Group #20243253 — subgroups

Subgroups — #20243253

Order 6016.

296 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 165 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×C₈×Dic₄₇	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	#15688877	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₄₇	C₄	1	2	yes	—
9	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
10	C₄×C₄₇⋊C₈	C₄	1	4	yes	complemented
11	C₈×Dic₄₇	—	2	4	no	—
12	C₄₇⋊#95	C₄	1	2	yes	—
13	C₂²×C₈×C₄₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₄₇	C₄	1	4	yes	complemented
16	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
19	C₂²×Dic₄₇	C₂×C₄	1	4	yes	—
20	C₂×C₈×C₄₇	—	2	2	no	—
21	C₂×C₈×C₄₇	—	2	2	no	—
22	C₂×C₈×C₄₇	C₂×C₄	1	4	yes	—
23	C₄²×C₄₇	C₂×C₄	1	2	yes	—
24	C₄²×C₄₇	C₂×C₄	1	2	yes	—
25	C₄⋊Dic₄₇	C₈	1	8	yes	complemented
26	C₄×Dic₄₇	—	2	2	no	—
27	C₄×Dic₄₇	—	2	2	no	—
28	C₂×C₄₇⋊C₈	—	2	1	no	—
29	C₂×C₄₇⋊C₈	—	2	1	no	—
30	C₂×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
31	C₂×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
32	C₂×C₄₇⋊C₈	D₄	1	2	yes	—
33	C₂×C₄₇⋊C₈	Q₈	1	2	yes	—
34	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₄₇	—	2	2	no	—
36	C₂×C₄×C₄₇	C₄²	1	2	yes	—
37	C₂×C₄×C₄₇	C₄²	1	2	yes	—
38	C₂×C₄×C₄₇	C₂×C₈	1	4	yes	—
39	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
44	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
45	C₂×Dic₄₇	—	2	4	no	—
46	C₂×Dic₄₇	—	2	4	no	—
47	C₂×Dic₄₇	C₂×C₈	1	8	yes	—
48	C₈×C₄₇	—	2	4	no	—
49	C₄₇⋊C₈	—	2	2	no	—
50	C₄₇⋊C₈	C₄⋊C₄	1	4	yes	complemented
51	C₂²×C₄₇	C₂×C₄²	1	1	yes	characteristic
52	C₂²×C₄₇	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₄₇	C₄×Q₈	1	1	yes	characteristic
54	C₂²×C₄₇	C₂²×C₈	1	2	yes	—
55	C₂²×C₄₇	#131	1	2	yes	—
56	C₄×C₄₇	—	2	2	no	—
57	C₄×C₄₇	C₄×C₈	1	4	yes	complemented
58	C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₄₇	#117	1	1	yes	characteristic
60	C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
61	C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
62	Dic₄₇	—	2	8	no	—
63	C₈×C₄⋊C₄	—	47	1	no	maximal, Sylow
64	C₂×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
66	C₂×C₄₇	#405	1	1	yes	characteristic
67	C₂×C₄₇	C₈×D₄	1	2	yes	—
68	C₂×C₄₇	C₈×Q₈	1	2	yes	—
69	C₂×C₄⋊C₈	D₄₇	1	1	yes	characteristic
70	C₂×C₄×C₈	—	47	1	no	—
71	C₂×C₄×C₈	—	47	2	no	—
72	C₄×C₄⋊C₄	—	47	1	no	—
73	#336	—	47	2	no	—
74	C₄₇	C₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₈	—	47	1	no	—
76	C₂²×C₈	—	47	1	no	—
77	C₂²×C₈	C₂×D₄₇	1	2	yes	—
78	C₂×C₄⋊C₄	—	47	2	no	—
79	C₂×C₄²	—	47	2	no	—
80	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
81	C₄⋊C₈	Dic₄₇	1	4	yes	complemented
82	C₄×C₈	—	47	4	no	—
83	C₄×C₈	—	94	4	no	—
84	#95	—	47	2	no	—
85	C₂²×C₄	—	47	4	no	—
86	C₂²×C₄	C₄×D₄₇	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	—	2	2	no	—
91	C₂×C₈	—	47	2	no	—
92	C₂×C₈	—	47	2	no	—
93	C₂×C₈	—	47	2	no	—
94	C₂×C₈	—	47	2	no	—
95	C₂×C₈	—	94	1	no	—
96	C₂×C₈	—	94	1	no	—
97	C₂×C₈	C₂×Dic₄₇	1	4	yes	—
98	C₄⋊C₄	—	47	8	no	—
99	C₄²	—	94	2	no	—
100	C₄²	—	94	2	no	—
101	C₄²	C₄×D₄₇	1	2	yes	—
102	C₄²	C₂×Dic₄₇	1	2	yes	—
103	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	47	8	no	—
106	C₂×C₄	—	94	4	no	—
107	C₂×C₄	—	94	4	no	—
108	C₂×C₄	C₈×D₄₇	1	4	yes	—
109	C₂×C₄	C₄×Dic₄₇	1	2	yes	—
110	C₂×C₄	C₄×Dic₄₇	1	2	yes	—
111	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
112	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
113	C₂×C₄	#10625352	1	1	yes	characteristic
114	C₂×C₄	Q₈×D₄₇	1	1	yes	characteristic
115	C₂×C₄	#10625354	1	1	yes	characteristic
116	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
117	C₈	—	2	4	no	—
118	C₈	—	47	4	no	—
119	C₈	—	94	2	no	—
120	C₂²	C₂×C₄×Dic₄₇	1	1	yes	characteristic
121	C₂²	#13093986	1	1	yes	characteristic
122	C₂²	#13093989	1	1	yes	characteristic
123	C₂²	C₂×C₈×D₄₇	1	2	yes	—
124	C₂²	#13094032	1	2	yes	—
125	C₄	—	2	2	no	—
126	C₄	—	94	8	no	—
127	C₄	C₈×Dic₄₇	1	4	yes	complemented
128	C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
129	C₄	#13093982	1	1	yes	characteristic
130	C₄	D₄×Dic₄₇	1	1	yes	characteristic
131	C₄	Q₈×Dic₄₇	1	1	yes	characteristic
132	C₂	C₄⋊C₄×Dic₄₇	1	1	yes	characteristic
133	C₂	C₂×C₈×Dic₄₇	1	1	yes	characteristic
134	C₂	#15689081	1	1	yes	characteristic
135	C₂	#15689188	1	2	yes	—
136	C₂	#15689247	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₄₇⋊C₈

C₂

yes

maximal, characteristic

C₂×C₈×Dic₄₇

C₂

yes

maximal

C₄×C₄⋊Dic₄₇

C₂

yes

maximal, characteristic

#15688877

C₂

yes

maximal

C₂×C₄⋊C₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₄

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄×C₄₇⋊C₈

C₄

yes

complemented

C₈×Dic₄₇

—

C₄₇⋊#95

C₄

yes

—

C₂²×C₈×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₄

yes

complemented

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄⋊Dic₄₇

C₈

yes

complemented

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

D₄

yes

—

C₂×C₄₇⋊C₈

Q₈

yes

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄²

yes

—

C₂×C₄×C₄₇

C₄²

yes

—

C₂×C₄×C₄₇

C₂×C₈

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×C₈

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₄⋊C₄

yes

complemented

C₂²×C₄₇

C₂×C₄²

yes

characteristic

C₂²×C₄₇

C₄×D₄

yes

characteristic

C₂²×C₄₇

C₄×Q₈

yes

characteristic

C₂²×C₄₇

C₂²×C₈

yes

—

C₂²×C₄₇

#131

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₄×C₈

yes

complemented

C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₄₇

#117

yes

characteristic

C₄×C₄₇

C₄×D₄

yes

characteristic

C₄×C₄₇

C₄×Q₈

yes

characteristic

Dic₄₇

—

C₈×C₄⋊C₄

—

maximal, Sylow

C₂×C₄₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₄₇

C₂×C₄×C₈

yes

characteristic

C₂×C₄₇

#405

yes

characteristic

C₂×C₄₇

C₈×D₄

yes

—

C₂×C₄₇

C₈×Q₈

yes

—

C₂×C₄⋊C₈

D₄₇

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄×C₄⋊C₄

—

#336

—

C₄₇

C₈×C₄⋊C₄

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

Dic₄₇

yes

complemented

C₄×C₈

—

C₄×C₈

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄₇

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

—

C₂×C₈

C₂×Dic₄₇

yes

—

C₄⋊C₄

—

C₄²

—

100

C₄²

—

101

C₄²

C₄×D₄₇

yes

—

102

C₄²

C₂×Dic₄₇

yes

—

103

C₂³

C₂×C₄×D₄₇

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₈×D₄₇

yes

—

109

C₂×C₄

C₄×Dic₄₇

yes

—

110

C₂×C₄

C₄×Dic₄₇

yes

—

111

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

112

C₂×C₄

D₄×D₄₇

yes

characteristic

113

C₂×C₄

#10625352

yes

characteristic

114

C₂×C₄

Q₈×D₄₇

yes

characteristic

115

C₂×C₄

#10625354

yes

characteristic

116

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

—

120

C₂²

C₂×C₄×Dic₄₇

yes

characteristic

121

C₂²

#13093986

yes

characteristic

122

C₂²

#13093989

yes

characteristic

123

C₂²

C₂×C₈×D₄₇

yes

—

124

C₂²

#13094032

yes

—

125

C₄

—

126

C₄

—

127

C₄

C₈×Dic₄₇

yes

complemented

128

C₄

C₄⋊C₄×D₄₇

yes

characteristic

129

C₄

#13093982

yes

characteristic

130

C₄

D₄×Dic₄₇

yes

characteristic

131

C₄

Q₈×Dic₄₇

yes

characteristic

132

C₂

C₄⋊C₄×Dic₄₇

yes

characteristic

133

C₂

C₂×C₈×Dic₄₇

yes

characteristic

134

C₂

#15689081

yes

characteristic

135

C₂

#15689188

yes

—

136

C₂

#15689247

yes

—

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

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₄₇

commutator

C₂×C₄₇

C₂×C₄×C₈

Fitting

C₂×C₄⋊C₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂²×C₄

hypercenter

C₂×C₄⋊C₈

D₄₇

hypocenter

C₄₇

C₈×C₄⋊C₄

LCS

G ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
10. C₄×C₄₇⋊C₈	C₄	semidirect
15. C₄⋊C₈×C₄₇	C₄	semidirect
25. C₄⋊Dic₄₇	C₈	semidirect
50. C₄₇⋊C₈	C₄⋊C₄	semidirect
57. C₄×C₄₇	C₄×C₈	semidirect
74. C₄₇	C₈×C₄⋊C₄	semidirect
81. C₄⋊C₈	Dic₄₇	semidirect
127. C₄	C₈×Dic₄₇	semidirect

H ≅ G/N

type

10. C₄×C₄₇⋊C₈

C₄

semidirect

15. C₄⋊C₈×C₄₇

C₄

semidirect

25. C₄⋊Dic₄₇

C₈

semidirect

50. C₄₇⋊C₈

C₄⋊C₄

semidirect

57. C₄×C₄₇

C₄×C₈

semidirect

74. C₄₇

C₈×C₄⋊C₄

semidirect

81. C₄⋊C₈

Dic₄₇

semidirect

127. C₄

C₈×Dic₄₇

semidirect

Subgroups — #20243253

Conjugacy classes of subgroups

Complements