Group #17994751 — subgroups

Subgroups — #17994751

Order 4736.

280 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 133 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×#11924468	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924467	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₃₇	C₂	1	1	yes	maximal, characteristic
4	#14557402	C₂	1	2	yes	maximal
5	C₂×#106×C₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64436	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
8	C₂×#10596936	—	2	4	no	—
9	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
10	#11924468	—	2	2	no	—
11	#11924467	—	2	4	no	—
12	C₈×Dic₃₇	C₄	1	4	yes	complemented
13	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
15	#106×C₃₇	—	2	2	no	—
16	C₂²×C₄×C₃₇	—	2	2	no	—
17	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
18	C₂²×Dic₃₇	—	2	2	no	—
19	C₂²×Dic₃₇	D₄	1	2	yes	—
20	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₃₇	D₄	1	2	yes	—
22	C₂×C₈×C₃₇	Q₈	1	2	yes	—
23	C₄⋊C₄×C₃₇	—	2	2	no	—
24	C₄⋊C₄×C₃₇	—	2	2	no	—
25	C₄⋊Dic₃₇	—	2	2	no	—
26	C₄⋊Dic₃₇	—	2	2	no	—
27	#10596936	—	4	8	no	—
28	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
29	C₄×Dic₃₇	Q₈	1	2	yes	—
30	C₂×#64436	C₂×C₄	1	2	yes	—
31	C₂×#64436	D₄	1	2	yes	—
32	C₂×#64436	Q₈	1	2	yes	—
33	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₃₇	—	4	2	no	—
35	C₂×C₄×C₃₇	—	4	4	no	—
36	C₂×C₄×C₃₇	C₂²×C₄	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×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×Dic₃₇	—	4	2	no	—
43	C₂×Dic₃₇	—	4	4	no	—
44	C₂×Dic₃₇	C₄⋊C₄	1	4	yes	—
45	C₂×Dic₃₇	SD₁₆	1	8	yes	—
46	C₈×C₃₇	C₄⋊C₄	1	4	yes	complemented
47	#64436	C₄⋊C₄	1	4	yes	complemented
48	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₃₇	#127	1	1	yes	characteristic
50	C₂²×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₃₇	C₂×SD₁₆	1	4	yes	—
52	C₄×C₃₇	—	8	2	no	—
53	C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
56	C₄×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
57	Dic₃₇	—	8	2	no	—
58	Dic₃₇	#106	1	8	yes	complemented
59	#1844	—	37	1	no	maximal, Sylow
60	C₂×C₃₇	#384	1	1	yes	characteristic
61	C₂×C₃₇	C₂×#106	1	2	yes	—
62	C₂×C₃₇	#492	1	2	yes	—
63	C₂×C₃₇	#498	1	2	yes	—
64	C₂×#106	—	37	1	no	—
65	C₂×#106	—	37	2	no	—
66	C₂×#106	D₃₇	1	1	yes	characteristic
67	C₂×C₄×C₈	—	37	1	no	—
68	#384	—	37	2	no	—
69	C₃₇	#1844	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₈	—	37	1	no	—
71	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
72	C₂×C₄⋊C₄	—	37	2	no	—
73	C₂×C₄⋊C₄	—	74	4	no	—
74	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
75	C₂×C₄²	—	37	1	no	—
76	#106	—	2	2	no	—
77	#106	—	74	4	no	—
78	#106	—	74	2	no	—
79	C₄×C₈	—	37	4	no	—
80	C₂²×C₄	—	2	2	no	—
81	C₂²×C₄	—	37	2	no	—
82	C₂²×C₄	—	74	2	no	—
83	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
84	C₂×C₈	—	37	2	no	—
85	C₂×C₈	—	37	2	no	—
86	C₂×C₈	—	37	2	no	—
87	C₂×C₈	Dic₇₄	1	2	yes	—
88	C₂×C₈	C₄×D₃₇	1	2	yes	—
89	C₂×C₈	C₂²⋊D₃₇	1	2	yes	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	74	2	no	—
93	C₄⋊C₄	—	74	2	no	—
94	C₄⋊C₄	—	148	8	no	—
95	C₄²	—	37	2	no	—
96	C₄²	—	37	2	no	—
97	C₂³	D₄×D₃₇	1	1	yes	characteristic
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	4	4	no	—
100	C₂×C₄	—	37	4	no	—
101	C₂×C₄	—	37	8	no	—
102	C₂×C₄	—	148	2	no	—
103	C₂×C₄	—	148	4	no	—
104	C₂×C₄	C₂×Dic₇₄	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	Q₈×D₃₇	1	1	yes	characteristic
109	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
110	C₈	—	37	4	no	—
111	C₈	#10596936	1	4	yes	complemented
112	C₂²	C₄⋊C₄×D₃₇	1	1	yes	characteristic
113	C₂²	#11924532	1	1	yes	characteristic
114	C₂²	#11924535	1	1	yes	characteristic
115	C₂²	SD₁₆×D₃₇	1	4	yes	—
116	C₄	—	8	2	no	—
117	C₄	—	37	8	no	—
118	C₄	—	296	2	no	—
119	C₄	C₄⋊C₄×D₃₇	1	1	yes	characteristic
120	C₄	C₂×#10596936	1	1	yes	characteristic
121	C₄	#11924534	1	1	yes	characteristic
122	C₄	#11924535	1	1	yes	characteristic
123	C₂	#14557402	1	1	yes	characteristic
124	C₂	#106×D₃₇	1	2	yes	—
125	C₂	#14557904	1	2	yes	—
126	C₂	#14557930	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924468

C₂

yes

maximal, characteristic

C₂×#11924467

C₂

yes

maximal

C₂×C₈×Dic₃₇

C₂

yes

maximal, characteristic

#14557402

C₂

yes

maximal

C₂×#106×C₃₇

C₂

yes

maximal, characteristic

C₂²×#64436

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×#10596936

—

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

#11924468

—

#11924467

—

C₈×Dic₃₇

C₄

yes

complemented

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

#106×C₃₇

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂²×Dic₃₇

D₄

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

D₄

yes

—

C₂×C₈×C₃₇

Q₈

yes

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

#10596936

—

C₄×Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

Q₈

yes

—

C₂×#64436

C₂×C₄

yes

—

C₂×#64436

D₄

yes

—

C₂×#64436

Q₈

yes

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

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×Q₈

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₄⋊C₄

yes

—

C₂×Dic₃₇

SD₁₆

yes

—

C₈×C₃₇

C₄⋊C₄

yes

complemented

#64436

C₄⋊C₄

yes

complemented

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₇

#127

yes

characteristic

C₂²×C₃₇

C₄⋊Q₈

yes

characteristic

C₂²×C₃₇

C₂×SD₁₆

yes

—

C₄×C₃₇

—

C₄×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₇

C₄⋊Q₈

yes

characteristic

C₄×C₃₇

C₄⋊Q₈

yes

characteristic

Dic₃₇

—

Dic₃₇

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₃₇

#492

yes

—

C₂×C₃₇

#498

yes

—

C₂×#106

—

C₂×#106

—

C₂×#106

D₃₇

yes

characteristic

C₂×C₄×C₈

—

#384

—

C₃₇

#1844

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₇

yes

—

C₂×C₄²

—

#106

—

#106

—

#106

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₇₄

yes

—

C₂×C₈

C₄×D₃₇

yes

—

C₂×C₈

C₂²⋊D₃₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

148

—

C₄²

—

C₄²

—

C₂³

D₄×D₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

148

—

103

C₂×C₄

—

148

—

104

C₂×C₄

C₂×Dic₇₄

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

Q₈×D₃₇

yes

characteristic

109

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

110

C₈

—

111

C₈

#10596936

yes

complemented

112

C₂²

C₄⋊C₄×D₃₇

yes

characteristic

113

C₂²

#11924532

yes

characteristic

114

C₂²

#11924535

yes

characteristic

115

C₂²

SD₁₆×D₃₇

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

296

—

119

C₄

C₄⋊C₄×D₃₇

yes

characteristic

120

C₄

C₂×#10596936

yes

characteristic

121

C₄

#11924534

yes

characteristic

122

C₄

#11924535

yes

characteristic

123

C₂

#14557402

yes

characteristic

124

C₂

#106×D₃₇

yes

—

125

C₂

#14557904

yes

—

126

C₂

#14557930

yes

—

	subgroup	G/H
center	C₂³	D₄×D₃₇
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₂×#106×C₃₇	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₂×#106	D₃₇
hypocenter	C₃₇	#1844
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₃₇

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₂×#106×C₃₇

C₂

socle, prime radical

C₂³×C₃₇

C₂×D₄

cosocle

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
12. C₈×Dic₃₇	C₄	semidirect
46. C₈×C₃₇	C₄⋊C₄	semidirect
47. #64436	C₄⋊C₄	semidirect
58. Dic₃₇	#106	semidirect
69. C₃₇	#1844	semidirect
111. C₈	#10596936	semidirect

H ≅ G/N

type

12. C₈×Dic₃₇

C₄

semidirect

46. C₈×C₃₇

C₄⋊C₄

semidirect

47. #64436

C₄⋊C₄

semidirect

58. Dic₃₇

#106

semidirect

69. C₃₇

#1844

semidirect

111. C₈

#10596936

semidirect

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

subgroup	index
C₂×#11924468	2
C₂×#11924467	2
C₂×C₈×Dic₃₇	2
#14557402	2
C₂×#106×C₃₇	2
#1844	37

subgroup

index

C₂×C₈×Dic₃₇

C₂×#106×C₃₇

Subgroups — #17994751

Conjugacy classes of subgroups

Complements

Maximal subgroups