Group #46858202 — subgroups

Subgroups — C₁₃₇×#8816 #46858202

Order 35072.

422 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 238 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	#1845×C₁₃₇	C₂	1	2	yes	maximal
2	#1834×C₁₃₇	C₂	1	4	yes	maximal
3	C₂×C₈²×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#107×C₁₃₇	—	2	4	no	—
5	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
6	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
7	#384×C₁₃₇	C₂²	1	4	yes	—
8	#340×C₁₃₇	—	2	8	no	—
9	C₈²×C₁₃₇	C₄	1	4	yes	complemented
10	C₂²×C₈×C₁₃₇	D₄	1	4	yes	—
11	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
12	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
13	C₂×C₄⋊C₄×C₁₃₇	—	2	8	no	—
14	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
15	#107×C₁₃₇	—	4	8	no	—
16	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₁₃₇	Q₈	1	4	yes	—
19	C₄×C₈×C₁₃₇	Q₈	1	2	yes	—
20	C₂²×C₄×C₁₃₇	—	4	4	no	—
21	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
23	C₂×C₈×C₁₃₇	—	2	8	no	—
24	C₂×C₈×C₁₃₇	C₄⋊C₄	1	8	yes	—
25	C₂×C₈×C₁₃₇	C₄⋊C₄	1	4	yes	—
26	C₂×C₈×C₁₃₇	D₈	1	4	yes	—
27	C₂×C₈×C₁₃₇	Q₁₆	1	4	yes	—
28	C₄⋊C₄×C₁₃₇	—	4	8	no	—
29	C₄⋊C₄×C₁₃₇	—	4	16	no	—
30	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
31	C₄²×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
32	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
33	C₂³×C₁₃₇	#127	1	1	yes	characteristic
34	C₂×C₄×C₁₃₇	—	8	4	no	—
35	C₂×C₄×C₁₃₇	—	8	8	no	—
36	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
39	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
40	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
41	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	2	yes	—
42	C₂×C₄×C₁₃₇	#135	1	8	yes	—
43	C₈×C₁₃₇	—	2	8	no	—
44	C₈×C₁₃₇	#107	1	8	yes	—
45	C₂²×C₁₃₇	#384	1	1	yes	characteristic
46	C₂²×C₁₃₇	#493	1	1	yes	characteristic
47	C₂²×C₁₃₇	#494	1	1	yes	characteristic
48	C₂²×C₁₃₇	#495	1	4	yes	—
49	C₄×C₁₃₇	—	16	4	no	—
50	C₄×C₁₃₇	C₂×#107	1	2	yes	—
51	C₄×C₁₃₇	#427	1	4	yes	—
52	C₄×C₁₃₇	#499	1	4	yes	—
53	C₄×C₁₃₇	#500	1	2	yes	—
54	C₂×C₁₃₇	#1709	1	2	yes	—
55	C₂×C₁₃₇	#1710	1	2	yes	—
56	C₂×C₁₃₇	#1845	1	1	yes	characteristic
57	C₂×C₁₃₇	#1846	1	2	yes	—
58	#8816	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₁₃₇	#8816	1	1	yes	characteristic, direct product factor, Sylow
60	#1845	C₂×C₁₃₇	1	2	yes	—
61	#1834	C₂×C₁₃₇	1	4	yes	—
62	C₂×C₈²	C₂×C₁₃₇	1	1	yes	characteristic
63	C₂×#107	—	2	4	no	—
64	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
65	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
66	#384	C₂²×C₁₃₇	1	4	yes	—
67	#340	—	2	8	no	—
68	C₈²	C₄×C₁₃₇	1	4	yes	complemented
69	C₂²×C₈	D₄×C₁₃₇	1	4	yes	—
70	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
71	C₂×C₄⋊C₄	—	2	8	no	—
72	C₂×C₄⋊C₄	—	2	4	no	—
73	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
74	#107	—	4	8	no	—
75	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
76	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
77	C₄×C₈	Q₈×C₁₃₇	1	4	yes	—
78	C₄×C₈	Q₈×C₁₃₇	1	2	yes	—
79	C₂²×C₄	—	4	4	no	—
80	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
81	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
82	C₂×C₈	—	2	8	no	—
83	C₂×C₈	C₄⋊C₄×C₁₃₇	1	8	yes	—
84	C₂×C₈	C₄⋊C₄×C₁₃₇	1	4	yes	—
85	C₂×C₈	D₈×C₁₃₇	1	4	yes	—
86	C₂×C₈	Q₁₆×C₁₃₇	1	4	yes	—
87	C₄⋊C₄	—	4	8	no	—
88	C₄⋊C₄	—	4	16	no	—
89	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
90	C₄²	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
91	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
92	C₂³	#127×C₁₃₇	1	1	yes	characteristic
93	C₂×C₄	—	8	4	no	—
94	C₂×C₄	—	8	8	no	—
95	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
96	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
97	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
98	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
99	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
100	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	2	yes	—
101	C₂×C₄	#135×C₁₃₇	1	8	yes	—
102	C₈	—	2	8	no	—
103	C₈	#107×C₁₃₇	1	8	yes	—
104	C₂²	#384×C₁₃₇	1	1	yes	characteristic
105	C₂²	#493×C₁₃₇	1	1	yes	characteristic
106	C₂²	#494×C₁₃₇	1	1	yes	characteristic
107	C₂²	#495×C₁₃₇	1	4	yes	—
108	C₄	—	16	4	no	—
109	C₄	C₂×#107×C₁₃₇	1	2	yes	—
110	C₄	#427×C₁₃₇	1	4	yes	—
111	C₄	#499×C₁₃₇	1	4	yes	—
112	C₄	#500×C₁₃₇	1	2	yes	—
113	C₂	#1709×C₁₃₇	1	2	yes	—
114	C₂	#1710×C₁₃₇	1	2	yes	—
115	C₂	#1845×C₁₃₇	1	1	yes	characteristic
116	C₂	#1846×C₁₃₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#1845×C₁₃₇

C₂

yes

maximal

#1834×C₁₃₇

C₂

yes

maximal

C₂×C₈²×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#107×C₁₃₇

—

C₂×C₄×C₈×C₁₃₇

C₂²

yes

—

C₂×C₄×C₈×C₁₃₇

C₂²

yes

characteristic

#384×C₁₃₇

C₂²

yes

—

#340×C₁₃₇

—

C₈²×C₁₃₇

C₄

yes

complemented

C₂²×C₈×C₁₃₇

D₄

yes

—

C₂²×C₈×C₁₃₇

D₄

yes

—

C₂×C₄⋊C₄×C₁₃₇

—

C₂×C₄⋊C₄×C₁₃₇

—

C₂×C₄²×C₁₃₇

C₂³

yes

characteristic

#107×C₁₃₇

—

C₄×C₈×C₁₃₇

C₂×C₄

yes

—

C₄×C₈×C₁₃₇

C₂×C₄

yes

—

C₄×C₈×C₁₃₇

Q₈

yes

—

C₄×C₈×C₁₃₇

Q₈

yes

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃₇

C₂×D₄

yes

—

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

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

C₂²×C₄

yes

characteristic

C₄²×C₁₃₇

C₂×Q₈

yes

characteristic

C₄²×C₁₃₇

C₂×Q₈

yes

—

C₂³×C₁₃₇

#127

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₄⋊Q₈

yes

—

C₂×C₄×C₁₃₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₈

yes

—

C₂×C₄×C₁₃₇

C₂×Q₁₆

yes

—

C₂×C₄×C₁₃₇

#135

yes

—

C₈×C₁₃₇

—

C₈×C₁₃₇

#107

yes

—

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×#107

yes

—

C₄×C₁₃₇

#427

yes

—

C₄×C₁₃₇

#499

yes

—

C₄×C₁₃₇

#500

yes

—

C₂×C₁₃₇

#1709

yes

—

C₂×C₁₃₇

#1710

yes

—

C₂×C₁₃₇

#1845

yes

characteristic

C₂×C₁₃₇

#1846

yes

—

#8816

C₁₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₇

#8816

yes

characteristic, direct product factor, Sylow

#1845

C₂×C₁₃₇

yes

—

#1834

C₂×C₁₃₇

yes

—

C₂×C₈²

C₂×C₁₃₇

yes

characteristic

C₂×#107

—

C₂×C₄×C₈

C₂²×C₁₃₇

yes

—

C₂×C₄×C₈

C₂²×C₁₃₇

yes

characteristic

#384

C₂²×C₁₃₇

yes

—

#340

—

C₈²

C₄×C₁₃₇

yes

complemented

C₂²×C₈

D₄×C₁₃₇

yes

—

C₂²×C₈

D₄×C₁₃₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³×C₁₃₇

yes

characteristic

#107

—

C₄×C₈

C₂×C₄×C₁₃₇

yes

—

C₄×C₈

C₂×C₄×C₁₃₇

yes

—

C₄×C₈

Q₈×C₁₃₇

yes

—

C₄×C₈

Q₈×C₁₃₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₁₃₇

yes

characteristic

C₂²×C₄

C₂×D₄×C₁₃₇

yes

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₁₃₇

yes

—

C₂×C₈

C₄⋊C₄×C₁₃₇

yes

—

C₂×C₈

D₈×C₁₃₇

yes

—

C₂×C₈

Q₁₆×C₁₃₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×C₄×C₁₃₇

yes

characteristic

C₄²

C₂×Q₈×C₁₃₇

yes

characteristic

C₄²

C₂×Q₈×C₁₃₇

yes

—

C₂³

#127×C₁₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄×C₁₃₇

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄×C₁₃₇

yes

—

C₂×C₄

C₄⋊Q₈×C₁₃₇

yes

—

C₂×C₄

C₄⋊Q₈×C₁₃₇

yes

characteristic

C₂×C₄

C₂×D₈×C₁₃₇

yes

—

100

C₂×C₄

C₂×Q₁₆×C₁₃₇

yes

—

101

C₂×C₄

#135×C₁₃₇

yes

—

102

C₈

—

103

C₈

#107×C₁₃₇

yes

—

104

C₂²

#384×C₁₃₇

yes

characteristic

105

C₂²

#493×C₁₃₇

yes

characteristic

106

C₂²

#494×C₁₃₇

yes

characteristic

107

C₂²

#495×C₁₃₇

yes

—

108

C₄

—

109

C₄

C₂×#107×C₁₃₇

yes

—

110

C₄

#427×C₁₃₇

yes

—

111

C₄

#499×C₁₃₇

yes

—

112

C₄

#500×C₁₃₇

yes

—

113

C₂

#1709×C₁₃₇

yes

—

114

C₂

#1710×C₁₃₇

yes

—

115

C₂

#1845×C₁₃₇

yes

characteristic

116

C₂

#1846×C₁₃₇

yes

—

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

subgroup

G/H

center, socle, prime radical

C₂³×C₁₃₇

#127

commutator

C₄²

C₂²×C₄×C₁₃₇

Frattini, cosocle

C₂×C₄²

C₂³×C₁₃₇

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

N	H ≅ G/N	type
9. C₈²×C₁₃₇	C₄	semidirect
58. #8816	C₁₃₇	direct (N × H)
59. C₁₃₇	#8816	direct (N × H)
68. C₈²	C₄×C₁₃₇	semidirect

H ≅ G/N

type

9. C₈²×C₁₃₇

C₄

semidirect

58. #8816

C₁₃₇

direct (N × H)

59. C₁₃₇

#8816

direct (N × H)

68. C₈²

C₄×C₁₃₇

semidirect

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

subgroup	index
#1845×C₁₃₇	2
#1834×C₁₃₇	2
C₂×C₈²×C₁₃₇	2
#8816	137

subgroup

index

#1845×C₁₃₇

#1834×C₁₃₇

C₂×C₈²×C₁₃₇

#8816

137

Subgroups — C₁₃₇×#8816 #46858202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C137×#8816 #46858202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8816 #46858202