Group #20242753 — subgroups

Subgroups — #20242753

Order 6016.

172 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 61 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	#15689503	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689403	C₂	1	2	yes	maximal
3	#15688876	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688870	C₂	1	2	yes	maximal, complemented
5	C₈×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₈₈	—	2	1	no	—
7	C₂×Dic₁₈₈	—	2	1	no	—
8	#13094024	C₂²	1	2	yes	—
9	#13093957	—	2	2	no	—
10	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	#13093945	—	2	2	no	—
12	C₂²×C₈×C₄₇	—	2	1	no	—
13	C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²⋊C₈×C₄₇	—	2	1	no	—
16	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₄₇	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₄₇	—	2	1	no	—
19	C₂×Dic₉₄	—	2	2	no	—
20	C₂×C₈×C₄₇	—	2	1	no	—
21	C₂×C₈×C₄₇	—	4	2	no	—
22	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
23	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
24	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
25	C₂²⋊C₄×C₄₇	—	2	1	no	—
26	C₄²×C₄₇	C₂³	1	1	yes	characteristic
27	C₄⋊Dic₄₇	—	2	2	no	—
28	C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄₇⋊C₈	—	2	2	no	—
30	Dic₁₈₈	—	4	2	no	—
31	Dic₁₈₈	—	4	2	no	—
32	C₂³×C₄₇	—	2	1	no	—
33	D₄×C₄₇	—	2	1	no	—
34	D₄×C₄₇	Q₁₆	1	2	yes	complemented
35	C₂×C₄×C₄₇	—	2	1	no	—
36	C₂×C₄×C₄₇	—	4	1	no	—
37	C₂×C₄×C₄₇	—	4	1	no	—
38	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
41	C₂×Dic₄₇	—	4	2	no	—
42	Dic₉₄	—	4	2	no	—
43	Dic₉₄	—	4	2	no	—
44	C₈×C₄₇	—	2	1	no	—
45	C₈×C₄₇	—	4	1	no	—
46	C₈×C₄₇	D₈	1	2	yes	—
47	C₄₇⋊C₈	—	4	2	no	—
48	C₂²×C₄₇	—	2	2	no	—
49	C₂²×C₄₇	—	4	1	no	—
50	C₂²×C₄₇	—	4	1	no	—
51	C₂²×C₄₇	#121	1	1	yes	characteristic
52	C₄×C₄₇	—	2	1	no	—
53	C₄×C₄₇	—	4	1	no	—
54	C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
55	C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
56	C₄×C₄₇	#135	1	1	yes	characteristic
57	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
58	Dic₄₇	—	8	2	no	—
59	#1669	—	47	1	no	maximal, Sylow
60	C₂×C₄₇	—	4	2	no	—
61	C₂×C₄₇	#459	1	1	yes	characteristic
62	C₂×C₄₇	#467	1	1	yes	characteristic
63	C₂×C₄₇	#473	1	1	yes	characteristic
64	#494	—	47	1	no	—
65	#474	—	47	2	no	—
66	C₈×D₄	D₄₇	1	1	yes	characteristic
67	#335	—	47	1	no	—
68	#332	—	47	2	no	—
69	C₄₇	#1669	1	1	yes	characteristic, complemented, Sylow
70	C₂×Q₁₆	—	94	1	no	—
71	C₂×Q₁₆	—	94	1	no	—
72	C₂²×C₈	—	2	1	no	—
73	C₄⋊Q₈	—	47	2	no	—
74	C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
75	#107	—	94	2	no	—
76	C₄⋊C₈	—	47	2	no	—
77	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
78	#102	—	94	2	no	—
79	C₂²⋊C₈	—	2	1	no	—
80	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
81	C₂×Q₈	—	94	2	no	—
82	C₂×D₄	D₁₈₈	1	1	yes	characteristic
83	C₂²×C₄	—	2	1	no	—
84	Q₁₆	—	188	2	no	—
85	Q₁₆	—	188	2	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	4	2	no	—
88	C₂×C₈	—	94	2	no	—
89	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
91	C₄⋊C₄	—	94	2	no	—
92	C₄⋊C₄	—	94	2	no	—
93	C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
94	C₂²⋊C₄	—	2	1	no	—
95	C₄²	C₂²×D₄₇	1	1	yes	characteristic
96	C₂³	—	2	1	no	—
97	Q₈	—	188	2	no	—
98	Q₈	—	188	2	no	—
99	D₄	—	2	1	no	—
100	D₄	Dic₁₈₈	1	2	yes	complemented
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	4	1	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	—	188	2	no	—
105	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
106	C₂×C₄	#10625350	1	1	yes	characteristic
107	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
108	C₈	—	2	1	no	—
109	C₈	—	4	1	no	—
110	C₈	—	188	2	no	—
111	C₈	#10625334	1	2	yes	—
112	C₂²	—	2	2	no	—
113	C₂²	—	4	1	no	—
114	C₂²	—	4	1	no	—
115	C₂²	#13093998	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	4	1	no	—
118	C₄	—	376	2	no	—
119	C₄	C₂×#10625334	1	1	yes	characteristic
120	C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
121	C₄	#13094046	1	1	yes	characteristic
122	C₄	#13094049	1	1	yes	characteristic
123	C₂	—	4	2	no	—
124	C₂	#15689318	1	1	yes	characteristic
125	C₂	#15689364	1	1	yes	characteristic
126	C₂	#15689398	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689503

C₂

yes

maximal, characteristic, complemented

#15689403

C₂

yes

maximal

#15688876

C₂

yes

maximal, characteristic, complemented

#15688870

C₂

yes

maximal, complemented

C₈×D₄×C₄₇

C₂

yes

maximal, characteristic

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

#13094024

C₂²

yes

—

#13093957

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093945

—

C₂²×C₈×C₄₇

—

C₄×D₄×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×D₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₄

yes

characteristic

C₂×C₈×C₄₇

D₄

yes

characteristic

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

Dic₁₈₈

—

Dic₁₈₈

—

C₂³×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

Q₁₆

yes

complemented

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

D₈

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#121

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₈

yes

characteristic

C₄×C₄₇

C₂×Q₁₆

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

#1669

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

C₈×D₄

D₄₇

yes

characteristic

#335

—

#332

—

C₄₇

#1669

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄×D₄

C₂×D₄₇

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

characteristic

#102

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×D₄

D₁₈₈

yes

characteristic

C₂²×C₄

—

Q₁₆

—

188

—

Q₁₆

—

188

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₁₈₈

yes

characteristic

C₂²⋊C₄

—

C₄²

C₂²×D₄₇

yes

characteristic

C₂³

—

Q₈

—

188

—

Q₈

—

188

—

D₄

—

100

D₄

Dic₁₈₈

yes

complemented

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

188

—

105

C₂×C₄

C₂×D₁₈₈

yes

characteristic

106

C₂×C₄

#10625350

yes

characteristic

107

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

188

—

111

C₈

#10625334

yes

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

#13093998

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

376

—

119

C₄

C₂×#10625334

yes

characteristic

120

C₄

C₂×Dic₁₈₈

yes

characteristic

121

C₄

#13094046

yes

characteristic

122

C₄

#13094049

yes

characteristic

123

C₂

—

124

C₂

#15689318

yes

characteristic

125

C₂

#15689364

yes

characteristic

126

C₂

#15689398

yes

characteristic

	subgroup	G/H
center	C₂²	#13093998
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₈×D₄×C₄₇	C₂
socle	C₂²×C₄₇	#121
hypercenter	C₈×D₄	D₄₇
hypocenter	C₄₇	#1669
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄

subgroup

G/H

center

C₂²

#13093998

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₈×D₄×C₄₇

C₂

socle

C₂²×C₄₇

#121

hypercenter

C₈×D₄

D₄₇

hypocenter

C₄₇

#1669

LCS

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄

N	H ≅ G/N	type
1. #15689503	C₂	semidirect
3. #15688876	C₂	semidirect
4. #15688870	C₂	semidirect
34. D₄×C₄₇	Q₁₆	semidirect
69. C₄₇	#1669	semidirect
100. D₄	Dic₁₈₈	semidirect

H ≅ G/N

type

1. #15689503

C₂

semidirect

3. #15688876

C₂

semidirect

4. #15688870

C₂

semidirect

34. D₄×C₄₇

Q₁₆

semidirect

69. C₄₇

#1669

semidirect

100. D₄

Dic₁₈₈

semidirect

Subgroups — #20242753

Conjugacy classes of subgroups

Complements