Group #20242033 — subgroups

Subgroups — C₂×#15688870 #20242033

Order 6016.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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×#13094024	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#15688870	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₉₄	—	2	1	no	—
6	C₂²×C₄₇⋊C₈	—	2	1	no	—
7	#13094024	—	2	1	no	—
8	#13094024	—	2	1	no	—
9	#13094024	C₄	1	2	yes	—
10	#13094024	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₄₇	—	2	1	no	—
12	C₂×C₄⋊Dic₄₇	C₄	1	2	yes	—
13	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
14	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
15	C₂²×C₈×C₄₇	—	2	1	no	—
16	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₄₇	—	2	1	no	—
18	C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
19	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
22	C₂²×Dic₄₇	—	2	2	no	—
23	C₂×Dic₉₄	—	2	2	no	—
24	C₂×Dic₉₄	—	2	2	no	—
25	C₂×Dic₉₄	—	4	2	no	—
26	C₂×Dic₉₄	—	4	2	no	—
27	C₂×C₈×C₄₇	—	2	2	no	—
28	C₂×C₈×C₄₇	—	4	2	no	—
29	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₄₇	D₄	1	1	yes	characteristic
31	C₄²×C₄₇	D₄	1	1	yes	characteristic
32	C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	C₄⋊Dic₄₇	—	2	2	no	—
34	C₄⋊Dic₄₇	—	2	2	no	—
35	C₄⋊Dic₄₇	—	2	2	no	—
36	C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₄₇⋊C₈	—	2	2	no	—
38	C₂×C₄₇⋊C₈	—	4	2	no	—
39	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₄₇	—	2	2	no	—
41	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₄₇	D₈	1	2	yes	—
45	C₂×C₄×C₄₇	SD₁₆	1	2	yes	—
46	C₂×C₄×C₄₇	SD₁₆	1	2	yes	—
47	C₂×C₄×C₄₇	Q₁₆	1	2	yes	—
48	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₄₇	—	2	4	no	—
52	C₂×Dic₄₇	—	4	4	no	—
53	Dic₉₄	—	4	4	no	—
54	Dic₉₄	—	4	2	no	—
55	Dic₉₄	—	4	2	no	—
56	C₈×C₄₇	—	4	2	no	—
57	C₄₇⋊C₈	—	4	2	no	—
58	C₂²×C₄₇	#101	1	2	yes	—
59	C₂²×C₄₇	#102	1	2	yes	—
60	C₂²×C₄₇	#103	1	2	yes	—
61	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₄×C₄₇	—	2	1	no	—
63	C₄×C₄₇	—	2	1	no	—
64	C₄×C₄₇	#102	1	2	yes	—
65	C₄×C₄₇	#103	1	2	yes	—
66	C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
67	C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
68	C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
70	Dic₄₇	—	4	4	no	—
71	C₂×#332	—	47	1	no	maximal, Sylow
72	C₂×C₄₇	#332	1	4	yes	complemented
73	C₂×C₄₇	C₂×#101	1	1	yes	characteristic
74	C₂×C₄₇	C₂×#102	1	1	yes	characteristic
75	C₂×C₄₇	C₂×#103	1	1	yes	characteristic
76	C₂×C₄⋊Q₈	—	47	1	no	—
77	C₂×C₄⋊C₈	—	47	1	no	—
78	C₂×C₄⋊C₈	D₄₇	1	1	yes	characteristic
79	#332	—	47	4	no	—
80	C₄₇	C₂×#332	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₈	—	94	1	no	—
82	C₂²×C₈	—	2	1	no	—
83	C₂²×C₈	—	94	1	no	—
84	C₄⋊Q₈	—	47	2	no	—
85	C₄⋊Q₈	—	47	2	no	—
86	C₄⋊Q₈	—	94	1	no	—
87	C₄⋊Q₈	—	94	1	no	—
88	C₂×C₄⋊C₄	—	47	2	no	—
89	C₂×C₄⋊C₄	—	94	1	no	—
90	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
91	C₄⋊C₈	—	2	1	no	—
92	C₄⋊C₈	—	47	2	no	—
93	C₄⋊C₈	—	94	1	no	—
94	C₄⋊C₈	C₂×D₄₇	1	2	yes	—
95	C₂×Q₈	—	94	2	no	—
96	C₂×Q₈	—	94	2	no	—
97	C₂×Q₈	—	188	2	no	—
98	C₂×Q₈	—	188	2	no	—
99	C₂²×C₄	—	94	2	no	—
100	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
101	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
102	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	4	2	no	—
105	C₂×C₈	—	94	2	no	—
106	C₂×C₈	—	188	2	no	—
107	C₄⋊C₄	—	47	4	no	—
108	C₄⋊C₄	—	94	2	no	—
109	C₄⋊C₄	—	94	2	no	—
110	C₄⋊C₄	—	94	2	no	—
111	C₄²	C₄×D₄₇	1	1	yes	characteristic
112	C₄²	D₁₈₈	1	1	yes	characteristic
113	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
114	C₄²	C₂²×D₄₇	1	1	yes	characteristic
115	C₂³	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
116	Q₈	—	188	4	no	—
117	Q₈	—	188	2	no	—
118	Q₈	—	188	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	94	4	no	—
121	C₂×C₄	—	188	4	no	—
122	C₂×C₄	#10625325	1	2	yes	—
123	C₂×C₄	Dic₁₈₈	1	2	yes	—
124	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
125	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
126	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
127	C₂×C₄	#10625334	1	2	yes	—
128	C₂×C₄	#10625336	1	2	yes	—
129	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
130	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
132	C₈	—	4	2	no	—
133	C₈	—	188	2	no	—
134	C₂²	#13093942	1	2	yes	—
135	C₂²	#13093944	1	2	yes	—
136	C₂²	#13093949	1	2	yes	—
137	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	188	4	no	—
141	C₄	#13093944	1	2	yes	—
142	C₄	#13093949	1	2	yes	—
143	C₄	C₂×#10625334	1	1	yes	characteristic
144	C₄	C₂×#10625336	1	1	yes	characteristic
145	C₄	C₂×#10625325	1	1	yes	characteristic
146	C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
147	C₂	#15688870	1	4	yes	direct product factor
148	C₂	C₂×#13093942	1	1	yes	characteristic
149	C₂	C₂×#13093944	1	1	yes	characteristic
150	C₂	C₂×#13093949	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094024

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic

#15688870

C₂

yes

maximal, direct product factor

C₂×C₄⋊C₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

—

C₂²×C₄₇⋊C₈

—

#13094024

—

#13094024

—

#13094024

C₄

yes

—

#13094024

C₂²

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₄

yes

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂²×C₈×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

C₂²

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

D₈

yes

—

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

Q₁₆

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

#101

yes

—

C₂²×C₄₇

#102

yes

—

C₂²×C₄₇

#103

yes

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#102

yes

—

C₄×C₄₇

#103

yes

—

C₄×C₄₇

C₂×D₈

yes

characteristic

C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₄×C₄₇

C₂×Q₁₆

yes

characteristic

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₄₇

yes

characteristic

#332

—

C₄₇

C₂×#332

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

188

—

C₂×Q₈

—

188

—

C₂²×C₄

—

100

C₂²×C₄

C₄×D₄₇

yes

characteristic

101

C₂²×C₄

D₁₈₈

yes

characteristic

102

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

188

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄²

C₄×D₄₇

yes

characteristic

112

C₄²

D₁₈₈

yes

characteristic

113

C₄²

C₂²⋊D₄₇

yes

characteristic

114

C₄²

C₂²×D₄₇

yes

characteristic

115

C₂³

(C₂×D₄₇)⋊C₄

yes

characteristic

116

Q₈

—

188

—

117

Q₈

—

188

—

118

Q₈

—

188

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

188

—

122

C₂×C₄

#10625325

yes

—

123

C₂×C₄

Dic₁₈₈

yes

—

124

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

125

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

126

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

127

C₂×C₄

#10625334

yes

—

128

C₂×C₄

#10625336

yes

—

129

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

130

C₂×C₄

C₂×D₁₈₈

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

132

C₈

—

133

C₈

—

188

—

134

C₂²

#13093942

yes

—

135

C₂²

#13093944

yes

—

136

C₂²

#13093949

yes

—

137

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

188

—

141

C₄

#13093944

yes

—

142

C₄

#13093949

yes

—

143

C₄

C₂×#10625334

yes

characteristic

144

C₄

C₂×#10625336

yes

characteristic

145

C₄

C₂×#10625325

yes

characteristic

146

C₄

C₂×Dic₁₈₈

yes

characteristic

147

C₂

#15688870

yes

direct product factor

148

C₂

C₂×#13093942

yes

characteristic

149

C₂

C₂×#13093944

yes

characteristic

150

C₂

C₂×#13093949

yes

characteristic

	subgroup	G/H
center	C₂³	(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₄²	C₂²×D₄₇
Fitting	C₂×C₄⋊C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²⋊C₄
hypercenter	C₂×C₄⋊C₈	D₄₇
hypocenter	C₄₇	C₂×#332
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

(C₂×D₄₇)⋊C₄

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₄²

C₂²×D₄₇

Fitting

C₂×C₄⋊C₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂²⋊C₄

hypercenter

C₂×C₄⋊C₈

D₄₇

hypocenter

C₄₇

C₂×#332

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #15688870	C₂	direct (N × H)
72. C₂×C₄₇	#332	semidirect
80. C₄₇	C₂×#332	semidirect
147. C₂	#15688870	direct (N × H)

H ≅ G/N

type

3. #15688870

C₂

direct (N × H)

72. C₂×C₄₇

#332

semidirect

80. C₄₇

C₂×#332

semidirect

147. C₂

#15688870

direct (N × H)

Subgroups — C₂×#15688870 #20242033

Conjugacy classes of subgroups

Complements

Subgroups — C2×#15688870 #20242033

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15688870 #20242033