Group #20242748 — subgroups

Subgroups — #20242748

Order 6016.

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

normal subgroups only

This group has 188 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	#15689500	C₂	1	1	yes	maximal, characteristic
2	#15689323	C₂	1	2	yes	maximal, complemented
3	#15688876	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688866	C₂	1	2	yes	maximal, complemented
5	C₈×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10625336	—	2	2	no	—
7	C₂×D₃₇₆	—	2	1	no	—
8	C₂×D₃₇₆	—	2	1	no	—
9	#13094021	C₂²	1	2	yes	—
10	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	#13093944	—	2	2	no	—
12	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₄₇	—	2	1	no	—
14	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₄₇	—	2	1	no	—
16	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
18	C₂×D₁₈₈	—	2	2	no	—
19	C₂×D₁₈₈	—	2	2	no	—
20	C₂×D₁₈₈	—	2	2	no	—
21	C₂×C₈×C₄₇	—	2	1	no	—
22	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
23	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
24	C₄⋊C₄×C₄₇	—	2	1	no	—
25	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
26	C₄²×C₄₇	—	2	1	no	—
27	C₄²×C₄₇	C₂³	1	1	yes	characteristic
28	#10625336	—	4	4	no	—
29	C₂×C₄₇⋊C₈	—	2	2	no	—
30	D₃₇₆	—	4	2	no	—
31	D₃₇₆	—	4	2	no	—
32	C₂²×D₄₇	—	4	2	no	—
33	Q₈×C₄₇	—	2	1	no	—
34	Q₈×C₄₇	D₈	1	2	yes	complemented
35	C₂×C₄×C₄₇	—	2	1	no	—
36	C₂×C₄×C₄₇	—	2	1	no	—
37	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
40	D₁₈₈	—	4	2	no	—
41	D₁₈₈	—	4	2	no	—
42	D₁₈₈	—	4	2	no	—
43	D₁₈₈	—	4	2	no	—
44	D₁₈₈	—	8	2	no	—
45	C₈×C₄₇	—	2	1	no	—
46	C₈×C₄₇	—	4	1	no	—
47	C₈×C₄₇	SD₁₆	1	2	yes	complemented
48	C₄₇⋊C₈	—	4	2	no	—
49	C₂²×C₄₇	#121	1	1	yes	characteristic
50	C₂×D₄₇	—	8	2	no	—
51	C₂×D₄₇	—	8	2	no	—
52	C₂×D₄₇	—	8	2	no	—
53	C₄×C₄₇	—	2	2	no	—
54	C₄×C₄₇	—	2	1	no	—
55	C₄×C₄₇	—	4	1	no	—
56	C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
57	C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
58	C₄×C₄₇	#135	1	1	yes	characteristic
59	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
60	#1668	—	47	1	no	maximal, Sylow
61	C₂×C₄₇	#460	1	1	yes	characteristic
62	C₂×C₄₇	#466	1	1	yes	characteristic
63	C₂×C₄₇	#472	1	1	yes	characteristic
64	D₄₇	—	16	2	no	—
65	#493	—	47	1	no	—
66	#460	—	47	2	no	—
67	C₈×Q₈	D₄₇	1	1	yes	characteristic, complemented
68	#335	—	47	1	no	—
69	#331	—	47	2	no	—
70	C₄₇	#1668	1	1	yes	characteristic, complemented, Sylow
71	C₂×SD₁₆	—	94	2	no	—
72	C₂×D₈	—	94	1	no	—
73	C₂×D₈	—	94	1	no	—
74	#127	—	47	2	no	—
75	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
76	C₄⋊C₈	—	2	1	no	—
77	C₄⋊C₈	—	47	2	no	—
78	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
79	#102	—	94	2	no	—
80	C₄×C₈	—	2	1	no	—
81	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
82	C₂×Q₈	D₁₈₈	1	1	yes	characteristic
83	C₂×D₄	—	94	2	no	—
84	C₂×D₄	—	94	2	no	—
85	C₂×D₄	—	94	2	no	—
86	SD₁₆	—	188	4	no	—
87	D₈	—	188	2	no	—
88	D₈	—	188	2	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	94	2	no	—
91	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
92	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
95	C₄²	—	2	1	no	—
96	C₄²	C₂²×D₄₇	1	1	yes	characteristic
97	C₂³	—	188	2	no	—
98	Q₈	—	2	1	no	—
99	Q₈	D₃₇₆	1	2	yes	complemented
100	D₄	—	376	2	no	—
101	D₄	—	188	2	no	—
102	D₄	—	188	2	no	—
103	D₄	—	188	2	no	—
104	D₄	—	188	2	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
108	C₂×C₄	#10625350	1	1	yes	characteristic
109	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
110	C₈	—	2	1	no	—
111	C₈	—	4	1	no	—
112	C₈	—	188	2	no	—
113	C₈	#10625336	1	2	yes	complemented
114	C₂²	—	376	2	no	—
115	C₂²	—	376	2	no	—
116	C₂²	—	376	2	no	—
117	C₂²	#13093998	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	1	no	—
120	C₄	—	4	1	no	—
121	C₄	C₂×D₃₇₆	1	1	yes	characteristic
122	C₄	C₂×#10625336	1	1	yes	characteristic
123	C₄	#13094046	1	1	yes	characteristic
124	C₄	#13094049	1	1	yes	characteristic
125	C₂	—	752	2	no	—
126	C₂	#15689323	1	1	yes	characteristic
127	C₂	#15689359	1	1	yes	characteristic
128	C₂	#15689393	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689500

C₂

yes

maximal, characteristic

#15689323

C₂

yes

maximal, complemented

#15688876

C₂

yes

maximal, characteristic, complemented

#15688866

C₂

yes

maximal, complemented

C₈×Q₈×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×#10625336

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

#13094021

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093944

—

C₄×Q₈×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

D₄

yes

characteristic

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₄

yes

characteristic

C₂×C₈×C₄₇

D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625336

—

C₂×C₄₇⋊C₈

—

D₃₇₆

—

D₃₇₆

—

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

D₈

yes

complemented

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

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

SD₁₆

yes

complemented

C₄₇⋊C₈

—

C₂²×C₄₇

#121

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₈

yes

characteristic

C₄×C₄₇

C₂×SD₁₆

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

#493

—

#460

—

C₈×Q₈

D₄₇

yes

characteristic, complemented

#335

—

#331

—

C₄₇

#1668

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

#127

—

C₄×Q₈

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

characteristic

#102

—

C₄×C₈

—

C₄×C₈

C₂×D₄₇

yes

characteristic

C₂×Q₈

D₁₈₈

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

188

—

D₈

—

188

—

D₈

—

188

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

D₁₈₈

yes

characteristic

C₄²

—

C₄²

C₂²×D₄₇

yes

characteristic

C₂³

—

188

—

Q₈

—

Q₈

D₃₇₆

yes

complemented

100

D₄

—

376

—

101

D₄

—

188

—

102

D₄

—

188

—

103

D₄

—

188

—

104

D₄

—

188

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂×D₁₈₈

yes

characteristic

108

C₂×C₄

#10625350

yes

characteristic

109

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

188

—

113

C₈

#10625336

yes

complemented

114

C₂²

—

376

—

115

C₂²

—

376

—

116

C₂²

—

376

—

117

C₂²

#13093998

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₂×D₃₇₆

yes

characteristic

122

C₄

C₂×#10625336

yes

characteristic

123

C₄

#13094046

yes

characteristic

124

C₄

#13094049

yes

characteristic

125

C₂

—

752

—

126

C₂

#15689323

yes

characteristic

127

C₂

#15689359

yes

characteristic

128

C₂

#15689393

yes

characteristic

	subgroup	G/H
center	C₂²	#13093998
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₈×Q₈×C₄₇	C₂
socle	C₂²×C₄₇	#121
hypercenter	C₈×Q₈	D₄₇
hypocenter	C₄₇	#1668
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#13093998

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₈×Q₈×C₄₇

C₂

socle

C₂²×C₄₇

#121

hypercenter

C₈×Q₈

D₄₇

hypocenter

C₄₇

#1668

LCS

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
2. #15689323	C₂	semidirect
3. #15688876	C₂	semidirect
4. #15688866	C₂	semidirect
5. C₈×Q₈×C₄₇	C₂	semidirect
34. Q₈×C₄₇	D₈	semidirect
47. C₈×C₄₇	SD₁₆	semidirect
67. C₈×Q₈	D₄₇	semidirect
70. C₄₇	#1668	semidirect
99. Q₈	D₃₇₆	semidirect
113. C₈	#10625336	semidirect

H ≅ G/N

type

2. #15689323

C₂

semidirect

3. #15688876

C₂

semidirect

4. #15688866

C₂

semidirect

5. C₈×Q₈×C₄₇

C₂

semidirect

34. Q₈×C₄₇

D₈

semidirect

47. C₈×C₄₇

SD₁₆

semidirect

67. C₈×Q₈

D₄₇

semidirect

70. C₄₇

#1668

semidirect

99. Q₈

D₃₇₆

semidirect

113. C₈

#10625336

semidirect

Subgroups — #20242748

Conjugacy classes of subgroups

Complements