2	4	no	—
33	Q₁₆×C₁₃₉	—	2	4	no	—
34	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
35	C₄⋊C₄×C₁₃₉	—	4	2	no	—
36	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
37	Dic₂₇₈	—	4	4	no	—
38	Dic₂₇₈	C₂×D₄	1	4	yes	—
39	D₅₅₆	—	4	4	no	—
40	D₅₅₆	C₂×D₄	1	4	yes	—
41	C₂²⋊D₁₃₉	—	2	4	no	—
42	C₄×D₁₃₉	—	2	4	no	—
43	C₄×D₁₃₉	—	4	8	no	—
44	C₁₃₉⋊C₈	—	2	4	no	—
45	Q₈×C₁₃₉	—	2	8	no	—
46	C₂×C₄×C₁₃₉	—	2	1	no	—
47	C₂×C₄×C₁₃₉	—	2	4	no	—
48	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
49	C₈×C₁₃₉	C₂×D₄	1	2	yes	—
50	C₈×C₁₃₉	C₂×D₄	1	2	yes	—
51	C₂²×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₁₃₉	—	2	4	no	—
53	C₄×C₁₃₉	—	2	2	no	—
54	C₄×C₁₃₉	—	4	4	no	—
55	C₄×C₁₃₉	C₂²×D₄	1	2	yes	—
56	Dic₁₃₉	—	2	4	no	—
57	C₂×C₁₃₉	—	2	1	no	—
58	C₂×C₁₃₉	D₄²	1	1	yes	characteristic
59	D₁₃₉	—	4	4	no	—
60	C₁₃₉	#3293	1	1	yes	characteristic, complemented, Sylow
61	#3293	—	139	1	no	maximal, Sylow
62	#578	—	139	4	no	—
63	#494	D₁₃₉	1	1	yes	characteristic, complemented
64	#473	—	139	4	no	—
65	#456	—	139	4	no	—
66	#443	—	139	2	no	—
67	2₋¹⁺⁴	—	139	4	no	—
68	#137	—	278	8	no	—
69	#137	—	139	8	no	—
70	#135	—	278	4	no	—
71	#135	—	139	2	no	—
72	C₂×Q₁₆	—	278	4	no	—
73	C₂×Q₁₆	C₂×D₁₃₉	1	4	yes	—
74	#131	—	139	4	no	—
75	C₄⋊Q₈	C₂×D₁₃₉	1	2	yes	—
76	#108	—	139	2	no	—
77	#104	—	139	4	no	—
78	#101	—	139	4	no	—
79	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
80	#41	—	278	8	no	—
81	#41	—	556	4	no	—
82	#41	—	139	4	no	—
83	C₂×Q₈	—	278	8	no	—
84	C₂×Q₈	C₂²×D₁₃₉	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	278	8	no	—
88	Q₁₆	—	139	2	no	—
89	SD₁₆	—	278	8	no	—
90	SD₁₆	—	139	4	no	—
91	D₈	—	139	2	no	—
92	M₁₆	—	278	4	no	—
93	M₁₆	—	139	4	no	—
94	C₂×C₈	—	278	4	no	—
95	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	556	4	no	—
100	Q₈	—	139	4	no	—
101	D₄	—	278	4	no	—
102	D₄	—	556	4	no	—
103	D₄	—	139	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	278	4	no	—
107	C₂×C₄	—	556	8	no	—
108	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
109	C₈	—	278	4	no	—
110	C₈	D₄×D₁₃₉	1	2	yes	—
111	C₈	D₄×D₁₃₉	1	2	yes	—
112	C₂²	—	278	4	no	—
113	C₂²	#17984887	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	278	4	no	—
117	C₄	C₂×D₄×D₁₃₉	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	556	4	no	—
120	C₂	#22581588	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581766

C₂

yes

maximal, complemented

#22581225

C₂

yes

maximal, complemented

#22581135

C₂

yes

maximal, complemented

#22581071

C₂

yes

maximal, complemented

#494×C₁₃₉

C₂

yes

maximal, characteristic, complemented

#17984890

C₂²

yes

—

#17984874

—

#17984873

C₂²

yes

—

#17984865

—

#17984864

C₂²

yes

—

Q₁₆×D₁₃₉

—

#17984848

C₂²

yes

—

#17984778

C₂²

yes

—

#17984770

C₂²

yes

—

#17984761

C₂²

yes

—

C₂×Q₁₆×C₁₃₉

C₂²

yes

—

C₄⋊Q₈×C₁₃₉

C₂²

yes

—

C₄×C₈×C₁₃₉

C₂²

yes

—

—

C₂³

yes

—

Q₈×D₁₃₉

—

Dic₅₅₆

D₄

yes

—

#14555775

—

#14555774

D₄

yes

—

#14555772

—

D₁₁₁₂

D₄

yes

—

#14555769

C₂³

yes

—

#14555768

—

C₈×D₁₃₉

—

C₂×Q₈×C₁₃₉

C₂³

yes

—

Q₁₆×C₁₃₉

—

Q₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂³

yes

—

C₄⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

C₂³

yes

characteristic

Dic₂₇₈

—

Dic₂₇₈

C₂×D₄

yes

—

D₅₅₆

—

D₅₅₆

C₂×D₄

yes

—

C₂²⋊D₁₃₉

—

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

—

Q₈×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂⁴

yes

characteristic

C₈×C₁₃₉

C₂×D₄

yes

—

C₈×C₁₃₉

C₂×D₄

yes

—

C₂²×C₁₃₉

H(5,𝔽₂)

yes

characteristic

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂²×D₄

yes

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

D₄²

yes

characteristic

D₁₃₉

—

C₁₃₉

#3293

yes

characteristic, complemented, Sylow

#3293

—

139

maximal, Sylow

#578

—

139

—

#494

D₁₃₉

yes

characteristic, complemented

#473

—

139

—

#456

—

139

—

#443

—

139

—

2₋¹⁺⁴

—

139

—

#137

—

278

—

#137

—

139

—

#135

—

278

—

#135

—

139

—

C₂×Q₁₆

—

278

—

C₂×Q₁₆

C₂×D₁₃₉

yes

—

#131

—

139

—

C₄⋊Q₈

C₂×D₁₃₉

yes

—

#108

—

139

—

#104

—

139

—

#101

—

139

—

C₄×C₈

C₂×D₁₃₉

yes

characteristic

#41

—

278

—

#41

—

556

—

#41

—

139

—

C₂×Q₈

—

278

—

C₂×Q₈

C₂²×D₁₃₉

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

278

—

Q₁₆

—

139

—

SD₁₆

—

278

—

SD₁₆

—

139

—

D₈

—

139

—

M₁₆

—

278

—

M₁₆

—

139

—

C₂×C₈

—

278

—

C₂×C₈

C₂²×D₁₃₉

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₁₃₉

yes

characteristic

Q₈

—

Q₈

—

556

—

100

Q₈

—

139

—

101

D₄

—

278

—

102

D₄

—

556

—

103

D₄

—

139

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

278

—

107

C₂×C₄

—

556

—

108

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

109

C₈

—

278

—

110

C₈

D₄×D₁₃₉

yes

—

111

C₈

D₄×D₁₃₉

yes

—

112

C₂²

—

278

—

113

C₂²

#17984887

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

278

—

117

C₄

C₂×D₄×D₁₃₉

yes

—

118

C₂

—

119

C₂

—

556

—

120

C₂

#22581588

yes

characteristic

	subgroup	G/H
center	C₂	#22581588
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#494×C₁₃₉	C₂
socle	C₂×C₁₃₉	D₄²
hypercenter	#494	D₁₃₉
hypocenter	C₁₃₉	#3293
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#22581588

commutator, cosocle

C₂×C₄×C₁₃₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₉

Fitting

#494×C₁₃₉

C₂

socle

C₂×C₁₃₉

D₄²

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #22581766	C₂	semidirect
2. #22581225	C₂	semidirect
3. #22581135	C₂	semidirect
4. #22581071	C₂	semidirect
5. #494×C₁₃₉	C₂	semidirect
60. C₁₃₉	#3293	semidirect
63. #494	D₁₃₉	semidirect

H ≅ G/N

type

1. #22581766

C₂

semidirect

2. #22581225

C₂

semidirect

3. #22581135

C₂

semidirect

4. #22581071

C₂

semidirect

5. #494×C₁₃₉

C₂

semidirect

60. C₁₃₉

#3293

semidirect

63. #494

D₁₃₉

semidirect

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

subgroup	index
#22581766	2
#22581225	2
#22581135	2
#22581071	2
#494×C₁₃₉	2
#3293	139

subgroup

index

#494×C₁₃₉

139

Subgroups — #31818208

Conjugacy classes of subgroups

Complements

Maximal subgroups