Group #20244337 — subgroups

Subgroups — #20244337

Order 6016.

292 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 129 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×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#15688885	C₂	1	4	yes	maximal
4	C₂×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
7	C₂×C₄×Dic₄₇	—	2	1	no	—
8	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
9	C₄₇⋊#95	—	2	1	no	—
10	C₂²×Q₈×C₄₇	—	2	1	no	—
11	C₄⋊Q₈×C₄₇	—	2	2	no	—
12	C₄⋊Q₈×C₄₇	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
14	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₄₇	—	2	4	no	—
17	C₂×Q₈×C₄₇	—	4	4	no	—
18	C₂²×C₄×C₄₇	—	2	2	no	—
19	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
22	C₂²×Dic₄₇	—	2	2	no	—
23	C₄⋊C₄×C₄₇	—	2	4	no	—
24	C₄⋊C₄×C₄₇	—	2	2	no	—
25	C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
26	C₄²×C₄₇	C₂×C₄	1	2	yes	—
27	C₄²×C₄₇	D₄	1	2	yes	—
28	C₄⋊Dic₄₇	—	2	2	no	—
29	C₄⋊Dic₄₇	Q₈	1	4	yes	complemented
30	C₄×Dic₄₇	—	4	2	no	—
31	C₂×C₄₇⋊C₈	—	2	2	no	—
32	C₂×C₄₇⋊C₈	—	2	4	no	—
33	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
34	Q₈×C₄₇	—	4	8	no	—
35	C₂×C₄×C₄₇	—	2	2	no	—
36	C₂×C₄×C₄₇	—	2	4	no	—
37	C₂×C₄×C₄₇	—	4	4	no	—
38	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₄₇	SD₁₆	1	2	yes	—
41	C₂×C₄×C₄₇	Q₁₆	1	2	yes	—
42	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₄₇	#41	1	2	yes	—
46	C₂×Dic₄₇	—	2	4	no	—
47	C₂×Dic₄₇	—	4	2	no	—
48	C₂×Dic₄₇	—	4	2	no	—
49	C₄₇⋊C₈	—	4	2	no	—
50	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄₇	C₂²⋊Q₈	1	2	yes	—
52	C₂²×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
54	C₂²×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₄₇	—	2	2	no	—
56	C₄×C₄₇	—	4	4	no	—
57	C₄×C₄₇	#103	1	4	yes	—
58	C₄×C₄₇	C₄×Q₈	1	2	yes	—
59	C₄×C₄₇	#124	1	1	yes	characteristic
60	C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
61	Dic₄₇	—	4	4	no	—
62	#1987	—	47	1	no	maximal, Sylow
63	C₂×C₄₇	#391	1	1	yes	characteristic
64	C₂×C₄₇	C₂×#103	1	1	yes	characteristic
65	C₂×C₄₇	#418	1	1	yes	characteristic
66	C₂×C₄₇	#476	1	2	yes	—
67	C₂×C₄₇	#477	1	2	yes	—
68	C₂×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
69	C₂×C₄⋊C₈	—	47	1	no	—
70	C₄×C₄⋊C₄	—	47	1	no	—
71	#340	—	47	4	no	—
72	C₄₇	#1987	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	—	2	1	no	—
74	C₂²×C₈	—	47	2	no	—
75	C₄⋊Q₈	—	2	2	no	—
76	C₄⋊Q₈	Dic₄₇	1	4	yes	complemented
77	C₂×C₄⋊C₄	—	2	1	no	—
78	C₂×C₄⋊C₄	—	47	2	no	—
79	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
80	C₂×C₄²	—	94	1	no	—
81	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
82	C₄⋊C₈	—	94	2	no	—
83	#95	—	94	1	no	—
84	C₂×Q₈	—	2	4	no	—
85	C₂×Q₈	—	4	4	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	94	2	no	—
88	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
89	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
91	C₂×C₈	—	94	4	no	—
92	C₂×C₈	—	94	2	no	—
93	C₄⋊C₄	—	2	4	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	47	4	no	—
96	C₄⋊C₄	—	94	2	no	—
97	C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
98	C₄²	—	188	2	no	—
99	C₄²	C₂×Dic₄₇	1	2	yes	—
100	C₄²	C₂²⋊D₄₇	1	2	yes	—
101	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
102	Q₈	—	4	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	4	4	no	—
106	C₂×C₄	—	94	4	no	—
107	C₂×C₄	—	188	2	no	—
108	C₂×C₄	—	188	2	no	—
109	C₂×C₄	#10625336	1	2	yes	—
110	C₂×C₄	#10625337	1	2	yes	—
111	C₂×C₄	C₂²⋊Dic₄₇	1	2	yes	—
112	C₂×C₄	C₂²⋊Dic₄₇	1	2	yes	—
113	C₂×C₄	Q₈×D₄₇	1	2	yes	—
114	C₂×C₄	#10625354	1	2	yes	—
115	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
116	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
117	C₈	—	188	2	no	—
118	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
119	C₂²	#13094000	1	2	yes	—
120	C₂²	C₂×#10625336	1	1	yes	characteristic
121	C₂²	C₂×#10625337	1	1	yes	characteristic
122	C₂²	#13094050	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	188	4	no	—
126	C₄	#13093948	1	4	yes	—
127	C₄	Q₈×Dic₄₇	1	2	yes	—
128	C₄	#13094013	1	1	yes	characteristic
129	C₄	#13094022	1	1	yes	characteristic
130	C₂	#15689030	1	1	yes	characteristic
131	C₂	C₂×#13093948	1	1	yes	characteristic
132	C₂	#15689128	1	1	yes	characteristic
133	C₂	#15689410	1	2	yes	—
134	C₂	#15689415	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₄₇

C₂

yes

maximal, characteristic

#15688885

C₂

yes

maximal

C₂×C₄⋊Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

—

C₄⋊(C₄₇⋊C₈)

—

C₄₇⋊#95

—

C₂²×Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

D₄

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

Q₈

yes

complemented

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

Q₁₆

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

C₂²⋊Q₈

yes

—

C₂²×C₄₇

C₂×SD₁₆

yes

characteristic

C₂²×C₄₇

C₂×Q₁₆

yes

characteristic

C₂²×C₄₇

AGL(1,ℤ/8)

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#103

yes

—

C₄×C₄₇

C₄×Q₈

yes

—

C₄×C₄₇

#124

yes

characteristic

C₄×C₄₇

C₄⋊Q₈

yes

characteristic

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#477

yes

—

C₂×C₄⋊Q₈

D₄₇

yes

characteristic

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

—

#340

—

C₄₇

#1987

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₄₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

C₂²×C₄

C₂²×D₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₄₇

yes

—

C₄²

—

188

—

C₄²

C₂×Dic₄₇

yes

—

100

C₄²

C₂²⋊D₄₇

yes

—

101

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

188

—

108

C₂×C₄

—

188

—

109

C₂×C₄

#10625336

yes

—

110

C₂×C₄

#10625337

yes

—

111

C₂×C₄

C₂²⋊Dic₄₇

yes

—

112

C₂×C₄

C₂²⋊Dic₄₇

yes

—

113

C₂×C₄

Q₈×D₄₇

yes

—

114

C₂×C₄

#10625354

yes

—

115

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

116

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

117

C₈

—

188

—

118

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

119

C₂²

#13094000

yes

—

120

C₂²

C₂×#10625336

yes

characteristic

121

C₂²

C₂×#10625337

yes

characteristic

122

C₂²

#13094050

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

188

—

126

C₄

#13093948

yes

—

127

C₄

Q₈×Dic₄₇

yes

—

128

C₄

#13094013

yes

characteristic

129

C₄

#13094022

yes

characteristic

130

C₂

#15689030

yes

characteristic

131

C₂

C₂×#13093948

yes

characteristic

132

C₂

#15689128

yes

characteristic

133

C₂

#15689410

yes

—

134

C₂

#15689415

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×C₄⋊Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#1987
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×C₄⋊Q₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₄₇

hypocenter

C₄₇

#1987

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₄₇	C₄	semidirect
29. C₄⋊Dic₄₇	Q₈	semidirect
72. C₄₇	#1987	semidirect
76. C₄⋊Q₈	Dic₄₇	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₄₇

C₄

semidirect

29. C₄⋊Dic₄₇

Q₈

semidirect

72. C₄₇

#1987

semidirect

76. C₄⋊Q₈

Dic₄₇

semidirect

Subgroups — #20244337

Conjugacy classes of subgroups

Complements