Group #18004936 — subgroups

Subgroups — C₂×#14557902 #18004936

Order 4736.

384 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 149 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×#11924536	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924528	C₂	1	1	yes	maximal, characteristic
3	#14557902	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11924461	C₂	1	4	yes	maximal
5	C₂×#97×C₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₇₄	C₂²	1	2	yes	—
7	#11924536	—	2	2	no	—
8	#11924536	C₂²	1	4	yes	—
9	#11924528	—	2	2	no	—
10	#11924528	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₃₇	—	2	2	no	—
12	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
13	C₂×#10596936	—	2	2	no	—
14	#11924461	C₂²	1	16	yes	—
15	C₂²×C₈×C₃₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
17	#97×C₃₇	C₂²	1	4	yes	—
18	C₂²×C₄×C₃₇	D₄	1	2	yes	—
19	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
20	C₂²×Dic₃₇	—	2	2	no	—
21	C₂²×Dic₃₇	—	2	2	no	—
22	C₂×Dic₇₄	—	2	8	no	—
23	C₂×Dic₇₄	—	2	2	no	—
24	C₂×Dic₇₄	C₂³	1	4	yes	—
25	C₂×C₈×C₃₇	—	2	4	no	—
26	C₂×C₈×C₃₇	C₂³	1	4	yes	—
27	C₄²×C₃₇	D₄	1	2	yes	—
28	C₄²×C₃₇	C₂³	1	2	yes	—
29	C₄⋊Dic₃₇	—	2	8	no	—
30	C₄⋊Dic₃₇	—	2	2	no	—
31	C₄⋊Dic₃₇	C₂³	1	4	yes	—
32	#10596936	—	2	8	no	—
33	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₃₇	—	2	4	no	—
35	C₂×C₄×C₃₇	C₂×D₄	1	4	yes	—
36	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₃₇	#41	1	4	yes	—
38	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
39	C₂×Dic₃₇	—	2	4	no	—
40	C₂×Dic₃₇	—	2	4	no	—
41	C₂×Dic₃₇	—	4	4	no	—
42	C₂×Dic₃₇	—	4	4	no	—
43	Dic₇₄	—	2	8	no	—
44	Dic₇₄	—	4	4	no	—
45	C₈×C₃₇	—	2	4	no	—
46	C₂²×C₃₇	#124	1	2	yes	—
47	C₂²×C₃₇	#137	1	4	yes	—
48	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
49	C₄×C₃₇	—	2	4	no	—
50	C₄×C₃₇	#124	1	2	yes	—
51	C₄×C₃₇	C₂×#41	1	2	yes	—
52	Dic₃₇	—	4	4	no	—
53	Dic₃₇	—	4	4	no	—
54	C₂×#491	—	37	1	no	maximal, Sylow
55	C₂×C₃₇	#491	1	4	yes	complemented
56	C₂×C₃₇	C₂×#124	1	1	yes	characteristic
57	C₂×C₃₇	C₂×#137	1	2	yes	—
58	C₂×C₄⋊Q₈	—	37	1	no	—
59	C₂×#125	—	37	1	no	—
60	#491	—	37	8	no	—
61	C₂×#103	—	37	4	no	—
62	C₂×#97	D₃₇	1	1	yes	characteristic
63	C₃₇	C₂×#491	1	1	yes	characteristic, complemented, Sylow
64	C₂²×Q₈	—	37	2	no	—
65	C₂²×C₈	C₂×D₃₇	1	2	yes	—
66	C₄⋊Q₈	—	37	4	no	—
67	C₄⋊Q₈	—	74	2	no	—
68	#125	—	37	4	no	—
69	#125	—	74	2	no	—
70	C₂×C₄⋊C₄	—	37	2	no	—
71	C₂×C₄⋊C₄	—	74	2	no	—
72	C₂×C₄⋊C₄	—	74	2	no	—
73	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
74	#103	—	37	16	no	—
75	#97	C₂×D₃₇	1	4	yes	—
76	C₂×Q₈	—	37	4	no	—
77	C₂×Q₈	—	74	2	no	—
78	C₂×Q₈	—	74	8	no	—
79	C₂²×C₄	—	74	2	no	—
80	C₂²×C₄	—	74	2	no	—
81	C₂²×C₄	D₁₄₈	1	2	yes	—
82	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
83	C₂×C₈	—	2	4	no	—
84	C₂×C₈	C₂²×D₃₇	1	4	yes	—
85	C₄⋊C₄	—	37	4	no	—
86	C₄⋊C₄	—	74	8	no	—
87	C₄⋊C₄	—	74	2	no	—
88	C₄⋊C₄	—	74	8	no	—
89	C₄²	D₁₄₈	1	2	yes	—
90	C₄²	C₂²×D₃₇	1	2	yes	—
91	C₂³	C₂×D₁₄₈	1	1	yes	characteristic
92	Q₈	—	74	8	no	—
93	Q₈	—	148	4	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	74	4	no	—
96	C₂×C₄	—	74	4	no	—
97	C₂×C₄	—	148	4	no	—
98	C₂×C₄	—	148	4	no	—
99	C₂×C₄	C₂×D₁₄₈	1	4	yes	—
100	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
101	C₂×C₄	#10596962	1	4	yes	—
102	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
103	C₈	—	2	4	no	—
104	C₂²	#11924524	1	2	yes	—
105	C₂²	#11924569	1	4	yes	—
106	C₂²	C₂²×D₁₄₈	1	1	yes	characteristic
107	C₄	—	2	4	no	—
108	C₄	—	148	4	no	—
109	C₄	—	148	4	no	—
110	C₄	#11924524	1	2	yes	—
111	C₄	C₂×#10596962	1	2	yes	—
112	C₂	#14557902	1	4	yes	direct product factor
113	C₂	C₂×#11924524	1	1	yes	characteristic
114	C₂	C₂×#11924569	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924536

C₂

yes

maximal, characteristic

C₂×#11924528

C₂

yes

maximal, characteristic

#14557902

C₂

yes

maximal, direct product factor

C₂×#11924461

C₂

yes

maximal

C₂×#97×C₃₇

C₂

yes

maximal, characteristic

C₂²×Dic₇₄

C₂²

yes

—

#11924536

—

#11924536

C₂²

yes

—

#11924528

—

#11924528

C₂²

yes

—

C₂×C₄⋊Dic₃₇

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×#10596936

—

#11924461

C₂²

yes

—

C₂²×C₈×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

#97×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂²×Dic₃₇

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

C₂³

yes

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

C₂³

yes

—

C₄²×C₃₇

D₄

yes

—

C₄²×C₃₇

C₂³

yes

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂³

yes

—

#10596936

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

Dic₇₄

—

Dic₇₄

—

C₈×C₃₇

—

C₂²×C₃₇

#124

yes

—

C₂²×C₃₇

#137

yes

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

#124

yes

—

C₄×C₃₇

C₂×#41

yes

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₄⋊Q₈

—

C₂×#125

—

#491

—

C₂×#103

—

C₂×#97

D₃₇

yes

characteristic

C₃₇

C₂×#491

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

C₂×D₃₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

#103

—

#97

C₂×D₃₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₄₈

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂²×D₃₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₁₄₈

yes

—

C₄²

C₂²×D₃₇

yes

—

C₂³

C₂×D₁₄₈

yes

characteristic

Q₈

—

Q₈

—

148

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

148

—

C₂×C₄

—

148

—

C₂×C₄

C₂×D₁₄₈

yes

—

100

C₂×C₄

C₂×D₁₄₈

yes

characteristic

101

C₂×C₄

#10596962

yes

—

102

C₂×C₄

C₂³×D₃₇

yes

characteristic

103

C₈

—

104

C₂²

#11924524

yes

—

105

C₂²

#11924569

yes

—

106

C₂²

C₂²×D₁₄₈

yes

characteristic

107

C₄

—

108

C₄

—

148

—

109

C₄

—

148

—

110

C₄

#11924524

yes

—

111

C₄

C₂×#10596962

yes

—

112

C₂

#14557902

yes

direct product factor

113

C₂

C₂×#11924524

yes

characteristic

114

C₂

C₂×#11924569

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₄₈
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₂×#97×C₃₇	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
hypercenter	C₂×#97	D₃₇
hypocenter	C₃₇	C₂×#491
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₁₄₈

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₂×#97×C₃₇

C₂

socle, prime radical

C₂³×C₃₇

C₂×D₄

hypercenter

C₂×#97

D₃₇

hypocenter

C₃₇

C₂×#491

LCS

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
3. #14557902	C₂	direct (N × H)
55. C₂×C₃₇	#491	semidirect
63. C₃₇	C₂×#491	semidirect
112. C₂	#14557902	direct (N × H)

H ≅ G/N

type

3. #14557902

C₂

direct (N × H)

55. C₂×C₃₇

#491

semidirect

63. C₃₇

C₂×#491

semidirect

112. C₂

#14557902

direct (N × H)

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

subgroup	index
C₂×#11924536	2
C₂×#11924528	2
#14557902	2
C₂×#11924461	2
C₂×#97×C₃₇	2
C₂×#491	37

subgroup

index

C₂×#97×C₃₇

Subgroups — C₂×#14557902 #18004936

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14557902 #18004936

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14557902 #18004936