Group #17996150 — subgroups

Subgroups — #17996150

Order 4736.

210 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 67 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	#14557582	C₂	1	1	yes	maximal, characteristic, complemented
2	#14557560	C₂	1	2	yes	maximal, complemented
3	#14557551	C₂	1	2	yes	maximal
4	#14557501	C₂	1	1	yes	maximal, characteristic
5	C₃₇×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10596962	C₂²	1	1	yes	characteristic
7	C₂×#10596934	C₂²	1	2	yes	—
8	C₂²×#64436	C₂²	1	1	yes	characteristic
9	#11924472	—	2	2	no	—
10	C₄⋊#64436	—	2	4	no	—
11	#11924464	—	2	4	no	—
12	#11924448	—	2	2	no	—
13	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
14	#117×C₃₇	C₂²	1	2	yes	—
15	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
16	C₂×C₂²⋊D₃₇	—	2	1	no	—
17	#10596962	—	2	2	no	—
18	#10596962	—	2	2	no	—
19	C₂×D₁₄₈	Q₈	1	1	yes	characteristic
20	C₂×C₄×D₃₇	—	2	1	no	—
21	C₂×Dic₇₄	Q₈	1	1	yes	characteristic
22	M₁₆×C₃₇	—	2	2	no	—
23	C₂×C₈×C₃₇	D₄	1	2	yes	—
24	C₄⋊C₄×C₃₇	—	2	2	no	—
25	C₂²⋊C₄×C₃₇	—	2	2	no	—
26	C₄²×C₃₇	D₄	1	4	yes	—
27	#10596934	—	2	2	no	—
28	#10596934	—	2	2	no	—
29	C₂×#64436	—	2	1	no	—
30	C₂×#64436	—	2	2	no	—
31	C₂×#64436	—	2	2	no	—
32	C₂³×C₃₇	#41	1	1	yes	characteristic
33	C₂²×D₃₇	—	2	1	no	—
34	C₂×C₄×C₃₇	—	2	4	no	—
35	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₃₇	#41	1	2	yes	—
39	C₂²⋊D₃₇	—	4	2	no	—
40	C₂×Dic₃₇	—	2	1	no	—
41	D₁₄₈	—	2	2	no	—
42	C₄×D₃₇	—	4	1	no	—
43	C₄×D₃₇	—	4	1	no	—
44	Dic₇₄	—	2	2	no	—
45	C₈×C₃₇	—	2	2	no	—
46	#64436	—	4	1	no	—
47	#64436	—	4	2	no	—
48	C₂²×C₃₇	—	4	1	no	—
49	C₂²×C₃₇	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₃₇	#123	1	2	yes	—
51	C₂×D₃₇	—	4	2	no	—
52	C₂×D₃₇	—	4	1	no	—
53	C₄×C₃₇	—	4	4	no	—
54	C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₃₇	#123	1	1	yes	characteristic
56	C₄×C₃₇	#127	1	1	yes	characteristic
57	Dic₃₇	—	4	1	no	—
58	#2098	—	37	1	no	maximal, Sylow
59	C₂×C₃₇	—	2	2	no	—
60	C₂×C₃₇	—	2	1	no	—
61	C₂×C₃₇	#399	1	1	yes	characteristic
62	D₃₇	—	8	1	no	—
63	#430	—	37	1	no	—
64	#424	—	37	2	no	—
65	#421	—	37	2	no	—
66	#408	—	37	1	no	—
67	H(3,ℤ/4)	D₃₇	1	1	yes	characteristic, complemented
68	C₃₇	#2098	1	1	yes	characteristic, complemented, Sylow
69	C₂×#41	—	37	1	no	—
70	C₂×M₁₆	—	37	2	no	—
71	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
72	C₂²×C₈	—	37	1	no	—
73	#117	C₂×D₃₇	1	2	yes	—
74	#108	—	74	2	no	—
75	C₄⋊C₈	—	74	4	no	—
76	#104	—	74	4	no	—
77	C₂²⋊C₈	—	74	2	no	—
78	#41	—	74	2	no	—
79	#41	—	74	2	no	—
80	C₂×Q₈	—	37	1	no	—
81	C₂×D₄	—	37	1	no	—
82	C₂×D₄	—	74	1	no	—
83	C₂²×C₄	—	74	1	no	—
84	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	74	2	no	—
87	M₁₆	—	74	2	no	—
88	C₂×C₈	—	74	1	no	—
89	C₂×C₈	—	74	2	no	—
90	C₂×C₈	—	74	2	no	—
91	C₂×C₈	D₁₄₈	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₁₄₈	1	4	yes	—
95	C₂³	—	74	1	no	—
96	C₂³	#10596964	1	1	yes	characteristic
97	Q₈	—	74	2	no	—
98	D₄	—	74	2	no	—
99	D₄	—	148	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	74	1	no	—
102	C₂×C₄	—	148	1	no	—
103	C₂×C₄	—	148	1	no	—
104	C₂×C₄	C₂×D₁₄₈	1	2	yes	—
105	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
106	C₂×C₄	#10596964	1	2	yes	—
107	C₂×C₄	Q₈×D₃₇	1	1	yes	characteristic
108	C₈	—	2	2	no	—
109	C₈	—	148	1	no	—
110	C₈	—	148	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	148	2	no	—
113	C₂²	—	148	1	no	—
114	C₂²	#11924516	1	1	yes	characteristic
115	C₂²	#11924520	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	148	1	no	—
118	C₄	#11924516	1	2	yes	—
119	C₄	#11924520	1	1	yes	characteristic
120	C₄	#11924533	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	296	1	no	—
124	C₂	#14557470	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14557582

C₂

yes

maximal, characteristic, complemented

#14557560

C₂

yes

maximal, complemented

#14557551

C₂

yes

maximal

#14557501

C₂

yes

maximal, characteristic

C₃₇×H(3,ℤ/4)

C₂

yes

maximal, characteristic, complemented

C₂×#10596962

C₂²

yes

characteristic

C₂×#10596934

C₂²

yes

—

C₂²×#64436

C₂²

yes

—

—

—

—

C₂×M₁₆×C₃₇

C₂²

yes

characteristic

#117×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₂²⋊D₃₇

—

#10596962

—

#10596962

—

C₂×D₁₄₈

Q₈

yes

characteristic

C₂×C₄×D₃₇

—

C₂×Dic₇₄

Q₈

yes

characteristic

M₁₆×C₃₇

—

C₂×C₈×C₃₇

D₄

yes

—

C₄⋊C₄×C₃₇

—

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

D₄

yes

—

—

—

—

—

—

C₂³×C₃₇

#41

yes

characteristic

C₂²×D₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂²⋊D₃₇

—

C₂×Dic₃₇

—

D₁₄₈

—

C₄×D₃₇

—

C₄×D₃₇

—

Dic₇₄

—

C₈×C₃₇

—

#64436

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₃₇

#123

yes

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂²⋊Q₈

yes

—

C₄×C₃₇

yes

C₄×C₃₇

yes

Dic₃₇

—

#2098

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

#399

yes

characteristic

D₃₇

—

—

—

—

—

D₃₇

yes

characteristic, complemented

C₃₇

#2098

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₇

yes

characteristic

C₂²×C₈

—

#117

C₂×D₃₇

yes

—

#108

—

C₄⋊C₈

—

#104

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₄₈

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₁₄₈

yes

—

C₂³

—

C₂³

#10596964

yes

characteristic

Q₈

—

D₄

—

D₄

—

148

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

148

—

103

C₂×C₄

—

148

—

104

C₂×C₄

C₂×D₁₄₈

yes

—

105

C₂×C₄

C₂×D₁₄₈

yes

characteristic

106

C₂×C₄

#10596964

yes

—

107

C₂×C₄

Q₈×D₃₇

yes

characteristic

108

C₈

—

109

C₈

—

148

—

110

C₈

—

148

—

111

C₂²

—

112

C₂²

—

148

—

113

C₂²

—

148

—

114

C₂²

#11924516

yes

characteristic

115

C₂²

#11924520

yes

—

116

C₄

—

117

C₄

—

148

—

118

C₄

#11924516

yes

—

119

C₄

#11924520

yes

characteristic

120

C₄

#11924533

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

296

—

124

C₂

#14557470

yes

characteristic

	subgroup	G/H
center	C₄	#11924533
commutator, cosocle	C₂²×C₄×C₃₇	C₂³
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×H(3,ℤ/4)	C₂
socle	C₂×C₃₇	#399
prime radical	C₂×#10596962	C₂²
hypercenter	H(3,ℤ/4)	D₃₇
hypocenter	C₃₇	#2098
LCS	G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#11924533

commutator, cosocle

C₂²×C₄×C₃₇

C₂³

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×H(3,ℤ/4)

C₂

C₂×C₃₇

C₂²

D₃₇

C₃₇

G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #14557582	C₂	semidirect
2. #14557560	C₂	semidirect
5. C₃₇×H(3,ℤ/4)	C₂	semidirect
67. H(3,ℤ/4)	D₃₇	semidirect
68. C₃₇	#2098	semidirect

H ≅ G/N

type

1. #14557582

C₂

semidirect

2. #14557560

C₂

semidirect

5. C₃₇×H(3,ℤ/4)

C₂

semidirect

67. H(3,ℤ/4)

D₃₇

semidirect

68. C₃₇

#2098

semidirect

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

subgroup	index
#14557582	2
#14557560	2
#14557551	2
#14557501	2
C₃₇×H(3,ℤ/4)	2
#2098	37

subgroup

index

C₃₇×H(3,ℤ/4)

Subgroups — #17996150

Conjugacy classes of subgroups

Complements

Maximal subgroups