Group #17994762 — subgroups

Subgroups — #17994762

Order 4736.

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

normal subgroups only

This group has 256 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×#11924469	C₂	1	2	yes	maximal
2	C₂×#11924467	C₂	1	2	yes	maximal
3	C₂×#11924447	C₂	1	1	yes	maximal, characteristic
4	#14557399	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#384	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64436	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
9	#11924469	—	2	4	no	—
10	#11924467	—	2	4	no	—
11	#11924447	C₄	1	4	yes	complemented
12	#11924443	—	2	2	no	—
13	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
14	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₃₇	—	2	2	no	—
17	C₂²×C₄×C₃₇	D₄	1	2	yes	—
18	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
19	C₂²×Dic₃₇	—	2	1	no	—
20	C₂²×Dic₃₇	—	2	1	no	—
21	C₄⋊C₄×C₃₇	—	2	2	no	—
22	C₄⋊C₄×C₃₇	—	2	2	no	—
23	C₄⋊C₄×C₃₇	—	4	4	no	—
24	C₄²×C₃₇	C₂×C₄	1	2	yes	—
25	C₄²×C₃₇	Q₈	1	2	yes	—
26	C₄⋊Dic₃₇	—	2	2	no	—
27	C₄⋊Dic₃₇	—	2	2	no	—
28	C₂×#64436	C₂×C₄	1	4	yes	—
29	C₂×#64436	D₄	1	4	yes	—
30	C₂×#64436	Q₈	1	4	yes	—
31	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₃₇	—	2	2	no	—
33	C₂×C₄×C₃₇	—	2	2	no	—
34	C₂×C₄×C₃₇	—	4	2	no	—
35	C₂×C₄×C₃₇	—	4	4	no	—
36	C₂×C₄×C₃₇	C₄⋊C₄	1	4	yes	—
37	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
41	C₂×Dic₃₇	—	4	1	no	—
42	C₂×Dic₃₇	—	4	1	no	—
43	C₂×Dic₃₇	—	4	2	no	—
44	C₂×Dic₃₇	—	4	2	no	—
45	#64436	C₄⋊C₄	1	8	yes	complemented
46	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₃₇	#127	1	1	yes	characteristic
48	C₂²×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
49	C₂²×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
50	C₂²×C₃₇	#137	1	2	yes	—
51	C₄×C₃₇	—	2	4	no	—
52	C₄×C₃₇	—	8	2	no	—
53	C₄×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
54	C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
55	Dic₃₇	—	8	1	no	—
56	Dic₃₇	—	8	1	no	—
57	#1847	—	37	1	no	maximal, Sylow
58	C₂×C₃₇	#384	1	1	yes	characteristic
59	C₂×C₃₇	#428	1	2	yes	—
60	C₂×C₃₇	#496	1	1	yes	characteristic
61	C₂×C₃₇	#497	1	1	yes	characteristic
62	C₂×C₃₇	#501	1	2	yes	—
63	C₂×#107	—	37	2	no	—
64	C₂×#106	—	37	2	no	—
65	C₂×#97	—	37	1	no	—
66	#384	D₃₇	1	1	yes	characteristic
67	#382	—	37	1	no	—
68	C₃₇	#1847	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₈	—	37	2	no	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	37	1	no	—
72	C₂×C₄⋊C₄	—	37	1	no	—
73	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
74	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
75	#107	—	74	4	no	—
76	#106	—	74	4	no	—
77	#97	—	37	4	no	—
78	#95	—	74	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	74	1	no	—
81	C₂²×C₄	—	74	1	no	—
82	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
83	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
84	C₂×C₈	—	37	4	no	—
85	C₂×C₈	—	37	4	no	—
86	C₂×C₈	—	37	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	4	4	no	—
90	C₄⋊C₄	—	74	2	no	—
91	C₄⋊C₄	—	74	2	no	—
92	C₄²	Dic₇₄	1	2	yes	—
93	C₄²	C₄×D₃₇	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	148	1	no	—
100	C₂×C₄	—	148	1	no	—
101	C₂×C₄	—	148	2	no	—
102	C₂×C₄	—	148	2	no	—
103	C₂×C₄	#10596936	1	4	yes	—
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	2	yes	—
107	C₂×C₄	Q₈×D₃₇	1	2	yes	—
108	C₈	—	37	8	no	—
109	C₂²	C₂×#10596936	1	1	yes	characteristic
110	C₂²	#11924532	1	1	yes	characteristic
111	C₂²	#11924534	1	1	yes	characteristic
112	C₂²	#11924562	1	2	yes	—
113	C₂²	#11924567	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	8	2	no	—
116	C₄	—	296	1	no	—
117	C₄	—	296	1	no	—
118	C₄	C₄⋊C₄×D₃₇	1	2	yes	—
119	C₄	#11924535	1	2	yes	—
120	C₂	#14557402	1	1	yes	characteristic
121	C₂	#14557573	1	2	yes	—
122	C₂	#14557920	1	1	yes	characteristic
123	C₂	#14557925	1	1	yes	characteristic
124	C₂	#14557939	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924469

C₂

yes

maximal

C₂×#11924467

C₂

yes

maximal

C₂×#11924447

C₂

yes

maximal, characteristic

#14557399

C₂

yes

maximal, characteristic

C₃₇×#384

C₂

yes

maximal, characteristic

C₂²×#64436

C₂²

yes

—

C₂×C₄⋊Dic₃₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₇

C₂²

yes

—

—

C₄

yes

complemented

#11924443

—

C₂×C₄⋊C₄×C₃₇

—

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂²×Dic₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

Q₈

yes

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

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

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

#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

AGL(1,ℤ/8)

yes

—

C₂²×C₃₇

#137

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×C₄⋊C₄

yes

—

C₄×C₃₇

C₄⋊Q₈

yes

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×#107

—

C₂×#106

—

C₂×#97

—

#384

D₃₇

yes

characteristic

#382

—

C₃₇

#1847

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₇

yes

—

C₂×C₄²

C₂×D₃₇

yes

—

—

—

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Dic₇₄

yes

—

C₄²

C₄×D₃₇

yes

—

C₂³

C₂×C₂²⋊D₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

148

—

100

C₂×C₄

—

148

—

101

C₂×C₄

—

148

—

102

C₂×C₄

—

148

—

103

C₂×C₄

#10596936

yes

—

104

C₂×C₄

C₂×Dic₇₄

yes

characteristic

105

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

106

C₂×C₄

D₄×D₃₇

yes

—

107

C₂×C₄

Q₈×D₃₇

yes

—

108

C₈

—

109

C₂²

C₂×#10596936

yes

characteristic

110

C₂²

#11924532

yes

characteristic

111

C₂²

#11924534

yes

characteristic

112

C₂²

#11924562

yes

—

113

C₂²

#11924567

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

296

—

117

C₄

—

296

—

118

C₄

C₄⋊C₄×D₃₇

yes

—

119

C₄

#11924535

yes

—

120

C₂

#14557402

yes

characteristic

121

C₂

#14557573

yes

—

122

C₂

#14557920

yes

characteristic

123

C₂

#14557925

yes

characteristic

124

C₂

#14557939

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₇
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#384	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#384	D₃₇
hypocenter	C₃₇	#1847
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₇

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#384

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 ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
11. #11924447	C₄	semidirect
45. #64436	C₄⋊C₄	semidirect
68. C₃₇	#1847	semidirect

H ≅ G/N

type

11. #11924447

C₄

semidirect

45. #64436

C₄⋊C₄

semidirect

68. C₃₇

#1847

semidirect

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

subgroup	index
C₂×#11924469	2
C₂×#11924467	2
C₂×#11924447	2
#14557399	2
C₃₇×#384	2
#1847	37

subgroup

index

C₃₇×#384

Subgroups — #17994762

Conjugacy classes of subgroups

Complements

Maximal subgroups