Group #18010526 — subgroups

Subgroups — #18010526

Order 4736.

162 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 52 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	#14558415	C₂	1	2	yes	maximal
2	#14558404	C₂	1	2	yes	maximal
3	#14558402	C₂	1	1	yes	maximal, characteristic
4	#14558401	C₂	1	1	yes	maximal, characteristic
5	C₄×#10596929	C₂	1	1	yes	maximal, characteristic
6	D₃₇⋊M₁₆	C₂²	1	2	yes	—
7	#11924615	C₂²	1	2	yes	—
8	C₂×#10596929	C₂²	1	1	yes	characteristic
9	C₂×#10596929	C₂²	1	1	yes	characteristic
10	C₄²×D₃₇	C₂²	1	1	yes	characteristic
11	#11924446	C₄	1	1	yes	characteristic, complemented
12	#11924446	C₄	1	1	yes	characteristic, complemented
13	C₄×#64436	C₄	1	1	yes	characteristic, complemented
14	C₄×C₈×C₃₇	C₄	1	1	yes	characteristic, complemented
15	C₂×C₄×D₃₇	—	2	1	no	—
16	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
17	C₂²⋊#3866	—	2	2	no	—
18	#10596957	—	2	2	no	—
19	C₄⋊#3866	—	2	2	no	—
20	C₄×#3866	—	2	2	no	—
21	#10596953	—	2	2	no	—
22	D₃₇⋊C₈	—	2	2	no	—
23	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₃₇	—	2	1	no	—
27	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂×#64436	C₂×C₄	1	1	yes	characteristic
29	C₂×#64436	C₂×C₄	1	1	yes	characteristic
30	#10596929	—	2	2	no	—
31	#10596929	—	2	2	no	—
32	C₂²×D₃₇	#41	1	1	yes	characteristic
33	C₂×#3866	—	4	2	no	—
34	C₂×C₄×C₃₇	—	2	1	no	—
35	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₃₇	—	2	1	no	—
37	C₂×Dic₃₇	#41	1	1	yes	characteristic
38	C₄×D₃₇	—	4	2	no	—
39	C₄×D₃₇	#41	1	2	yes	—
40	C₄×D₃₇	#41	1	2	yes	—
41	#64438	—	4	2	no	—
42	C₈×C₃₇	—	2	1	no	—
43	C₈×C₃₇	—	2	1	no	—
44	#64436	—	2	1	no	—
45	#64436	—	2	1	no	—
46	C₂²×C₃₇	#117	1	1	yes	characteristic
47	C₂×D₃₇	—	4	1	no	—
48	C₂×D₃₇	#126	1	2	yes	—
49	#3866	—	8	2	no	—
50	C₄×C₃₇	—	2	2	no	—
51	C₄×C₃₇	#117	1	1	yes	characteristic
52	C₄×C₃₇	#117	1	1	yes	characteristic
53	Dic₃₇	—	4	1	no	—
54	Dic₃₇	#126	1	2	yes	—
55	#1841	—	37	1	no	maximal, Sylow
56	C₂×C₃₇	—	2	1	no	—
57	C₂×C₃₇	#383	1	1	yes	characteristic
58	D₃₇	—	2	2	no	—
59	C₄×M₁₆	—	37	1	no	—
60	#344	—	37	2	no	—
61	#339	—	37	2	no	—
62	#338	—	37	1	no	—
63	H(3,ℤ/4)	—	37	1	no	—
64	C₃₇	#1841	1	1	yes	characteristic, complemented, Sylow
65	C₂×M₁₆	—	37	1	no	—
66	C₂×M₁₆	—	37	1	no	—
67	C₂×M₁₆	—	37	2	no	—
68	#117	—	37	2	no	—
69	C₂×C₄²	—	37	1	no	—
70	#97	—	37	1	no	—
71	#97	—	37	1	no	—
72	C₄×C₈	—	37	1	no	—
73	C₄×C₈	#3866	1	1	yes	characteristic, complemented
74	C₂²×C₄	—	37	1	no	—
75	C₂²×C₄	—	74	1	no	—
76	M₁₆	—	74	2	no	—
77	M₁₆	—	74	2	no	—
78	M₁₆	—	74	2	no	—
79	M₁₆	—	74	2	no	—
80	C₂×C₈	—	37	1	no	—
81	C₂×C₈	—	37	1	no	—
82	C₂×C₈	—	74	2	no	—
83	C₂×C₈	C₂×#3866	1	1	yes	characteristic
84	C₂×C₈	C₂×#3866	1	1	yes	characteristic
85	C₄⋊C₄	—	74	2	no	—
86	C₂²⋊C₄	—	74	2	no	—
87	C₄²	—	37	1	no	—
88	C₄²	—	74	1	no	—
89	C₄²	—	74	2	no	—
90	C₄²	C₂×#3866	1	1	yes	characteristic
91	C₂³	—	37	1	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	37	1	no	—
94	C₂×C₄	—	37	2	no	—
95	C₂×C₄	—	37	2	no	—
96	C₂×C₄	—	74	1	no	—
97	C₂×C₄	—	148	2	no	—
98	C₂×C₄	—	148	2	no	—
99	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
100	C₈	—	2	1	no	—
101	C₈	—	2	1	no	—
102	C₈	—	74	1	no	—
103	C₈	—	74	1	no	—
104	C₈	—	148	2	no	—
105	C₂²	—	37	2	no	—
106	C₂²	—	148	1	no	—
107	C₂²	#11924615	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	37	2	no	—
110	C₄	—	296	2	no	—
111	C₄	—	148	1	no	—
112	C₄	#11924615	1	1	yes	characteristic
113	C₄	#11924615	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	74	2	no	—
116	C₂	#14558478	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558415

C₂

yes

maximal

#14558404

C₂

yes

maximal

#14558402

C₂

yes

maximal, characteristic

#14558401

C₂

yes

maximal, characteristic

C₄×#10596929

C₂

yes

maximal, characteristic

D₃₇⋊M₁₆

C₂²

yes

—

#11924615

C₂²

yes

—

C₂×#10596929

C₂²

yes

characteristic

C₂×#10596929

C₂²

yes

characteristic

C₄²×D₃₇

C₂²

yes

characteristic

#11924446

C₄

yes

characteristic, complemented

#11924446

C₄

yes

characteristic, complemented

C₄×#64436

C₄

yes

characteristic, complemented

C₄×C₈×C₃₇

C₄

yes

characteristic, complemented

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂²⋊#3866

—

—

—

—

—

D₃₇⋊C₈

—

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

#10596929

—

#10596929

—

C₂²×D₃₇

#41

yes

characteristic

C₂×#3866

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

#41

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

#41

yes

—

C₄×D₃₇

#41

yes

—

#64438

—

C₈×C₃₇

—

C₈×C₃₇

—

#64436

—

#64436

—

C₂²×C₃₇

#117

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

#126

yes

—

#3866

—

C₄×C₃₇

—

C₄×C₃₇

yes

C₄×C₃₇

yes

Dic₃₇

—

Dic₃₇

#126

yes

—

#1841

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

#383

yes

characteristic

D₃₇

—

C₄×M₁₆

—

—

—

—

—

C₃₇

#1841

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

#117

—

C₂×C₄²

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

#3866

yes

characteristic, complemented

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#3866

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

148

—

C₂×C₄

—

148

—

C₂×C₄

C₂²×#3866

yes

characteristic

100

C₈

—

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

—

148

—

105

C₂²

—

106

C₂²

—

148

—

107

C₂²

#11924615

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

296

—

111

C₄

—

148

—

112

C₄

#11924615

yes

characteristic

113

C₄

#11924615

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

#14558478

yes

characteristic

	subgroup	G/H
center	C₄	#11924615
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	C₄×C₈×C₃₇	C₄
socle	C₂×C₃₇	#383
prime radical	C₂²×D₃₇	#41
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	C₄×C₈	#3866
hypocenter	C₃₇	#1841
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈

subgroup

G/H

center

C₄

#11924615

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#3866

Fitting

C₄×C₈×C₃₇

C₄

C₂×C₃₇

C₂²×D₃₇

C₂×C₄×D₃₇

C₂³

C₄×C₈

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈

N	H ≅ G/N	type
11. #11924446	C₄	semidirect
12. #11924446	C₄	semidirect
13. C₄×#64436	C₄	semidirect
14. C₄×C₈×C₃₇	C₄	semidirect
64. C₃₇	#1841	semidirect
73. C₄×C₈	#3866	semidirect

H ≅ G/N

type

11. #11924446

C₄

semidirect

12. #11924446

C₄

semidirect

13. C₄×#64436

C₄

semidirect

14. C₄×C₈×C₃₇

C₄

semidirect

64. C₃₇

#1841

semidirect

73. C₄×C₈

#3866

semidirect

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

subgroup	index
#14558415	2
#14558404	2
#14558402	2
#14558401	2
C₄×#10596929	2
#1841	37

subgroup

index

Subgroups — #18010526

Conjugacy classes of subgroups

Complements

Maximal subgroups