Group #17995569 — subgroups

Subgroups — #17995569

Order 4736.

276 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 109 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×#11924536	C₂	1	1	yes	maximal, characteristic
2	#14557399	C₂	1	1	yes	maximal, characteristic
3	#14557296	C₂	1	4	yes	maximal
4	C₂×#97×C₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₇₄	—	2	1	no	—
6	#11924536	—	2	2	no	—
7	#11924536	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₃₇	—	2	1	no	—
9	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
11	#11924443	—	2	1	no	—
12	#11924443	—	2	1	no	—
13	C₂²×C₈×C₃₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
15	#97×C₃₇	—	2	2	no	—
16	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
19	C₂²×Dic₃₇	—	2	2	no	—
20	C₂²×Dic₃₇	—	2	2	no	—
21	C₂×Dic₇₄	—	2	4	no	—
22	C₂×Dic₇₄	—	4	4	no	—
23	C₂×C₈×C₃₇	—	2	2	no	—
24	C₂×C₈×C₃₇	—	2	2	no	—
25	C₂×C₈×C₃₇	Q₈	1	4	yes	complemented
26	C₄²×C₃₇	C₂×C₄	1	2	yes	—
27	C₄²×C₃₇	D₄	1	2	yes	—
28	C₄⋊Dic₃₇	—	2	4	no	—
29	C₄⋊Dic₃₇	—	2	2	no	—
30	C₄⋊Dic₃₇	—	2	4	no	—
31	C₄⋊Dic₃₇	C₂×C₄	1	4	yes	—
32	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₃₇	—	2	2	no	—
34	C₂×C₄×C₃₇	—	2	2	no	—
35	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₃₇	#41	1	2	yes	—
41	C₂×Dic₃₇	—	2	4	no	—
42	C₂×Dic₃₇	—	4	4	no	—
43	C₂×Dic₃₇	—	4	2	no	—
44	C₂×Dic₃₇	—	4	2	no	—
45	C₂×Dic₃₇	—	4	2	no	—
46	Dic₇₄	—	4	8	no	—
47	C₈×C₃₇	—	2	4	no	—
48	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₃₇	#124	1	1	yes	characteristic
50	C₂²×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
52	C₂²×C₃₇	#137	1	2	yes	—
53	C₄×C₃₇	—	2	2	no	—
54	C₄×C₃₇	—	2	2	no	—
55	C₄×C₃₇	C₄×Q₈	1	2	yes	—
56	C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
57	Dic₃₇	—	4	4	no	—
58	Dic₃₇	—	8	2	no	—
59	#1993	—	37	1	no	maximal, Sylow
60	C₂×C₃₇	#391	1	1	yes	characteristic
61	C₂×C₃₇	#418	1	1	yes	characteristic
62	C₂×C₃₇	#419	1	1	yes	characteristic
63	C₂×C₃₇	#489	1	2	yes	—
64	C₂×C₃₇	#501	1	2	yes	—
65	C₂×C₄⋊Q₈	—	37	1	no	—
66	C₂×#97	D₃₇	1	1	yes	characteristic
67	#382	—	37	1	no	—
68	#340	—	37	4	no	—
69	C₃₇	#1993	1	1	yes	characteristic, complemented, Sylow
70	C₂²×Q₈	—	74	1	no	—
71	C₂²×C₈	C₂×D₃₇	1	2	yes	—
72	C₄⋊Q₈	—	37	4	no	—
73	C₄⋊Q₈	—	74	2	no	—
74	C₂×C₄⋊C₄	—	37	2	no	—
75	C₂×C₄⋊C₄	—	37	2	no	—
76	C₂×C₄⋊C₄	—	74	1	no	—
77	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
78	#97	—	2	2	no	—
79	#95	—	74	1	no	—
80	#95	—	74	1	no	—
81	C₂×Q₈	—	74	4	no	—
82	C₂×Q₈	—	148	4	no	—
83	C₂²×C₄	—	74	2	no	—
84	C₂²×C₄	—	74	2	no	—
85	C₂²×C₄	D₁₄₈	1	1	yes	characteristic
86	C₂²×C₄	D₁₄₈	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	Dic₇₄	1	4	yes	complemented
91	C₄⋊C₄	—	37	4	no	—
92	C₄⋊C₄	—	74	4	no	—
93	C₄⋊C₄	—	74	2	no	—
94	C₄⋊C₄	—	74	4	no	—
95	C₄²	C₄×D₃₇	1	2	yes	—
96	C₄²	C₂²⋊D₃₇	1	2	yes	—
97	C₂³	C₂×D₁₄₈	1	1	yes	characteristic
98	Q₈	—	148	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	74	4	no	—
102	C₂×C₄	—	148	2	no	—
103	C₂×C₄	—	148	2	no	—
104	C₂×C₄	—	148	2	no	—
105	C₂×C₄	—	148	4	no	—
106	C₂×C₄	#10596938	1	2	yes	—
107	C₂×C₄	#10596938	1	2	yes	—
108	C₂×C₄	C₂×Dic₇₄	1	2	yes	—
109	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
110	C₂×C₄	#10596962	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
112	C₈	—	2	4	no	—
113	C₂²	C₂×#10596938	1	1	yes	characteristic
114	C₂²	#11924524	1	1	yes	characteristic
115	C₂²	#11924536	1	1	yes	characteristic
116	C₂²	#11924564	1	2	yes	—
117	C₂²	#11924569	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	296	2	no	—
121	C₄	—	148	4	no	—
122	C₄	C₄×Dic₇₄	1	2	yes	—
123	C₄	#11924515	1	2	yes	—
124	C₂	#14557439	1	1	yes	characteristic
125	C₂	#14557539	1	1	yes	characteristic
126	C₂	#14557543	1	1	yes	characteristic
127	C₂	#14557893	1	2	yes	—
128	C₂	#14557942	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924536

C₂

yes

maximal, characteristic

#14557399

C₂

yes

maximal, characteristic

#14557296

C₂

yes

maximal

C₂×#97×C₃₇

C₂

yes

maximal, characteristic

C₂²×Dic₇₄

—

#11924536

—

#11924536

C₄

yes

complemented

C₂×C₄⋊Dic₃₇

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

#11924443

—

#11924443

—

C₂²×C₈×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

#97×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

characteristic

C₂²×C₄×C₃₇

D₄

yes

characteristic

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂²×Dic₃₇

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

Q₈

yes

complemented

C₄²×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

D₄

yes

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

#41

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

Dic₇₄

—

C₈×C₃₇

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₇

#124

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

C₄×Q₈

yes

—

C₄×C₃₇

C₂²⋊Q₈

yes

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₃₇

#501

yes

—

C₂×C₄⋊Q₈

—

C₂×#97

D₃₇

yes

characteristic

#382

—

#340

—

C₃₇

#1993

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

C₂×D₃₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₇

yes

—

—

—

C₂×Q₈

—

C₂×Q₈

—

148

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₄₈

yes

characteristic

C₂²×C₄

D₁₄₈

yes

characteristic

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₇₄

yes

complemented

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₃₇

yes

—

C₄²

C₂²⋊D₃₇

yes

—

C₂³

C₂×D₁₄₈

yes

characteristic

Q₈

—

148

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

148

—

103

C₂×C₄

—

148

—

104

C₂×C₄

—

148

—

105

C₂×C₄

—

148

—

106

C₂×C₄

#10596938

yes

—

107

C₂×C₄

#10596938

yes

—

108

C₂×C₄

C₂×Dic₇₄

yes

—

109

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

110

C₂×C₄

#10596962

yes

—

111

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

112

C₈

—

113

C₂²

C₂×#10596938

yes

characteristic

114

C₂²

#11924524

yes

characteristic

115

C₂²

#11924536

yes

characteristic

116

C₂²

#11924564

yes

—

117

C₂²

#11924569

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

296

—

121

C₄

—

148

—

122

C₄

C₄×Dic₇₄

yes

—

123

C₄

#11924515

yes

—

124

C₂

#14557439

yes

characteristic

125

C₂

#14557539

yes

characteristic

126

C₂

#14557543

yes

characteristic

127

C₂

#14557893

yes

—

128

C₂

#14557942

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₄₈
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₂×#97×C₃₇	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₂×#97	D₃₇
hypocenter	C₃₇	#1993
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₁₄₈

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₂×#97×C₃₇

C₂

socle, prime radical

C₂³×C₃₇

C₂×D₄

cosocle

C₂²×C₄×C₃₇

C₂³

hypercenter

C₂×#97

D₃₇

hypocenter

C₃₇

#1993

LCS

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
7. #11924536	C₄	semidirect
25. C₂×C₈×C₃₇	Q₈	semidirect
69. C₃₇	#1993	semidirect
90. C₂×C₈	Dic₇₄	semidirect

H ≅ G/N

type

7. #11924536

C₄

semidirect

25. C₂×C₈×C₃₇

Q₈

semidirect

69. C₃₇

#1993

semidirect

90. C₂×C₈

Dic₇₄

semidirect

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

subgroup	index
C₂×#11924536	2
#14557399	2
#14557296	2
C₂×#97×C₃₇	2
#1993	37

subgroup

index

C₂×#97×C₃₇

Subgroups — #17995569

Conjugacy classes of subgroups

Complements

Maximal subgroups