Group #17994477 — subgroups

Subgroups — #17994477

Order 4736.

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

normal subgroups only

This group has 232 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	C₂×#11924472	C₂	1	1	yes	maximal, characteristic
2	#14557489	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₃₇	C₂	1	1	yes	maximal, characteristic
4	#14557292	C₂	1	2	yes	maximal
5	C₃₇×#427	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	C₄	1	4	yes	complemented
10	#11924447	—	2	2	no	—
11	C₄×#64436	—	2	2	no	—
12	C₈×Dic₃₇	C₄	1	4	yes	complemented
13	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
14	#117×C₃₇	C₂²	1	2	yes	—
15	#107×C₃₇	C₄	1	2	yes	complemented
16	#106×C₃₇	C₄	1	2	yes	complemented
17	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
18	C₂²×Dic₃₇	—	2	1	no	—
19	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
24	C₂²⋊C₄×C₃₇	—	2	2	no	—
25	C₄²×C₃₇	—	2	2	no	—
26	C₄×Dic₃₇	—	2	1	no	—
27	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
28	#10596934	—	2	2	no	—
29	#10596934	C₂×C₄	1	4	yes	—
30	C₂×#64436	—	2	2	no	—
31	C₂×#64436	C₂×C₄	1	2	yes	—
32	C₂×#64436	D₄	1	1	yes	characteristic
33	C₂×#64436	D₄	1	1	yes	characteristic
34	C₂×#64436	Q₈	1	2	yes	—
35	C₂³×C₃₇	#41	1	1	yes	characteristic
36	C₂×C₄×C₃₇	—	2	4	no	—
37	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
43	C₂×Dic₃₇	—	2	4	no	—
44	C₂×Dic₃₇	—	2	2	no	—
45	C₈×C₃₇	C₄²	1	2	yes	—
46	C₈×C₃₇	C₄²	1	2	yes	—
47	#64436	—	2	4	no	—
48	#64436	C₄⋊C₄	1	2	yes	—
49	#64436	C₄⋊C₄	1	2	yes	—
50	C₂²×C₃₇	—	2	2	no	—
51	C₂²×C₃₇	#117	1	1	yes	characteristic
52	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₃₇	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₃₇	—	4	4	no	—
55	C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
56	C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
58	C₄×C₃₇	C₄×Q₈	1	1	yes	characteristic
59	Dic₃₇	—	2	4	no	—
60	#1772	—	37	1	no	maximal, Sylow
61	C₂×C₃₇	—	2	1	no	—
62	C₂×C₃₇	—	2	1	no	—
63	C₂×C₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₃₇	#443	1	2	yes	—
65	C₂×#108	—	37	1	no	—
66	#427	D₃₇	1	1	yes	characteristic
67	#405	—	37	2	no	—
68	C₂×C₄×C₈	—	37	1	no	—
69	#339	—	37	2	no	—
70	C₃₇	#1772	1	1	yes	characteristic, complemented, Sylow
71	C₂×M₁₆	—	37	2	no	—
72	C₂²×C₈	—	37	1	no	—
73	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
74	#117	C₂×D₃₇	1	2	yes	—
75	C₂×C₄²	—	37	1	no	—
76	#108	—	37	4	no	—
77	#107	Dic₃₇	1	2	yes	complemented
78	#106	Dic₃₇	1	2	yes	complemented
79	#97	—	74	2	no	—
80	C₄×C₈	—	37	4	no	—
81	C₄×C₈	—	74	2	no	—
82	C₂²×C₄	—	74	1	no	—
83	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
84	M₁₆	—	37	4	no	—
85	M₁₆	—	74	2	no	—
86	C₂×C₈	—	37	2	no	—
87	C₂×C₈	—	37	1	no	—
88	C₂×C₈	—	37	1	no	—
89	C₂×C₈	—	37	2	no	—
90	C₂×C₈	—	74	2	no	—
91	C₂×C₈	C₄×D₃₇	1	2	yes	—
92	C₂×C₈	C₄×D₃₇	1	2	yes	—
93	C₂×C₈	C₂×Dic₃₇	1	1	yes	characteristic
94	C₂×C₈	C₂×Dic₃₇	1	1	yes	characteristic
95	C₄⋊C₄	C₂×Dic₃₇	1	4	yes	—
96	C₂²⋊C₄	—	2	2	no	—
97	C₄²	—	2	2	no	—
98	C₄²	—	37	2	no	—
99	C₄²	—	74	1	no	—
100	C₂³	#10596964	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	74	4	no	—
103	C₂×C₄	—	74	2	no	—
104	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
106	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
107	C₂×C₄	Q₈×D₃₇	1	1	yes	characteristic
108	C₂×C₄	#10596966	1	1	yes	characteristic
109	C₂×C₄	C₂²×Dic₃₇	1	1	yes	characteristic
110	C₈	—	37	2	no	—
111	C₈	—	37	2	no	—
112	C₈	—	74	4	no	—
113	C₈	C₄×Dic₃₇	1	2	yes	—
114	C₈	C₄×Dic₃₇	1	2	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	#11924494	1	1	yes	characteristic
117	C₂²	D₄×Dic₃₇	1	1	yes	characteristic
118	C₂²	#11924501	1	1	yes	characteristic
119	C₄	—	4	4	no	—
120	C₄	—	74	4	no	—
121	C₄	C₂×C₄×Dic₃₇	1	1	yes	characteristic
122	C₄	C₄⋊C₄×D₃₇	1	1	yes	characteristic
123	C₄	#11924498	1	1	yes	characteristic
124	C₄	Q₈×Dic₃₇	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	C₄⋊C₄×Dic₃₇	1	1	yes	characteristic
128	C₂	#14557650	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924472

C₂

yes

maximal, characteristic

#14557489

C₂

yes

maximal

C₂×C₈×Dic₃₇

C₂

yes

maximal, characteristic

#14557292

C₂

yes

maximal

C₃₇×#427

C₂

yes

maximal, characteristic

C₂×#10596934

C₂²

yes

—

C₂²×#64436

C₂²

yes

characteristic

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

#11924472

C₄

yes

complemented

#11924447

—

C₄×#64436

—

C₈×Dic₃₇

C₄

yes

complemented

C₂²×C₈×C₃₇

C₂²

yes

characteristic

#117×C₃₇

C₂²

yes

—

#107×C₃₇

C₄

yes

complemented

#106×C₃₇

C₄

yes

complemented

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

C₂×C₄

yes

—

#10596934

—

#10596934

C₂×C₄

yes

—

C₂×#64436

—

C₂×#64436

C₂×C₄

yes

—

C₂×#64436

D₄

yes

characteristic

C₂×#64436

D₄

yes

characteristic

C₂×#64436

Q₈

yes

—

C₂³×C₃₇

#41

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

#41

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₈×C₃₇

C₄²

yes

—

C₈×C₃₇

C₄²

yes

—

#64436

—

#64436

C₄⋊C₄

yes

—

#64436

C₄⋊C₄

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

#117

yes

characteristic

C₂²×C₃₇

C₄×D₄

yes

characteristic

C₂²×C₃₇

C₄×Q₈

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

C₂×C₄²

yes

characteristic

C₄×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₇

C₄×D₄

yes

characteristic

C₄×C₃₇

C₄×Q₈

yes

characteristic

Dic₃₇

—

#1772

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₃₇

#443

yes

—

C₂×#108

—

#427

D₃₇

yes

characteristic

#405

—

C₂×C₄×C₈

—

#339

—

C₃₇

#1772

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

characteristic

#117

C₂×D₃₇

yes

—

C₂×C₄²

—

#108

—

#107

Dic₃₇

yes

complemented

#106

Dic₃₇

yes

complemented

#97

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₃₇

yes

—

C₂×C₈

C₄×D₃₇

yes

—

C₂×C₈

C₂×Dic₃₇

yes

characteristic

C₂×C₈

C₂×Dic₃₇

yes

characteristic

C₄⋊C₄

C₂×Dic₃₇

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

100

C₂³

#10596964

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

105

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

106

C₂×C₄

D₄×D₃₇

yes

characteristic

107

C₂×C₄

Q₈×D₃₇

yes

characteristic

108

C₂×C₄

#10596966

yes

characteristic

109

C₂×C₄

C₂²×Dic₃₇

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

C₄×Dic₃₇

yes

—

114

C₈

C₄×Dic₃₇

yes

—

115

C₂²

—

116

C₂²

#11924494

yes

characteristic

117

C₂²

D₄×Dic₃₇

yes

characteristic

118

C₂²

#11924501

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

C₂×C₄×Dic₃₇

yes

characteristic

122

C₄

C₄⋊C₄×D₃₇

yes

characteristic

123

C₄

#11924498

yes

characteristic

124

C₄

Q₈×Dic₃₇

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

C₄⋊C₄×Dic₃₇

yes

characteristic

128

C₂

#14557650

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₃₇
commutator	C₄×C₃₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#427	C₂
socle	C₂²×C₃₇	C₄×D₄
prime radical	C₂³×C₃₇	#41
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#427	D₃₇
hypocenter	C₃₇	#1772
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₃₇

commutator

C₄×C₃₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#427

C₂

socle

C₂²×C₃₇

C₄×D₄

prime radical

C₂³×C₃₇

#41

cosocle

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
9. #11924472	C₄	semidirect
12. C₈×Dic₃₇	C₄	semidirect
15. #107×C₃₇	C₄	semidirect
16. #106×C₃₇	C₄	semidirect
70. C₃₇	#1772	semidirect
77. #107	Dic₃₇	semidirect
78. #106	Dic₃₇	semidirect

H ≅ G/N

type

9. #11924472

C₄

semidirect

12. C₈×Dic₃₇

C₄

semidirect

15. #107×C₃₇

C₄

semidirect

16. #106×C₃₇

C₄

semidirect

70. C₃₇

#1772

semidirect

77. #107

Dic₃₇

semidirect

78. #106

Dic₃₇

semidirect

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

subgroup	index
C₂×#11924472	2
#14557489	2
C₂×C₈×Dic₃₇	2
#14557292	2
C₃₇×#427	2
#1772	37

subgroup

index

C₂×C₈×Dic₃₇

C₃₇×#427

Subgroups — #17994477

Conjugacy classes of subgroups

Complements

Maximal subgroups