Group #31805185 — subgroups

Subgroups — #31805185

Order 17792.

150 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 73 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	#22581006	C₂	1	1	yes	maximal, characteristic
2	#22581006	C₂	1	1	yes	maximal, characteristic
3	#22580705	C₂	1	2	yes	maximal, complemented
4	#22580703	C₂	1	2	yes	maximal, complemented
5	#432×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₁₃₉⋊C₈)	C₄	1	2	yes	—
7	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	2	yes	—
8	C₂²⋊(C₁₃₉⋊C₈)	—	2	1	no	—
9	C₂²⋊(C₁₃₉⋊C₈)	C₄	1	2	yes	—
10	C₄×C₁₃₉⋊C₈	—	2	1	no	—
11	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
15	C₂²⋊C₈×C₁₃₉	—	2	1	no	—
16	#97×C₁₃₉	—	2	1	no	—
17	C₂×C₁₃₉⋊C₈	—	2	2	no	—
18	C₂×C₁₃₉⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₁₃₉⋊C₈	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₃₉	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₃₉	Q₈	1	1	yes	characteristic
23	C₂×C₈×C₁₃₉	—	2	2	no	—
24	C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₁₃₉	D₄	1	1	yes	characteristic
26	C₄²×C₁₃₉	Q₈	1	1	yes	characteristic
27	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
28	C₁₃₉⋊C₈	—	2	2	no	—
29	C₁₃₉⋊C₈	—	2	2	no	—
30	C₁₃₉⋊C₈	—	4	2	no	—
31	C₂³×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₃₉	—	2	2	no	—
33	C₂×C₄×C₁₃₉	—	2	2	no	—
34	C₂×C₄×C₁₃₉	—	4	1	no	—
35	C₂×C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₃₉	C₄⋊C₄	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×Q₈	1	1	yes	characteristic
41	C₈×C₁₃₉	—	4	2	no	—
42	C₂²×C₁₃₉	—	2	1	no	—
43	C₂²×C₁₃₉	—	2	1	no	—
44	C₂²×C₁₃₉	—	2	1	no	—
45	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₁₃₉	—	2	1	no	—
47	C₄×C₁₃₉	—	2	1	no	—
48	C₄×C₁₃₉	—	2	1	no	—
49	C₄×C₁₃₉	—	2	1	no	—
50	C₄×C₁₃₉	#131	1	2	yes	—
51	C₄×C₁₃₉	#135	1	2	yes	—
52	C₂×C₁₃₉	—	4	1	no	—
53	C₂×C₁₃₉	#424	1	1	yes	characteristic
54	C₂×C₁₃₉	#427	1	1	yes	characteristic
55	C₂×C₁₃₉	#430	1	1	yes	characteristic
56	C₁₃₉	#1572	1	1	yes	characteristic, complemented, Sylow
57	#1572	—	139	1	no	maximal, Sylow
58	#432	D₁₃₉	1	1	yes	characteristic
59	#431	—	139	1	no	—
60	#431	—	139	1	no	—
61	#335	—	139	2	no	—
62	#334	—	139	2	no	—
63	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
64	C₄⋊C₈	—	139	2	no	—
65	C₄⋊C₈	—	139	2	no	—
66	C₄⋊C₈	C₂×D₁₃₉	1	2	yes	—
67	C₂²⋊C₈	—	2	1	no	—
68	C₂²⋊C₈	—	278	1	no	—
69	C₂²⋊C₈	—	139	2	no	—
70	#97	—	2	1	no	—
71	C₄×C₈	—	278	1	no	—
72	C₄×C₈	—	139	1	no	—
73	C₄×C₈	—	139	1	no	—
74	C₂²×C₄	C₄×D₁₃₉	1	1	yes	characteristic
75	C₂²×C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
76	C₂²×C₄	Dic₂₇₈	1	1	yes	characteristic
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	278	2	no	—
79	C₂×C₈	—	139	2	no	—
80	C₂×C₈	—	139	2	no	—
81	C₄²	C₄×D₁₃₉	1	1	yes	characteristic
82	C₄²	C₂²⋊D₁₃₉	1	1	yes	characteristic
83	C₄²	Dic₂₇₈	1	1	yes	characteristic
84	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
85	C₂³	Dic₁₃₉⋊C₄	1	1	yes	characteristic
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	4	1	no	—
89	C₂×C₄	Dic₁₃₉⋊C₄	1	1	yes	characteristic
90	C₂×C₄	Dic₁₃₉⋊C₄	1	1	yes	characteristic
91	C₂×C₄	Dic₁₃₉⋊C₄	1	1	yes	characteristic
92	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
93	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₂₇₈	1	1	yes	characteristic
95	C₈	—	4	2	no	—
96	C₈	—	278	2	no	—
97	C₈	—	278	2	no	—
98	C₈	—	556	2	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	—	2	1	no	—
102	C₂²	C₂×Dic₁₃₉⋊C₄	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	#17984850	1	2	yes	—
108	C₄	#17984862	1	2	yes	—
109	C₂	—	4	1	no	—
110	C₂	#22580981	1	1	yes	characteristic
111	C₂	#22580989	1	1	yes	characteristic
112	C₂	#22581001	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581006

C₂

yes

maximal, characteristic

#22581006

C₂

yes

maximal, characteristic

#22580705

C₂

yes

maximal, complemented

#22580703

C₂

yes

maximal, complemented

#432×C₁₃₉

C₂

yes

maximal, characteristic

C₄⋊(C₁₃₉⋊C₈)

C₄

yes

—

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

—

C₂²⋊(C₁₃₉⋊C₈)

—

C₂²⋊(C₁₃₉⋊C₈)

C₄

yes

—

C₄×C₁₃₉⋊C₈

—

C₄×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₄×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₉

C₂²

yes

—

C₂²⋊C₈×C₁₃₉

—

#97×C₁₃₉

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₉

D₄

yes

characteristic

C₂²×C₄×C₁₃₉

Q₈

yes

characteristic

C₂×C₈×C₁₃₉

—

C₄²×C₁₃₉

C₂×C₄

yes

characteristic

C₄²×C₁₃₉

D₄

yes

characteristic

C₄²×C₁₃₉

Q₈

yes

characteristic

C₄²×C₁₃₉

C₂³

yes

characteristic

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

characteristic

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#131

yes

—

C₄×C₁₃₉

#135

yes

—

C₂×C₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#1572

—

139

maximal, Sylow

#432

D₁₃₉

yes

characteristic

#431

—

139

—

#431

—

139

—

#335

—

139

—

#334

—

139

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

C₄⋊C₈

—

139

—

C₄⋊C₈

—

139

—

C₄⋊C₈

C₂×D₁₃₉

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

278

—

C₂²⋊C₈

—

139

—

#97

—

C₄×C₈

—

278

—

C₄×C₈

—

139

—

C₄×C₈

—

139

—

C₂²×C₄

C₄×D₁₃₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₃₉

yes

characteristic

C₂²×C₄

Dic₂₇₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

278

—

C₂×C₈

—

139

—

C₂×C₈

—

139

—

C₄²

C₄×D₁₃₉

yes

characteristic

C₄²

C₂²⋊D₁₃₉

yes

characteristic

C₄²

Dic₂₇₈

yes

characteristic

C₄²

C₂²×D₁₃₉

yes

characteristic

C₂³

Dic₁₃₉⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

Dic₁₃₉⋊C₄

yes

characteristic

C₂×C₄

Dic₁₃₉⋊C₄

yes

characteristic

C₂×C₄

Dic₁₃₉⋊C₄

yes

characteristic

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

C₂×C₄

C₂×Dic₂₇₈

yes

characteristic

C₈

—

C₈

—

278

—

C₈

—

278

—

C₈

—

556

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

C₂×Dic₁₃₉⋊C₄

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

#17984850

yes

—

108

C₄

#17984862

yes

—

109

C₂

—

110

C₂

#22580981

yes

characteristic

111

C₂

#22580989

yes

characteristic

112

C₂

#22581001

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₃₉
Fitting	#432×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×C₄⋊C₄
prime radical	C₂³×C₁₃₉	C₄⋊C₄
cosocle	C₄²×C₁₃₉	C₂³
hypercenter	#432	D₁₃₉
hypocenter	C₁₃₉	#1572
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₃₉

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₄²

C₂²×D₁₃₉

Fitting

#432×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂×C₄⋊C₄

prime radical

C₂³×C₁₃₉

C₄⋊C₄

cosocle

C₄²×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
3. #22580705	C₂	semidirect
4. #22580703	C₂	semidirect
56. C₁₃₉	#1572	semidirect

H ≅ G/N

type

3. #22580705

C₂

semidirect

4. #22580703

C₂

semidirect

56. C₁₃₉

#1572

semidirect

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

subgroup	index
#22581006	2
#22580705	2
#22580703	2
#432×C₁₃₉	2
#1572	139

subgroup

index

#432×C₁₃₉

139

Subgroups — #31805185

Conjugacy classes of subgroups

Complements

Maximal subgroups