Group #31807914 — subgroups

Subgroups — #31807914

Order 17792.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#22580996	C₂	1	2	yes	maximal
2	C₂×#17984774	C₂	1	1	yes	maximal, characteristic
3	#22580721	C₂	1	2	yes	maximal
4	#22580718	C₂	1	1	yes	maximal, characteristic
5	#424×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555769	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
8	#17984775	—	2	2	no	—
9	#17984774	—	2	2	no	—
10	#17984773	—	2	2	no	—
11	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
13	#117×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₃₉	—	2	2	no	—
15	C₂²×Dic₁₃₉	—	2	2	no	—
16	#14555769	—	2	2	no	—
17	#14555769	Q₈	1	4	yes	—
18	C₂×C₁₃₉⋊C₈	—	2	2	no	—
19	C₄⋊Dic₁₃₉	—	2	2	no	—
20	C₄⋊Dic₁₃₉	—	2	2	no	—
21	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
22	M₁₆×C₁₃₉	—	2	2	no	—
23	C₂×C₈×C₁₃₉	—	2	2	no	—
24	C₂×C₈×C₁₃₉	—	2	1	no	—
25	C₂×C₈×C₁₃₉	D₄	1	2	yes	—
26	C₄⋊C₄×C₁₃₉	D₄	1	2	yes	—
27	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
28	C₄²×C₁₃₉	—	2	1	no	—
29	C₂×Dic₁₃₉	—	4	2	no	—
30	C₂×Dic₁₃₉	—	8	2	no	—
31	C₁₃₉⋊C₈	—	2	4	no	—
32	C₂³×C₁₃₉	#41	1	1	yes	characteristic
33	C₂×C₄×C₁₃₉	—	2	2	no	—
34	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
37	C₂×C₄×C₁₃₉	#41	1	2	yes	—
38	C₈×C₁₃₉	—	2	2	no	—
39	C₈×C₁₃₉	—	4	1	no	—
40	C₂²×C₁₃₉	—	2	2	no	—
41	C₂²×C₁₃₉	#121	1	1	yes	characteristic
42	C₂²×C₁₃₉	#125	1	2	yes	—
43	C₄×C₁₃₉	—	4	2	no	—
44	C₄×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
45	C₄×C₁₃₉	#123	1	1	yes	characteristic
46	C₄×C₁₃₉	C₄⋊Q₈	1	1	yes	characteristic
47	Dic₁₃₉	—	8	2	no	—
48	C₂×C₁₃₉	—	2	2	no	—
49	C₂×C₁₃₉	#398	1	1	yes	characteristic
50	C₂×C₁₃₉	#471	1	2	yes	—
51	C₁₃₉	#2090	1	1	yes	characteristic, complemented, Sylow
52	#2090	—	139	1	no	maximal, Sylow
53	#428	—	139	2	no	—
54	C₂×#106	—	139	1	no	—
55	#424	D₁₃₉	1	1	yes	characteristic
56	#343	—	139	2	no	—
57	#341	—	139	1	no	—
58	C₂×M₁₆	—	139	2	no	—
59	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
61	#117	C₂×D₁₃₉	1	1	yes	characteristic
62	C₂×C₄⋊C₄	—	139	2	no	—
63	#107	—	278	2	no	—
64	#106	—	278	2	no	—
65	#106	—	278	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₂²×C₄	—	278	2	no	—
68	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	278	2	no	—
71	M₁₆	—	139	4	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	278	2	no	—
75	C₂×C₈	C₂²⋊D₁₃₉	1	2	yes	—
76	C₄⋊C₄	—	278	2	no	—
77	C₄⋊C₄	—	278	2	no	—
78	C₄⋊C₄	D₅₅₆	1	2	yes	—
79	C₂²⋊C₄	—	2	1	no	—
80	C₄²	—	2	1	no	—
81	C₂³	#14555785	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	556	2	no	—
84	C₂×C₄	—	1112	2	no	—
85	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
86	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
87	C₂×C₄	Q₈×D₁₃₉	1	2	yes	—
88	C₂×C₄	#14555784	1	2	yes	—
89	C₈	—	2	2	no	—
90	C₈	—	4	1	no	—
91	C₈	—	278	4	no	—
92	C₂²	—	2	2	no	—
93	C₂²	#17984816	1	1	yes	characteristic
94	C₂²	#17984832	1	2	yes	—
95	C₄	—	4	2	no	—
96	C₄	—	1112	2	no	—
97	C₄	#17984822	1	2	yes	—
98	C₄	#17984823	1	1	yes	characteristic
99	C₄	#17984840	1	1	yes	characteristic
100	C₂	—	2	2	no	—
101	C₂	#22580883	1	1	yes	characteristic
102	C₂	#22581216	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22580996

C₂

yes

maximal

C₂×#17984774

C₂

yes

maximal, characteristic

#22580721

C₂

yes

maximal

#22580718

C₂

yes

maximal, characteristic

#424×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555769

C₂²

yes

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

#17984775

—

#17984774

—

#17984773

—

C₂×M₁₆×C₁₃₉

C₂²

yes

characteristic

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

#117×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₉

—

C₂²×Dic₁₃₉

—

#14555769

—

#14555769

Q₈

yes

—

C₂×C₁₃₉⋊C₈

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

D₄

yes

—

C₄⋊C₄×C₁₃₉

D₄

yes

—

C₂²⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

#41

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

—

C₂×C₄×C₁₃₉

#41

yes

—

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

#121

yes

characteristic

C₂²×C₁₃₉

#125

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂²⋊Q₈

yes

—

C₄×C₁₃₉

#123

yes

characteristic

C₄×C₁₃₉

C₄⋊Q₈

yes

characteristic

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#398

yes

characteristic

C₂×C₁₃₉

#471

yes

—

C₁₃₉

#2090

yes

characteristic, complemented, Sylow

#2090

—

139

maximal, Sylow

#428

—

139

—

C₂×#106

—

139

—

#424

D₁₃₉

yes

characteristic

#343

—

139

—

#341

—

139

—

C₂×M₁₆

—

139

—

C₂×M₁₆

C₂×D₁₃₉

yes

characteristic

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

#117

C₂×D₁₃₉

yes

characteristic

C₂×C₄⋊C₄

—

139

—

#107

—

278

—

#106

—

278

—

#106

—

278

—

C₄⋊C₈

—

C₂²×C₄

—

278

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

M₁₆

—

M₁₆

—

278

—

M₁₆

—

139

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

278

—

C₂×C₈

C₂²⋊D₁₃₉

yes

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

D₅₅₆

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#14555785

yes

characteristic

C₂×C₄

—

C₂×C₄

—

556

—

C₂×C₄

—

1112

—

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

C₂×C₄

C₂×D₅₅₆

yes

characteristic

C₂×C₄

Q₈×D₁₃₉

yes

—

C₂×C₄

#14555784

yes

—

C₈

—

C₈

—

C₈

—

278

—

C₂²

—

C₂²

#17984816

yes

characteristic

C₂²

#17984832

yes

—

C₄

—

C₄

—

1112

—

C₄

#17984822

yes

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

#22580883

yes

characteristic

102

C₂

#22581216

yes

—

	subgroup	G/H
center	C₂²	#17984816
commutator, cosocle	C₂²×C₄×C₁₃₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#424×C₁₃₉	C₂
socle	C₂²×C₁₃₉	#121
prime radical	C₂³×C₁₃₉	#41
hypercenter	#424	D₁₃₉
hypocenter	C₁₃₉	#2090
LCS	G ▷ C₂²×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#17984816

commutator, cosocle

C₂²×C₄×C₁₃₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#424×C₁₃₉

C₂

C₂²×C₁₃₉

C₂³×C₁₃₉

D₁₃₉

C₁₃₉

G ▷ C₂²×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
51. C₁₃₉	#2090	semidirect

H ≅ G/N

type

51. C₁₃₉

#2090

semidirect

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

subgroup	index
#22580996	2
C₂×#17984774	2
#22580721	2
#22580718	2
#424×C₁₃₉	2
#2090	139

subgroup

index

#424×C₁₃₉

139

Subgroups — #31807914

Conjugacy classes of subgroups

Complements

Maximal subgroups