Group #18010129 — subgroups

Subgroups — #18010129

Order 4736.

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

normal subgroups only

This group has 228 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	#14558553	C₂	1	2	yes	maximal
2	#14558393	C₂	1	4	yes	maximal, complemented
3	C₂×#11924534	C₂	1	1	yes	maximal, characteristic
4	#11924603	C₂²	1	4	yes	—
5	C₂²⋊#64438	—	2	2	no	—
6	#11924591	—	2	2	no	—
7	C₂²×Dic₇₄	C₄	1	1	yes	characteristic
8	#11924534	—	2	2	no	—
9	#11924534	C₄	1	2	yes	—
10	#11924534	C₂²	1	2	yes	—
11	C₂×#10596936	—	2	2	no	—
12	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₃₇	C₄	1	1	yes	characteristic
14	C₂×Q₈×C₃₇	—	2	1	no	—
15	C₂×Q₈×C₃₇	—	4	2	no	—
16	C₂×Q₈×C₃₇	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₃₇	—	2	1	no	—
18	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₃₇	—	2	1	no	—
20	C₂²×Dic₃₇	D₄	1	2	yes	—
21	C₂×Dic₇₄	—	2	1	no	—
22	C₂×Dic₇₄	—	4	2	no	—
23	C₂×Dic₇₄	C₂×C₄	1	2	yes	—
24	C₂×#64438	—	2	4	no	—
25	#10596936	—	2	4	no	—
26	#10596936	—	2	4	no	—
27	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₃₇	D₄	1	2	yes	—
29	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
30	C₂³×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
31	Q₈×C₃₇	—	4	2	no	—
32	Q₈×C₃₇	—	4	2	no	—
33	C₂×C₄×C₃₇	—	2	2	no	—
34	C₂×C₄×C₃₇	—	2	2	no	—
35	C₂×C₄×C₃₇	—	4	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	4	no	—
39	C₂×Dic₃₇	—	2	2	no	—
40	C₂×Dic₃₇	—	4	2	no	—
41	C₂×Dic₃₇	C₂²⋊C₄	1	2	yes	—
42	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
43	Dic₇₄	—	4	2	no	—
44	Dic₇₄	—	4	2	no	—
45	#64438	—	4	4	no	—
46	C₂²×C₃₇	—	2	2	no	—
47	C₂²×C₃₇	#100	1	2	yes	—
48	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	#137	1	4	yes	—
55	#1558	—	37	1	no	maximal, Sylow
56	C₂×C₃₇	—	2	2	no	—
57	C₂×C₃₇	C₂×#100	1	1	yes	characteristic
58	C₂×C₃₇	#419	1	2	yes	—
59	C₂×C₄⋊Q₈	—	37	1	no	—
60	#432	—	37	2	no	—
61	#333	—	37	4	no	—
62	C₃₇	#1558	1	1	yes	characteristic, complemented, Sylow
63	C₂²×Q₈	—	37	1	no	—
64	C₂²×Q₈	#3866	1	1	yes	characteristic
65	C₄⋊Q₈	—	37	2	no	—
66	C₄⋊Q₈	—	37	2	no	—
67	C₄⋊Q₈	—	74	2	no	—
68	C₂×C₄⋊C₄	—	74	2	no	—
69	C₂×C₄²	—	37	1	no	—
70	C₄⋊C₈	—	37	4	no	—
71	C₂²⋊C₈	—	74	2	no	—
72	#97	—	74	2	no	—
73	C₂×Q₈	—	2	1	no	—
74	C₂×Q₈	—	4	2	no	—
75	C₂×Q₈	—	37	2	no	—
76	C₂×Q₈	—	74	1	no	—
77	C₂×Q₈	—	148	2	no	—
78	C₂×Q₈	C₂×#3866	1	2	yes	—
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	—	37	2	no	—
81	C₂²×C₄	—	74	1	no	—
82	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
83	C₂×C₈	—	74	4	no	—
84	C₄⋊C₄	—	74	4	no	—
85	C₄⋊C₄	—	74	4	no	—
86	C₄²	—	37	2	no	—
87	C₄²	—	37	1	no	—
88	C₄²	—	37	1	no	—
89	C₂³	C₂²⋊#3866	1	1	yes	characteristic
90	Q₈	—	4	2	no	—
91	Q₈	—	4	2	no	—
92	Q₈	—	148	2	no	—
93	Q₈	—	148	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	37	2	no	—
98	C₂×C₄	—	37	2	no	—
99	C₂×C₄	—	74	4	no	—
100	C₂×C₄	—	74	2	no	—
101	C₂×C₄	—	148	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	—	2	2	no	—
106	C₂²	#11924596	1	2	yes	—
107	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
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	2	no	—
115	C₂	C₂×#11924596	1	1	yes	characteristic
116	C₂	#14558526	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558553

C₂

yes

maximal

#14558393

C₂

yes

maximal, complemented

C₂×#11924534

C₂

yes

maximal, characteristic

#11924603

C₂²

yes

—

C₂²⋊#64438

—

#11924591

—

C₂²×Dic₇₄

C₄

yes

characteristic

#11924534

—

#11924534

C₄

yes

—

#11924534

C₂²

yes

—

C₂×#10596936

—

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

C₂²×Q₈×C₃₇

C₄

yes

characteristic

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

—

C₂²×Dic₃₇

D₄

yes

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

C₂×C₄

yes

—

C₂×#64438

—

#10596936

—

#10596936

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

D₄

yes

—

C₄×Dic₃₇

C₂³

yes

characteristic

C₂³×C₃₇

C₂²⋊C₄

yes

characteristic

Q₈×C₃₇

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

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

—

C₂×Dic₃₇

C₂²⋊C₄

yes

—

C₂×Dic₃₇

C₂×D₄

yes

—

Dic₇₄

—

Dic₇₄

—

#64438

—

C₂²×C₃₇

—

C₂²×C₃₇

#100

yes

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

#137

yes

—

#1558

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂×#100

yes

characteristic

C₂×C₃₇

#419

yes

—

C₂×C₄⋊Q₈

—

#432

—

#333

—

C₃₇

#1558

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

#3866

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

148

—

C₂×Q₈

C₂×#3866

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#3866

yes

characteristic

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₂³

C₂²⋊#3866

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

148

—

Q₈

—

148

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

148

—

102

C₂×C₄

C₂²⋊#3866

yes

characteristic

103

C₂×C₄

C₂²×#3866

yes

characteristic

104

C₈

—

148

—

105

C₂²

—

106

C₂²

#11924596

yes

—

107

C₂²

C₂×C₂²⋊#3866

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

148

—

114

C₂

—

115

C₂

C₂×#11924596

yes

characteristic

116

C₂

#14558526

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#3866
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	C₂²×Q₈×C₃₇	C₄
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₃₇	C₂²⋊C₄
cosocle	C₄×Dic₃₇	C₂³
hypercenter	C₂²×Q₈	#3866
hypocenter	C₃₇	#1558
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#3866

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#3866

Fitting

C₂²×Q₈×C₃₇

C₄

socle

C₂²×C₃₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₃₇

C₂²⋊C₄

cosocle

C₄×Dic₃₇

C₂³

C₂²×Q₈

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

N	H ≅ G/N	type
2. #14558393	C₂	semidirect
62. C₃₇	#1558	semidirect

H ≅ G/N

type

2. #14558393

C₂

semidirect

62. C₃₇

#1558

semidirect

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

subgroup	index
#14558553	2
#14558393	2
C₂×#11924534	2
#1558	37

subgroup

index

Subgroups — #18010129

Conjugacy classes of subgroups

Complements

Maximal subgroups