Group #20243554 — subgroups

Subgroups — #20243554

Order 6016.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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×#13093957	C₂	1	2	yes	maximal
2	C₂×#13093955	C₂	1	2	yes	maximal
3	C₂×#13093935	C₂	1	1	yes	maximal, characteristic
4	#15688989	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#384	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
9	#13093957	—	2	4	no	—
10	#13093955	—	2	4	no	—
11	#13093935	C₄	1	4	yes	complemented
12	C₄₇⋊#95	—	2	2	no	—
13	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
14	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₄₇	—	2	2	no	—
17	C₂²×C₄×C₄₇	D₄	1	2	yes	—
18	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
19	C₂²×Dic₄₇	—	2	1	no	—
20	C₂²×Dic₄₇	—	2	1	no	—
21	C₄⋊C₄×C₄₇	—	2	2	no	—
22	C₄⋊C₄×C₄₇	—	2	2	no	—
23	C₄⋊C₄×C₄₇	—	4	4	no	—
24	C₄²×C₄₇	C₂×C₄	1	2	yes	—
25	C₄²×C₄₇	Q₈	1	2	yes	—
26	C₄⋊Dic₄₇	—	2	2	no	—
27	C₄⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
29	C₂×C₄₇⋊C₈	D₄	1	4	yes	—
30	C₂×C₄₇⋊C₈	Q₈	1	4	yes	—
31	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₄₇	—	2	2	no	—
33	C₂×C₄×C₄₇	—	2	2	no	—
34	C₂×C₄×C₄₇	—	4	2	no	—
35	C₂×C₄×C₄₇	—	4	4	no	—
36	C₂×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
37	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
41	C₂×Dic₄₇	—	4	1	no	—
42	C₂×Dic₄₇	—	4	1	no	—
43	C₂×Dic₄₇	—	4	2	no	—
44	C₂×Dic₄₇	—	4	2	no	—
45	C₄₇⋊C₈	C₄⋊C₄	1	8	yes	complemented
46	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄₇	#127	1	1	yes	characteristic
48	C₂²×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
49	C₂²×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
50	C₂²×C₄₇	#137	1	2	yes	—
51	C₄×C₄₇	—	2	4	no	—
52	C₄×C₄₇	—	8	2	no	—
53	C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
54	C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
55	Dic₄₇	—	8	1	no	—
56	Dic₄₇	—	8	1	no	—
57	#1847	—	47	1	no	maximal, Sylow
58	C₂×C₄₇	#384	1	1	yes	characteristic
59	C₂×C₄₇	#428	1	2	yes	—
60	C₂×C₄₇	#496	1	1	yes	characteristic
61	C₂×C₄₇	#497	1	1	yes	characteristic
62	C₂×C₄₇	#501	1	2	yes	—
63	C₂×#107	—	47	2	no	—
64	C₂×#106	—	47	2	no	—
65	C₂×#97	—	47	1	no	—
66	#384	D₄₇	1	1	yes	characteristic
67	#382	—	47	1	no	—
68	C₄₇	#1847	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₈	—	47	2	no	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	47	1	no	—
72	C₂×C₄⋊C₄	—	47	1	no	—
73	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
74	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
75	#107	—	94	4	no	—
76	#106	—	94	4	no	—
77	#97	—	47	4	no	—
78	#95	—	94	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	94	1	no	—
81	C₂²×C₄	—	94	1	no	—
82	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
83	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
84	C₂×C₈	—	47	4	no	—
85	C₂×C₈	—	47	4	no	—
86	C₂×C₈	—	47	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	4	4	no	—
90	C₄⋊C₄	—	94	2	no	—
91	C₄⋊C₄	—	94	2	no	—
92	C₄²	Dic₉₄	1	2	yes	—
93	C₄²	C₄×D₄₇	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	188	1	no	—
100	C₂×C₄	—	188	1	no	—
101	C₂×C₄	—	188	2	no	—
102	C₂×C₄	—	188	2	no	—
103	C₂×C₄	Dic₄₇⋊C₄	1	4	yes	—
104	C₂×C₄	C₂×Dic₉₄	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
106	C₂×C₄	D₄×D₄₇	1	2	yes	—
107	C₂×C₄	Q₈×D₄₇	1	2	yes	—
108	C₈	—	47	8	no	—
109	C₂²	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
110	C₂²	#13094020	1	1	yes	characteristic
111	C₂²	#13094022	1	1	yes	characteristic
112	C₂²	#13094050	1	2	yes	—
113	C₂²	#13094055	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	8	2	no	—
116	C₄	—	376	1	no	—
117	C₄	—	376	1	no	—
118	C₄	C₄⋊C₄×D₄₇	1	2	yes	—
119	C₄	#13094023	1	2	yes	—
120	C₂	#15688992	1	1	yes	characteristic
121	C₂	#15689163	1	2	yes	—
122	C₂	#15689510	1	1	yes	characteristic
123	C₂	#15689515	1	1	yes	characteristic
124	C₂	#15689529	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093957

C₂

yes

maximal

C₂×#13093955

C₂

yes

maximal

C₂×#13093935

C₂

yes

maximal, characteristic

#15688989

C₂

yes

maximal, characteristic

C₄₇×#384

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

—

C₄

yes

complemented

C₄₇⋊#95

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

Q₈

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

D₄

yes

—

C₂×C₄₇⋊C₈

Q₈

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

C₄⋊C₄

yes

complemented

C₂²×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄₇

#127

yes

characteristic

C₂²×C₄₇

C₄⋊Q₈

yes

characteristic

C₂²×C₄₇

AGL(1,ℤ/8)

yes

—

C₂²×C₄₇

#137

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₄×C₄₇

C₄⋊Q₈

yes

—

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×#107

—

C₂×#106

—

C₂×#97

—

#384

D₄₇

yes

characteristic

#382

—

C₄₇

#1847

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

—

—

—

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Dic₉₄

yes

—

C₄²

C₄×D₄₇

yes

—

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

188

—

100

C₂×C₄

—

188

—

101

C₂×C₄

—

188

—

102

C₂×C₄

—

188

—

103

C₂×C₄

Dic₄₇⋊C₄

yes

—

104

C₂×C₄

C₂×Dic₉₄

yes

characteristic

105

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

106

C₂×C₄

D₄×D₄₇

yes

—

107

C₂×C₄

Q₈×D₄₇

yes

—

108

C₈

—

109

C₂²

C₂×Dic₄₇⋊C₄

yes

characteristic

110

C₂²

#13094020

yes

characteristic

111

C₂²

#13094022

yes

characteristic

112

C₂²

#13094050

yes

—

113

C₂²

#13094055

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

376

—

117

C₄

—

376

—

118

C₄

C₄⋊C₄×D₄₇

yes

—

119

C₄

#13094023

yes

—

120

C₂

#15688992

yes

characteristic

121

C₂

#15689163

yes

—

122

C₂

#15689510

yes

characteristic

123

C₂

#15689515

yes

characteristic

124

C₂

#15689529

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#384	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	#384	D₄₇
hypocenter	C₄₇	#1847
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#384

C₂

socle

C₂³×C₄₇

C₂×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
11. #13093935	C₄	semidirect
45. C₄₇⋊C₈	C₄⋊C₄	semidirect
68. C₄₇	#1847	semidirect

H ≅ G/N

type

11. #13093935

C₄

semidirect

45. C₄₇⋊C₈

C₄⋊C₄

semidirect

68. C₄₇

#1847

semidirect

Subgroups — #20243554

Conjugacy classes of subgroups

Complements