Group #20241222 — subgroups

Subgroups — #20241222

Order 6016.

180 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₇⋊C₈	—	2	1	no	—
5	C₂²×C₄₇⋊C₈	C₄	1	2	yes	—
6	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₄⋊(C₄₇⋊C₈)	C₄	1	2	yes	—
8	C₄×C₄₇⋊C₈	—	2	1	no	—
9	C₄×C₄₇⋊C₈	—	2	1	no	—
10	C₂²×C₈×C₄₇	—	2	1	no	—
11	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₄₇	—	2	1	no	—
13	C₄⋊C₈×C₄₇	C₄	1	2	yes	—
14	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₄₇	Q₈	1	1	yes	characteristic
17	C₂×C₈×C₄₇	—	2	2	no	—
18	C₂×C₈×C₄₇	—	4	1	no	—
19	C₂×C₈×C₄₇	—	4	1	no	—
20	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₄₇	D₄	1	1	yes	characteristic
23	C₄²×C₄₇	D₄	1	1	yes	characteristic
24	C₂×C₄₇⋊C₈	—	2	2	no	—
25	C₂×C₄₇⋊C₈	—	2	2	no	—
26	C₂×C₄₇⋊C₈	—	2	2	no	—
27	C₂×C₄₇⋊C₈	—	4	1	no	—
28	C₂×C₄₇⋊C₈	—	4	1	no	—
29	C₂×C₄₇⋊C₈	C₈	1	4	yes	complemented
30	C₂³×C₄₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₄₇	—	2	2	no	—
32	C₂×C₄×C₄₇	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₄₇	C₂×C₈	1	2	yes	—
39	C₂×C₄×C₄₇	M₁₆	1	2	yes	—
40	C₂×C₄×C₄₇	D₈	1	1	yes	characteristic
41	C₂×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
42	C₂×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
43	C₂×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
44	C₈×C₄₇	—	4	2	no	—
45	C₄₇⋊C₈	—	2	4	no	—
46	C₄₇⋊C₈	—	4	2	no	—
47	C₂²×C₄₇	#95	1	1	yes	characteristic
48	C₂²×C₄₇	C₄⋊C₈	1	2	yes	—
49	C₂²×C₄₇	#106	1	1	yes	characteristic
50	C₂²×C₄₇	#107	1	1	yes	characteristic
51	C₂²×C₄₇	#108	1	2	yes	—
52	C₄×C₄₇	—	2	1	no	—
53	C₄×C₄₇	—	2	1	no	—
54	C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
55	C₄×C₄₇	#97	1	1	yes	characteristic
56	C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
57	C₄×C₄₇	#102	1	1	yes	characteristic
58	C₄×C₄₇	#102	1	1	yes	characteristic
59	C₄×C₄₇	#103	1	1	yes	characteristic
60	C₄×C₄₇	#103	1	1	yes	characteristic
61	#1277	—	47	1	no	maximal, Sylow
62	C₂×C₄₇	#334	1	2	yes	—
63	C₂×C₄₇	#335	1	2	yes	—
64	C₂×C₄₇	#336	1	1	yes	characteristic
65	C₂×C₄₇	#340	1	1	yes	characteristic
66	C₂×C₄₇	#341	1	1	yes	characteristic
67	C₂×C₄⋊C₈	—	47	1	no	—
68	C₂×C₄⋊C₈	D₄₇	1	1	yes	characteristic
69	C₂×C₄×C₈	—	47	1	no	—
70	C₄₇	#1277	1	1	yes	characteristic, complemented, Sylow
71	C₂²×C₈	—	2	1	no	—
72	C₂²×C₈	—	47	2	no	—
73	C₂²×C₈	—	94	1	no	—
74	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
75	C₄⋊C₈	—	2	1	no	—
76	C₄⋊C₈	—	47	2	no	—
77	C₄⋊C₈	—	94	1	no	—
78	C₄⋊C₈	Dic₄₇	1	2	yes	—
79	C₄×C₈	—	94	1	no	—
80	C₄×C₈	—	94	1	no	—
81	C₂²×C₄	Dic₉₄	1	1	yes	characteristic
82	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
83	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	4	1	no	—
86	C₂×C₈	—	4	1	no	—
87	C₂×C₈	—	47	4	no	—
88	C₂×C₈	—	94	2	no	—
89	C₂×C₈	—	94	2	no	—
90	C₂×C₈	—	94	2	no	—
91	C₂×C₈	—	188	1	no	—
92	C₂×C₈	—	188	1	no	—
93	C₄²	C₄×D₄₇	1	1	yes	characteristic
94	C₄²	D₁₈₈	1	1	yes	characteristic
95	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
96	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
97	C₂³	Dic₄₇⋊C₄	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	C₈×D₄₇	1	2	yes	—
100	C₂×C₄	#10625324	1	2	yes	—
101	C₂×C₄	C₄×Dic₄₇	1	1	yes	characteristic
102	C₂×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
103	C₂×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
104	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
105	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
106	C₂×C₄	#10625334	1	1	yes	characteristic
107	C₂×C₄	#10625335	1	1	yes	characteristic
108	C₂×C₄	#10625336	1	1	yes	characteristic
109	C₂×C₄	#10625337	1	1	yes	characteristic
110	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
111	C₈	—	4	2	no	—
112	C₈	—	94	4	no	—
113	C₈	—	188	2	no	—
114	C₂²	C₄₇⋊#95	1	1	yes	characteristic
115	C₂²	Dic₄₇⋊C₈	1	2	yes	—
116	C₂²	#13093955	1	1	yes	characteristic
117	C₂²	#13093957	1	1	yes	characteristic
118	C₂²	#13093959	1	2	yes	—
119	C₄	—	2	1	no	—
120	C₄	—	2	1	no	—
121	C₄	C₈×Dic₄₇	1	1	yes	characteristic
122	C₄	#13093934	1	1	yes	characteristic
123	C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
124	C₄	#13093944	1	1	yes	characteristic
125	C₄	#13093945	1	1	yes	characteristic
126	C₄	#13093947	1	1	yes	characteristic
127	C₄	#13093948	1	1	yes	characteristic
128	C₂	#15688873	1	2	yes	—
129	C₂	#15688875	1	2	yes	—
130	C₂	#15688877	1	1	yes	characteristic
131	C₂	#15688885	1	1	yes	characteristic
132	C₂	#15688887	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic

C₂×C₄×C₄₇⋊C₈

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

—

C₂²×C₄₇⋊C₈

C₄

yes

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

C₄

yes

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

Q₈

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₈

yes

complemented

C₂³×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄²

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×C₈

yes

—

C₂×C₄×C₄₇

M₁₆

yes

—

C₂×C₄×C₄₇

D₈

yes

characteristic

C₂×C₄×C₄₇

SD₁₆

yes

characteristic

C₂×C₄×C₄₇

SD₁₆

yes

characteristic

C₂×C₄×C₄₇

Q₁₆

yes

characteristic

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

#95

yes

characteristic

C₂²×C₄₇

C₄⋊C₈

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×C₈

yes

characteristic

C₄×C₄₇

#97

yes

characteristic

C₄×C₄₇

C₂²⋊C₈

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₂×C₄₇

yes

—

C₂×C₄₇

#335

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₄₇

yes

characteristic

C₂×C₄×C₈

—

C₄₇

#1277

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

Dic₄₇

yes

—

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

—

188

—

C₂×C₈

—

188

—

C₄²

C₄×D₄₇

yes

characteristic

C₄²

D₁₈₈

yes

characteristic

C₄²

C₂×Dic₄₇

yes

characteristic

C₄²

C₂²⋊D₄₇

yes

characteristic

C₂³

Dic₄₇⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

C₈×D₄₇

yes

—

100

C₂×C₄

#10625324

yes

—

101

C₂×C₄

C₄×Dic₄₇

yes

characteristic

102

C₂×C₄

Dic₄₇⋊C₄

yes

characteristic

103

C₂×C₄

C₄⋊Dic₄₇

yes

characteristic

104

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

105

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

106

C₂×C₄

#10625334

yes

characteristic

107

C₂×C₄

#10625335

yes

characteristic

108

C₂×C₄

#10625336

yes

characteristic

109

C₂×C₄

#10625337

yes

characteristic

110

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

188

—

114

C₂²

C₄₇⋊#95

yes

characteristic

115

C₂²

Dic₄₇⋊C₈

yes

—

116

C₂²

#13093955

yes

characteristic

117

C₂²

#13093957

yes

characteristic

118

C₂²

#13093959

yes

—

119

C₄

—

120

C₄

—

121

C₄

C₈×Dic₄₇

yes

characteristic

122

C₄

#13093934

yes

characteristic

123

C₄

(C₂×D₄₇)⋊C₈

yes

—

124

C₄

#13093944

yes

characteristic

125

C₄

#13093945

yes

characteristic

126

C₄

#13093947

yes

characteristic

127

C₄

#13093948

yes

characteristic

128

C₂

#15688873

yes

—

129

C₂

#15688875

yes

—

130

C₂

#15688877

yes

characteristic

131

C₂

#15688885

yes

characteristic

132

C₂

#15688887

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊D₄₇
commutator	C₄×C₄₇	C₄×C₈
Frattini	C₂×C₄²	C₂×D₄₇
Fitting	C₂×C₄⋊C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₄⋊C₄
hypercenter	C₂×C₄⋊C₈	D₄₇
hypocenter	C₄₇	#1277
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂²×C₄

C₂²⋊D₄₇

commutator

C₄×C₄₇

C₄×C₈

Frattini

C₂×C₄²

C₂×D₄₇

Fitting

C₂×C₄⋊C₈×C₄₇

C₂

socle

C₂³×C₄₇

C₄⋊C₄

hypercenter

C₂×C₄⋊C₈

D₄₇

hypocenter

C₄₇

#1277

LCS

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
29. C₂×C₄₇⋊C₈	C₈	semidirect
70. C₄₇	#1277	semidirect

H ≅ G/N

type

29. C₂×C₄₇⋊C₈

C₈

semidirect

70. C₄₇

#1277

semidirect

Subgroups — #20241222

Conjugacy classes of subgroups

Complements