Group #18010519 — subgroups

Subgroups — #18010519

Order 4736.

224 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 74 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	#14558479	C₂	1	1	yes	maximal, characteristic
2	#14558477	C₂	1	1	yes	maximal, characteristic
3	#14558407	C₂	1	4	yes	maximal
4	#97×D₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#3866	—	2	2	no	—
6	C₂×C₄⋊#3866	C₂²	1	2	yes	—
7	C₂×C₄⋊#3866	C₂²	1	2	yes	—
8	(C₂×#3866)⋊C₄	—	2	2	no	—
9	C₂×C₈×D₃₇	C₂²	1	2	yes	—
10	C₄²×D₃₇	C₂²	1	1	yes	characteristic
11	#11924447	C₄	1	1	yes	characteristic, complemented
12	#11924446	C₄	1	2	yes	complemented
13	#97×C₃₇	C₄	1	1	yes	characteristic, complemented
14	C₂²×#3866	—	2	2	no	—
15	C₂²×#3866	—	2	2	no	—
16	C₂×C₄×D₃₇	D₄	1	2	yes	—
17	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
18	C₄⋊#3866	—	2	4	no	—
19	C₄⋊#3866	—	2	4	no	—
20	C₄⋊#3866	—	4	4	no	—
21	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
22	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
23	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₃₇	Q₈	1	2	yes	—
25	C₂×#64436	C₂×C₄	1	2	yes	—
26	C₈×D₃₇	—	2	2	no	—
27	C₈×D₃₇	—	2	2	no	—
28	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
29	C₂×#3866	—	4	2	no	—
30	C₂×#3866	—	4	2	no	—
31	C₂×#3866	—	4	4	no	—
32	C₂×#3866	—	4	2	no	—
33	C₂×#3866	—	4	2	no	—
34	C₂×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₃₇	C₄⋊C₄	1	2	yes	—
37	C₂×Dic₃₇	C₂×Q₈	1	1	yes	characteristic
38	C₄×D₃₇	—	2	2	no	—
39	C₄×D₃₇	—	2	2	no	—
40	C₄×D₃₇	#41	1	2	yes	—
41	C₄×D₃₇	#41	1	2	yes	—
42	C₈×C₃₇	—	2	2	no	—
43	#64436	—	2	2	no	—
44	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×D₃₇	#124	1	1	yes	characteristic
46	C₂×D₃₇	#124	1	1	yes	characteristic
47	C₂×D₃₇	AGL(1,ℤ/8)	1	2	yes	—
48	C₂×D₃₇	#137	1	2	yes	—
49	#3866	—	8	2	no	—
50	#3866	—	8	2	no	—
51	C₄×C₃₇	—	2	2	no	—
52	C₄×C₃₇	#117	1	2	yes	—
53	Dic₃₇	—	2	2	no	—
54	Dic₃₇	#125	1	2	yes	—
55	#1836	—	37	1	no	maximal, Sylow
56	C₂×C₃₇	#382	1	1	yes	characteristic
57	C₂×C₃₇	#428	1	2	yes	—
58	D₃₇	#489	1	2	yes	—
59	D₃₇	#490	1	1	yes	characteristic
60	D₃₇	#491	1	1	yes	characteristic
61	C₂×#97	—	37	1	no	—
62	#384	—	37	1	no	—
63	#382	—	37	1	no	—
64	#340	—	37	4	no	—
65	C₃₇	#1836	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₈	—	37	2	no	—
67	C₂×C₄⋊C₄	—	37	2	no	—
68	C₂×C₄⋊C₄	—	37	2	no	—
69	C₂×C₄⋊C₄	—	74	2	no	—
70	C₂×C₄²	—	37	1	no	—
71	#97	—	37	2	no	—
72	#97	—	37	1	no	—
73	#97	#3866	1	1	yes	characteristic, complemented
74	#95	—	74	2	no	—
75	C₂²×C₄	—	37	1	no	—
76	C₂²×C₄	—	37	2	no	—
77	C₂²×C₄	—	74	2	no	—
78	C₂²×C₄	—	74	2	no	—
79	C₂×C₈	—	37	2	no	—
80	C₂×C₈	—	74	2	no	—
81	C₂×C₈	—	74	2	no	—
82	C₂×C₈	C₂×#3866	1	2	yes	—
83	C₄⋊C₄	—	74	4	no	—
84	C₄⋊C₄	—	74	4	no	—
85	C₄⋊C₄	—	148	4	no	—
86	C₄²	—	37	2	no	—
87	C₄²	—	37	1	no	—
88	C₄²	C₂×#3866	1	1	yes	characteristic
89	C₂³	—	37	1	no	—
90	C₂×C₄	—	37	1	no	—
91	C₂×C₄	—	37	2	no	—
92	C₂×C₄	—	37	2	no	—
93	C₂×C₄	—	37	2	no	—
94	C₂×C₄	—	74	2	no	—
95	C₂×C₄	—	74	2	no	—
96	C₂×C₄	—	148	2	no	—
97	C₂×C₄	—	148	2	no	—
98	C₂×C₄	—	148	4	no	—
99	C₂×C₄	—	148	2	no	—
100	C₂×C₄	—	148	2	no	—
101	C₂×C₄	C₄⋊#3866	1	2	yes	—
102	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
103	C₈	—	2	2	no	—
104	C₈	—	74	2	no	—
105	C₂²	—	37	1	no	—
106	C₂²	—	37	1	no	—
107	C₂²	—	37	2	no	—
108	C₂²	—	37	2	no	—
109	C₂²	C₂×C₄⋊#3866	1	1	yes	characteristic
110	C₄	—	2	2	no	—
111	C₄	—	37	2	no	—
112	C₄	—	296	2	no	—
113	C₄	—	296	2	no	—
114	C₄	—	74	2	no	—
115	C₄	#11924615	1	2	yes	—
116	C₂	—	37	1	no	—
117	C₂	—	37	2	no	—
118	C₂	—	37	1	no	—
119	C₂	#14558477	1	1	yes	characteristic
120	C₂	#14558544	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558479

C₂

yes

maximal, characteristic

#14558477

C₂

yes

maximal, characteristic

#14558407

C₂

yes

maximal

#97×D₃₇

C₂

yes

maximal, characteristic

C₂×C₄⋊#3866

—

C₂×C₄⋊#3866

C₂²

yes

—

C₂×C₄⋊#3866

C₂²

yes

—

(C₂×#3866)⋊C₄

—

C₂×C₈×D₃₇

C₂²

yes

—

C₄²×D₃₇

C₂²

yes

characteristic

#11924447

C₄

yes

characteristic, complemented

#11924446

C₄

yes

complemented

#97×C₃₇

C₄

yes

characteristic, complemented

C₂²×#3866

—

C₂²×#3866

—

C₂×C₄×D₃₇

D₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

—

—

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

Q₈

yes

—

C₂×#64436

C₂×C₄

yes

—

C₈×D₃₇

—

C₈×D₃₇

—

C₂²×D₃₇

C₂×D₄

yes

—

—

—

—

—

C₂×C₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₄⋊C₄

yes

—

C₂×Dic₃₇

C₂×Q₈

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

#41

yes

—

C₄×D₃₇

#41

yes

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

C₂×D₃₇

yes

C₂×D₃₇

yes

C₂×D₃₇

yes

—

C₂×D₃₇

#137

yes

—

#3866

—

#3866

—

C₄×C₃₇

—

C₄×C₃₇

#117

yes

—

Dic₃₇

—

Dic₃₇

#125

yes

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

#489

yes

—

D₃₇

yes

D₃₇

yes

C₂×#97

—

#384

—

#382

—

#340

—

C₃₇

#1836

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

—

—

yes

characteristic, complemented

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#3866

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

148

—

C₄²

—

C₄²

—

C₄²

C₂×#3866

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

148

—

C₂×C₄

—

148

—

C₂×C₄

—

148

—

C₂×C₄

—

148

—

100

C₂×C₄

—

148

—

101

C₂×C₄

C₄⋊#3866

yes

—

102

C₂×C₄

C₂²×#3866

yes

characteristic

103

C₈

—

104

C₈

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂×C₄⋊#3866

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

296

—

113

C₄

—

296

—

114

C₄

—

115

C₄

#11924615

yes

—

116

C₂

—

117

C₂

—

118

C₂

—

119

C₂

#14558477

yes

characteristic

120

C₂

#14558544

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#3866
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	#97×C₃₇	C₄
socle	C₂²×C₃₇	C₂×C₄⋊C₄
prime radical	C₂²×D₃₇	C₂×D₄
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	#97	#3866
hypocenter	C₃₇	#1836
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊#3866

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#3866

Fitting

#97×C₃₇

C₄

socle

C₂²×C₃₇

C₂×C₄⋊C₄

prime radical

C₂²×D₃₇

C₂×D₄

cosocle

C₂×C₄×D₃₇

C₂³

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
11. #11924447	C₄	semidirect
12. #11924446	C₄	semidirect
13. #97×C₃₇	C₄	semidirect
65. C₃₇	#1836	semidirect
73. #97	#3866	semidirect

H ≅ G/N

type

11. #11924447

C₄

semidirect

12. #11924446

C₄

semidirect

13. #97×C₃₇

C₄

semidirect

65. C₃₇

#1836

semidirect

73. #97

#3866

semidirect

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

subgroup	index
#14558479	2
#14558477	2
#14558407	2
#97×D₃₇	2
#1836	37

subgroup

index

#97×D₃₇

Subgroups — #18010519

Conjugacy classes of subgroups

Complements

Maximal subgroups