Group #18010053 — subgroups

Subgroups — #18010053

Order 4736.

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

normal subgroups only

This group has 226 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	C₄×#10596957	C₂	1	2	yes	maximal
2	#14558390	C₂	1	4	yes	maximal, complemented
3	#14558086	C₂	1	1	yes	maximal, characteristic
4	C₂×#10596957	—	2	2	no	—
5	#11924591	C₂²	1	4	yes	—
6	C₄×#64438	—	2	2	no	—
7	#11924535	C₄	1	2	yes	—
8	#11924527	C₂²	1	2	yes	—
9	#11924513	—	2	2	no	—
10	C₄×Dic₇₄	—	2	2	no	—
11	#11924495	C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
13	#117×C₃₇	C₄	1	1	yes	characteristic
14	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₃₇	D₄	1	2	yes	—
16	C₂×Dic₇₄	—	2	2	no	—
17	#10596957	—	2	8	no	—
18	C₂×#64438	—	2	4	no	—
19	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
20	C₂²⋊C₄×C₃₇	—	2	1	no	—
21	C₄²×C₃₇	—	2	1	no	—
22	C₂²⋊Dic₃₇	—	2	1	no	—
23	C₄⋊Dic₃₇	—	2	2	no	—
24	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
25	#10596936	—	2	2	no	—
26	#10596936	—	2	2	no	—
27	C₄×Dic₃₇	—	2	1	no	—
28	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₃₇	D₄	1	2	yes	—
30	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
31	C₂³×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₃₇	—	2	2	no	—
33	C₂×C₄×C₃₇	—	2	2	no	—
34	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₃₇	—	2	4	no	—
37	C₂×Dic₃₇	—	2	2	no	—
38	C₂×Dic₃₇	C₂²⋊C₄	1	2	yes	—
39	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
40	Dic₇₄	—	4	4	no	—
41	#64438	—	2	8	no	—
42	C₂²×C₃₇	—	2	1	no	—
43	C₂²×C₃₇	—	2	2	no	—
44	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₄×C₃₇	—	2	1	no	—
46	C₄×C₃₇	—	4	2	no	—
47	C₄×C₃₇	#101	1	2	yes	—
48	Dic₃₇	—	2	2	no	—
49	Dic₃₇	—	4	2	no	—
50	Dic₃₇	#104	1	4	yes	—
51	#1526	—	37	1	no	maximal, Sylow
52	C₂×C₃₇	—	4	1	no	—
53	C₂×C₃₇	C₂×#101	1	1	yes	characteristic
54	C₂×C₃₇	C₂×#104	1	2	yes	—
55	#533	—	37	1	no	—
56	C₄×M₁₆	—	37	2	no	—
57	#330	—	37	4	no	—
58	C₃₇	#1526	1	1	yes	characteristic, complemented, Sylow
59	C₂×M₁₆	—	74	2	no	—
60	C₄⋊Q₈	—	37	2	no	—
61	#125	—	37	2	no	—
62	C₂²⋊Q₈	—	74	2	no	—
63	C₄×Q₈	—	74	2	no	—
64	#117	—	37	1	no	—
65	#117	#3866	1	1	yes	characteristic
66	C₂×C₄²	—	37	1	no	—
67	#97	—	37	4	no	—
68	C₄×C₈	—	74	2	no	—
69	C₂×Q₈	—	74	2	no	—
70	C₂²×C₄	—	37	2	no	—
71	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
72	M₁₆	—	74	8	no	—
73	C₂×C₈	—	74	4	no	—
74	C₄⋊C₄	—	37	2	no	—
75	C₄⋊C₄	—	74	2	no	—
76	C₄⋊C₄	—	74	2	no	—
77	C₄⋊C₄	—	74	2	no	—
78	C₄⋊C₄	C₂×#3866	1	2	yes	—
79	C₂²⋊C₄	—	2	1	no	—
80	C₂²⋊C₄	—	74	1	no	—
81	C₄²	—	2	1	no	—
82	C₄²	—	37	2	no	—
83	C₄²	—	37	1	no	—
84	C₄²	—	37	1	no	—
85	C₄²	—	74	1	no	—
86	C₂³	C₂²⋊#3866	1	1	yes	characteristic
87	Q₈	—	148	4	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	37	2	no	—
91	C₂×C₄	—	37	2	no	—
92	C₂×C₄	—	74	4	no	—
93	C₂×C₄	—	74	2	no	—
94	C₂×C₄	C₂²⋊#3866	1	1	yes	characteristic
95	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
96	C₈	—	74	8	no	—
97	C₂²	—	2	1	no	—
98	C₂²	—	2	2	no	—
99	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	4	2	no	—
102	C₄	—	37	4	no	—
103	C₄	—	74	2	no	—
104	C₄	—	148	2	no	—
105	C₄	#11924597	1	2	yes	—
106	C₂	—	4	1	no	—
107	C₂	C₂×#11924597	1	1	yes	characteristic
108	C₂	D₃₇⋊#104	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#10596957

C₂

yes

maximal

#14558390

C₂

yes

maximal, complemented

#14558086

C₂

yes

maximal, characteristic

C₂×#10596957

—

#11924591

C₂²

yes

—

C₄×#64438

—

#11924535

C₄

yes

—

#11924527

C₂²

yes

—

#11924513

—

C₄×Dic₇₄

—

#11924495

C₄

yes

characteristic

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

#117×C₃₇

C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

D₄

yes

—

C₂×Dic₇₄

—

#10596957

—

C₂×#64438

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

—

C₂²⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

#10596936

—

#10596936

—

C₄×Dic₃₇

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

D₄

yes

—

C₄×Dic₃₇

C₂³

yes

characteristic

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

—

Dic₇₄

—

#64438

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#101

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

#104

yes

—

#1526

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂×#101

yes

characteristic

C₂×C₃₇

C₂×#104

yes

—

#533

—

C₄×M₁₆

—

#330

—

C₃₇

#1526

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

—

yes

C₂×C₄²

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×#3866

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×#3866

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂³

C₂²⋊#3866

yes

characteristic

Q₈

—

148

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊#3866

yes

characteristic

C₂×C₄

C₂²×#3866

yes

characteristic

C₈

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊#3866

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

148

—

105

C₄

#11924597

yes

—

106

C₂

—

107

C₂

C₂×#11924597

yes

characteristic

108

C₂

D₃₇⋊#104

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²⋊#3866
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	#117×C₃₇	C₄
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₃₇	C₂²⋊C₄
cosocle	C₄×Dic₃₇	C₂³
hypercenter	#117	#3866
hypocenter	C₃₇	#1526
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

subgroup

G/H

center

C₂×C₄

C₂²⋊#3866

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#3866

Fitting

#117×C₃₇

C₄

socle

C₂²×C₃₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₃₇

C₂²⋊C₄

cosocle

C₄×Dic₃₇

C₂³

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

N	H ≅ G/N	type
2. #14558390	C₂	semidirect
58. C₃₇	#1526	semidirect

H ≅ G/N

type

2. #14558390

C₂

semidirect

58. C₃₇

#1526

semidirect

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

subgroup	index
C₄×#10596957	2
#14558390	2
#14558086	2
#1526	37

subgroup

index

Subgroups — #18010053

Conjugacy classes of subgroups

Complements

Maximal subgroups