Group #31804312 — subgroups

Subgroups — #31804312

Order 17792.

192 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 57 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	#22580917	C₂	1	2	yes	maximal
2	#381×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#14555769	—	2	2	no	—
4	C₂²⋊(C₁₃₉⋊C₈)	—	2	2	no	—
5	C₂²⋊(C₁₃₉⋊C₈)	C₄	1	4	yes	complemented
6	C₂³×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊C₄×C₁₃₉	C₄	1	2	yes	—
8	#95×C₁₃₉	—	2	2	no	—
9	#14555769	—	4	4	no	—
10	C₂×C₁₃₉⋊C₈	—	2	4	no	—
11	C₂⁴×C₁₃₉	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₁₃₉	—	2	2	no	—
13	C₂²×C₄×C₁₃₉	—	4	2	no	—
14	C₂²×C₄×C₁₃₉	C₂×C₄	1	2	yes	—
15	C₂²×C₄×C₁₃₉	D₄	1	2	yes	—
16	C₂²×C₄×C₁₃₉	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₃₉	Q₈	1	1	yes	characteristic
18	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
19	C₁₃₉⋊C₈	—	4	4	no	—
20	C₂³×C₁₃₉	—	2	2	no	—
21	C₂³×C₁₃₉	—	4	2	no	—
22	C₂³×C₁₃₉	C₄²	1	1	yes	characteristic
23	C₂³×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
24	C₂³×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₃₉	—	2	4	no	—
26	C₂×C₄×C₁₃₉	—	4	2	no	—
27	C₂×C₄×C₁₃₉	—	4	1	no	—
28	C₂×C₄×C₁₃₉	—	4	1	no	—
29	C₂×C₄×C₁₃₉	—	4	4	no	—
30	C₂×C₄×C₁₃₉	—	4	2	no	—
31	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
32	C₂×C₄×C₁₃₉	C₄⋊C₄	1	2	yes	—
33	C₂²×C₁₃₉	—	2	2	no	—
34	C₂²×C₁₃₉	—	2	1	no	—
35	C₂²×C₁₃₉	—	2	1	no	—
36	C₂²×C₁₃₉	—	2	2	no	—
37	C₂²×C₁₃₉	—	2	2	no	—
38	C₂²×C₁₃₉	—	4	4	no	—
39	C₂²×C₁₃₉	#95	1	1	yes	characteristic
40	C₂²×C₁₃₉	#99	1	2	yes	—
41	C₄×C₁₃₉	—	2	2	no	—
42	C₄×C₁₃₉	—	2	2	no	—
43	C₄×C₁₃₉	—	8	2	no	—
44	C₂×C₁₃₉	—	2	2	no	—
45	C₂×C₁₃₉	—	4	1	no	—
46	C₂×C₁₃₉	—	4	1	no	—
47	C₂×C₁₃₉	#342	1	1	yes	characteristic
48	C₂×C₁₃₉	#344	1	2	yes	—
49	C₁₃₉	#1306	1	1	yes	characteristic, complemented, Sylow
50	#1306	—	139	1	no	maximal, Sylow
51	#407	—	139	2	no	—
52	#381	D₁₃₉	1	1	yes	characteristic
53	C₂³×C₄	C₂×D₁₃₉	1	1	yes	characteristic
54	C₂×M₁₆	—	278	2	no	—
55	C₂×C₂²⋊C₄	Dic₁₃₉	1	2	yes	—
56	C₂²⋊C₈	—	278	2	no	—
57	C₂²⋊C₈	—	139	4	no	—
58	#95	—	2	2	no	—
59	C₂⁴	C₂×Dic₁₃₉	1	1	yes	characteristic
60	C₂²×C₄	—	2	2	no	—
61	C₂²×C₄	—	4	2	no	—
62	C₂²×C₄	C₄×D₁₃₉	1	2	yes	—
63	C₂²×C₄	C₂²⋊D₁₃₉	1	2	yes	—
64	C₂²×C₄	D₅₅₆	1	1	yes	characteristic
65	C₂²×C₄	Dic₂₇₈	1	1	yes	characteristic
66	M₁₆	—	556	4	no	—
67	C₂×C₈	—	278	4	no	—
68	C₂²⋊C₄	—	4	2	no	—
69	C₂³	—	2	2	no	—
70	C₂³	—	4	2	no	—
71	C₂³	C₄×Dic₁₃₉	1	1	yes	characteristic
72	C₂³	C₂²⋊Dic₁₃₉	1	1	yes	characteristic
73	C₂³	C₄⋊Dic₁₃₉	1	1	yes	characteristic
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	—	4	1	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	—	4	4	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	(C₂×D₁₃₉)⋊C₄	1	2	yes	—
81	C₂×C₄	Dic₁₃₉⋊C₄	1	2	yes	—
82	C₈	—	556	4	no	—
83	C₂²	—	2	2	no	—
84	C₂²	—	2	1	no	—
85	C₂²	—	2	1	no	—
86	C₂²	—	2	2	no	—
87	C₂²	—	2	2	no	—
88	C₂²	—	4	4	no	—
89	C₂²	C₁₃₉⋊#95	1	1	yes	characteristic
90	C₂²	#17984757	1	2	yes	—
91	C₄	—	2	2	no	—
92	C₄	—	2	2	no	—
93	C₄	—	8	2	no	—
94	C₂	—	2	2	no	—
95	C₂	—	4	1	no	—
96	C₂	—	4	1	no	—
97	C₂	#22580720	1	1	yes	characteristic
98	C₂	#22580723	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22580917

C₂

yes

maximal

#381×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555769

—

C₂²⋊(C₁₃₉⋊C₈)

—

C₂²⋊(C₁₃₉⋊C₈)

C₄

yes

complemented

C₂³×C₄×C₁₃₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₁₃₉

C₄

yes

—

#95×C₁₃₉

—

#14555769

—

C₂×C₁₃₉⋊C₈

—

C₂⁴×C₁₃₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

D₄

yes

characteristic

C₂²×C₄×C₁₃₉

Q₈

yes

characteristic

C₂²⋊C₄×C₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

C₄²

yes

characteristic

C₂³×C₁₃₉

C₂²⋊C₄

yes

characteristic

C₂³×C₁₃₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₉

C₄⋊C₄

yes

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

#95

yes

characteristic

C₂²×C₁₃₉

#99

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#342

yes

characteristic

C₂×C₁₃₉

#344

yes

—

C₁₃₉

#1306

yes

characteristic, complemented, Sylow

#1306

—

139

maximal, Sylow

#407

—

139

—

#381

D₁₃₉

yes

characteristic

C₂³×C₄

C₂×D₁₃₉

yes

characteristic

C₂×M₁₆

—

278

—

C₂×C₂²⋊C₄

Dic₁₃₉

yes

—

C₂²⋊C₈

—

278

—

C₂²⋊C₈

—

139

—

#95

—

C₂⁴

C₂×Dic₁₃₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₁₃₉

yes

—

C₂²×C₄

C₂²⋊D₁₃₉

yes

—

C₂²×C₄

D₅₅₆

yes

characteristic

C₂²×C₄

Dic₂₇₈

yes

characteristic

M₁₆

—

556

—

C₂×C₈

—

278

—

C₂²⋊C₄

—

C₂³

—

C₂³

—

C₂³

C₄×Dic₁₃₉

yes

characteristic

C₂³

C₂²⋊Dic₁₃₉

yes

characteristic

C₂³

C₄⋊Dic₁₃₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

(C₂×D₁₃₉)⋊C₄

yes

—

C₂×C₄

Dic₁₃₉⋊C₄

yes

—

C₈

—

556

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₁₃₉⋊#95

yes

characteristic

C₂²

#17984757

yes

—

C₄

—

C₄

—

C₄

—

C₂

—

C₂

—

C₂

—

C₂

#22580720

yes

characteristic

C₂

#22580723

yes

—

	subgroup	G/H
center	C₂²	C₁₃₉⋊#95
commutator	C₂³×C₁₃₉	C₄²
Frattini	C₂³×C₄	C₂×D₁₃₉
Fitting	#381×C₁₃₉	C₂
socle	C₂²×C₁₃₉	#95
prime radical	C₂⁴×C₁₃₉	C₂×C₄
cosocle	C₂³×C₄×C₁₃₉	C₂²
hypercenter	#381	D₁₃₉
hypocenter	C₁₃₉	#1306
LCS	G ▷ C₂³×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #381

subgroup

G/H

center

C₂²

C₁₃₉⋊#95

commutator

C₂³×C₁₃₉

C₄²

Frattini

C₂³×C₄

C₂×D₁₃₉

Fitting

#381×C₁₃₉

C₂

socle

C₂²×C₁₃₉

#95

prime radical

C₂⁴×C₁₃₉

C₂×C₄

cosocle

C₂³×C₄×C₁₃₉

C₂²

D₁₃₉

C₁₃₉

G ▷ C₂³×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #381

N	H ≅ G/N	type
5. C₂²⋊(C₁₃₉⋊C₈)	C₄	semidirect
49. C₁₃₉	#1306	semidirect

H ≅ G/N

type

5. C₂²⋊(C₁₃₉⋊C₈)

C₄

semidirect

49. C₁₃₉

#1306

semidirect

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

subgroup	index
#22580917	2
#381×C₁₃₉	2
#1306	139

subgroup

index

#22580917

#381×C₁₃₉

#1306

139

Subgroups — #31804312

Conjugacy classes of subgroups

Complements

Maximal subgroups