Group #17994721 — subgroups

Subgroups — #17994721

Order 4736.

172 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 81 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	#14557571	C₂	1	1	yes	maximal, characteristic
2	#14557488	C₂	1	1	yes	maximal, characteristic
3	#14557305	C₂	1	2	yes	maximal
4	#14557290	C₂	1	2	yes	maximal
5	C₂×#108×C₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596934	C₂²	1	1	yes	characteristic
7	#11924495	C₂²	1	2	yes	—
8	#11924495	C₂²	1	1	yes	characteristic
9	#11924470	—	2	1	no	—
10	#11924468	—	2	1	no	—
11	#11924446	C₄	1	2	yes	complemented
12	C₈×Dic₃₇	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
15	#108×C₃₇	—	2	1	no	—
16	#108×C₃₇	—	2	1	no	—
17	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
18	M₁₆×C₃₇	—	2	2	no	—
19	M₁₆×C₃₇	—	2	2	no	—
20	C₂×C₈×C₃₇	—	2	2	no	—
21	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₃₇	D₄	1	1	yes	characteristic
24	C₂×C₈×C₃₇	D₄	1	1	yes	characteristic
25	C₂×C₈×C₃₇	Q₈	1	1	yes	characteristic
26	C₂×C₈×C₃₇	Q₈	1	1	yes	characteristic
27	C₂²⋊Dic₃₇	—	2	1	no	—
28	C₂²⋊Dic₃₇	—	2	2	no	—
29	C₄⋊Dic₃₇	—	2	1	no	—
30	C₄⋊Dic₃₇	—	2	2	no	—
31	C₄×Dic₃₇	—	2	2	no	—
32	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
33	#10596934	—	2	1	no	—
34	#10596934	—	2	1	no	—
35	C₂×#64436	C₂×C₄	1	2	yes	—
36	C₂³×C₃₇	#41	1	1	yes	characteristic
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	1	yes	characteristic
40	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
41	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
42	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
43	C₂×Dic₃₇	—	2	2	no	—
44	C₂×Dic₃₇	—	4	2	no	—
45	C₈×C₃₇	—	4	2	no	—
46	C₈×C₃₇	C₄⋊C₄	1	1	yes	characteristic
47	C₈×C₃₇	C₄⋊C₄	1	1	yes	characteristic
48	C₈×C₃₇	C₄⋊C₄	1	1	yes	characteristic
49	C₈×C₃₇	C₄⋊C₄	1	1	yes	characteristic
50	#64436	—	2	2	no	—
51	C₂²×C₃₇	—	4	1	no	—
52	C₂²×C₃₇	#117	1	1	yes	characteristic
53	C₂²×C₃₇	#124	1	1	yes	characteristic
54	C₂²×C₃₇	#125	1	1	yes	characteristic
55	C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₃₇	#117	1	1	yes	characteristic
57	C₄×C₃₇	#124	1	1	yes	characteristic
58	C₄×C₃₇	#125	1	1	yes	characteristic
59	Dic₃₇	—	4	2	no	—
60	Dic₃₇	—	8	2	no	—
61	#1833	—	37	1	no	maximal, Sylow
62	C₂×C₃₇	—	2	1	no	—
63	C₂×C₃₇	—	2	1	no	—
64	C₂×C₃₇	—	2	1	no	—
65	C₂×C₃₇	#382	1	1	yes	characteristic
66	C₂×#108	D₃₇	1	1	yes	characteristic
67	#427	—	37	1	no	—
68	#405	—	37	1	no	—
69	#344	—	37	2	no	—
70	H(3,ℤ/4)	—	37	2	no	—
71	C₃₇	#1833	1	1	yes	characteristic, complemented, Sylow
72	C₂×M₁₆	—	37	1	no	—
73	C₂×M₁₆	C₂×D₃₇	1	2	yes	—
74	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
75	#117	—	37	1	no	—
76	#117	—	37	2	no	—
77	#108	—	2	1	no	—
78	#108	—	2	1	no	—
79	#107	—	74	1	no	—
80	#106	—	74	1	no	—
81	#97	—	37	2	no	—
82	C₄×C₈	—	37	2	no	—
83	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	2	2	no	—
86	M₁₆	—	74	1	no	—
87	M₁₆	—	74	1	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	37	2	no	—
90	C₂×C₈	Dic₇₄	1	1	yes	characteristic
91	C₂×C₈	Dic₇₄	1	1	yes	characteristic
92	C₂×C₈	C₄×D₃₇	1	1	yes	characteristic
93	C₂×C₈	C₄×D₃₇	1	1	yes	characteristic
94	C₂×C₈	C₂²⋊D₃₇	1	1	yes	characteristic
95	C₂×C₈	C₂²⋊D₃₇	1	1	yes	characteristic
96	C₄⋊C₄	—	74	1	no	—
97	C₄⋊C₄	—	74	2	no	—
98	C₂²⋊C₄	—	74	1	no	—
99	C₂²⋊C₄	—	74	2	no	—
100	C₄²	—	37	2	no	—
101	C₄²	—	74	2	no	—
102	C₂³	#10596966	1	1	yes	characteristic
103	C₂×C₄	—	74	2	no	—
104	C₂×C₄	—	148	2	no	—
105	C₂×C₄	C₂×Dic₇₄	1	1	yes	characteristic
106	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
107	C₂×C₄	#10596964	1	1	yes	characteristic
108	C₂×C₄	#10596964	1	1	yes	characteristic
109	C₂×C₄	#10596966	1	1	yes	characteristic
110	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
111	C₈	—	4	2	no	—
112	C₈	—	74	2	no	—
113	C₈	#10596936	1	1	yes	characteristic
114	C₈	#10596936	1	1	yes	characteristic
115	C₈	#10596936	1	1	yes	characteristic
116	C₈	#10596936	1	1	yes	characteristic
117	C₂²	—	4	1	no	—
118	C₂²	#11924494	1	1	yes	characteristic
119	C₂²	#11924525	1	1	yes	characteristic
120	C₂²	#11924527	1	1	yes	characteristic
121	C₄	—	296	2	no	—
122	C₄	—	148	2	no	—
123	C₄	C₂×#10596936	1	1	yes	characteristic
124	C₄	#11924494	1	1	yes	characteristic
125	C₄	#11924523	1	1	yes	characteristic
126	C₄	#11924527	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#14557398	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14557571

C₂

yes

maximal, characteristic

#14557488

C₂

yes

maximal, characteristic

#14557305

C₂

yes

maximal

#14557290

C₂

yes

maximal

C₂×#108×C₃₇

C₂

yes

maximal, characteristic

C₂×#10596934

C₂²

yes

characteristic

#11924495

C₂²

yes

—

#11924495

C₂²

yes

—

—

C₄

yes

complemented

C₈×Dic₃₇

C₄

yes

complemented

C₂×M₁₆×C₃₇

C₂²

yes

—

C₂²×C₈×C₃₇

C₂²

yes

characteristic

#108×C₃₇

—

#108×C₃₇

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

M₁₆×C₃₇

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₃₇

D₄

yes

characteristic

C₂×C₈×C₃₇

D₄

yes

characteristic

C₂×C₈×C₃₇

Q₈

yes

characteristic

C₂×C₈×C₃₇

Q₈

yes

characteristic

C₂²⋊Dic₃₇

—

C₂²⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

C₂×C₄

yes

—

#10596934

—

#10596934

—

C₂×#64436

C₂×C₄

yes

—

C₂³×C₃₇

#41

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

C₂×C₄×C₃₇

yes

C₂×C₄×C₃₇

yes

C₂×C₄×C₃₇

yes

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₈×C₃₇

—

C₈×C₃₇

C₄⋊C₄

yes

characteristic

C₈×C₃₇

C₄⋊C₄

yes

characteristic

C₈×C₃₇

C₄⋊C₄

yes

characteristic

C₈×C₃₇

C₄⋊C₄

yes

characteristic

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₄×C₃₇

C₂×C₄⋊C₄

yes

C₄×C₃₇

yes

C₄×C₃₇

yes

C₄×C₃₇

yes

Dic₃₇

—

Dic₃₇

—

#1833

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

#382

yes

characteristic

C₂×#108

D₃₇

yes

—

—

—

—

C₃₇

#1833

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₇

yes

—

C₂²×C₈

C₂×D₃₇

yes

—

—

—

—

—

—

—

C₄×C₈

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₇₄

yes

characteristic

C₂×C₈

Dic₇₄

yes

characteristic

C₂×C₈

C₄×D₃₇

yes

characteristic

C₂×C₈

C₄×D₃₇

yes

characteristic

C₂×C₈

C₂²⋊D₃₇

yes

characteristic

C₂×C₈

C₂²⋊D₃₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₄²

—

101

C₄²

—

102

C₂³

#10596966

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

148

—

105

C₂×C₄

C₂×Dic₇₄

yes

characteristic

106

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

107

C₂×C₄

#10596964

yes

characteristic

108

C₂×C₄

#10596964

yes

characteristic

109

C₂×C₄

#10596966

yes

characteristic

110

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

#10596936

yes

characteristic

114

C₈

#10596936

yes

characteristic

115

C₈

#10596936

yes

characteristic

116

C₈

#10596936

yes

characteristic

117

C₂²

—

118

C₂²

#11924494

yes

characteristic

119

C₂²

#11924525

yes

characteristic

120

C₂²

#11924527

yes

characteristic

121

C₄

—

296

—

122

C₄

—

148

—

123

C₄

C₂×#10596936

yes

characteristic

124

C₄

#11924494

yes

characteristic

125

C₄

#11924523

yes

characteristic

126

C₄

#11924527

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#14557398

yes

characteristic

	subgroup	G/H
center	C₄	#11924494
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₂×#108×C₃₇	C₂
socle	C₂×C₃₇	#382
prime radical	C₂³×C₃₇	#41
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₂×#108	D₃₇
hypocenter	C₃₇	#1833
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#11924494

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₂×#108×C₃₇

C₂

C₂×C₃₇

C₂³×C₃₇

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
11. #11924446	C₄	semidirect
12. C₈×Dic₃₇	C₄	semidirect
71. C₃₇	#1833	semidirect

H ≅ G/N

type

11. #11924446

C₄

semidirect

12. C₈×Dic₃₇

C₄

semidirect

71. C₃₇

#1833

semidirect

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

subgroup	index
#14557571	2
#14557488	2
#14557305	2
#14557290	2
C₂×#108×C₃₇	2
#1833	37

subgroup

index

C₂×#108×C₃₇

Subgroups — #17994721

Conjugacy classes of subgroups

Complements

Maximal subgroups