Group #18010510 — subgroups

Subgroups — #18010510

Order 4736.

236 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 62 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×#11924597	C₂	1	2	yes	maximal
2	#14558414	C₂	1	4	yes	maximal
3	#14558010	C₂	1	1	yes	maximal, characteristic
4	C₂×#10596957	C₂²	1	4	yes	—
5	#11924597	—	2	2	no	—
6	#11924597	—	2	2	no	—
7	C₂²×Dic₇₄	C₂²	1	1	yes	characteristic
8	C₄×Dic₇₄	—	2	2	no	—
9	#11924501	—	2	2	no	—
10	#11924495	C₄	1	1	yes	characteristic
11	#11924492	C₄	1	2	yes	—
12	#11924492	C₂²	1	2	yes	—
13	#117×C₃₇	C₄	1	1	yes	characteristic
14	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
17	C₂×Dic₇₄	—	2	4	no	—
18	C₂×Dic₇₄	D₄	1	2	yes	—
19	C₂×Dic₇₄	D₄	1	2	yes	—
20	#10596957	—	2	4	no	—
21	#10596957	—	2	4	no	—
22	C₂×#64438	—	2	4	no	—
23	C₄⋊C₄×C₃₇	—	2	1	no	—
24	C₂²⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
25	C₄²×C₃₇	—	2	1	no	—
26	C₂²⋊Dic₃₇	C₂×C₄	1	2	yes	—
27	C₄⋊Dic₃₇	—	2	1	no	—
28	#10596936	—	2	2	no	—
29	#10596936	—	2	2	no	—
30	C₄×Dic₃₇	—	2	1	no	—
31	C₄×Dic₃₇	—	2	2	no	—
32	C₄×Dic₃₇	—	2	2	no	—
33	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₃₇	—	2	2	no	—
35	C₂×C₄×C₃₇	—	2	2	no	—
36	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×Dic₃₇	—	2	2	no	—
39	C₂×Dic₃₇	—	2	2	no	—
40	C₂×Dic₃₇	C₂²⋊C₄	1	2	yes	—
41	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
42	C₂×Dic₃₇	#41	1	4	yes	—
43	Dic₇₄	—	2	4	no	—
44	Dic₇₄	—	2	4	no	—
45	#64438	—	4	4	no	—
46	C₂²×C₃₇	—	4	1	no	—
47	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₃₇	#117	1	2	yes	—
49	C₄×C₃₇	—	2	1	no	—
50	C₄×C₃₇	—	2	1	no	—
51	C₄×C₃₇	—	4	2	no	—
52	Dic₃₇	—	2	2	no	—
53	Dic₃₇	—	4	2	no	—
54	Dic₃₇	#123	1	4	yes	—
55	#1830	—	37	1	no	maximal, Sylow
56	C₂×C₃₇	—	2	1	no	—
57	C₂×C₃₇	—	2	2	no	—
58	C₂×C₃₇	#381	1	1	yes	characteristic
59	#519	—	37	1	no	—
60	C₂×#101	—	37	2	no	—
61	#344	—	37	4	no	—
62	C₃₇	#1830	1	1	yes	characteristic, complemented, Sylow
63	C₂²×Q₈	—	37	1	no	—
64	C₂×M₁₆	—	37	4	no	—
65	C₄×Q₈	—	74	2	no	—
66	C₄×Q₈	—	74	2	no	—
67	#117	—	37	1	no	—
68	#117	—	37	2	no	—
69	#117	—	37	2	no	—
70	#117	#3866	1	1	yes	characteristic
71	#101	—	74	2	no	—
72	#101	—	74	2	no	—
73	C₂×Q₈	—	37	2	no	—
74	C₂×Q₈	—	37	2	no	—
75	C₂×Q₈	—	74	4	no	—
76	C₂²×C₄	—	37	1	no	—
77	C₂²×C₄	—	37	1	no	—
78	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
79	M₁₆	—	74	4	no	—
80	M₁₆	—	74	4	no	—
81	C₂×C₈	—	74	4	no	—
82	C₄⋊C₄	—	2	1	no	—
83	C₄⋊C₄	—	74	2	no	—
84	C₄⋊C₄	—	74	2	no	—
85	C₄⋊C₄	—	74	1	no	—
86	C₂²⋊C₄	—	37	2	no	—
87	C₂²⋊C₄	C₂×#3866	1	2	yes	—
88	C₄²	—	2	1	no	—
89	C₄²	—	74	2	no	—
90	C₄²	—	74	2	no	—
91	C₄²	—	74	1	no	—
92	C₂³	C₂²×#3866	1	1	yes	characteristic
93	Q₈	—	74	4	no	—
94	Q₈	—	74	4	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	37	2	no	—
98	C₂×C₄	—	37	4	no	—
99	C₂×C₄	—	37	2	no	—
100	C₂×C₄	—	74	2	no	—
101	C₂×C₄	—	74	2	no	—
102	C₂×C₄	C₂²⋊#3866	1	1	yes	characteristic
103	C₂×C₄	C₂²⋊#3866	1	1	yes	characteristic
104	C₈	—	148	4	no	—
105	C₂²	—	4	1	no	—
106	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
107	C₂²	#11924615	1	2	yes	—
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	4	2	no	—
111	C₄	—	37	4	no	—
112	C₄	—	74	2	no	—
113	C₄	—	148	2	no	—
114	C₂	—	2	1	no	—
115	C₂	—	2	2	no	—
116	C₂	#14558476	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924597

C₂

yes

maximal

#14558414

C₂

yes

maximal

#14558010

C₂

yes

maximal, characteristic

C₂×#10596957

C₂²

yes

—

#11924597

—

#11924597

—

C₂²×Dic₇₄

C₂²

yes

characteristic

C₄×Dic₇₄

—

#11924501

—

#11924495

C₄

yes

characteristic

#11924492

C₄

yes

—

#11924492

C₂²

yes

—

#117×C₃₇

C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

C₂³

yes

characteristic

C₂×Dic₇₄

—

C₂×Dic₇₄

D₄

yes

—

C₂×Dic₇₄

D₄

yes

—

#10596957

—

#10596957

—

C₂×#64438

—

C₄⋊C₄×C₃₇

—

C₂²⋊C₄×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

—

C₂²⋊Dic₃₇

C₂×C₄

yes

—

C₄⋊Dic₃₇

—

#10596936

—

#10596936

—

C₄×Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂²⋊C₄

yes

—

C₂×Dic₃₇

C₂×D₄

yes

—

C₂×Dic₃₇

#41

yes

—

Dic₇₄

—

Dic₇₄

—

#64438

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₇

#117

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

#123

yes

—

#1830

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

yes

—

—

—

C₃₇

#1830

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

—

—

—

—

yes

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#3866

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×#3866

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂³

C₂²×#3866

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₂²⋊#3866

yes

characteristic

103

C₂×C₄

C₂²⋊#3866

yes

characteristic

104

C₈

—

148

—

105

C₂²

—

106

C₂²

C₂×C₂²⋊#3866

yes

characteristic

107

C₂²

#11924615

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

148

—

114

C₂

—

115

C₂

—

116

C₂

#14558476

yes

characteristic

	subgroup	G/H
center	C₂	#14558476
commutator, prime radical	C₂³×C₃₇	C₂²×C₄
Frattini	C₂³	C₂²×#3866
Fitting	#117×C₃₇	C₄
socle	C₂×C₃₇	#381
cosocle	C₂²×Dic₃₇	C₂³
hypercenter	#117	#3866
hypocenter	C₃₇	#1830
LCS	G ▷ C₂³×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #117

subgroup

G/H

center

C₂

#14558476

commutator, prime radical

C₂³×C₃₇

C₂²×C₄

Frattini

C₂³

C₂²×#3866

Fitting

#117×C₃₇

C₄

socle

C₂×C₃₇

#381

cosocle

C₂²×Dic₃₇

C₂³

C₃₇

G ▷ C₂³×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂ ◁ C₂³ ◁ #117

N	H ≅ G/N	type
62. C₃₇	#1830	semidirect

H ≅ G/N

type

62. C₃₇

#1830

semidirect

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

subgroup	index
C₂×#11924597	2
#14558414	2
#14558010	2
#1830	37

subgroup

index

Subgroups — #18010510

Conjugacy classes of subgroups

Complements

Maximal subgroups