Group #17994725 — subgroups

Subgroups — #17994725

Order 4736.

216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 93 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×C₈×Dic₃₇	C₂	1	1	yes	maximal, characteristic
2	#14557398	C₂	1	2	yes	maximal
3	#14557296	C₂	1	1	yes	maximal, characteristic
4	#14557295	C₂	1	2	yes	maximal
5	C₃₇×#340	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64436	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
8	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
9	C₈×Dic₃₇	C₄	1	4	yes	complemented
10	#11924443	—	2	2	no	—
11	#11924443	—	2	2	no	—
12	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
14	C₂²×C₄×C₃₇	—	2	2	no	—
15	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
16	C₂²×Dic₃₇	—	2	2	no	—
17	C₂²×Dic₃₇	D₄	1	1	yes	characteristic
18	C₂²×Dic₃₇	D₄	1	1	yes	characteristic
19	C₂×C₈×C₃₇	—	2	2	no	—
20	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
21	C₄⋊C₄×C₃₇	—	2	2	no	—
22	C₄⋊C₄×C₃₇	—	2	2	no	—
23	C₄⋊Dic₃₇	—	2	2	no	—
24	C₄⋊Dic₃₇	—	2	2	no	—
25	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
26	C₄×Dic₃₇	Q₈	1	2	yes	—
27	C₂×#64436	—	2	2	no	—
28	C₂×#64436	C₂×C₄	1	2	yes	—
29	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₃₇	—	4	2	no	—
31	C₂×C₄×C₃₇	—	4	2	no	—
32	C₂×C₄×C₃₇	—	4	2	no	—
33	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₃₇	#41	1	2	yes	—
36	C₂×C₄×C₃₇	#41	1	2	yes	—
37	C₂×Dic₃₇	—	2	1	no	—
38	C₂×Dic₃₇	—	2	2	no	—
39	C₂×Dic₃₇	—	2	1	no	—
40	C₂×Dic₃₇	—	4	2	no	—
41	C₂×Dic₃₇	—	4	2	no	—
42	C₂×Dic₃₇	—	4	2	no	—
43	C₂×Dic₃₇	C₄⋊C₄	1	2	yes	—
44	C₂×Dic₃₇	C₄⋊C₄	1	2	yes	—
45	C₈×C₃₇	—	2	2	no	—
46	#64436	—	2	2	no	—
47	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₃₇	#124	1	2	yes	—
49	C₂²×C₃₇	#135	1	1	yes	characteristic
50	C₂²×C₃₇	#135	1	2	yes	—
51	C₂²×C₃₇	#135	1	1	yes	characteristic
52	C₄×C₃₇	—	8	2	no	—
53	C₄×C₃₇	#117	1	1	yes	characteristic
54	C₄×C₃₇	#117	1	1	yes	characteristic
55	C₄×C₃₇	#125	1	2	yes	—
56	Dic₃₇	—	2	2	no	—
57	Dic₃₇	—	2	2	no	—
58	Dic₃₇	—	8	2	no	—
59	#1835	—	37	1	no	maximal, Sylow
60	C₂×C₃₇	#382	1	1	yes	characteristic
61	C₂×C₃₇	#427	1	1	yes	characteristic
62	C₂×C₃₇	#427	1	1	yes	characteristic
63	C₂×C₃₇	#488	1	2	yes	—
64	C₂×C₃₇	#488	1	2	yes	—
65	C₂×C₄×C₈	—	37	1	no	—
66	#382	—	37	2	no	—
67	#340	—	37	1	no	—
68	#340	—	37	2	no	—
69	#340	D₃₇	1	1	yes	characteristic
70	C₃₇	#1835	1	1	yes	characteristic, complemented, Sylow
71	C₂²×C₈	—	37	1	no	—
72	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
73	C₂×C₄⋊C₄	—	37	2	no	—
74	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
75	C₂×C₄²	—	37	1	no	—
76	C₄×C₈	—	37	4	no	—
77	#95	—	74	2	no	—
78	#95	—	74	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	37	1	no	—
81	C₂²×C₄	—	37	1	no	—
82	C₂²×C₄	—	74	2	no	—
83	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	37	2	no	—
86	C₂×C₈	—	74	2	no	—
87	C₂×C₈	C₄×D₃₇	1	2	yes	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	74	2	no	—
91	C₄⋊C₄	—	74	2	no	—
92	C₄²	—	37	2	no	—
93	C₄²	—	37	2	no	—
94	C₂³	D₄×D₃₇	1	1	yes	characteristic
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	37	2	no	—
99	C₂×C₄	—	37	2	no	—
100	C₂×C₄	—	74	1	no	—
101	C₂×C₄	—	74	2	no	—
102	C₂×C₄	—	74	1	no	—
103	C₂×C₄	—	148	2	no	—
104	C₂×C₄	—	148	2	no	—
105	C₂×C₄	—	148	2	no	—
106	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
107	C₂×C₄	#10596962	1	2	yes	—
108	C₂×C₄	#10596964	1	2	yes	—
109	C₂×C₄	Q₈×D₃₇	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	—	74	2	no	—
112	C₂²	C₄⋊C₄×D₃₇	1	1	yes	characteristic
113	C₂²	#11924522	1	2	yes	—
114	C₂²	#11924555	1	1	yes	characteristic
115	C₂²	#11924557	1	2	yes	—
116	C₂²	#11924559	1	1	yes	characteristic
117	C₄	—	8	2	no	—
118	C₄	—	296	2	no	—
119	C₄	—	74	2	no	—
120	C₄	—	74	2	no	—
121	C₄	#11924492	1	1	yes	characteristic
122	C₄	#11924493	1	1	yes	characteristic
123	C₄	#11924526	1	2	yes	—
124	C₂	#14557397	1	1	yes	characteristic
125	C₂	#14557568	1	1	yes	characteristic
126	C₂	#14557570	1	1	yes	characteristic
127	C₂	#14557886	1	2	yes	—
128	C₂	#14557889	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₃₇

C₂

yes

maximal, characteristic

#14557398

C₂

yes

maximal

#14557296

C₂

yes

maximal, characteristic

#14557295

C₂

yes

maximal

C₃₇×#340

C₂

yes

maximal, characteristic

C₂²×#64436

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

C₈×Dic₃₇

C₄

yes

complemented

#11924443

—

#11924443

—

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂²×Dic₃₇

D₄

yes

characteristic

C₂²×Dic₃₇

D₄

yes

characteristic

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄×Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

Q₈

yes

—

C₂×#64436

—

C₂×#64436

C₂×C₄

yes

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

#41

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₄⋊C₄

yes

—

C₂×Dic₃₇

C₄⋊C₄

yes

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₇

#124

yes

—

C₂²×C₃₇

#135

yes

characteristic

C₂²×C₃₇

#135

yes

—

C₂²×C₃₇

#135

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

yes

C₄×C₃₇

yes

C₄×C₃₇

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₃₇

#488

yes

—

C₂×C₄×C₈

—

—

—

—

D₃₇

yes

characteristic

C₃₇

#1835

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₇

yes

—

C₂×C₄²

—

C₄×C₈

—

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₃₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₂³

D₄×D₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

148

—

104

C₂×C₄

—

148

—

105

C₂×C₄

—

148

—

106

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

107

C₂×C₄

#10596962

yes

—

108

C₂×C₄

#10596964

yes

—

109

C₂×C₄

Q₈×D₃₇

yes

characteristic

110

C₈

—

111

C₈

—

112

C₂²

C₄⋊C₄×D₃₇

yes

characteristic

113

C₂²

#11924522

yes

—

114

C₂²

#11924555

yes

characteristic

115

C₂²

#11924557

yes

—

116

C₂²

#11924559

yes

characteristic

117

C₄

—

118

C₄

—

296

—

119

C₄

—

120

C₄

—

121

C₄

#11924492

yes

characteristic

122

C₄

#11924493

yes

characteristic

123

C₄

#11924526

yes

—

124

C₂

#14557397

yes

characteristic

125

C₂

#14557568

yes

characteristic

126

C₂

#14557570

yes

characteristic

127

C₂

#14557886

yes

—

128

C₂

#14557889

yes

—

	subgroup	G/H
center	C₂³	D₄×D₃₇
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#340	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#340	D₃₇
hypocenter	C₃₇	#1835
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₃₇

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#340

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 ◁ #340

N	H ≅ G/N	type
9. C₈×Dic₃₇	C₄	semidirect
70. C₃₇	#1835	semidirect

H ≅ G/N

type

9. C₈×Dic₃₇

C₄

semidirect

70. C₃₇

#1835

semidirect

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

subgroup	index
C₂×C₈×Dic₃₇	2
#14557398	2
#14557296	2
#14557295	2
C₃₇×#340	2
#1835	37

subgroup

index

C₂×C₈×Dic₃₇

C₃₇×#340

Subgroups — #17994725

Conjugacy classes of subgroups

Complements

Maximal subgroups