Group #46858212 — subgroups

Subgroups — C₁₃₇×#8826 #46858212

Order 35072.

338 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 198 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	#1848×C₁₃₇	C₂	1	2	yes	maximal
2	#1567×C₁₃₇	C₂	1	4	yes	maximal
3	C₂×#322×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#108×C₁₃₇	—	2	4	no	—
5	#423×C₁₃₇	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
8	#335×C₁₃₇	—	2	8	no	—
9	#322×C₁₃₇	C₄	1	4	yes	—
10	C₂×M₁₆×C₁₃₇	—	2	4	no	—
11	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
12	C₂²×C₈×C₁₃₇	Q₈	1	4	yes	—
13	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
14	#108×C₁₃₇	—	4	8	no	—
15	C₄⋊C₈×C₁₃₇	—	2	8	no	—
16	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₁₃₇	D₄	1	4	yes	—
19	C₄×C₈×C₁₃₇	Q₈	1	2	yes	—
20	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
22	M₁₆×C₁₃₇	—	4	8	no	—
23	C₂×C₈×C₁₃₇	—	4	4	no	—
24	C₂×C₈×C₁₃₇	—	4	4	no	—
25	C₂×C₈×C₁₃₇	C₄⋊C₄	1	8	yes	—
26	C₂×C₈×C₁₃₇	C₄⋊C₄	1	4	yes	—
27	C₂×C₈×C₁₃₇	D₈	1	4	yes	—
28	C₂×C₈×C₁₃₇	Q₁₆	1	4	yes	—
29	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
30	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
31	C₄²×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
32	C₂³×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
33	C₂×C₄×C₁₃₇	—	2	4	no	—
34	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
37	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
38	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
39	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	2	yes	—
40	C₈×C₁₃₇	—	2	8	no	—
41	C₈×C₁₃₇	—	8	4	no	—
42	C₈×C₁₃₇	#107	1	8	yes	—
43	C₂²×C₁₃₇	—	2	2	no	—
44	C₂²×C₁₃₇	#384	1	1	yes	characteristic
45	C₂²×C₁₃₇	#500	1	2	yes	—
46	C₄×C₁₃₇	—	2	2	no	—
47	C₄×C₁₃₇	C₂×#107	1	2	yes	—
48	C₄×C₁₃₇	#430	1	4	yes	—
49	C₄×C₁₃₇	#493	1	1	yes	characteristic
50	C₄×C₁₃₇	#494	1	1	yes	characteristic
51	C₂×C₁₃₇	—	2	2	no	—
52	C₂×C₁₃₇	#1845	1	1	yes	characteristic
53	C₂×C₁₃₇	#1849	1	2	yes	—
54	#8826	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₁₃₇	#8826	1	1	yes	characteristic, direct product factor, Sylow
56	#1848	C₂×C₁₃₇	1	2	yes	—
57	#1567	C₂×C₁₃₇	1	4	yes	—
58	C₂×#322	C₂×C₁₃₇	1	1	yes	characteristic
59	C₂×#108	—	2	4	no	—
60	#423	C₂²×C₁₃₇	1	4	yes	—
61	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
62	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
63	#335	—	2	8	no	—
64	#322	C₄×C₁₃₇	1	4	yes	—
65	C₂×M₁₆	—	2	4	no	—
66	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
67	C₂²×C₈	Q₈×C₁₃₇	1	4	yes	—
68	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
69	#108	—	4	8	no	—
70	C₄⋊C₈	—	2	8	no	—
71	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
72	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
73	C₄×C₈	D₄×C₁₃₇	1	4	yes	—
74	C₄×C₈	Q₈×C₁₃₇	1	2	yes	—
75	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
76	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
77	M₁₆	—	4	8	no	—
78	C₂×C₈	—	4	4	no	—
79	C₂×C₈	—	4	4	no	—
80	C₂×C₈	C₄⋊C₄×C₁₃₇	1	8	yes	—
81	C₂×C₈	C₄⋊C₄×C₁₃₇	1	4	yes	—
82	C₂×C₈	D₈×C₁₃₇	1	4	yes	—
83	C₂×C₈	Q₁₆×C₁₃₇	1	4	yes	—
84	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
85	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
86	C₄²	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
87	C₂³	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
91	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
92	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
93	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
94	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	2	yes	—
95	C₈	—	2	8	no	—
96	C₈	—	8	4	no	—
97	C₈	#107×C₁₃₇	1	8	yes	—
98	C₂²	—	2	2	no	—
99	C₂²	#384×C₁₃₇	1	1	yes	characteristic
100	C₂²	#500×C₁₃₇	1	2	yes	—
101	C₄	—	2	2	no	—
102	C₄	C₂×#107×C₁₃₇	1	2	yes	—
103	C₄	#430×C₁₃₇	1	4	yes	—
104	C₄	#493×C₁₃₇	1	1	yes	characteristic
105	C₄	#494×C₁₃₇	1	1	yes	characteristic
106	C₂	—	2	2	no	—
107	C₂	#1845×C₁₃₇	1	1	yes	characteristic
108	C₂	#1849×C₁₃₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#1848×C₁₃₇

C₂

yes

maximal

#1567×C₁₃₇

C₂

yes

maximal

C₂×#322×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#108×C₁₃₇

—

#423×C₁₃₇

C₂²

yes

—

C₂×C₄×C₈×C₁₃₇

C₂²

yes

—

C₂×C₄×C₈×C₁₃₇

C₂²

yes

characteristic

#335×C₁₃₇

—

#322×C₁₃₇

C₄

yes

—

C₂×M₁₆×C₁₃₇

—

C₂²×C₈×C₁₃₇

D₄

yes

—

C₂²×C₈×C₁₃₇

Q₈

yes

—

C₂×C₄²×C₁₃₇

C₂³

yes

characteristic

#108×C₁₃₇

—

C₄⋊C₈×C₁₃₇

—

C₄×C₈×C₁₃₇

C₂×C₄

yes

—

C₄×C₈×C₁₃₇

C₂×C₄

yes

—

C₄×C₈×C₁₃₇

D₄

yes

—

C₄×C₈×C₁₃₇

Q₈

yes

—

C₂²×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃₇

C₂×Q₈

yes

—

M₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₃₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₃₇

D₈

yes

—

C₂×C₈×C₁₃₇

Q₁₆

yes

—

C₄²×C₁₃₇

C₂²×C₄

yes

characteristic

C₄²×C₁₃₇

C₂×D₄

yes

—

C₄²×C₁₃₇

C₂×Q₈

yes

characteristic

C₂³×C₁₃₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₃₇

#127

yes

characteristic

C₂×C₄×C₁₃₇

C₄⋊Q₈

yes

—

C₂×C₄×C₁₃₇

C₂×D₈

yes

—

C₂×C₄×C₁₃₇

C₂×Q₁₆

yes

—

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₈×C₁₃₇

#107

yes

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

#384

yes

characteristic

C₂²×C₁₃₇

#500

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×#107

yes

—

C₄×C₁₃₇

#430

yes

—

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

C₂×C₁₃₇

—

C₂×C₁₃₇

#1845

yes

characteristic

C₂×C₁₃₇

#1849

yes

—

#8826

C₁₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₇

#8826

yes

characteristic, direct product factor, Sylow

#1848

C₂×C₁₃₇

yes

—

#1567

C₂×C₁₃₇

yes

—

C₂×#322

C₂×C₁₃₇

yes

characteristic

C₂×#108

—

#423

C₂²×C₁₃₇

yes

—

C₂×C₄×C₈

C₂²×C₁₃₇

yes

—

C₂×C₄×C₈

C₂²×C₁₃₇

yes

characteristic

#335

—

#322

C₄×C₁₃₇

yes

—

C₂×M₁₆

—

C₂²×C₈

D₄×C₁₃₇

yes

—

C₂²×C₈

Q₈×C₁₃₇

yes

—

C₂×C₄²

C₂³×C₁₃₇

yes

characteristic

#108

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₁₃₇

yes

—

C₄×C₈

C₂×C₄×C₁₃₇

yes

—

C₄×C₈

D₄×C₁₃₇

yes

—

C₄×C₈

Q₈×C₁₃₇

yes

—

C₂²×C₄

C₂×D₄×C₁₃₇

yes

characteristic

C₂²×C₄

C₂×Q₈×C₁₃₇

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₁₃₇

yes

—

C₂×C₈

C₄⋊C₄×C₁₃₇

yes

—

C₂×C₈

D₈×C₁₃₇

yes

—

C₂×C₈

Q₁₆×C₁₃₇

yes

—

C₄²

C₂²×C₄×C₁₃₇

yes

characteristic

C₄²

C₂×D₄×C₁₃₇

yes

—

C₄²

C₂×Q₈×C₁₃₇

yes

characteristic

C₂³

C₄⋊Q₈×C₁₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄×C₁₃₇

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄×C₁₃₇

yes

—

C₂×C₄

#127×C₁₃₇

yes

characteristic

C₂×C₄

C₄⋊Q₈×C₁₃₇

yes

—

C₂×C₄

C₂×D₈×C₁₃₇

yes

—

C₂×C₄

C₂×Q₁₆×C₁₃₇

yes

—

C₈

—

C₈

—

C₈

#107×C₁₃₇

yes

—

C₂²

—

C₂²

#384×C₁₃₇

yes

characteristic

100

C₂²

#500×C₁₃₇

yes

—

101

C₄

—

102

C₄

C₂×#107×C₁₃₇

yes

—

103

C₄

#430×C₁₃₇

yes

—

104

C₄

#493×C₁₃₇

yes

characteristic

105

C₄

#494×C₁₃₇

yes

characteristic

106

C₂

—

107

C₂

#1845×C₁₃₇

yes

characteristic

108

C₂

#1849×C₁₃₇

yes

—

	subgroup	G/H
center	C₂×C₄×C₁₃₇	#127
commutator	C₄²	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	#384
prime radical	C₂³×C₁₃₇	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₃₇

#127

commutator

C₄²

C₂²×C₄×C₁₃₇

Frattini, cosocle

C₂×C₄²

C₂³×C₁₃₇

socle

C₂²×C₁₃₇

#384

prime radical

C₂³×C₁₃₇

C₄⋊Q₈

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

N	H ≅ G/N	type
54. #8826	C₁₃₇	direct (N × H)
55. C₁₃₇	#8826	direct (N × H)

H ≅ G/N

type

54. #8826

C₁₃₇

direct (N × H)

55. C₁₃₇

#8826

direct (N × H)

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

subgroup	index
#1848×C₁₃₇	2
#1567×C₁₃₇	2
C₂×#322×C₁₃₇	2
#8826	137

subgroup

index

#1848×C₁₃₇

#1567×C₁₃₇

C₂×#322×C₁₃₇

#8826

137

Subgroups — C₁₃₇×#8826 #46858212

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C137×#8826 #46858212

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8826 #46858212