Group #31805993 — subgroups

Subgroups — #31805993

Order 17792.

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

normal subgroups only

This group has 224 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	#22581358	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581325	C₂	1	2	yes	maximal, complemented
3	#22581299	C₂	1	2	yes	maximal, complemented
4	C₈×C₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#493×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555773	—	2	4	no	—
7	#17984842	C₂²	1	2	yes	—
8	#17984776	—	2	2	no	—
9	#17984763	—	2	4	no	—
10	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×D₈×C₁₃₉	—	2	2	no	—
13	#127×C₁₃₉	C₂²	1	2	yes	—
14	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₇₈	—	2	2	no	—
16	#14555773	—	4	8	no	—
17	C₂×C₁₃₉⋊C₈	D₄	1	2	yes	—
18	C₂×C₁₃₉⋊C₈	D₄	1	2	yes	—
19	C₄⋊Dic₁₃₉	—	2	2	no	—
20	C₄⋊Dic₁₃₉	—	2	2	no	—
21	C₂×D₄×C₁₃₉	—	2	2	no	—
22	C₂×D₄×C₁₃₉	—	2	2	no	—
23	C₂×D₄×C₁₃₉	—	2	2	no	—
24	D₈×C₁₃₉	—	4	4	no	—
25	C₂×C₈×C₁₃₉	D₄	1	2	yes	—
26	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
27	Dic₂₇₈	—	4	4	no	—
28	C₂×Dic₁₃₉	—	4	2	no	—
29	C₁₃₉⋊C₈	—	2	2	no	—
30	C₁₃₉⋊C₈	D₈	1	4	yes	complemented
31	C₂³×C₁₃₉	—	4	2	no	—
32	D₄×C₁₃₉	—	4	4	no	—
33	D₄×C₁₃₉	—	4	4	no	—
34	D₄×C₁₃₉	—	4	4	no	—
35	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
38	C₈×C₁₃₉	SD₁₆	1	4	yes	complemented
39	C₂²×C₁₃₉	—	8	2	no	—
40	C₂²×C₁₃₉	—	8	2	no	—
41	C₂²×C₁₃₉	—	8	2	no	—
42	C₂²×C₁₃₉	#127	1	1	yes	characteristic
43	C₄×C₁₃₉	C₂×D₈	1	2	yes	—
44	C₄×C₁₃₉	C₂×SD₁₆	1	2	yes	—
45	C₄×C₁₃₉	#135	1	2	yes	—
46	Dic₁₃₉	—	8	2	no	—
47	C₂×C₁₃₉	—	16	2	no	—
48	C₂×C₁₃₉	#492	1	1	yes	characteristic
49	C₂×C₁₃₉	#493	1	1	yes	characteristic
50	C₂×C₁₃₉	#495	1	1	yes	characteristic
51	C₁₃₉	#1706	1	1	yes	characteristic, complemented, Sylow
52	#1706	—	139	1	no	maximal, Sylow
53	#500	—	139	1	no	—
54	#493	D₁₃₉	1	1	yes	characteristic
55	#492	—	139	2	no	—
56	#486	—	139	2	no	—
57	C₈²	—	139	1	no	—
58	C₂×SD₁₆	—	278	4	no	—
59	C₂×D₈	—	2	2	no	—
60	C₄⋊Q₈	—	139	2	no	—
61	#127	C₂×D₁₃₉	1	2	yes	—
62	#107	—	278	2	no	—
63	#102	—	278	4	no	—
64	C₄×C₈	—	139	1	no	—
65	C₄×C₈	—	139	1	no	—
66	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
67	C₂×Q₈	—	278	2	no	—
68	C₂×D₄	—	2	2	no	—
69	C₂×D₄	—	2	2	no	—
70	C₂×D₄	—	2	2	no	—
71	SD₁₆	—	556	8	no	—
72	D₈	—	4	4	no	—
73	C₂×C₈	—	139	2	no	—
74	C₂×C₈	—	139	2	no	—
75	C₂×C₈	C₂²⋊D₁₃₉	1	2	yes	—
76	C₄⋊C₄	—	278	2	no	—
77	C₄⋊C₄	—	278	2	no	—
78	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
79	C₂³	—	4	2	no	—
80	Q₈	—	556	4	no	—
81	D₄	—	4	4	no	—
82	D₄	—	4	4	no	—
83	D₄	—	4	4	no	—
84	C₂×C₄	—	556	2	no	—
85	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
86	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
87	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
88	C₈	—	278	2	no	—
89	C₈	—	139	4	no	—
90	C₈	#14555773	1	4	yes	complemented
91	C₂²	—	8	2	no	—
92	C₂²	—	8	2	no	—
93	C₂²	—	8	2	no	—
94	C₂²	#17984838	1	1	yes	characteristic
95	C₄	—	1112	2	no	—
96	C₄	D₈×D₁₃₉	1	2	yes	—
97	C₄	C₂×#14555773	1	2	yes	—
98	C₄	#17984861	1	2	yes	—
99	C₂	—	16	2	no	—
100	C₂	#22581325	1	1	yes	characteristic
101	C₂	#22581328	1	1	yes	characteristic
102	C₂	#22581336	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581358

C₂

yes

maximal, characteristic, complemented

#22581325

C₂

yes

maximal, complemented

#22581299

C₂

yes

maximal, complemented

C₈×C₁₃₉⋊C₈

C₂

yes

maximal, characteristic, complemented

#493×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555773

—

#17984842

C₂²

yes

—

#17984776

—

#17984763

—

C₄×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₄×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₂×D₈×C₁₃₉

—

#127×C₁₃₉

C₂²

yes

—

C₄×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

#14555773

—

C₂×C₁₃₉⋊C₈

D₄

yes

—

C₂×C₁₃₉⋊C₈

D₄

yes

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

—

D₈×C₁₃₉

—

C₂×C₈×C₁₃₉

D₄

yes

—

C₄²×C₁₃₉

C₂³

yes

characteristic

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

D₈

yes

complemented

C₂³×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₈×C₁₃₉

SD₁₆

yes

complemented

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

#127

yes

characteristic

C₄×C₁₃₉

C₂×D₈

yes

—

C₄×C₁₃₉

C₂×SD₁₆

yes

—

C₄×C₁₃₉

#135

yes

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#1706

—

139

maximal, Sylow

#500

—

139

—

#493

D₁₃₉

yes

characteristic

#492

—

139

—

#486

—

139

—

C₈²

—

139

—

C₂×SD₁₆

—

278

—

C₂×D₈

—

C₄⋊Q₈

—

139

—

#127

C₂×D₁₃₉

yes

—

#107

—

278

—

#102

—

278

—

C₄×C₈

—

139

—

C₄×C₈

—

139

—

C₄×C₈

C₂×D₁₃₉

yes

characteristic

C₂×Q₈

—

278

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

556

—

D₈

—

C₂×C₈

—

139

—

C₂×C₈

—

139

—

C₂×C₈

C₂²⋊D₁₃₉

yes

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄²

C₂²×D₁₃₉

yes

characteristic

C₂³

—

Q₈

—

556

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

556

—

C₂×C₄

D₄×D₁₃₉

yes

characteristic

C₂×C₄

D₄×D₁₃₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

C₈

—

278

—

C₈

—

139

—

C₈

#14555773

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

#17984838

yes

characteristic

C₄

—

1112

—

C₄

D₈×D₁₃₉

yes

—

C₄

C₂×#14555773

yes

—

C₄

#17984861

yes

—

C₂

—

100

C₂

#22581325

yes

characteristic

101

C₂

#22581328

yes

characteristic

102

C₂

#22581336

yes

characteristic

	subgroup	G/H
center	C₂²	#17984838
commutator, cosocle	C₄²×C₁₃₉	C₂³
Frattini	C₄²	C₂²×D₁₃₉
Fitting, prime radical	#493×C₁₃₉	C₂
socle	C₂²×C₁₃₉	#127
hypercenter	#493	D₁₃₉
hypocenter	C₁₃₉	#1706
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#17984838

commutator, cosocle

C₄²×C₁₃₉

C₂³

Frattini

C₄²

C₂²×D₁₃₉

Fitting, prime radical

#493×C₁₃₉

C₂

C₂²×C₁₃₉

D₁₃₉

C₁₃₉

G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #22581358	C₂	semidirect
2. #22581325	C₂	semidirect
3. #22581299	C₂	semidirect
4. C₈×C₁₃₉⋊C₈	C₂	semidirect
30. C₁₃₉⋊C₈	D₈	semidirect
38. C₈×C₁₃₉	SD₁₆	semidirect
51. C₁₃₉	#1706	semidirect
90. C₈	#14555773	semidirect

H ≅ G/N

type

1. #22581358

C₂

semidirect

2. #22581325

C₂

semidirect

3. #22581299

C₂

semidirect

4. C₈×C₁₃₉⋊C₈

C₂

semidirect

30. C₁₃₉⋊C₈

D₈

semidirect

38. C₈×C₁₃₉

SD₁₆

semidirect

51. C₁₃₉

#1706

semidirect

90. C₈

#14555773

semidirect

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

subgroup	index
#22581358	2
#22581325	2
#22581299	2
C₈×C₁₃₉⋊C₈	2
#493×C₁₃₉	2
#1706	139

subgroup

index

#22581358

#22581325

#22581299

C₈×C₁₃₉⋊C₈

#493×C₁₃₉

#1706

139

Subgroups — #31805993

Conjugacy classes of subgroups

Complements

Maximal subgroups