Group #20242023 — subgroups

Subgroups — #20242023

Order 6016.

238 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 77 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	#15689180	C₂	1	2	yes	maximal, complemented
2	#15688863	C₂	1	4	yes	maximal, complemented
3	C₄₇×#532	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
5	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
6	#13093935	C₂²	1	4	yes	—
7	C₂×#41×C₄₇	C₄	1	2	yes	—
8	C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic
9	#127×C₄₇	C₄	1	1	yes	characteristic
10	#124×C₄₇	C₂²	1	2	yes	—
11	#121×C₄₇	—	2	2	no	—
12	C₄×D₄×C₄₇	—	2	2	no	—
13	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	#41×C₄₇	—	4	4	no	—
15	C₂×Q₈×C₄₇	C₂×C₄	1	2	yes	—
16	C₂×D₄×C₄₇	—	2	2	no	—
17	C₂×D₄×C₄₇	—	2	2	no	—
18	C₂×D₄×C₄₇	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₄₇	—	2	2	no	—
20	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	D₄	1	2	yes	—
22	C₄⋊C₄×C₄₇	—	2	2	no	—
23	C₂²⋊C₄×C₄₇	—	2	4	no	—
24	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₄₇	D₄	1	2	yes	—
26	C₄²×C₄₇	C₂³	1	1	yes	characteristic
27	C₂×C₄₇⋊C₈	—	2	4	no	—
28	C₂³×C₄₇	—	2	2	no	—
29	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
30	Q₈×C₄₇	—	4	2	no	—
31	D₄×C₄₇	—	4	2	no	—
32	D₄×C₄₇	—	4	4	no	—
33	D₄×C₄₇	—	4	4	no	—
34	C₂×C₄×C₄₇	—	2	2	no	—
35	C₂×C₄×C₄₇	—	2	2	no	—
36	C₂×C₄×C₄₇	—	4	2	no	—
37	C₂×C₄×C₄₇	—	4	4	no	—
38	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
42	C₄₇⋊C₈	—	4	4	no	—
43	C₂²×C₄₇	—	2	1	no	—
44	C₂²×C₄₇	—	2	2	no	—
45	C₂²×C₄₇	—	4	2	no	—
46	C₂²×C₄₇	—	4	4	no	—
47	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₄×C₄₇	—	2	1	no	—
49	C₄×C₄₇	—	2	2	no	—
50	C₄×C₄₇	—	2	2	no	—
51	C₄×C₄₇	—	4	2	no	—
52	C₄×C₄₇	#100	1	2	yes	—
53	#1536	—	47	1	no	maximal, Sylow
54	C₂×C₄₇	—	4	1	no	—
55	C₂×C₄₇	—	8	2	no	—
56	C₂×C₄₇	C₂×#100	1	1	yes	characteristic
57	C₂×C₄₇	#421	1	2	yes	—
58	#532	D₄₇	1	1	yes	characteristic
59	#432	—	47	2	no	—
60	#329	—	47	4	no	—
61	C₄₇	#1536	1	1	yes	characteristic, complemented, Sylow
62	C₂×#41	Dic₄₇	1	2	yes	—
63	C₄⋊Q₈	Dic₄₇	1	1	yes	characteristic
64	#127	Dic₄₇	1	1	yes	characteristic
65	#124	C₂×D₄₇	1	2	yes	—
66	#121	—	2	2	no	—
67	C₄×D₄	—	2	2	no	—
68	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
69	C₄⋊C₈	—	94	2	no	—
70	C₂²⋊C₈	—	94	2	no	—
71	#97	—	47	4	no	—
72	#41	—	4	4	no	—
73	C₂×Q₈	C₂×Dic₄₇	1	2	yes	—
74	C₂×D₄	—	2	2	no	—
75	C₂×D₄	—	2	2	no	—
76	C₂×D₄	C₂×Dic₄₇	1	2	yes	—
77	C₂²×C₄	—	2	2	no	—
78	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
80	C₂×C₈	—	94	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₂²⋊C₄	—	2	4	no	—
83	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
84	C₄²	C₂²⋊D₄₇	1	2	yes	—
85	C₄²	C₂²×D₄₇	1	1	yes	characteristic
86	C₂³	—	2	2	no	—
87	C₂³	C₂²⋊Dic₄₇	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	D₄	—	4	2	no	—
90	D₄	—	4	4	no	—
91	D₄	—	4	4	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	4	no	—
96	C₂×C₄	C₂²⋊Dic₄₇	1	2	yes	—
97	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
98	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
99	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
100	C₈	—	188	4	no	—
101	C₂²	—	2	1	no	—
102	C₂²	—	2	2	no	—
103	C₂²	—	4	2	no	—
104	C₂²	—	4	4	no	—
105	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	2	2	no	—
108	C₄	—	2	2	no	—
109	C₄	—	4	2	no	—
110	C₄	#13093941	1	2	yes	—
111	C₂	—	4	1	no	—
112	C₂	—	8	2	no	—
113	C₂	C₂×#13093941	1	1	yes	characteristic
114	C₂	#15689140	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689180

C₂

yes

maximal, complemented

#15688863

C₂

yes

maximal, complemented

C₄₇×#532

C₂

yes

maximal, characteristic

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

C₂²

yes

—

C₂×#41×C₄₇

C₄

yes

—

C₄⋊Q₈×C₄₇

C₄

yes

characteristic

#127×C₄₇

C₄

yes

characteristic

#124×C₄₇

C₂²

yes

—

#121×C₄₇

—

C₄×D₄×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#41×C₄₇

—

C₂×Q₈×C₄₇

C₂×C₄

yes

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

—

C₄²×C₄₇

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#100

yes

—

#1536

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂×#100

yes

characteristic

C₂×C₄₇

#421

yes

—

#532

D₄₇

yes

characteristic

#432

—

#329

—

C₄₇

#1536

yes

characteristic, complemented, Sylow

C₂×#41

Dic₄₇

yes

—

C₄⋊Q₈

Dic₄₇

yes

characteristic

#127

Dic₄₇

yes

characteristic

#124

C₂×D₄₇

yes

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#41

—

C₂×Q₈

C₂×Dic₄₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₄₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×Dic₄₇

yes

characteristic

C₄²

C₂²⋊D₄₇

yes

—

C₄²

C₂²×D₄₇

yes

characteristic

C₂³

—

C₂³

C₂²⋊Dic₄₇

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₄₇

yes

—

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

100

C₈

—

188

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

#13093941

yes

—

111

C₂

—

112

C₂

—

113

C₂

C₂×#13093941

yes

characteristic

114

C₂

#15689140

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#532	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#532	D₄₇
hypocenter	C₄₇	#1536
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#532

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #15689180	C₂	semidirect
2. #15688863	C₂	semidirect
61. C₄₇	#1536	semidirect

H ≅ G/N

type

1. #15689180

C₂

semidirect

2. #15688863

C₂

semidirect

61. C₄₇

#1536

semidirect

Subgroups — #20242023

Conjugacy classes of subgroups

Complements