Group #17994766 — subgroups

Subgroups — #17994766

Order 4736.

220 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 105 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×#11924472	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924471	C₂	1	2	yes	maximal
3	#14557556	C₂	1	2	yes	maximal
4	C₂×C₈×Dic₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#108×C₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596934	C₂²	1	2	yes	—
7	C₂²×#64436	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
9	#11924472	—	2	2	no	—
10	#11924471	—	2	4	no	—
11	#11924465	—	2	4	no	—
12	C₈×Dic₃₇	C₄	1	2	yes	—
13	C₈×Dic₃₇	C₄	1	2	yes	—
14	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
15	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
16	#108×C₃₇	—	2	2	no	—
17	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
18	C₂²×Dic₃₇	Q₈	1	2	yes	—
19	M₁₆×C₃₇	—	2	2	no	—
20	M₁₆×C₃₇	—	2	2	no	—
21	C₂×C₈×C₃₇	—	2	2	no	—
22	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₃₇	D₄	1	2	yes	—
24	C₂×C₈×C₃₇	Q₈	1	2	yes	—
25	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₃₇	D₄	1	2	yes	—
28	#10596934	—	2	2	no	—
29	#10596934	—	2	2	no	—
30	C₂×#64436	—	2	2	no	—
31	C₂×#64436	C₂×C₄	1	2	yes	—
32	C₂×#64436	D₄	1	2	yes	—
33	C₂×#64436	Q₈	1	2	yes	—
34	C₂³×C₃₇	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×Dic₃₇	—	2	4	no	—
42	C₂×Dic₃₇	C₄⋊C₄	1	2	yes	—
43	C₂×Dic₃₇	C₄⋊C₄	1	2	yes	—
44	C₈×C₃₇	—	4	2	no	—
45	C₈×C₃₇	C₄⋊C₄	1	4	yes	—
46	#64436	—	4	2	no	—
47	#64436	C₄⋊C₄	1	4	yes	—
48	C₂²×C₃₇	—	2	2	no	—
49	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
52	C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₃₇	#127	1	1	yes	characteristic
55	C₄×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₃₇	—	2	2	no	—
57	Dic₃₇	#108	1	4	yes	complemented
58	#1848	—	37	1	no	maximal, Sylow
59	C₂×C₃₇	—	2	1	no	—
60	C₂×C₃₇	—	2	1	no	—
61	C₂×C₃₇	#384	1	1	yes	characteristic
62	C₂×C₃₇	C₂×#108	1	2	yes	—
63	C₂×#108	—	37	2	no	—
64	C₂×#108	—	37	1	no	—
65	C₂×#108	D₃₇	1	1	yes	characteristic
66	#423	—	37	2	no	—
67	C₂×C₄×C₈	—	37	1	no	—
68	C₃₇	#1848	1	1	yes	characteristic, complemented, Sylow
69	C₂×M₁₆	—	37	2	no	—
70	C₂×M₁₆	C₂×D₃₇	1	2	yes	—
71	C₂²×C₈	—	37	1	no	—
72	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
73	C₂×C₄²	—	37	1	no	—
74	#108	—	2	2	no	—
75	#108	—	74	4	no	—
76	#108	—	74	2	no	—
77	C₄⋊C₈	—	74	4	no	—
78	C₄×C₈	—	37	2	no	—
79	C₄×C₈	—	37	2	no	—
80	C₂²×C₄	—	37	2	no	—
81	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
82	M₁₆	—	2	2	no	—
83	M₁₆	—	2	2	no	—
84	M₁₆	—	74	2	no	—
85	M₁₆	—	74	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	37	2	no	—
88	C₂×C₈	—	37	2	no	—
89	C₂×C₈	—	37	2	no	—
90	C₂×C₈	—	74	2	no	—
91	C₂×C₈	Dic₇₄	1	2	yes	—
92	C₂×C₈	C₄×D₃₇	1	2	yes	—
93	C₂×C₈	C₂²⋊D₃₇	1	2	yes	—
94	C₄²	—	37	1	no	—
95	C₄²	—	37	1	no	—
96	C₄²	—	37	2	no	—
97	C₂³	Q₈×D₃₇	1	1	yes	characteristic
98	C₂×C₄	—	37	2	no	—
99	C₂×C₄	—	37	2	no	—
100	C₂×C₄	—	74	4	no	—
101	C₂×C₄	C₂×Dic₇₄	1	1	yes	characteristic
102	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
103	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
104	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
105	C₂×C₄	Q₈×D₃₇	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
107	C₈	—	4	2	no	—
108	C₈	—	37	4	no	—
109	C₈	—	148	2	no	—
110	C₈	#10596936	1	4	yes	—
111	C₂²	—	2	2	no	—
112	C₂²	C₄⋊C₄×D₃₇	1	1	yes	characteristic
113	C₂²	#11924534	1	1	yes	characteristic
114	C₂²	#11924535	1	1	yes	characteristic
115	C₄	—	37	4	no	—
116	C₄	—	74	2	no	—
117	C₄	C₄⋊C₄×D₃₇	1	1	yes	characteristic
118	C₄	C₂×#10596936	1	1	yes	characteristic
119	C₄	#11924532	1	1	yes	characteristic
120	C₄	#11924535	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	2	1	no	—
123	C₂	#14557402	1	1	yes	characteristic
124	C₂	#108×D₃₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924472

C₂

yes

maximal, characteristic

C₂×#11924471

C₂

yes

maximal

#14557556

C₂

yes

maximal

C₂×C₈×Dic₃₇

C₂

yes

maximal, characteristic

C₂×#108×C₃₇

C₂

yes

maximal, characteristic

C₂×#10596934

C₂²

yes

—

C₂²×#64436

C₂²

yes

characteristic

C₂×C₄×Dic₃₇

C₂²

yes

—

—

—

C₈×Dic₃₇

C₄

yes

—

C₈×Dic₃₇

C₄

yes

—

C₂×M₁₆×C₃₇

C₂²

yes

—

C₂²×C₈×C₃₇

C₂²

yes

characteristic

#108×C₃₇

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

Q₈

yes

—

M₁₆×C₃₇

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

D₄

yes

—

C₂×C₈×C₃₇

Q₈

yes

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

D₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×#64436

D₄

yes

—

C₂×#64436

Q₈

yes

—

C₂³×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

C₄⋊C₄

yes

—

C₂×Dic₃₇

C₄⋊C₄

yes

—

C₈×C₃₇

—

C₈×C₃₇

C₄⋊C₄

yes

—

#64436

—

#64436

C₄⋊C₄

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₇

C₄⋊Q₈

yes

characteristic

C₂²×C₃₇

C₄⋊Q₈

yes

characteristic

C₄×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₇

#127

yes

characteristic

C₄×C₃₇

C₄⋊Q₈

yes

characteristic

Dic₃₇

—

Dic₃₇

yes

—

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

#384

yes

characteristic

C₂×C₃₇

C₂×#108

yes

—

C₂×#108

—

C₂×#108

—

C₂×#108

D₃₇

yes

characteristic

#423

—

C₂×C₄×C₈

—

C₃₇

#1848

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₇

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₂×C₄²

—

#108

—

#108

—

#108

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₇₄

yes

—

C₂×C₈

C₄×D₃₇

yes

—

C₂×C₈

C₂²⋊D₃₇

yes

—

C₄²

—

C₄²

—

C₄²

—

C₂³

Q₈×D₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂×Dic₇₄

yes

characteristic

102

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

103

C₂×C₄

D₄×D₃₇

yes

characteristic

104

C₂×C₄

D₄×D₃₇

yes

characteristic

105

C₂×C₄

Q₈×D₃₇

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

148

—

110

C₈

#10596936

yes

—

111

C₂²

—

112

C₂²

C₄⋊C₄×D₃₇

yes

characteristic

113

C₂²

#11924534

yes

characteristic

114

C₂²

#11924535

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

C₄⋊C₄×D₃₇

yes

characteristic

118

C₄

C₂×#10596936

yes

characteristic

119

C₄

#11924532

yes

characteristic

120

C₄

#11924535

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

#14557402

yes

characteristic

124

C₂

#108×D₃₇

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₃₇
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₂×#108×C₃₇	C₂
socle	C₂²×C₃₇	C₂×C₄⋊C₄
prime radical	C₂³×C₃₇	C₂×Q₈
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₂×#108	D₃₇
hypocenter	C₃₇	#1848
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₃₇

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₂×#108×C₃₇

C₂

socle

C₂²×C₃₇

C₂×C₄⋊C₄

prime radical

C₂³×C₃₇

C₂×Q₈

cosocle

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
57. Dic₃₇	#108	semidirect
68. C₃₇	#1848	semidirect

H ≅ G/N

type

57. Dic₃₇

#108

semidirect

68. C₃₇

#1848

semidirect

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

subgroup	index
C₂×#11924472	2
C₂×#11924471	2
#14557556	2
C₂×C₈×Dic₃₇	2
C₂×#108×C₃₇	2
#1848	37

subgroup

index

C₂×#11924472

C₂×#11924471

#14557556

C₂×C₈×Dic₃₇

C₂×#108×C₃₇

#1848

Subgroups — #17994766

Conjugacy classes of subgroups

Complements

Maximal subgroups