Group #20249508 — subgroups

Subgroups — #20249508

Order 6016.

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

normal subgroups only

This group has 448 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×#13094016	C₂	1	1	yes	maximal, characteristic
2	#15689054	C₂	1	4	yes	maximal
3	#15689042	C₂	1	4	yes	maximal
4	#15688992	C₂	1	2	yes	maximal
5	#15688988	C₂	1	2	yes	maximal
6	#15688969	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
8	#13094016	—	2	4	no	—
9	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
12	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
13	C₂×Dic₄₇⋊C₄	C₂²	1	8	yes	—
14	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
15	C₄₇⋊#95	C₂²	1	2	yes	—
16	C₄₇⋊#95	C₂²	1	2	yes	—
17	C₄₇⋊#95	C₂²	1	8	yes	—
18	C₂²×Q₈×C₄₇	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₄₇	—	2	4	no	—
20	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₄₇	—	2	4	no	—
24	C₂×Q₈×C₄₇	—	4	4	no	—
25	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
26	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
29	C₂²×Dic₄₇	C₂³	1	2	yes	—
30	C₂²×Dic₄₇	C₂³	1	2	yes	—
31	C₂²×Dic₄₇	C₂³	1	4	yes	—
32	C₄⋊C₄×C₄₇	—	2	4	no	—
33	C₄⋊C₄×C₄₇	—	2	4	no	—
34	C₄²×C₄₇	D₄	1	4	yes	—
35	C₄⋊Dic₄₇	—	2	2	no	—
36	C₄⋊Dic₄₇	—	2	2	no	—
37	Dic₄₇⋊C₄	—	2	4	no	—
38	Dic₄₇⋊C₄	—	2	4	no	—
39	Dic₄₇⋊C₄	—	4	8	no	—
40	C₄×Dic₄₇	Q₈	1	8	yes	complemented
41	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
42	Q₈×C₄₇	—	4	8	no	—
43	C₂×C₄×C₄₇	—	2	2	no	—
44	C₂×C₄×C₄₇	—	2	2	no	—
45	C₂×C₄×C₄₇	—	2	4	no	—
46	C₂×C₄×C₄₇	—	2	8	no	—
47	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	—
51	C₂×Dic₄₇	—	2	4	no	—
52	C₂×Dic₄₇	—	2	4	no	—
53	C₂×Dic₄₇	—	2	2	no	—
54	C₂×Dic₄₇	—	2	2	no	—
55	C₂×Dic₄₇	—	2	4	no	—
56	C₂×Dic₄₇	—	2	4	no	—
57	C₂×Dic₄₇	C₂×Q₈	1	8	yes	—
58	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
59	C₂²×C₄₇	C₂²×Q₈	1	2	yes	—
60	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
61	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
62	C₄×C₄₇	—	2	2	no	—
63	C₄×C₄₇	—	2	2	no	—
64	C₄×C₄₇	—	4	4	no	—
65	C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
66	Dic₄₇	—	2	8	no	—
67	Dic₄₇	—	4	2	no	—
68	Dic₄₇	—	4	2	no	—
69	#2660	—	47	1	no	maximal, Sylow
70	C₂×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₄₇	#535	1	1	yes	characteristic
72	C₂×C₄₇	#536	1	1	yes	characteristic
73	C₂×C₄₇	#544	1	4	yes	—
74	C₂×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
75	C₂×#125	—	47	1	no	—
76	#398	—	47	4	no	—
77	#395	—	47	4	no	—
78	#384	—	47	2	no	—
79	#382	—	47	2	no	—
80	#376	—	47	1	no	—
81	C₄₇	#2660	1	1	yes	characteristic, complemented, Sylow
82	C₂²×Q₈	C₂×D₄₇	1	2	yes	—
83	C₄⋊Q₈	—	2	4	no	—
84	#125	—	94	4	no	—
85	C₂×C₄⋊C₄	—	47	2	no	—
86	C₂×C₄⋊C₄	—	47	2	no	—
87	C₂×C₄⋊C₄	—	47	1	no	—
88	C₂×C₄⋊C₄	—	47	1	no	—
89	C₂×C₄⋊C₄	—	47	8	no	—
90	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
91	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
92	C₂×C₄²	—	47	2	no	—
93	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
94	#95	—	47	2	no	—
95	#95	—	47	2	no	—
96	#95	—	47	8	no	—
97	C₂×Q₈	—	2	4	no	—
98	C₂×Q₈	—	4	4	no	—
99	C₂²×C₄	—	47	4	no	—
100	C₂²×C₄	—	47	2	no	—
101	C₂²×C₄	—	47	2	no	—
102	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
103	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
106	C₄⋊C₄	—	2	4	no	—
107	C₄⋊C₄	—	2	4	no	—
108	C₄⋊C₄	—	94	4	no	—
109	C₄⋊C₄	—	94	4	no	—
110	C₄⋊C₄	—	94	2	no	—
111	C₄⋊C₄	—	94	2	no	—
112	C₄⋊C₄	—	188	8	no	—
113	C₄²	—	47	8	no	—
114	C₄²	C₂²⋊D₄₇	1	4	yes	—
115	C₂³	C₂³×D₄₇	1	1	yes	characteristic
116	Q₈	—	4	8	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	8	no	—
121	C₂×C₄	—	47	8	no	—
122	C₂×C₄	—	94	4	no	—
123	C₂×C₄	—	94	4	no	—
124	C₂×C₄	—	94	2	no	—
125	C₂×C₄	—	94	2	no	—
126	C₂×C₄	—	94	4	no	—
127	C₂×C₄	—	94	4	no	—
128	C₂×C₄	Q₈×D₄₇	1	4	yes	—
129	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
130	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
131	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
132	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
133	C₂²	C₂×Q₈×D₄₇	1	2	yes	—
134	C₂²	#13094069	1	2	yes	—
135	C₂²	#13094072	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	4	4	no	—
139	C₄	—	94	8	no	—
140	C₄	—	188	2	no	—
141	C₄	—	188	2	no	—
142	C₄	#13094022	1	4	yes	—
143	C₂	C₂×#13094022	1	1	yes	characteristic
144	C₂	#15689690	1	1	yes	characteristic
145	C₂	#15689697	1	1	yes	characteristic
146	C₂	#15689750	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094016

C₂

yes

maximal, characteristic

#15689054

C₂

yes

maximal

#15689042

C₂

yes

maximal

#15688992

C₂

yes

maximal

#15688988

C₂

yes

maximal

#15688969

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₄₇

C₂

yes

maximal, characteristic

#13094016

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₄₇

C₂²

yes

—

C₄⋊Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

D₄

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

Q₈

yes

complemented

C₂³×C₄₇

C₂⁴

yes

characteristic

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×Q₈

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

—

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄⋊Q₈

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#2660

—

maximal, Sylow

C₂×C₄₇

C₂×C₄⋊Q₈

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄⋊Q₈

D₄₇

yes

—

—

—

—

—

—

C₄₇

#2660

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₄₇

yes

—

C₄⋊Q₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²×D₄₇

yes

—

103

C₂²×C₄

C₂²×D₄₇

yes

characteristic

104

C₂²×C₄

C₂²×D₄₇

yes

characteristic

105

C₂²×C₄

C₂²×D₄₇

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

188

—

113

C₄²

—

114

C₄²

C₂²⋊D₄₇

yes

—

115

C₂³

C₂³×D₄₇

yes

characteristic

116

Q₈

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

Q₈×D₄₇

yes

—

129

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

130

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

131

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

132

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

133

C₂²

C₂×Q₈×D₄₇

yes

—

134

C₂²

#13094069

yes

—

135

C₂²

#13094072

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

188

—

141

C₄

—

188

—

142

C₄

#13094022

yes

—

143

C₂

C₂×#13094022

yes

characteristic

144

C₂

#15689690

yes

characteristic

145

C₂

#15689697

yes

characteristic

146

C₂

#15689750

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₄₇	C₂
hypercenter	C₂×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#2660
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₄₇

C₂

hypercenter

C₂×C₄⋊Q₈

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
40. C₄×Dic₄₇	Q₈	semidirect
81. C₄₇	#2660	semidirect

H ≅ G/N

type

40. C₄×Dic₄₇

Q₈

semidirect

81. C₄₇

#2660

semidirect

Subgroups — #20249508

Conjugacy classes of subgroups

Complements