Group #46858168 — subgroups

Subgroups — C₁₃₇×#8782 #46858168

Order 35072.

262 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 126 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	#1746×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1564×C₁₃₇	C₂	1	2	yes	maximal
3	#1281×C₁₃₇	C₂	1	2	yes	maximal
4	#1274×C₁₃₇	C₂	1	2	yes	maximal
5	#431×C₁₃₇	C₂²	1	4	yes	—
6	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
8	#376×C₁₃₇	C₂²	1	1	yes	characteristic
9	#336×C₁₃₇	C₄	1	4	yes	—
10	#335×C₁₃₇	—	2	2	no	—
11	#334×C₁₃₇	—	2	2	no	—
12	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
13	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
16	C₄⋊C₈×C₁₃₇	—	2	4	no	—
17	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
18	#97×C₁₃₇	—	2	2	no	—
19	C₄×C₈×C₁₃₇	—	2	4	no	—
20	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
21	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
22	C₄×C₈×C₁₃₇	Q₈	1	2	yes	—
23	#95×C₁₃₇	—	2	2	no	—
24	C₂²×C₄×C₁₃₇	—	2	2	no	—
25	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
27	C₂×C₈×C₁₃₇	—	2	4	no	—
28	C₂×C₈×C₁₃₇	—	2	2	no	—
29	C₂×C₈×C₁₃₇	—	2	4	no	—
30	C₂×C₈×C₁₃₇	—	4	4	no	—
31	C₂×C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
32	C₂×C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
33	C₄²×C₁₃₇	—	2	4	no	—
34	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
35	C₄²×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
36	C₄²×C₁₃₇	#41	1	2	yes	—
37	C₂³×C₁₃₇	#117	1	1	yes	characteristic
38	C₂×C₄×C₁₃₇	—	4	2	no	—
39	C₂×C₄×C₁₃₇	—	4	4	no	—
40	C₂×C₄×C₁₃₇	—	4	2	no	—
41	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
43	C₂×C₄×C₁₃₇	#124	1	2	yes	—
44	C₂×C₄×C₁₃₇	#125	1	2	yes	—
45	C₂×C₄×C₁₃₇	#131	1	4	yes	—
46	C₈×C₁₃₇	—	2	2	no	—
47	C₈×C₁₃₇	—	2	2	no	—
48	C₈×C₁₃₇	—	4	2	no	—
49	C₈×C₁₃₇	—	8	4	no	—
50	C₂²×C₁₃₇	—	2	2	no	—
51	C₂²×C₁₃₇	#382	1	1	yes	characteristic
52	C₂²×C₁₃₇	#433	1	2	yes	—
53	C₄×C₁₃₇	—	2	2	no	—
54	C₄×C₁₃₇	—	2	2	no	—
55	C₄×C₁₃₇	—	8	2	no	—
56	C₄×C₁₃₇	#424	1	2	yes	—
57	C₄×C₁₃₇	#433	1	2	yes	—
58	C₂×C₁₃₇	—	2	2	no	—
59	C₂×C₁₃₇	#1832	1	1	yes	characteristic
60	C₂×C₁₃₇	#1833	1	2	yes	—
61	#8782	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₁₃₇	#8782	1	1	yes	characteristic, direct product factor, Sylow
63	#1746	C₂×C₁₃₇	1	1	yes	characteristic
64	#1564	C₂×C₁₃₇	1	2	yes	—
65	#1281	C₂×C₁₃₇	1	2	yes	—
66	#1274	C₂×C₁₃₇	1	2	yes	—
67	#431	C₂²×C₁₃₇	1	4	yes	—
68	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
69	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
70	#376	C₂²×C₁₃₇	1	1	yes	characteristic
71	#336	C₄×C₁₃₇	1	4	yes	—
72	#335	—	2	2	no	—
73	#334	—	2	2	no	—
74	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
75	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
76	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
77	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
78	C₄⋊C₈	—	2	4	no	—
79	C₂²⋊C₈	—	2	4	no	—
80	#97	—	2	2	no	—
81	C₄×C₈	—	2	4	no	—
82	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
83	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
84	C₄×C₈	Q₈×C₁₃₇	1	2	yes	—
85	#95	—	2	2	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
88	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
89	C₂×C₈	—	2	4	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	—	4	4	no	—
93	C₂×C₈	C₄⋊C₄×C₁₃₇	1	2	yes	—
94	C₂×C₈	C₄⋊C₄×C₁₃₇	1	2	yes	—
95	C₄²	—	2	4	no	—
96	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
97	C₄²	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
98	C₄²	#41×C₁₃₇	1	2	yes	—
99	C₂³	#117×C₁₃₇	1	1	yes	characteristic
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	4	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
104	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
105	C₂×C₄	#124×C₁₃₇	1	2	yes	—
106	C₂×C₄	#125×C₁₃₇	1	2	yes	—
107	C₂×C₄	#131×C₁₃₇	1	4	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	2	2	no	—
110	C₈	—	4	2	no	—
111	C₈	—	8	4	no	—
112	C₂²	—	2	2	no	—
113	C₂²	#382×C₁₃₇	1	1	yes	characteristic
114	C₂²	#433×C₁₃₇	1	2	yes	—
115	C₄	—	2	2	no	—
116	C₄	—	2	2	no	—
117	C₄	—	8	2	no	—
118	C₄	#424×C₁₃₇	1	2	yes	—
119	C₄	#433×C₁₃₇	1	2	yes	—
120	C₂	—	2	2	no	—
121	C₂	#1832×C₁₃₇	1	1	yes	characteristic
122	C₂	#1833×C₁₃₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#1746×C₁₃₇

C₂

yes

maximal, characteristic

#1564×C₁₃₇

C₂

yes

maximal

#1281×C₁₃₇

C₂

yes

maximal

#1274×C₁₃₇

C₂

yes

maximal

#431×C₁₃₇

C₂²

yes

—

C₂×#97×C₁₃₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₃₇

C₂²

yes

characteristic

#376×C₁₃₇

C₂²

yes

characteristic

#336×C₁₃₇

C₄

yes

—

#335×C₁₃₇

—

#334×C₁₃₇

—

C₂²×C₈×C₁₃₇

C₂×C₄

yes

—

C₂²×C₈×C₁₃₇

C₂×C₄

yes

—

C₂×C₄²×C₁₃₇

C₂×C₄

yes

—

C₂×C₄²×C₁₃₇

C₂³

yes

characteristic

C₄⋊C₈×C₁₃₇

—

C₂²⋊C₈×C₁₃₇

—

#97×C₁₃₇

—

C₄×C₈×C₁₃₇

—

C₄×C₈×C₁₃₇

D₄

yes

characteristic

C₄×C₈×C₁₃₇

D₄

yes

characteristic

C₄×C₈×C₁₃₇

Q₈

yes

—

#95×C₁₃₇

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃₇

#41

yes

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₃₇

C₄⋊C₄

yes

—

C₄²×C₁₃₇

—

C₄²×C₁₃₇

C₂×D₄

yes

characteristic

C₄²×C₁₃₇

C₂×Q₈

yes

characteristic

C₄²×C₁₃₇

#41

yes

—

C₂³×C₁₃₇

#117

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×C₄⋊C₄

yes

C₂×C₄×C₁₃₇

yes

C₂×C₄×C₁₃₇

yes

—

C₂×C₄×C₁₃₇

#125

yes

—

C₂×C₄×C₁₃₇

#131

yes

—

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

#382

yes

characteristic

C₂²×C₁₃₇

#433

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#424

yes

—

C₄×C₁₃₇

#433

yes

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#1832

yes

characteristic

C₂×C₁₃₇

#1833

yes

—

#8782

C₁₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₇

#8782

yes

characteristic, direct product factor, Sylow

#1746

C₂×C₁₃₇

yes

characteristic

#1564

C₂×C₁₃₇

yes

—

#1281

C₂×C₁₃₇

yes

—

#1274

C₂×C₁₃₇

yes

—

#431

C₂²×C₁₃₇

yes

—

C₂×#97

C₂²×C₁₃₇

yes

characteristic

C₂×C₄×C₈

C₂²×C₁₃₇

yes

characteristic

#376

C₂²×C₁₃₇

yes

characteristic

#336

C₄×C₁₃₇

yes

—

#335

—

#334

—

C₂²×C₈

C₂×C₄×C₁₃₇

yes

—

C₂²×C₈

C₂×C₄×C₁₃₇

yes

—

C₂×C₄²

C₂×C₄×C₁₃₇

yes

—

C₂×C₄²

C₂³×C₁₃₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

D₄×C₁₃₇

yes

characteristic

C₄×C₈

D₄×C₁₃₇

yes

characteristic

C₄×C₈

Q₈×C₁₃₇

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₂²×C₄×C₁₃₇

yes

characteristic

C₂²×C₄

#41×C₁₃₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₁₃₇

yes

—

C₂×C₈

C₄⋊C₄×C₁₃₇

yes

—

C₄²

—

C₄²

C₂×D₄×C₁₃₇

yes

characteristic

C₄²

C₂×Q₈×C₁₃₇

yes

characteristic

C₄²

#41×C₁₃₇

yes

—

C₂³

#117×C₁₃₇

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂×C₄⋊C₄×C₁₃₇

yes

characteristic

104

C₂×C₄

#117×C₁₃₇

yes

characteristic

105

C₂×C₄

#124×C₁₃₇

yes

—

106

C₂×C₄

#125×C₁₃₇

yes

—

107

C₂×C₄

#131×C₁₃₇

yes

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₂²

—

113

C₂²

#382×C₁₃₇

yes

characteristic

114

C₂²

#433×C₁₃₇

yes

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

#424×C₁₃₇

yes

—

119

C₄

#433×C₁₃₇

yes

—

120

C₂

—

121

C₂

#1832×C₁₃₇

yes

characteristic

122

C₂

#1833×C₁₃₇

yes

—

	subgroup	G/H
center	C₂×C₄×C₁₃₇	#117
commutator	C₂²×C₄	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	#382
prime radical	C₂³×C₁₃₇	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ #376×C₁₃₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₃₇

#117

commutator

C₂²×C₄

C₂²×C₄×C₁₃₇

Frattini, cosocle

C₂×C₄²

C₂³×C₁₃₇

C₂²×C₁₃₇

C₂³×C₁₃₇

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₄×C₁₃₇ ◁ #376×C₁₃₇ ◁ G

N	H ≅ G/N	type
61. #8782	C₁₃₇	direct (N × H)
62. C₁₃₇	#8782	direct (N × H)

H ≅ G/N

type

61. #8782

C₁₃₇

direct (N × H)

62. C₁₃₇

#8782

direct (N × H)

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

subgroup	index
#1746×C₁₃₇	2
#1564×C₁₃₇	2
#1281×C₁₃₇	2
#1274×C₁₃₇	2
#8782	137

subgroup

index

#1746×C₁₃₇

#1564×C₁₃₇

#1281×C₁₃₇

#1274×C₁₃₇

#8782

137

Subgroups — C₁₃₇×#8782 #46858168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C137×#8782 #46858168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8782 #46858168