Group #18009474 — subgroups

Subgroups — #18009474

Order 4736.

188 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 76 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×#11924603	C₂	1	2	yes	maximal
2	#14557402	C₂	1	1	yes	maximal, characteristic
3	C₂²×#64438	—	2	2	no	—
4	#11924603	—	2	2	no	—
5	#11924603	C₄	1	4	yes	complemented
6	C₂×C₄⋊Dic₃₇	C₄	1	1	yes	characteristic
7	C₂×#10596936	—	2	2	no	—
8	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₃₇	C₄	1	1	yes	characteristic
10	C₂²×C₄×C₃₇	—	2	1	no	—
11	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
12	C₂²×Dic₃₇	—	2	1	no	—
13	C₂²×Dic₃₇	D₄	1	2	yes	—
14	C₂×#64438	—	2	4	no	—
15	C₂×#64438	—	4	2	no	—
16	C₂×#64438	—	4	2	no	—
17	C₄⋊C₄×C₃₇	—	4	1	no	—
18	C₄⋊Dic₃₇	—	4	1	no	—
19	#10596936	—	4	2	no	—
20	#10596936	—	4	2	no	—
21	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
22	C₄×Dic₃₇	D₄	1	1	yes	characteristic
23	C₄×Dic₃₇	Q₈	1	1	yes	characteristic
24	C₂³×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₃₇	—	2	1	no	—
26	C₂×C₄×C₃₇	—	2	1	no	—
27	C₂×C₄×C₃₇	—	4	1	no	—
28	C₂×C₄×C₃₇	—	4	2	no	—
29	C₂×C₄×C₃₇	C₄²	1	1	yes	characteristic
30	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×Dic₃₇	—	4	1	no	—
32	C₂×Dic₃₇	—	4	2	no	—
33	C₂×Dic₃₇	C₂²⋊C₄	1	2	yes	—
34	C₂×Dic₃₇	C₄⋊C₄	1	2	yes	—
35	C₂×Dic₃₇	D₈	1	2	yes	—
36	C₂×Dic₃₇	SD₁₆	1	2	yes	—
37	C₂×Dic₃₇	SD₁₆	1	2	yes	—
38	C₂×Dic₃₇	Q₁₆	1	2	yes	—
39	#64438	—	4	4	no	—
40	C₂²×C₃₇	#95	1	1	yes	characteristic
41	C₂²×C₃₇	#100	1	1	yes	characteristic
42	C₂²×C₃₇	#101	1	1	yes	characteristic
43	C₂²×C₃₇	#102	1	2	yes	—
44	C₂²×C₃₇	#103	1	2	yes	—
45	C₄×C₃₇	—	2	1	no	—
46	C₄×C₃₇	—	2	1	no	—
47	C₄×C₃₇	—	8	1	no	—
48	Dic₃₇	—	8	1	no	—
49	Dic₃₇	#102	1	2	yes	—
50	Dic₃₇	#103	1	2	yes	—
51	Dic₃₇	#106	1	2	yes	—
52	Dic₃₇	#107	1	2	yes	—
53	#1296	—	37	1	no	maximal, Sylow
54	C₂×C₃₇	#331	1	1	yes	characteristic
55	C₂×C₃₇	#332	1	2	yes	—
56	C₂×C₃₇	#333	1	1	yes	characteristic
57	C₂×C₃₇	#340	1	2	yes	—
58	C₂×C₃₇	#343	1	1	yes	characteristic
59	C₂×C₄⋊C₈	—	37	2	no	—
60	#384	—	37	1	no	—
61	C₃₇	#1296	1	1	yes	characteristic, complemented, Sylow
62	C₂²×C₈	—	74	2	no	—
63	C₂×C₄⋊C₄	—	37	1	no	—
64	C₂×C₄⋊C₄	—	74	2	no	—
65	C₂×C₄⋊C₄	#3866	1	1	yes	characteristic
66	C₂×C₄²	—	37	1	no	—
67	C₄⋊C₈	—	37	4	no	—
68	C₄⋊C₈	—	74	2	no	—
69	C₂²×C₄	—	2	1	no	—
70	C₂²×C₄	—	37	2	no	—
71	C₂²×C₄	—	74	1	no	—
72	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
73	C₂×C₈	—	74	4	no	—
74	C₂×C₈	—	148	2	no	—
75	C₂×C₈	—	148	2	no	—
76	C₄⋊C₄	—	4	1	no	—
77	C₄⋊C₄	—	148	1	no	—
78	C₄⋊C₄	—	148	2	no	—
79	C₄⋊C₄	—	148	2	no	—
80	C₄²	—	37	2	no	—
81	C₄²	—	37	1	no	—
82	C₄²	—	37	1	no	—
83	C₂³	C₂²⋊#3866	1	1	yes	characteristic
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	37	2	no	—
89	C₂×C₄	—	37	2	no	—
90	C₂×C₄	—	37	2	no	—
91	C₂×C₄	—	37	2	no	—
92	C₂×C₄	—	37	2	no	—
93	C₂×C₄	—	37	2	no	—
94	C₂×C₄	—	148	1	no	—
95	C₂×C₄	—	148	2	no	—
96	C₂×C₄	C₄×#3866	1	1	yes	characteristic
97	C₂×C₄	C₄⋊#3866	1	1	yes	characteristic
98	C₈	—	148	4	no	—
99	C₂²	(C₂×#3866)⋊C₄	1	1	yes	characteristic
100	C₂²	#11924596	1	1	yes	characteristic
101	C₂²	#11924597	1	1	yes	characteristic
102	C₂²	#11924598	1	2	yes	—
103	C₂²	#11924599	1	2	yes	—
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	8	1	no	—
107	C₄	—	37	2	no	—
108	C₄	—	37	2	no	—
109	C₄	—	37	2	no	—
110	C₄	—	37	2	no	—
111	C₄	—	296	1	no	—
112	C₂	#14558391	1	1	yes	characteristic
113	C₂	#14558392	1	2	yes	—
114	C₂	#14558393	1	1	yes	characteristic
115	C₂	#14558406	1	2	yes	—
116	C₂	#14558413	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924603

C₂

yes

maximal

#14557402

C₂

yes

maximal, characteristic

C₂²×#64438

—

#11924603

—

#11924603

C₄

yes

complemented

C₂×C₄⋊Dic₃₇

C₄

yes

characteristic

C₂×#10596936

—

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₇

C₄

yes

characteristic

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

—

C₂²×Dic₃₇

D₄

yes

—

C₂×#64438

—

C₂×#64438

—

C₂×#64438

—

C₄⋊C₄×C₃₇

—

C₄⋊Dic₃₇

—

#10596936

—

#10596936

—

C₄×Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

D₄

yes

characteristic

C₄×Dic₃₇

Q₈

yes

characteristic

C₂³×C₃₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₄²

yes

characteristic

C₂×C₄×C₃₇

C₄⋊C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂²⋊C₄

yes

—

C₂×Dic₃₇

C₄⋊C₄

yes

—

C₂×Dic₃₇

D₈

yes

—

C₂×Dic₃₇

SD₁₆

yes

—

C₂×Dic₃₇

SD₁₆

yes

—

C₂×Dic₃₇

Q₁₆

yes

—

#64438

—

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

—

C₂²×C₃₇

#103

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

Dic₃₇

—

Dic₃₇

#102

yes

—

Dic₃₇

#103

yes

—

Dic₃₇

#106

yes

—

Dic₃₇

#107

yes

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₃₇

#333

yes

characteristic

C₂×C₃₇

#340

yes

—

C₂×C₃₇

#343

yes

characteristic

C₂×C₄⋊C₈

—

#384

—

C₃₇

#1296

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

#3866

yes

characteristic

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#3866

yes

characteristic

C₂×C₈

—

C₂×C₈

—

148

—

C₂×C₈

—

148

—

C₄⋊C₄

—

C₄⋊C₄

—

148

—

C₄⋊C₄

—

148

—

C₄⋊C₄

—

148

—

C₄²

—

C₄²

—

C₄²

—

C₂³

C₂²⋊#3866

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

148

—

C₂×C₄

—

148

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

148

—

C₂²

(C₂×#3866)⋊C₄

yes

characteristic

100

C₂²

#11924596

yes

characteristic

101

C₂²

#11924597

yes

characteristic

102

C₂²

#11924598

yes

—

103

C₂²

#11924599

yes

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

296

—

112

C₂

#14558391

yes

characteristic

113

C₂

#14558392

yes

—

114

C₂

#14558393

yes

characteristic

115

C₂

#14558406

yes

—

116

C₂

#14558413

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊#3866
commutator	C₂×C₄×C₃₇	C₄²
Frattini	C₂²×C₄	C₂×#3866
Fitting	C₂×C₄⋊C₄×C₃₇	C₄
socle, prime radical	C₂³×C₃₇	C₂²⋊C₄
cosocle	C₂×C₄×Dic₃₇	C₂²
hypercenter	C₂×C₄⋊C₄	#3866
hypocenter	C₃₇	#1296
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂²⋊#3866

commutator

C₂×C₄×C₃₇

C₄²

Frattini

C₂²×C₄

C₂×#3866

Fitting

C₂×C₄⋊C₄×C₃₇

C₄

socle, prime radical

C₂³×C₃₇

C₂²⋊C₄

cosocle

C₂×C₄×Dic₃₇

C₂²

C₂×C₄⋊C₄

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
5. #11924603	C₄	semidirect
61. C₃₇	#1296	semidirect

H ≅ G/N

type

5. #11924603

C₄

semidirect

61. C₃₇

#1296

semidirect

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

subgroup	index
C₂×#11924603	2
#14557402	2
#1296	37

subgroup

index

C₂×#11924603

#14557402

#1296

Subgroups — #18009474

Conjugacy classes of subgroups

Complements

Maximal subgroups