Group #18006400 — subgroups

Subgroups — #18006400

Order 4736.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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	#14558346	C₂	1	4	yes	maximal, complemented
2	#14557805	C₂	1	4	yes	maximal, complemented
3	#14557715	C₂	1	4	yes	maximal, complemented
4	#14557651	C₂	1	2	yes	maximal, complemented
5	C₃₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
6	#11924584	C₂²	1	4	yes	—
7	#11924568	—	2	8	no	—
8	#11924567	C₂²	1	8	yes	—
9	#11924559	—	2	4	no	—
10	#11924558	C₂²	1	2	yes	—
11	Q₁₆×D₃₇	—	2	4	no	—
12	#11924542	C₂²	1	4	yes	—
13	#11924472	C₂²	1	2	yes	—
14	#11924464	C₂²	1	4	yes	—
15	#11924455	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₃₇	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
18	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₃₇	C₂³	1	4	yes	—
20	#10596966	—	2	8	no	—
21	Q₈×D₃₇	—	2	8	no	—
22	#10596962	—	4	4	no	—
23	#10596962	C₂³	1	4	yes	—
24	Q₁₆×C₃₇	—	2	4	no	—
25	Q₁₆×C₃₇	—	2	4	no	—
26	C₂×C₈×C₃₇	C₂³	1	2	yes	—
27	C₄⋊C₄×C₃₇	—	4	2	no	—
28	C₄²×C₃₇	C₂³	1	1	yes	characteristic
29	#10596942	—	2	8	no	—
30	#10596941	—	2	8	no	—
31	#10596934	C₂³	1	4	yes	—
32	Dic₁₄₈	D₄	1	2	yes	—
33	D₂₉₆	D₄	1	2	yes	—
34	#10596930	D₄	1	4	yes	—
35	#10596929	—	2	4	no	—
36	C₈×D₃₇	—	2	4	no	—
37	Q₈×C₃₇	—	2	8	no	—
38	C₂×C₄×C₃₇	—	2	1	no	—
39	C₂×C₄×C₃₇	—	2	4	no	—
40	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
41	C₂²⋊D₃₇	—	2	4	no	—
42	D₁₄₈	—	4	4	no	—
43	D₁₄₈	C₂×D₄	1	4	yes	—
44	C₄×D₃₇	—	2	4	no	—
45	C₄×D₃₇	—	4	8	no	—
46	Dic₇₄	—	4	4	no	—
47	Dic₇₄	C₂×D₄	1	4	yes	—
48	C₈×C₃₇	C₂×D₄	1	2	yes	—
49	C₈×C₃₇	C₂×D₄	1	2	yes	—
50	#64436	—	2	4	no	—
51	C₂²×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₃₇	—	2	4	no	—
53	C₄×C₃₇	—	2	2	no	—
54	C₄×C₃₇	—	4	4	no	—
55	C₄×C₃₇	C₂²×D₄	1	2	yes	—
56	Dic₃₇	—	2	4	no	—
57	#3293	—	37	1	no	maximal, Sylow
58	C₂×C₃₇	—	2	1	no	—
59	C₂×C₃₇	D₄²	1	1	yes	characteristic
60	D₃₇	—	4	4	no	—
61	#578	—	37	4	no	—
62	#494	D₃₇	1	1	yes	characteristic, complemented
63	#473	—	37	4	no	—
64	#456	—	37	4	no	—
65	#443	—	37	2	no	—
66	C₃₇	#3293	1	1	yes	characteristic, complemented, Sylow
67	2₋¹⁺⁴	—	37	4	no	—
68	#137	—	37	8	no	—
69	#137	—	74	8	no	—
70	#135	—	37	2	no	—
71	#135	—	74	4	no	—
72	C₂×Q₁₆	—	74	4	no	—
73	C₂×Q₁₆	C₂×D₃₇	1	4	yes	—
74	#131	—	37	4	no	—
75	C₄⋊Q₈	C₂×D₃₇	1	2	yes	—
76	#108	—	37	2	no	—
77	#104	—	37	4	no	—
78	#101	—	37	4	no	—
79	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
80	#41	—	37	4	no	—
81	#41	—	74	8	no	—
82	#41	—	148	4	no	—
83	C₂×Q₈	—	74	8	no	—
84	C₂×Q₈	C₂²×D₃₇	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	37	2	no	—
88	Q₁₆	—	74	8	no	—
89	SD₁₆	—	37	4	no	—
90	SD₁₆	—	74	8	no	—
91	D₈	—	37	2	no	—
92	M₁₆	—	37	4	no	—
93	M₁₆	—	74	4	no	—
94	C₂×C₈	—	74	4	no	—
95	C₂×C₈	C₂²×D₃₇	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₃₇	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	37	4	no	—
100	Q₈	—	148	4	no	—
101	D₄	—	37	4	no	—
102	D₄	—	74	4	no	—
103	D₄	—	148	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	74	4	no	—
107	C₂×C₄	—	148	8	no	—
108	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
109	C₈	—	74	4	no	—
110	C₈	D₄×D₃₇	1	2	yes	—
111	C₈	D₄×D₃₇	1	2	yes	—
112	C₂²	—	74	4	no	—
113	C₂²	#11924581	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	74	4	no	—
117	C₄	C₂×D₄×D₃₇	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	148	4	no	—
120	C₂	#14558168	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558346

C₂

yes

maximal, complemented

#14557805

C₂

yes

maximal, complemented

#14557715

C₂

yes

maximal, complemented

#14557651

C₂

yes

maximal, complemented

C₃₇×#494

C₂

yes

maximal, characteristic, complemented

#11924584

C₂²

yes

—

#11924568

—

#11924567

C₂²

yes

—

#11924559

—

#11924558

C₂²

yes

—

Q₁₆×D₃₇

—

#11924542

C₂²

yes

—

#11924472

C₂²

yes

—

#11924464

C₂²

yes

—

#11924455

C₂²

yes

—

C₂×Q₁₆×C₃₇

C₂²

yes

—

C₄⋊Q₈×C₃₇

C₂²

yes

—

C₄×C₈×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

C₂³

yes

—

#10596966

—

Q₈×D₃₇

—

#10596962

—

#10596962

C₂³

yes

—

Q₁₆×C₃₇

—

Q₁₆×C₃₇

—

C₂×C₈×C₃₇

C₂³

yes

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

C₂³

yes

—

—

C₂³

yes

—

Dic₁₄₈

D₄

yes

—

D₂₉₆

D₄

yes

—

#10596930

D₄

yes

—

#10596929

—

C₈×D₃₇

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂²⋊D₃₇

—

D₁₄₈

—

D₁₄₈

C₂×D₄

yes

—

C₄×D₃₇

—

C₄×D₃₇

—

Dic₇₄

—

Dic₇₄

C₂×D₄

yes

—

C₈×C₃₇

C₂×D₄

yes

—

C₈×C₃₇

C₂×D₄

yes

—

#64436

—

C₂²×C₃₇

H(5,𝔽₂)

yes

characteristic

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂²×D₄

yes

—

Dic₃₇

—

#3293

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

D₄²

yes

characteristic

D₃₇

—

#578

—

#494

D₃₇

yes

characteristic, complemented

#473

—

#456

—

#443

—

C₃₇

#3293

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

—

—

—

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₃₇

yes

—

#131

—

C₄⋊Q₈

C₂×D₃₇

yes

—

#108

—

#104

—

#101

—

C₄×C₈

C₂×D₃₇

yes

—

—

—

148

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₃₇

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²×D₃₇

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₃₇

yes

characteristic

Q₈

—

Q₈

—

100

Q₈

—

148

—

101

D₄

—

102

D₄

—

103

D₄

—

148

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

148

—

108

C₂×C₄

C₂³×D₃₇

yes

characteristic

109

C₈

—

110

C₈

D₄×D₃₇

yes

—

111

C₈

D₄×D₃₇

yes

—

112

C₂²

—

113

C₂²

#11924581

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

C₂×D₄×D₃₇

yes

—

118

C₂

—

119

C₂

—

148

—

120

C₂

#14558168

yes

characteristic

	subgroup	G/H
center	C₂	#14558168
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#494	C₂
socle	C₂×C₃₇	D₄²
hypercenter	#494	D₃₇
hypocenter	C₃₇	#3293
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#14558168

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₃₇×#494

C₂

socle

C₂×C₃₇

D₄²

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #14558346	C₂	semidirect
2. #14557805	C₂	semidirect
3. #14557715	C₂	semidirect
4. #14557651	C₂	semidirect
5. C₃₇×#494	C₂	semidirect
62. #494	D₃₇	semidirect
66. C₃₇	#3293	semidirect

H ≅ G/N

type

1. #14558346

C₂

semidirect

2. #14557805

C₂

semidirect

3. #14557715

C₂

semidirect

4. #14557651

C₂

semidirect

5. C₃₇×#494

C₂

semidirect

62. #494

D₃₇

semidirect

66. C₃₇

#3293

semidirect

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

subgroup	index
#14558346	2
#14557805	2
#14557715	2
#14557651	2
C₃₇×#494	2
#3293	37

subgroup

index

C₃₇×#494

Subgroups — #18006400

Conjugacy classes of subgroups

Complements

Maximal subgroups