Group #46870176 — subgroups

Subgroups — C₂×#1843×C₁₃₇ #46870176

Order 35072.

1230 conjugacy classes of subgroups; only the 702 normal subgroups are shown.

normal subgroups only

This group has 1230 conjugacy classes of subgroups — too many to draw. Only its 702 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	C₂²×C₄⋊C₈×C₁₃₇	C₂	1	6	yes	maximal
2	C₂×#376×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1843×C₁₃₇	C₂	1	8	yes	maximal, direct product factor
4	C₂³×C₈×C₁₃₇	C₂²	1	4	yes	—
5	C₂²×C₄²×C₁₃₇	C₂²	1	3	yes	—
6	C₂×C₄⋊C₈×C₁₃₇	—	2	24	no	—
7	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	24	yes	—
8	#376×C₁₃₇	C₄	1	4	yes	—
9	#376×C₁₃₇	C₂²	1	4	yes	—
10	C₂×#95×C₁₃₇	C₄	1	4	yes	—
11	C₂³×C₄×C₁₃₇	C₂×C₄	1	6	yes	—
12	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
13	C₂²×C₈×C₁₃₇	—	2	24	no	—
14	C₂²×C₈×C₁₃₇	C₂³	1	8	yes	—
15	C₂×C₄²×C₁₃₇	C₂×C₄	1	6	yes	—
16	C₂×C₄²×C₁₃₇	D₄	1	12	yes	—
17	C₂×C₄²×C₁₃₇	Q₈	1	12	yes	—
18	C₂×C₄²×C₁₃₇	C₂³	1	6	yes	—
19	C₄⋊C₈×C₁₃₇	—	2	48	no	—
20	#95×C₁₃₇	C₂×C₄	1	16	yes	—
21	C₂⁴×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₃₇	—	2	24	no	—
23	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	24	yes	—
24	C₂²×C₄×C₁₃₇	C₂²×C₄	1	12	yes	—
25	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₃₇	C₂×D₄	1	6	yes	—
27	C₂²×C₄×C₁₃₇	C₂×Q₈	1	6	yes	—
28	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
29	C₂×C₈×C₁₃₇	—	2	24	no	—
30	C₂×C₈×C₁₃₇	—	4	16	no	—
31	C₄²×C₁₃₇	C₄⋊C₄	1	24	yes	—
32	C₄²×C₁₃₇	C₂×D₄	1	12	yes	—
33	C₄²×C₁₃₇	C₂×Q₈	1	12	yes	—
34	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	6	yes	—
35	C₂³×C₁₃₇	#131	1	8	yes	—
36	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₃₇	—	2	24	no	—
38	C₂×C₄×C₁₃₇	—	2	48	no	—
39	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	24	yes	—
40	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	6	yes	—
41	C₂×C₄×C₁₃₇	#127	1	4	yes	—
42	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	12	yes	—
43	C₂×C₄×C₁₃₇	C₂²×D₄	1	3	yes	—
44	C₂×C₄×C₁₃₇	C₂²×Q₈	1	3	yes	—
45	C₈×C₁₃₇	—	4	8	no	—
46	C₂²×C₁₃₇	#384	1	4	yes	—
47	C₂²×C₁₃₇	#424	1	24	yes	—
48	C₂²×C₁₃₇	C₂²×C₄⋊C₄	1	3	yes	—
49	C₂²×C₁₃₇	C₂×#131	1	4	yes	—
50	C₄×C₁₃₇	—	2	24	no	—
51	C₄×C₁₃₇	#384	1	4	yes	—
52	C₄×C₁₃₇	C₂×#127	1	1	yes	characteristic
53	C₄×C₁₃₇	C₂×C₄⋊Q₈	1	3	yes	—
54	C₂×C₁₃₇	#1843	1	8	yes	direct product factor
55	C₂×C₁₃₇	C₂×#384	1	1	yes	characteristic
56	C₂×C₁₃₇	C₂×#424	1	6	yes	—
57	C₂×#1843	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₁₃₇	C₂×#1843	1	1	yes	characteristic, direct product factor, Sylow
59	C₂²×C₄⋊C₈	C₂×C₁₃₇	1	6	yes	—
60	C₂×#376	C₂×C₁₃₇	1	1	yes	characteristic
61	#1843	C₂×C₁₃₇	1	8	yes	direct product factor
62	C₂³×C₈	C₂²×C₁₃₇	1	4	yes	—
63	C₂²×C₄²	C₂²×C₁₃₇	1	3	yes	—
64	C₂×C₄⋊C₈	—	2	24	no	—
65	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	24	yes	—
66	#376	C₄×C₁₃₇	1	4	yes	—
67	#376	C₂²×C₁₃₇	1	4	yes	—
68	C₂×#95	C₄×C₁₃₇	1	4	yes	—
69	C₂³×C₄	C₂×C₄×C₁₃₇	1	6	yes	—
70	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
71	C₂²×C₈	—	2	24	no	—
72	C₂²×C₈	C₂³×C₁₃₇	1	8	yes	—
73	C₂×C₄²	C₂×C₄×C₁₃₇	1	6	yes	—
74	C₂×C₄²	D₄×C₁₃₇	1	12	yes	—
75	C₂×C₄²	Q₈×C₁₃₇	1	12	yes	—
76	C₂×C₄²	C₂³×C₁₃₇	1	6	yes	—
77	C₄⋊C₈	—	2	48	no	—
78	#95	C₂×C₄×C₁₃₇	1	16	yes	—
79	C₂⁴	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
80	C₂²×C₄	—	2	24	no	—
81	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	24	yes	—
82	C₂²×C₄	C₂²×C₄×C₁₃₇	1	12	yes	—
83	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
84	C₂²×C₄	C₂×D₄×C₁₃₇	1	6	yes	—
85	C₂²×C₄	C₂×Q₈×C₁₃₇	1	6	yes	—
86	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
87	C₂×C₈	—	2	24	no	—
88	C₂×C₈	—	4	16	no	—
89	C₄²	C₄⋊C₄×C₁₃₇	1	24	yes	—
90	C₄²	C₂×D₄×C₁₃₇	1	12	yes	—
91	C₄²	C₂×Q₈×C₁₃₇	1	12	yes	—
92	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	6	yes	—
93	C₂³	#131×C₁₃₇	1	8	yes	—
94	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
95	C₂×C₄	—	2	48	no	—
96	C₂×C₄	—	2	24	no	—
97	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	24	yes	—
98	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	6	yes	—
99	C₂×C₄	#127×C₁₃₇	1	4	yes	—
100	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	12	yes	—
101	C₂×C₄	C₂²×D₄×C₁₃₇	1	3	yes	—
102	C₂×C₄	C₂²×Q₈×C₁₃₇	1	3	yes	—
103	C₈	—	4	8	no	—
104	C₂²	#384×C₁₃₇	1	4	yes	—
105	C₂²	#424×C₁₃₇	1	24	yes	—
106	C₂²	C₂²×C₄⋊C₄×C₁₃₇	1	3	yes	—
107	C₂²	C₂×#131×C₁₃₇	1	4	yes	—
108	C₄	—	2	24	no	—
109	C₄	#384×C₁₃₇	1	4	yes	—
110	C₄	C₂×#127×C₁₃₇	1	1	yes	characteristic
111	C₄	C₂×C₄⋊Q₈×C₁₃₇	1	3	yes	—
112	C₂	#1843×C₁₃₇	1	8	yes	direct product factor
113	C₂	C₂×#384×C₁₃₇	1	1	yes	characteristic
114	C₂	C₂×#424×C₁₃₇	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊C₈×C₁₃₇

C₂

yes

maximal

C₂×#376×C₁₃₇

C₂

yes

maximal, characteristic

#1843×C₁₃₇

C₂

yes

maximal, direct product factor

C₂³×C₈×C₁₃₇

C₂²

yes

—

C₂²×C₄²×C₁₃₇

C₂²

yes

—

C₂×C₄⋊C₈×C₁₃₇

—

C₂×C₄⋊C₈×C₁₃₇

C₂²

yes

—

#376×C₁₃₇

C₄

yes

—

#376×C₁₃₇

C₂²

yes

—

C₂×#95×C₁₃₇

C₄

yes

—

C₂³×C₄×C₁₃₇

C₂×C₄

yes

—

C₂³×C₄×C₁₃₇

C₂³

yes

characteristic

C₂²×C₈×C₁₃₇

—

C₂²×C₈×C₁₃₇

C₂³

yes

—

C₂×C₄²×C₁₃₇

C₂×C₄

yes

—

C₂×C₄²×C₁₃₇

D₄

yes

—

C₂×C₄²×C₁₃₇

Q₈

yes

—

C₂×C₄²×C₁₃₇

C₂³

yes

—

C₄⋊C₈×C₁₃₇

—

#95×C₁₃₇

C₂×C₄

yes

—

C₂⁴×C₁₃₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

C₄⋊C₄

yes

—

C₂²×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂²×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃₇

C₂×D₄

yes

—

C₂²×C₄×C₁₃₇

C₂×Q₈

yes

—

C₂²×C₄×C₁₃₇

C₂⁴

yes

characteristic

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₄²×C₁₃₇

C₄⋊C₄

yes

—

C₄²×C₁₃₇

C₂×D₄

yes

—

C₄²×C₁₃₇

C₂×Q₈

yes

—

C₂³×C₁₃₇

C₂×C₄⋊C₄

yes

—

C₂³×C₁₃₇

#131

yes

—

C₂³×C₁₃₇

C₂³×C₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₃₇

#127

yes

—

C₂×C₄×C₁₃₇

C₄⋊Q₈

yes

—

C₂×C₄×C₁₃₇

C₂²×D₄

yes

—

C₂×C₄×C₁₃₇

C₂²×Q₈

yes

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

#384

yes

—

C₂²×C₁₃₇

#424

yes

—

C₂²×C₁₃₇

C₂²×C₄⋊C₄

yes

—

C₂²×C₁₃₇

C₂×#131

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#384

yes

—

C₄×C₁₃₇

C₂×#127

yes

characteristic

C₄×C₁₃₇

C₂×C₄⋊Q₈

yes

—

C₂×C₁₃₇

#1843

yes

direct product factor

C₂×C₁₃₇

C₂×#384

yes

characteristic

C₂×C₁₃₇

C₂×#424

yes

—

C₂×#1843

C₁₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₇

C₂×#1843

yes

characteristic, direct product factor, Sylow

C₂²×C₄⋊C₈

C₂×C₁₃₇

yes

—

C₂×#376

C₂×C₁₃₇

yes

characteristic

#1843

C₂×C₁₃₇

yes

direct product factor

C₂³×C₈

C₂²×C₁₃₇

yes

—

C₂²×C₄²

C₂²×C₁₃₇

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²×C₁₃₇

yes

—

#376

C₄×C₁₃₇

yes

—

#376

C₂²×C₁₃₇

yes

—

C₂×#95

C₄×C₁₃₇

yes

—

C₂³×C₄

C₂×C₄×C₁₃₇

yes

—

C₂³×C₄

C₂³×C₁₃₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂³×C₁₃₇

yes

—

C₂×C₄²

C₂×C₄×C₁₃₇

yes

—

C₂×C₄²

D₄×C₁₃₇

yes

—

C₂×C₄²

Q₈×C₁₃₇

yes

—

C₂×C₄²

C₂³×C₁₃₇

yes

—

C₄⋊C₈

—

#95

C₂×C₄×C₁₃₇

yes

—

C₂⁴

C₂²×C₄×C₁₃₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄⋊C₄×C₁₃₇

yes

—

C₂²×C₄

C₂²×C₄×C₁₃₇

yes

—

C₂²×C₄

C₂²×C₄×C₁₃₇

yes

characteristic

C₂²×C₄

C₂×D₄×C₁₃₇

yes

—

C₂²×C₄

C₂×Q₈×C₁₃₇

yes

—

C₂²×C₄

C₂⁴×C₁₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄²

C₄⋊C₄×C₁₃₇

yes

—

C₄²

C₂×D₄×C₁₃₇

yes

—

C₄²

C₂×Q₈×C₁₃₇

yes

—

C₂³

C₂×C₄⋊C₄×C₁₃₇

yes

—

C₂³

#131×C₁₃₇

yes

—

C₂³

C₂³×C₄×C₁₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄×C₁₃₇

yes

—

C₂×C₄

C₂×C₄⋊C₄×C₁₃₇

yes

—

C₂×C₄

#127×C₁₃₇

yes

—

100

C₂×C₄

C₄⋊Q₈×C₁₃₇

yes

—

101

C₂×C₄

C₂²×D₄×C₁₃₇

yes

—

102

C₂×C₄

C₂²×Q₈×C₁₃₇

yes

—

103

C₈

—

104

C₂²

#384×C₁₃₇

yes

—

105

C₂²

#424×C₁₃₇

yes

—

106

C₂²

C₂²×C₄⋊C₄×C₁₃₇

yes

—

107

C₂²

C₂×#131×C₁₃₇

yes

—

108

C₄

—

109

C₄

#384×C₁₃₇

yes

—

110

C₄

C₂×#127×C₁₃₇

yes

characteristic

111

C₄

C₂×C₄⋊Q₈×C₁₃₇

yes

—

112

C₂

#1843×C₁₃₇

yes

direct product factor

113

C₂

C₂×#384×C₁₃₇

yes

characteristic

114

C₂

C₂×#424×C₁₃₇

yes

—

	subgroup	G/H
center	C₂³×C₄×C₁₃₇	C₂³
commutator	C₂³	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle, prime radical	C₂⁴×C₁₃₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₃₇ ◁ G

subgroup

G/H

center

C₂³×C₄×C₁₃₇

C₂³

commutator

C₂³

C₂³×C₄×C₁₃₇

Frattini, cosocle

C₂²×C₄

C₂⁴×C₁₃₇

socle, prime radical

C₂⁴×C₁₃₇

C₂²×C₄

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₄×C₁₃₇ ◁ G

N	H ≅ G/N	type
3. #1843×C₁₃₇	C₂	direct (N × H)
54. C₂×C₁₃₇	#1843	direct (N × H)
57. C₂×#1843	C₁₃₇	direct (N × H)
58. C₁₃₇	C₂×#1843	direct (N × H)
61. #1843	C₂×C₁₃₇	direct (N × H)
112. C₂	#1843×C₁₃₇	direct (N × H)

H ≅ G/N

type

3. #1843×C₁₃₇

C₂

direct (N × H)

54. C₂×C₁₃₇

#1843

direct (N × H)

57. C₂×#1843

C₁₃₇

direct (N × H)

58. C₁₃₇

C₂×#1843

direct (N × H)

61. #1843

C₂×C₁₃₇

direct (N × H)

112. C₂

#1843×C₁₃₇

direct (N × H)

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

subgroup	index
C₂²×C₄⋊C₈×C₁₃₇	2
C₂×#376×C₁₃₇	2
#1843×C₁₃₇	2
C₂×#1843	137

subgroup

index

C₂²×C₄⋊C₈×C₁₃₇

C₂×#376×C₁₃₇

#1843×C₁₃₇

C₂×#1843

137

Subgroups — C₂×#1843×C₁₃₇ #46870176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#1843×C137 #46870176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1843×C₁₃₇ #46870176