Group #31816908 — subgroups

Subgroups — C₂×#22581355 #31816908

Order 17792.

360 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 181 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×#17984834	C₂	1	2	yes	maximal
2	#22581355	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17984776	C₂	1	2	yes	maximal
4	C₂×#17984774	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	#17984834	—	2	4	no	—
7	#17984834	C₂²	1	8	yes	—
8	C₂×C₄⋊Dic₁₃₉	C₂²	1	4	yes	—
9	C₂×Dic₁₃₉⋊C₄	—	2	4	no	—
10	#17984776	C₂²	1	8	yes	—
11	#17984774	C₂²	1	8	yes	—
12	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₃₉	C₂²	1	4	yes	—
16	C₂²×Dic₁₃₉	—	2	4	no	—
17	C₄⋊Dic₁₃₉	—	2	4	no	—
18	C₄⋊Dic₁₃₉	C₂³	1	8	yes	—
19	Dic₁₃₉⋊C₄	—	2	16	no	—
20	C₂²×C₄×C₁₃₉	D₄	1	2	yes	—
21	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₁₃₉	Q₈	1	4	yes	—
23	C₂×C₈×C₁₃₉	Q₈	1	4	yes	—
24	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
25	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
26	C₄²×C₁₃₉	D₄	1	2	yes	—
27	C₄²×C₁₃₉	C₂³	1	2	yes	—
28	C₂×Dic₁₃₉	—	2	8	no	—
29	C₂×Dic₁₃₉	—	4	8	no	—
30	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₃₉	—	2	4	no	—
32	C₂×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
33	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
36	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
37	C₈×C₁₃₉	C₂×Q₈	1	4	yes	—
38	C₈×C₁₃₉	C₂×Q₈	1	4	yes	—
39	C₂²×C₁₃₉	C₄⋊Q₈	1	2	yes	—
40	C₂²×C₁₃₉	#135	1	4	yes	—
41	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
42	C₄×C₁₃₉	—	2	4	no	—
43	C₄×C₁₃₉	C₄⋊Q₈	1	2	yes	—
44	C₄×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
45	C₄×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
46	Dic₁₃₉	—	4	8	no	—
47	C₂×C₁₃₉	#499	1	4	yes	complemented
48	C₂×C₁₃₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₁₃₉	C₂×#135	1	2	yes	—
50	C₁₃₉	C₂×#499	1	1	yes	characteristic, complemented, Sylow
51	C₂×#499	—	139	1	no	maximal, Sylow
52	C₂×#125	—	139	2	no	—
53	#499	—	139	8	no	—
54	C₂×#107	—	139	2	no	—
55	C₂×#106	—	139	2	no	—
56	C₂×C₄×C₈	D₁₃₉	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
59	#125	—	278	4	no	—
60	#125	—	139	8	no	—
61	C₂×C₄⋊C₄	—	278	4	no	—
62	C₂×C₄⋊C₄	—	139	4	no	—
63	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
64	#107	—	139	8	no	—
65	#106	—	139	8	no	—
66	C₄×C₈	C₂×D₁₃₉	1	4	yes	—
67	C₂²×C₄	—	278	4	no	—
68	C₂²×C₄	D₅₅₆	1	2	yes	—
69	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
70	C₂×C₈	Dic₂₇₈	1	4	yes	—
71	C₂×C₈	Dic₂₇₈	1	4	yes	—
72	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
73	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
74	C₄⋊C₄	—	278	16	no	—
75	C₄⋊C₄	—	278	4	no	—
76	C₄⋊C₄	—	139	8	no	—
77	C₄²	D₅₅₆	1	2	yes	—
78	C₄²	C₂²×D₁₃₉	1	2	yes	—
79	C₂³	C₂×D₅₅₆	1	1	yes	characteristic
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	278	8	no	—
82	C₂×C₄	—	556	8	no	—
83	C₂×C₄	C₂×D₅₅₆	1	4	yes	—
84	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
85	C₂×C₄	C₂×Dic₂₇₈	1	2	yes	—
86	C₂×C₄	C₂×Dic₂₇₈	1	2	yes	—
87	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
88	C₈	C₂×Dic₂₇₈	1	4	yes	—
89	C₈	C₂×Dic₂₇₈	1	4	yes	—
90	C₂²	#17984842	1	2	yes	—
91	C₂²	#17984864	1	4	yes	—
92	C₂²	C₂²×D₅₅₆	1	1	yes	characteristic
93	C₄	—	2	4	no	—
94	C₄	—	556	8	no	—
95	C₄	#17984842	1	2	yes	—
96	C₄	C₂²×Dic₂₇₈	1	1	yes	characteristic
97	C₄	C₂²×Dic₂₇₈	1	1	yes	characteristic
98	C₂	#22581355	1	4	yes	direct product factor
99	C₂	C₂×#17984842	1	1	yes	characteristic
100	C₂	C₂×#17984864	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984834

C₂

yes

maximal

#22581355

C₂

yes

maximal, direct product factor

C₂×#17984776

C₂

yes

maximal

C₂×#17984774

C₂

yes

maximal

C₂×C₄×C₈×C₁₃₉

C₂

yes

maximal, characteristic

#17984834

—

#17984834

C₂²

yes

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

C₂×Dic₁₃₉⋊C₄

—

#17984776

C₂²

yes

—

#17984774

C₂²

yes

—

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₄×C₈×C₁₃₉

C₂²

yes

—

C₂²×Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

C₂³

yes

—

Dic₁₃₉⋊C₄

—

C₂²×C₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

Q₈

yes

—

C₂×C₈×C₁₃₉

Q₈

yes

—

C₂×C₈×C₁₃₉

C₂³

yes

—

C₂×C₈×C₁₃₉

C₂³

yes

—

C₄²×C₁₃₉

D₄

yes

—

C₄²×C₁₃₉

C₂³

yes

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

—

C₂×C₄×C₁₃₉

C₂×Q₈

yes

—

C₂×C₄×C₁₃₉

C₂⁴

yes

characteristic

C₈×C₁₃₉

C₂×Q₈

yes

—

C₈×C₁₃₉

C₂×Q₈

yes

—

C₂²×C₁₃₉

C₄⋊Q₈

yes

—

C₂²×C₁₃₉

#135

yes

—

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

C₄⋊Q₈

yes

—

C₄×C₁₃₉

C₂²×Q₈

yes

characteristic

C₄×C₁₃₉

C₂²×Q₈

yes

characteristic

Dic₁₃₉

—

C₂×C₁₃₉

#499

yes

complemented

C₂×C₁₃₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₃₉

C₂×#135

yes

—

C₁₃₉

C₂×#499

yes

characteristic, complemented, Sylow

C₂×#499

—

139

maximal, Sylow

C₂×#125

—

139

—

#499

—

139

—

C₂×#107

—

139

—

C₂×#106

—

139

—

C₂×C₄×C₈

D₁₃₉

yes

characteristic

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

#125

—

278

—

#125

—

139

—

C₂×C₄⋊C₄

—

278

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

#107

—

139

—

#106

—

139

—

C₄×C₈

C₂×D₁₃₉

yes

—

C₂²×C₄

—

278

—

C₂²×C₄

D₅₅₆

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

C₂×C₈

Dic₂₇₈

yes

—

C₂×C₈

Dic₂₇₈

yes

—

C₂×C₈

C₂²×D₁₃₉

yes

—

C₂×C₈

C₂²×D₁₃₉

yes

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

139

—

C₄²

D₅₅₆

yes

—

C₄²

C₂²×D₁₃₉

yes

—

C₂³

C₂×D₅₅₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

—

556

—

C₂×C₄

C₂×D₅₅₆

yes

—

C₂×C₄

C₂×D₅₅₆

yes

characteristic

C₂×C₄

C₂×Dic₂₇₈

yes

—

C₂×C₄

C₂×Dic₂₇₈

yes

—

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

C₈

C₂×Dic₂₇₈

yes

—

C₈

C₂×Dic₂₇₈

yes

—

C₂²

#17984842

yes

—

C₂²

#17984864

yes

—

C₂²

C₂²×D₅₅₆

yes

characteristic

C₄

—

C₄

—

556

—

C₄

#17984842

yes

—

C₄

C₂²×Dic₂₇₈

yes

characteristic

C₄

C₂²×Dic₂₇₈

yes

characteristic

C₂

#22581355

yes

direct product factor

C₂

C₂×#17984842

yes

characteristic

100

C₂

C₂×#17984864

yes

—

	subgroup	G/H
center	C₂³	C₂×D₅₅₆
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	C₂×C₄×C₈×C₁₃₉	C₂
socle, prime radical	C₂³×C₁₃₉	C₂×D₄
hypercenter	C₂×C₄×C₈	D₁₃₉
hypocenter	C₁₃₉	C₂×#499
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₅₅₆

commutator, cosocle

C₂×C₄×C₁₃₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₉

Fitting

C₂×C₄×C₈×C₁₃₉

C₂

socle, prime radical

C₂³×C₁₃₉

C₂×D₄

hypercenter

C₂×C₄×C₈

D₁₃₉

hypocenter

C₁₃₉

C₂×#499

LCS

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #22581355	C₂	direct (N × H)
47. C₂×C₁₃₉	#499	semidirect
50. C₁₃₉	C₂×#499	semidirect
98. C₂	#22581355	direct (N × H)

H ≅ G/N

type

2. #22581355

C₂

direct (N × H)

47. C₂×C₁₃₉

#499

semidirect

50. C₁₃₉

C₂×#499

semidirect

98. C₂

#22581355

direct (N × H)

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

subgroup	index
C₂×#17984834	2
#22581355	2
C₂×#17984776	2
C₂×#17984774	2
C₂×C₄×C₈×C₁₃₉	2
C₂×#499	139

subgroup

index

C₂×C₄×C₈×C₁₃₉

139

Subgroups — C₂×#22581355 #31816908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22581355 #31816908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22581355 #31816908