Group #20242127 — subgroups

Subgroups — #334×D₄₇ #20242127

Order 6016.

200 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₄⋊C₈×D₄₇	C₂	1	2	yes	maximal
2	C₄×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#15688874	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688873	C₂	1	2	yes	maximal, complemented
5	C₄₇×#334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₄₇	—	2	2	no	—
7	C₂×C₈×D₄₇	C₄	1	1	yes	characteristic
8	C₂×C₈×D₄₇	C₄	1	1	yes	characteristic
9	C₄²×D₄₇	C₂²	1	1	yes	characteristic
10	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	Dic₄₇⋊C₈	—	2	2	no	—
12	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₈×Dic₄₇	C₄	1	1	yes	characteristic
14	C₈×Dic₄₇	C₄	1	1	yes	characteristic
15	C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
16	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×D₄₇	D₄	1	1	yes	characteristic
19	C₂×C₄×D₄₇	Q₈	1	1	yes	characteristic
20	C₂×C₈×C₄₇	—	2	2	no	—
21	C₂×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₄₇	C₂³	1	1	yes	characteristic
24	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
25	C₄×Dic₄₇	D₄	1	1	yes	characteristic
26	C₄×Dic₄₇	Q₈	1	1	yes	characteristic
27	C₂×C₄₇⋊C₈	—	2	2	no	—
28	C₂×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₈×D₄₇	—	2	2	no	—
31	C₈×D₄₇	—	4	4	no	—
32	C₈×D₄₇	C₈	1	4	yes	complemented
33	C₂²×D₄₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
37	C₂×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
40	C₄×D₄₇	—	2	2	no	—
41	C₄×D₄₇	C₂×C₈	1	2	yes	—
42	C₄×D₄₇	M₁₆	1	2	yes	—
43	C₄×D₄₇	SD₁₆	1	4	yes	—
44	C₈×C₄₇	—	2	1	no	—
45	C₈×C₄₇	—	4	2	no	—
46	C₈×C₄₇	C₂×C₈	1	2	yes	complemented
47	C₄₇⋊C₈	—	2	1	no	—
48	C₄₇⋊C₈	—	4	2	no	—
49	C₄₇⋊C₈	C₂×C₈	1	2	yes	complemented
50	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×D₄₇	C₄⋊C₈	1	2	yes	—
52	C₂×D₄₇	#106	1	2	yes	—
53	C₂×D₄₇	#108	1	2	yes	—
54	C₄×C₄₇	—	2	1	no	—
55	C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
56	C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
57	C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
58	Dic₄₇	—	2	1	no	—
59	Dic₄₇	C₄⋊C₈	1	2	yes	—
60	Dic₄₇	#106	1	2	yes	—
61	C₂×#334	—	47	1	no	maximal, Sylow
62	C₂×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄₇	C₂×#106	1	1	yes	characteristic
64	C₂×C₄₇	C₂×#108	1	1	yes	characteristic
65	D₄₇	#334	1	4	yes	direct product factor
66	C₂×C₄⋊C₈	—	47	2	no	—
67	C₂×C₄×C₈	—	47	1	no	—
68	#334	—	47	2	no	—
69	#334	—	47	1	no	—
70	#334	D₄₇	1	1	yes	characteristic, direct product factor
71	C₄₇	C₂×#334	1	1	yes	characteristic, complemented, Sylow
72	C₂²×C₈	—	47	1	no	—
73	C₂²×C₈	—	47	1	no	—
74	C₂²×C₈	—	94	2	no	—
75	C₂×C₄²	—	47	1	no	—
76	C₄⋊C₈	—	47	2	no	—
77	C₄⋊C₈	—	94	2	no	—
78	C₄⋊C₈	C₂×D₄₇	1	2	yes	—
79	C₄×C₈	—	47	1	no	—
80	C₄×C₈	—	47	1	no	—
81	C₄×C₈	—	47	1	no	—
82	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
83	C₂²×C₄	—	47	1	no	—
84	C₂²×C₄	—	47	1	no	—
85	C₂²×C₄	—	47	1	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	47	1	no	—
88	C₂×C₈	—	47	4	no	—
89	C₂×C₈	—	47	1	no	—
90	C₂×C₈	—	94	2	no	—
91	C₂×C₈	—	94	2	no	—
92	C₂×C₈	—	188	4	no	—
93	C₂×C₈	C₄×D₄₇	1	1	yes	characteristic
94	C₂×C₈	C₄×D₄₇	1	1	yes	characteristic
95	C₄²	—	47	1	no	—
96	C₄²	—	47	1	no	—
97	C₄²	—	47	1	no	—
98	C₄²	C₂²×D₄₇	1	1	yes	characteristic
99	C₂³	—	47	1	no	—
100	C₂×C₄	—	47	1	no	—
101	C₂×C₄	—	47	1	no	—
102	C₂×C₄	—	47	2	no	—
103	C₂×C₄	—	47	2	no	—
104	C₂×C₄	—	47	4	no	—
105	C₂×C₄	—	47	1	no	—
106	C₂×C₄	—	94	2	no	—
107	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
108	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
109	C₂×C₄	Q₈×D₄₇	1	1	yes	characteristic
110	C₈	—	2	1	no	—
111	C₈	—	4	2	no	—
112	C₈	—	47	2	no	—
113	C₈	—	94	1	no	—
114	C₈	—	188	2	no	—
115	C₈	C₈×D₄₇	1	2	yes	complemented
116	C₂²	—	47	2	no	—
117	C₂²	—	47	2	no	—
118	C₂²	—	47	2	no	—
119	C₂²	C₄⋊C₄×D₄₇	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	47	2	no	—
122	C₄	—	47	2	no	—
123	C₄	—	94	1	no	—
124	C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
125	C₄	M₁₆×D₄₇	1	1	yes	characteristic
126	C₄	SD₁₆×D₄₇	1	2	yes	—
127	C₂	—	47	4	no	—
128	C₂	C₄⋊C₈×D₄₇	1	1	yes	characteristic
129	C₂	#106×D₄₇	1	1	yes	characteristic
130	C₂	#108×D₄₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊C₈×D₄₇

C₂

yes

maximal

C₄×C₈×D₄₇

C₂

yes

maximal, characteristic

#15688874

C₂

yes

maximal, characteristic, complemented

#15688873

C₂

yes

maximal, complemented

C₄₇×#334

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₄₇

—

C₂×C₈×D₄₇

C₄

yes

characteristic

C₂×C₈×D₄₇

C₄

yes

characteristic

C₄²×D₄₇

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

Dic₄₇⋊C₈

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₈×Dic₄₇

C₄

yes

characteristic

C₈×Dic₄₇

C₄

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×C₄×D₄₇

C₂×C₄

yes

characteristic

C₂×C₄×D₄₇

D₄

yes

characteristic

C₂×C₄×D₄₇

Q₈

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₄×Dic₄₇

C₂×C₄

yes

characteristic

C₄×Dic₄₇

D₄

yes

characteristic

C₄×Dic₄₇

Q₈

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₈×D₄₇

—

C₈×D₄₇

—

C₈×D₄₇

C₈

yes

complemented

C₂²×D₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×Dic₄₇

C₄⋊C₄

yes

characteristic

C₂×Dic₄₇

C₄⋊C₄

yes

characteristic

C₂×Dic₄₇

C₄⋊C₄

yes

characteristic

C₄×D₄₇

—

C₄×D₄₇

C₂×C₈

yes

—

C₄×D₄₇

M₁₆

yes

—

C₄×D₄₇

SD₁₆

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂×C₈

yes

complemented

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₂×C₈

yes

complemented

C₂²×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×D₄₇

C₄⋊C₈

yes

—

C₂×D₄₇

#106

yes

—

C₂×D₄₇

#108

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂²×C₈

yes

characteristic

C₄×C₄₇

C₂×M₁₆

yes

characteristic

C₄×C₄₇

C₂×SD₁₆

yes

—

Dic₄₇

—

Dic₄₇

C₄⋊C₈

yes

—

Dic₄₇

#106

yes

—

C₂×#334

—

maximal, Sylow

C₂×C₄₇

C₂×C₄⋊C₈

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

yes

direct product factor

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

#334

—

#334

—

#334

D₄₇

yes

characteristic, direct product factor

C₄₇

C₂×#334

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

C₄×D₄₇

yes

characteristic

C₂×C₈

C₄×D₄₇

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×D₄₇

yes

characteristic

C₂³

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

108

C₂×C₄

D₄×D₄₇

yes

characteristic

109

C₂×C₄

Q₈×D₄₇

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

188

—

115

C₈

C₈×D₄₇

yes

complemented

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

C₄⋊C₄×D₄₇

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

C₂×C₈×D₄₇

yes

characteristic

125

C₄

M₁₆×D₄₇

yes

characteristic

126

C₄

SD₁₆×D₄₇

yes

—

127

C₂

—

128

C₂

C₄⋊C₈×D₄₇

yes

characteristic

129

C₂

#106×D₄₇

yes

characteristic

130

C₂

#108×D₄₇

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₄₇
commutator	C₄×C₄₇	C₂²×C₈
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#334	C₂
socle	C₂²×C₄₇	C₂×C₄⋊C₄
hypercenter	#334	D₄₇
hypocenter	C₄₇	C₂×#334
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×D₄₇

commutator

C₄×C₄₇

C₂²×C₈

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#334

C₂

socle

C₂²×C₄₇

C₂×C₄⋊C₄

D₄₇

C₄₇

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
3. #15688874	C₂	semidirect
4. #15688873	C₂	semidirect
5. C₄₇×#334	C₂	semidirect
32. C₈×D₄₇	C₈	semidirect
46. C₈×C₄₇	C₂×C₈	semidirect
49. C₄₇⋊C₈	C₂×C₈	semidirect
65. D₄₇	#334	direct (N × H)
70. #334	D₄₇	direct (N × H)
71. C₄₇	C₂×#334	semidirect
115. C₈	C₈×D₄₇	semidirect

H ≅ G/N

type

3. #15688874

C₂

semidirect

4. #15688873

C₂

semidirect

5. C₄₇×#334

C₂

semidirect

32. C₈×D₄₇

C₈

semidirect

46. C₈×C₄₇

C₂×C₈

semidirect

49. C₄₇⋊C₈

C₂×C₈

semidirect

65. D₄₇

#334

direct (N × H)

70. #334

D₄₇

direct (N × H)

71. C₄₇

C₂×#334

semidirect

115. C₈

C₈×D₄₇

semidirect

Subgroups — #334×D₄₇ #20242127

Conjugacy classes of subgroups

Complements

Subgroups — #334×D47 #20242127

Conjugacy classes of subgroups

Complements

Subgroups — #334×D₄₇ #20242127