Group #20245949 — subgroups

Subgroups — #125×Dic₄₇ #20245949

Order 6016.

484 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 249 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	#15689022	C₂	1	2	yes	maximal
2	#15689009	C₂	1	4	yes	maximal
3	#15688989	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×Dic₄₇	C₂	1	4	yes	maximal
5	C₄⋊C₄×Dic₄₇	C₂	1	2	yes	maximal
6	C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#125×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₄₇	C₂²	1	4	yes	—
10	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
11	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
12	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₄₇	C₂²	1	4	yes	—
14	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
15	C₄₇⋊#95	C₂²	1	4	yes	—
16	C₄₇⋊#95	C₂²	1	4	yes	—
17	C₄₇⋊#95	C₂²	1	4	yes	—
18	#125×C₄₇	C₄	1	8	yes	complemented
19	C₂×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
23	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
24	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₂²×Dic₄₇	C₂³	1	4	yes	—
26	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
27	C₂²×Dic₄₇	C₂³	1	2	yes	—
28	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₄₇	C₂×C₄	1	16	yes	—
30	C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
31	C₄²×C₄₇	C₂×C₄	1	4	yes	—
32	C₄⋊Dic₄₇	—	2	4	no	—
33	C₄⋊Dic₄₇	—	2	8	no	—
34	C₄×Dic₄₇	—	2	2	no	—
35	C₄×Dic₄₇	—	2	4	no	—
36	C₄×Dic₄₇	—	2	4	no	—
37	C₄×Dic₄₇	—	2	8	no	—
38	C₄×Dic₄₇	Q₈	1	4	yes	—
39	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
40	C₂×C₄×C₄₇	—	2	2	no	—
41	C₂×C₄×C₄₇	—	2	4	no	—
42	C₂×C₄×C₄₇	—	2	2	no	—
43	C₂×C₄×C₄₇	—	2	4	no	—
44	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
45	C₂×C₄×C₄₇	C₂²×C₄	1	8	yes	—
46	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₄₇	#41	1	2	yes	—
49	C₂×Dic₄₇	—	2	2	no	—
50	C₂×Dic₄₇	—	2	2	no	—
51	C₂×Dic₄₇	—	2	2	no	—
52	C₂×Dic₄₇	—	2	4	no	—
53	C₂×Dic₄₇	—	2	4	no	—
54	C₂×Dic₄₇	—	2	4	no	—
55	C₂×Dic₄₇	C₂×Q₈	1	2	yes	—
56	C₂×Dic₄₇	C₂×Q₈	1	2	yes	—
57	C₂×Dic₄₇	#41	1	4	yes	—
58	C₂×Dic₄₇	#41	1	4	yes	—
59	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
60	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
61	C₂²×C₄₇	C₂×#41	1	2	yes	—
62	C₂²×C₄₇	C₂×#41	1	2	yes	—
63	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
64	C₄×C₄₇	—	2	4	no	—
65	C₄×C₄₇	—	2	8	no	—
66	C₄×C₄₇	C₄×Q₈	1	4	yes	—
67	Dic₄₇	—	2	4	no	—
68	Dic₄₇	—	2	2	no	—
69	Dic₄₇	—	4	4	no	—
70	Dic₄₇	#125	1	4	yes	direct product factor
71	C₄×#125	—	47	1	no	maximal, Sylow
72	C₂×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄₇	C₄×#41	1	2	yes	—
74	C₂×C₄₇	C₂×#125	1	1	yes	characteristic
75	C₂×C₄₇	#529	1	2	yes	—
76	C₂×C₄₇	#533	1	1	yes	characteristic
77	C₂×#125	D₄₇	1	1	yes	characteristic
78	#389	—	47	2	no	—
79	#387	—	47	4	no	—
80	#382	—	47	1	no	—
81	C₄×C₄⋊C₄	—	47	4	no	—
82	C₄×C₄⋊C₄	—	47	2	no	—
83	C₄³	—	47	1	no	—
84	C₄₇	C₄×#125	1	1	yes	characteristic, complemented, Sylow
85	#125	Dic₄₇	1	8	yes	direct product factor
86	C₂×C₄⋊C₄	—	47	2	no	—
87	C₂×C₄⋊C₄	—	47	4	no	—
88	C₂×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
89	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
90	C₂×C₄²	—	47	1	no	—
91	C₂×C₄²	—	47	2	no	—
92	C₂×C₄²	—	47	2	no	—
93	C₂×C₄²	—	47	1	no	—
94	C₂×C₄²	—	47	4	no	—
95	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
96	#95	—	47	4	no	—
97	#95	—	47	4	no	—
98	#95	—	47	4	no	—
99	C₂²×C₄	—	47	1	no	—
100	C₂²×C₄	—	47	1	no	—
101	C₂²×C₄	—	47	2	no	—
102	C₂²×C₄	—	47	4	no	—
103	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
104	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
105	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
106	C₄⋊C₄	—	94	8	no	—
107	C₄⋊C₄	—	94	4	no	—
108	C₄⋊C₄	C₂×Dic₄₇	1	16	yes	—
109	C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
110	C₄²	—	47	4	no	—
111	C₄²	—	94	4	no	—
112	C₄²	—	94	8	no	—
113	C₄²	—	94	4	no	—
114	C₄²	—	94	2	no	—
115	C₄²	C₂×Dic₄₇	1	4	yes	—
116	C₂³	C₂³×D₄₇	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	47	2	no	—
122	C₂×C₄	—	47	4	no	—
123	C₂×C₄	—	47	2	no	—
124	C₂×C₄	—	47	4	no	—
125	C₂×C₄	—	94	2	no	—
126	C₂×C₄	—	94	2	no	—
127	C₂×C₄	—	94	2	no	—
128	C₂×C₄	—	94	4	no	—
129	C₂×C₄	—	94	4	no	—
130	C₂×C₄	—	94	4	no	—
131	C₂×C₄	Q₈×D₄₇	1	2	yes	—
132	C₂×C₄	#10625354	1	2	yes	—
133	C₂×C₄	C₂²×Dic₄₇	1	4	yes	—
134	C₂×C₄	C₂²×Dic₄₇	1	8	yes	—
135	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
136	C₂²	C₂³×Dic₄₇	1	1	yes	characteristic
137	C₂²	C₂×Q₈×D₄₇	1	1	yes	characteristic
138	C₂²	#41×D₄₇	1	2	yes	—
139	C₂²	#41×D₄₇	1	2	yes	—
140	C₂²	C₂×#10625354	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	2	8	no	—
143	C₄	—	47	4	no	—
144	C₄	—	94	4	no	—
145	C₄	—	94	2	no	—
146	C₄	—	188	4	no	—
147	C₄	Q₈×Dic₄₇	1	4	yes	—
148	C₂	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
149	C₂	#41×Dic₄₇	1	2	yes	—
150	C₂	#125×D₄₇	1	1	yes	characteristic
151	C₂	#15689658	1	2	yes	—
152	C₂	#15689679	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689022

C₂

yes

maximal

#15689009

C₂

yes

maximal

#15688989

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₄₇

C₂

yes

maximal

C₄⋊C₄×Dic₄₇

C₂

yes

maximal

C₄²×Dic₄₇

C₂

yes

maximal, characteristic

C₂×#125×C₄₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

C₂×C₄×Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

#125×C₄₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

Q₈

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×Q₈

yes

—

C₂×Dic₄₇

C₂×Q₈

yes

—

C₂×Dic₄₇

#41

yes

—

C₂×Dic₄₇

#41

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

C₂×#41

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×Q₈

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

#125

yes

direct product factor

C₄×#125

—

maximal, Sylow

C₂×C₄₇

C₂×C₄×Q₈

yes

characteristic

C₂×C₄₇

C₄×#41

yes

—

C₂×C₄₇

C₂×#125

yes

characteristic

C₂×C₄₇

#529

yes

—

C₂×C₄₇

#533

yes

characteristic

C₂×#125

D₄₇

yes

—

—

—

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

C₄³

—

C₄₇

C₄×#125

yes

characteristic, complemented, Sylow

#125

Dic₄₇

yes

direct product factor

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

—

—

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²×D₄₇

yes

—

104

C₂²×C₄

C₂²×D₄₇

yes

—

105

C₂²×C₄

C₂²×D₄₇

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

C₂×Dic₄₇

yes

—

109

C₄⋊C₄

C₂×Dic₄₇

yes

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

C₂×Dic₄₇

yes

—

116

C₂³

C₂³×D₄₇

yes

characteristic

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

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

Q₈×D₄₇

yes

—

132

C₂×C₄

#10625354

yes

—

133

C₂×C₄

C₂²×Dic₄₇

yes

—

134

C₂×C₄

C₂²×Dic₄₇

yes

—

135

C₂×C₄

C₂²×Dic₄₇

yes

—

136

C₂²

C₂³×Dic₄₇

yes

characteristic

137

C₂²

C₂×Q₈×D₄₇

yes

characteristic

138

C₂²

#41×D₄₇

yes

—

139

C₂²

#41×D₄₇

yes

—

140

C₂²

C₂×#10625354

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

188

—

147

C₄

Q₈×Dic₄₇

yes

—

148

C₂

C₂×Q₈×Dic₄₇

yes

characteristic

149

C₂

#41×Dic₄₇

yes

—

150

C₂

#125×D₄₇

yes

characteristic

151

C₂

#15689658

yes

—

152

C₂

#15689679

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₂×#125×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂×#125	D₄₇
hypocenter	C₄₇	C₄×#125
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator

C₂²×C₄₇

C₂³×C₄

Fitting

C₂×#125×C₄₇

C₂

socle

C₂³×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
18. #125×C₄₇	C₄	semidirect
70. Dic₄₇	#125	direct (N × H)
84. C₄₇	C₄×#125	semidirect
85. #125	Dic₄₇	direct (N × H)

H ≅ G/N

type

18. #125×C₄₇

C₄

semidirect

70. Dic₄₇

#125

direct (N × H)

84. C₄₇

C₄×#125

semidirect

85. #125

Dic₄₇

direct (N × H)

Subgroups — #125×Dic₄₇ #20245949

Conjugacy classes of subgroups

Complements

Subgroups — #125×Dic47 #20245949

Conjugacy classes of subgroups

Complements

Subgroups — #125×Dic₄₇ #20245949