Group #20242149 — subgroups

Subgroups — #20242149

Order 6016.

168 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 81 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	#15689180	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#15688875	C₂	1	2	yes	maximal, complemented
4	#15688873	C₂	1	2	yes	maximal, complemented
5	C₄₇×#432	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂²×C₄₇⋊C₈	C₄	1	1	yes	characteristic
8	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
10	#13093935	—	2	1	no	—
11	C₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
12	C₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
13	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
17	C₂²⋊C₈×C₄₇	—	2	1	no	—
18	#97×C₄₇	—	2	1	no	—
19	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	Q₈	1	1	yes	characteristic
22	C₂×C₈×C₄₇	—	2	2	no	—
23	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₄₇	D₄	1	1	yes	characteristic
25	C₄²×C₄₇	Q₈	1	1	yes	characteristic
26	C₄²×C₄₇	C₂³	1	1	yes	characteristic
27	C₂×C₄₇⋊C₈	—	2	2	no	—
28	C₂×C₄₇⋊C₈	—	2	2	no	—
29	C₂×C₄₇⋊C₈	—	2	2	no	—
30	C₂×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₄₇	—	2	2	no	—
36	C₂×C₄×C₄₇	—	2	1	no	—
37	C₂×C₄×C₄₇	—	2	1	no	—
38	C₂×C₄×C₄₇	—	2	2	no	—
39	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
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	1	yes	characteristic
45	C₈×C₄₇	—	4	2	no	—
46	C₄₇⋊C₈	—	2	1	no	—
47	C₄₇⋊C₈	—	2	1	no	—
48	C₄₇⋊C₈	—	4	2	no	—
49	C₄₇⋊C₈	M₁₆	1	2	yes	—
50	C₄₇⋊C₈	M₁₆	1	2	yes	—
51	C₂²×C₄₇	—	2	1	no	—
52	C₂²×C₄₇	—	2	1	no	—
53	C₂²×C₄₇	#108	1	2	yes	—
54	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₄₇	—	2	1	no	—
56	C₄×C₄₇	—	2	1	no	—
57	C₄×C₄₇	—	2	1	no	—
58	C₄×C₄₇	—	2	1	no	—
59	C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
60	C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
61	C₄×C₄₇	#135	1	2	yes	—
62	#1568	—	47	1	no	maximal, Sylow
63	C₂×C₄₇	—	2	2	no	—
64	C₂×C₄₇	#423	1	1	yes	characteristic
65	C₂×C₄₇	#427	1	1	yes	characteristic
66	C₂×C₄₇	C₂×#108	1	1	yes	characteristic
67	#432	—	47	1	no	—
68	#432	D₄₇	1	1	yes	characteristic
69	C₂×C₄×C₈	—	47	1	no	—
70	#335	—	47	2	no	—
71	#334	—	47	2	no	—
72	C₄₇	#1568	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₈	—	47	1	no	—
74	C₂²×C₈	—	47	1	no	—
75	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
76	C₄⋊C₈	—	47	2	no	—
77	C₄⋊C₈	C₂×D₄₇	1	2	yes	—
78	C₂²⋊C₈	—	2	1	no	—
79	C₂²⋊C₈	—	94	1	no	—
80	#97	—	2	1	no	—
81	#97	—	94	1	no	—
82	C₄×C₈	—	47	1	no	—
83	C₄×C₈	—	47	1	no	—
84	C₄×C₈	—	47	1	no	—
85	C₄×C₈	—	47	1	no	—
86	C₂²×C₄	Dic₉₄	1	1	yes	characteristic
87	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
88	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	47	1	no	—
91	C₂×C₈	—	47	1	no	—
92	C₂×C₈	—	47	1	no	—
93	C₂×C₈	—	47	1	no	—
94	C₂×C₈	—	94	2	no	—
95	C₂×C₈	—	94	2	no	—
96	C₂×C₈	—	94	2	no	—
97	C₄²	Dic₉₄	1	1	yes	characteristic
98	C₄²	C₄×D₄₇	1	1	yes	characteristic
99	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
100	C₄²	C₂²×D₄₇	1	1	yes	characteristic
101	C₂³	Dic₄₇⋊C₄	1	1	yes	characteristic
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
107	C₂×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
108	C₂×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
109	C₂×C₄	C₂×Dic₉₄	1	1	yes	characteristic
110	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
111	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
112	C₈	—	4	2	no	—
113	C₈	—	47	2	no	—
114	C₈	—	47	2	no	—
115	C₈	—	94	1	no	—
116	C₈	—	94	1	no	—
117	C₈	—	188	2	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	#13093959	1	2	yes	—
121	C₂²	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	M₁₆×D₄₇	1	1	yes	characteristic
127	C₄	M₁₆×D₄₇	1	1	yes	characteristic
128	C₄	#13094044	1	2	yes	—
129	C₂	—	2	2	no	—
130	C₂	#15689146	1	1	yes	characteristic
131	C₂	#15689159	1	1	yes	characteristic
132	C₂	C₂×#13093959	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689180

C₂

yes

maximal, characteristic

C₂×C₄×C₄₇⋊C₈

C₂

yes

maximal, characteristic

#15688875

C₂

yes

maximal, complemented

#15688873

C₂

yes

maximal, complemented

C₄₇×#432

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₄

yes

characteristic

C₂²×C₄₇⋊C₈

C₄

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

C₄×C₄₇⋊C₈

C₄

yes

characteristic

C₄×C₄₇⋊C₈

C₄

yes

characteristic

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

—

C₂²⋊C₈×C₄₇

—

#97×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

Q₈

yes

characteristic

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

Q₈

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂³×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

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

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

M₁₆

yes

—

C₄₇⋊C₈

M₁₆

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#108

yes

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×M₁₆

yes

characteristic

C₄×C₄₇

C₂×M₁₆

yes

characteristic

C₄×C₄₇

#135

yes

—

#1568

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

D₄₇

yes

characteristic

C₂×C₄×C₈

—

#335

—

#334

—

C₄₇

#1568

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

Dic₉₄

yes

characteristic

C₂²×C₄

C₄×D₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

Dic₉₄

yes

characteristic

C₄²

C₄×D₄₇

yes

characteristic

C₄²

C₂²⋊D₄₇

yes

characteristic

100

C₄²

C₂²×D₄₇

yes

characteristic

101

C₂³

Dic₄₇⋊C₄

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

Dic₄₇⋊C₄

yes

characteristic

107

C₂×C₄

Dic₄₇⋊C₄

yes

characteristic

108

C₂×C₄

Dic₄₇⋊C₄

yes

characteristic

109

C₂×C₄

C₂×Dic₉₄

yes

characteristic

110

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

111

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

188

—

118

C₂²

—

119

C₂²

—

120

C₂²

#13093959

yes

—

121

C₂²

C₂×Dic₄₇⋊C₄

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

M₁₆×D₄₇

yes

characteristic

127

C₄

M₁₆×D₄₇

yes

characteristic

128

C₄

#13094044

yes

—

129

C₂

—

130

C₂

#15689146

yes

characteristic

131

C₂

#15689159

yes

characteristic

132

C₂

C₂×#13093959

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#432	C₂
socle	C₂²×C₄₇	C₂×C₄⋊C₄
hypercenter	#432	D₄₇
hypocenter	C₄₇	#1568
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#432

C₂

socle

C₂²×C₄₇

C₂×C₄⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
3. #15688875	C₂	semidirect
4. #15688873	C₂	semidirect
72. C₄₇	#1568	semidirect

H ≅ G/N

type

3. #15688875

C₂

semidirect

4. #15688873

C₂

semidirect

72. C₄₇

#1568

semidirect

Subgroups — #20242149

Conjugacy classes of subgroups

Complements