Group #20243121 — subgroups

Subgroups — #20243121

Order 6016.

420 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 189 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×#13093983	C₂	1	2	yes	maximal
2	#15688890	C₂	1	4	yes	maximal
3	C₂×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
4	#13093983	—	2	4	no	—
5	#13093983	C₄	1	8	yes	complemented
6	C₂×C₄⋊Dic₄₇	—	2	2	no	—
7	C₂×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
8	C₂×C₄×Dic₄₇	—	2	2	no	—
9	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
10	#117×C₄₇	—	2	2	no	—
11	#117×C₄₇	—	2	2	no	—
12	C₂×C₄⋊C₄×C₄₇	C₄	1	2	yes	complemented
13	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₄₇	C₄	1	2	yes	complemented
15	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
16	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	D₄	1	2	yes	—
21	C₂²×C₄×C₄₇	D₄	1	2	yes	—
22	C₂²×C₄×C₄₇	D₄	1	2	yes	—
23	C₂²×C₄×C₄₇	Q₈	1	2	yes	—
24	C₂²×Dic₄₇	—	2	4	no	—
25	C₄⋊C₄×C₄₇	—	2	4	no	—
26	C₂²⋊C₄×C₄₇	—	2	4	no	—
27	C₄²×C₄₇	—	2	2	no	—
28	C₄²×C₄₇	—	2	2	no	—
29	C₂²⋊Dic₄₇	—	2	4	no	—
30	C₂²⋊Dic₄₇	C₂×C₄	1	8	yes	—
31	C₄⋊Dic₄₇	—	2	8	no	—
32	C₄×Dic₄₇	—	2	8	no	—
33	C₂³×C₄₇	—	4	2	no	—
34	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
35	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
36	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂³×C₄₇	C₂×D₄	1	2	yes	—
38	C₂³×C₄₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₄₇	—	2	4	no	—
40	C₂×C₄×C₄₇	—	4	4	no	—
41	C₂×C₄×C₄₇	C₄²	1	2	yes	—
42	C₂×C₄×C₄₇	C₄²	1	2	yes	—
43	C₂×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
44	C₂×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
45	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
49	C₂×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
50	C₂×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
51	C₂×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
52	C₂×Dic₄₇	—	2	8	no	—
53	C₂×Dic₄₇	—	4	8	no	—
54	C₂²×C₄₇	—	2	4	no	—
55	C₂²×C₄₇	—	2	2	no	—
56	C₂²×C₄₇	—	4	2	no	—
57	C₂²×C₄₇	—	4	1	no	—
58	C₂²×C₄₇	—	4	1	no	—
59	C₂²×C₄₇	C₂×C₄²	1	1	yes	characteristic
60	C₂²×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
63	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₄×C₄₇	—	4	4	no	—
65	C₄×C₄₇	#95	1	4	yes	—
66	C₄×C₄₇	#95	1	2	yes	—
67	C₄×C₄₇	#95	1	2	yes	—
68	Dic₄₇	—	4	8	no	—
69	#1740	—	47	1	no	maximal, Sylow
70	C₂×C₄₇	—	2	2	no	—
71	C₂×C₄₇	—	2	1	no	—
72	C₂×C₄₇	—	2	2	no	—
73	C₂×C₄₇	—	2	1	no	—
74	C₂×C₄₇	C₂×#95	1	1	yes	characteristic
75	C₂×C₄₇	#410	1	2	yes	—
76	C₂×#117	—	47	2	no	—
77	C₂×#117	D₄₇	1	1	yes	characteristic
78	#342	—	47	4	no	—
79	C₄₇	#1740	1	1	yes	characteristic, complemented, Sylow
80	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
81	#117	—	2	2	no	—
82	#117	—	2	2	no	—
83	#117	—	47	8	no	—
84	#117	—	94	4	no	—
85	C₂×C₄⋊C₄	—	94	2	no	—
86	C₂×C₄⋊C₄	Dic₄₇	1	2	yes	complemented
87	C₂×C₂²⋊C₄	—	47	4	no	—
88	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
89	C₂×C₄²	—	94	2	no	—
90	C₂×C₄²	Dic₄₇	1	2	yes	complemented
91	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
92	C₂²×C₄	—	94	4	no	—
93	C₂²×C₄	Dic₉₄	1	2	yes	—
94	C₂²×C₄	C₄×D₄₇	1	4	yes	—
95	C₂²×C₄	D₁₈₈	1	2	yes	—
96	C₂²×C₄	C₂×Dic₄₇	1	4	yes	—
97	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
98	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
100	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
101	C₄⋊C₄	—	2	4	no	—
102	C₄⋊C₄	—	94	8	no	—
103	C₂²⋊C₄	—	2	4	no	—
104	C₂²⋊C₄	—	47	8	no	—
105	C₂²⋊C₄	—	94	4	no	—
106	C₄²	—	2	2	no	—
107	C₄²	—	2	2	no	—
108	C₄²	—	94	8	no	—
109	C₂³	—	4	2	no	—
110	C₂³	C₂×Dic₉₄	1	1	yes	characteristic
111	C₂³	C₂×C₄×D₄₇	1	2	yes	—
112	C₂³	C₂×D₁₈₈	1	1	yes	characteristic
113	C₂³	C₂²×Dic₄₇	1	1	yes	characteristic
114	C₂³	C₂×C₂²⋊D₄₇	1	2	yes	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	94	8	no	—
118	C₂×C₄	—	188	8	no	—
119	C₂×C₄	C₄×Dic₄₇	1	2	yes	—
120	C₂×C₄	C₄×Dic₄₇	1	2	yes	—
121	C₂×C₄	Dic₄₇⋊C₄	1	4	yes	—
122	C₂×C₄	Dic₄₇⋊C₄	1	4	yes	—
123	C₂×C₄	C₄⋊Dic₄₇	1	2	yes	—
124	C₂×C₄	C₄⋊Dic₄₇	1	2	yes	—
125	C₂×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
126	C₂×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
127	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
128	C₂×C₄	C₂²⋊Dic₄₇	1	2	yes	—
129	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
130	C₂²	—	2	2	no	—
131	C₂²	—	2	4	no	—
132	C₂²	—	4	1	no	—
133	C₂²	—	4	2	no	—
134	C₂²	—	4	1	no	—
135	C₂²	C₂×C₄×Dic₄₇	1	1	yes	characteristic
136	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
137	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
138	C₂²	C₂×Dic₄₇⋊C₄	1	2	yes	—
139	C₂²	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
140	C₄	—	4	4	no	—
141	C₄	—	188	8	no	—
142	C₄	C₄₇⋊#95	1	4	yes	—
143	C₄	C₄₇⋊#95	1	2	yes	—
144	C₄	C₄₇⋊#95	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	2	2	no	—
148	C₂	—	2	1	no	—
149	C₂	C₂×C₄₇⋊#95	1	1	yes	characteristic
150	C₂	#15689097	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093983

C₂

yes

maximal

#15688890

C₂

yes

maximal

C₂×#117×C₄₇

C₂

yes

maximal, characteristic

#13093983

—

#13093983

C₄

yes

complemented

C₂×C₄⋊Dic₄₇

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

—

C₂³×C₄×C₄₇

C₂²

yes

characteristic

#117×C₄₇

—

#117×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₄

yes

complemented

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₄

yes

complemented

C₂⁴×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

Q₈

yes

—

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

C₂×C₄

yes

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²×C₄

yes

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄²

yes

—

C₂×C₄×C₄₇

C₄²

yes

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄²

yes

characteristic

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

C₂×C₄⋊C₄

yes

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

#95

yes

—

C₄×C₄₇

#95

yes

—

C₄×C₄₇

#95

yes

—

Dic₄₇

—

#1740

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂×#95

yes

characteristic

C₂×C₄₇

#410

yes

—

C₂×#117

—

C₂×#117

D₄₇

yes

characteristic

#342

—

C₄₇

#1740

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₄₇

yes

—

—

—

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₄₇

yes

complemented

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₄₇

yes

—

C₂×C₄²

—

C₂×C₄²

Dic₄₇

yes

complemented

C₂⁴

C₂²×D₄₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

Dic₉₄

yes

—

C₂²×C₄

C₄×D₄₇

yes

—

C₂²×C₄

D₁₈₈

yes

—

C₂²×C₄

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

—

100

C₂²×C₄

C₂²⋊D₄₇

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

—

105

C₂²⋊C₄

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₂³

—

110

C₂³

C₂×Dic₉₄

yes

characteristic

111

C₂³

C₂×C₄×D₄₇

yes

—

112

C₂³

C₂×D₁₈₈

yes

characteristic

113

C₂³

C₂²×Dic₄₇

yes

characteristic

114

C₂³

C₂×C₂²⋊D₄₇

yes

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

188

—

119

C₂×C₄

C₄×Dic₄₇

yes

—

120

C₂×C₄

C₄×Dic₄₇

yes

—

121

C₂×C₄

Dic₄₇⋊C₄

yes

—

122

C₂×C₄

Dic₄₇⋊C₄

yes

—

123

C₂×C₄

C₄⋊Dic₄₇

yes

—

124

C₂×C₄

C₄⋊Dic₄₇

yes

—

125

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

—

126

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

—

127

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

128

C₂×C₄

C₂²⋊Dic₄₇

yes

—

129

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

C₂×C₄×Dic₄₇

yes

characteristic

136

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

—

137

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

138

C₂²

C₂×Dic₄₇⋊C₄

yes

—

139

C₂²

C₂×C₄⋊Dic₄₇

yes

characteristic

140

C₄

—

141

C₄

—

188

—

142

C₄

C₄₇⋊#95

yes

—

143

C₄

C₄₇⋊#95

yes

—

144

C₄

C₄₇⋊#95

yes

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

—

149

C₂

C₂×C₄₇⋊#95

yes

characteristic

150

C₂

#15689097

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₄₇
commutator	C₂²×C₄₇	C₂×C₄²
Frattini	C₂⁴	C₂²×D₄₇
Fitting	C₂×#117×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	C₂×#117	D₄₇
hypocenter	C₄₇	#1740
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₄₇

commutator

C₂²×C₄₇

C₂×C₄²

Frattini

C₂⁴

C₂²×D₄₇

Fitting

C₂×#117×C₄₇

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

N	H ≅ G/N	type
5. #13093983	C₄	semidirect
12. C₂×C₄⋊C₄×C₄₇	C₄	semidirect
14. C₂×C₄²×C₄₇	C₄	semidirect
79. C₄₇	#1740	semidirect
86. C₂×C₄⋊C₄	Dic₄₇	semidirect
90. C₂×C₄²	Dic₄₇	semidirect

H ≅ G/N

type

5. #13093983

C₄

semidirect

12. C₂×C₄⋊C₄×C₄₇

C₄

semidirect

14. C₂×C₄²×C₄₇

C₄

semidirect

79. C₄₇

#1740

semidirect

86. C₂×C₄⋊C₄

Dic₄₇

semidirect

90. C₂×C₄²

Dic₄₇

semidirect

Subgroups — #20243121

Conjugacy classes of subgroups

Complements