Group #31806539 — subgroups

Subgroups — #31806539

Order 17792.

224 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂×#17984753	C₂	1	1	yes	maximal, characteristic
2	#22580819	C₂	1	1	yes	maximal, characteristic
3	#22580715	C₂	1	4	yes	maximal
4	#384×C₁₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
7	#17984753	C₄	1	4	yes	complemented
8	C₁₃₉⋊#95	—	2	2	no	—
9	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
10	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
12	C₂²×Dic₁₃₉	—	2	2	no	—
13	C₂×C₁₃₉⋊C₈	—	2	2	no	—
14	C₂×C₁₃₉⋊C₈	—	2	2	no	—
15	C₂×C₁₃₉⋊C₈	C₂×C₄	1	4	yes	—
16	C₄⋊Dic₁₃₉	—	2	4	no	—
17	C₂²×C₄×C₁₃₉	—	2	2	no	—
18	C₂²×C₄×C₁₃₉	D₄	1	2	yes	—
19	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₃₉	—	2	4	no	—
21	C₄⋊C₄×C₁₃₉	—	4	4	no	—
22	C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
23	C₄²×C₁₃₉	Q₈	1	2	yes	—
24	C₂×Dic₁₃₉	—	4	2	no	—
25	C₂×Dic₁₃₉	—	4	2	no	—
26	C₂×Dic₁₃₉	—	4	2	no	—
27	C₁₃₉⋊C₈	—	2	4	no	—
28	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₃₉	—	2	2	no	—
30	C₂×C₄×C₁₃₉	—	2	2	no	—
31	C₂×C₄×C₁₃₉	—	4	2	no	—
32	C₂×C₄×C₁₃₉	—	4	4	no	—
33	C₂×C₄×C₁₃₉	C₄⋊C₄	1	4	yes	—
34	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₁₃₉	#41	1	2	yes	—
37	C₂×C₄×C₁₃₉	#41	1	2	yes	—
38	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₁₃₉	#124	1	1	yes	characteristic
40	C₂²×C₁₃₉	#124	1	1	yes	characteristic
41	C₂²×C₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
42	C₂²×C₁₃₉	#137	1	2	yes	—
43	C₄×C₁₃₉	—	2	4	no	—
44	C₄×C₁₃₉	—	8	2	no	—
45	C₄×C₁₃₉	#117	1	2	yes	—
46	C₄×C₁₃₉	#125	1	2	yes	—
47	Dic₁₃₉	—	8	2	no	—
48	C₂×C₁₃₉	#382	1	1	yes	characteristic
49	C₂×C₁₃₉	#428	1	2	yes	—
50	C₂×C₁₃₉	#489	1	2	yes	—
51	C₂×C₁₃₉	#490	1	1	yes	characteristic
52	C₂×C₁₃₉	#491	1	1	yes	characteristic
53	C₁₃₉	#1836	1	1	yes	characteristic, complemented, Sylow
54	#1836	—	139	1	no	maximal, Sylow
55	C₂×#97	—	139	1	no	—
56	#384	D₁₃₉	1	1	yes	characteristic
57	#382	—	139	1	no	—
58	#340	—	139	4	no	—
59	C₂²×C₈	—	139	2	no	—
60	C₂×C₄⋊C₄	—	2	2	no	—
61	C₂×C₄⋊C₄	—	139	2	no	—
62	C₂×C₄⋊C₄	C₂×D₁₃₉	1	2	yes	—
63	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
64	#97	—	139	4	no	—
65	#95	—	278	2	no	—
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	278	2	no	—
68	C₂²×C₄	C₂²⋊D₁₃₉	1	2	yes	—
69	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
70	C₂×C₈	—	278	2	no	—
71	C₂×C₈	—	278	2	no	—
72	C₂×C₈	—	139	4	no	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	4	4	no	—
75	C₄⋊C₄	—	278	4	no	—
76	C₄²	C₄×D₁₃₉	1	2	yes	—
77	C₄²	Dic₂₇₈	1	2	yes	—
78	C₂³	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	4	4	no	—
83	C₂×C₄	—	556	2	no	—
84	C₂×C₄	—	556	2	no	—
85	C₂×C₄	—	556	2	no	—
86	C₂×C₄	Dic₁₃₉⋊C₄	1	4	yes	—
87	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₂₇₈	1	1	yes	characteristic
89	C₂×C₄	#14555784	1	2	yes	—
90	C₂×C₄	#14555786	1	2	yes	—
91	C₈	—	278	4	no	—
92	C₂²	C₂×Dic₁₃₉⋊C₄	1	1	yes	characteristic
93	C₂²	#17984829	1	1	yes	characteristic
94	C₂²	#17984831	1	1	yes	characteristic
95	C₂²	#17984868	1	2	yes	—
96	C₂²	#17984873	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	8	2	no	—
99	C₄	—	1112	2	no	—
100	C₄	#17984800	1	2	yes	—
101	C₄	#17984833	1	2	yes	—
102	C₂	#22580818	1	1	yes	characteristic
103	C₂	#22580993	1	2	yes	—
104	C₂	#22581315	1	2	yes	—
105	C₂	#22581317	1	1	yes	characteristic
106	C₂	#22581321	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984753

C₂

yes

maximal, characteristic

#22580819

C₂

yes

maximal, characteristic

#22580715

C₂

yes

maximal

#384×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×C₁₃₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

#17984753

C₄

yes

complemented

C₁₃₉⋊#95

—

C₂×C₄⋊C₄×C₁₃₉

—

C₂×C₄⋊C₄×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₂²×Dic₁₃₉

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₃₉

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

C₂×C₄

yes

—

C₄²×C₁₃₉

Q₈

yes

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₂×C₄×C₁₃₉

#41

yes

—

C₂²×C₁₃₉

C₂×C₄⋊C₄

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

—

C₂²×C₁₃₉

#137

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#117

yes

—

C₄×C₁₃₉

#125

yes

—

Dic₁₃₉

—

C₂×C₁₃₉

#382

yes

characteristic

C₂×C₁₃₉

#428

yes

—

C₂×C₁₃₉

#489

yes

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#1836

—

139

maximal, Sylow

C₂×#97

—

139

—

#384

D₁₃₉

yes

characteristic

#382

—

139

—

#340

—

139

—

C₂²×C₈

—

139

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄⋊C₄

C₂×D₁₃₉

yes

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

#97

—

139

—

#95

—

278

—

C₂²×C₄

—

C₂²×C₄

—

278

—

C₂²×C₄

C₂²⋊D₁₃₉

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

C₂×C₈

—

278

—

C₂×C₈

—

278

—

C₂×C₈

—

139

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

278

—

C₄²

C₄×D₁₃₉

yes

—

C₄²

Dic₂₇₈

yes

—

C₂³

C₂×C₂²⋊D₁₃₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

556

—

C₂×C₄

—

556

—

C₂×C₄

—

556

—

C₂×C₄

Dic₁₃₉⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

C₂×C₄

C₂×Dic₂₇₈

yes

characteristic

C₂×C₄

#14555784

yes

—

C₂×C₄

#14555786

yes

—

C₈

—

278

—

C₂²

C₂×Dic₁₃₉⋊C₄

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#17984873

yes

—

C₄

—

C₄

—

C₄

—

1112

—

100

C₄

#17984800

yes

—

101

C₄

#17984833

yes

—

102

C₂

#22580818

yes

characteristic

103

C₂

#22580993

yes

—

104

C₂

#22581315

yes

—

105

C₂

#22581317

yes

characteristic

106

C₂

#22581321

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#384×C₁₃₉	C₂
socle, prime radical	C₂³×C₁₃₉	C₂×D₄
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	#384	D₁₃₉
hypocenter	C₁₃₉	#1836
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃₉

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#384×C₁₃₉

C₂

socle, prime radical

C₂³×C₁₃₉

C₂×D₄

cosocle

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
7. #17984753	C₄	semidirect
53. C₁₃₉	#1836	semidirect

H ≅ G/N

type

7. #17984753

C₄

semidirect

53. C₁₃₉

#1836

semidirect

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

subgroup	index
C₂×#17984753	2
#22580819	2
#22580715	2
#384×C₁₃₉	2
#1836	139

subgroup

index

#384×C₁₃₉

139

Subgroups — #31806539

Conjugacy classes of subgroups

Complements

Maximal subgroups