Group #44503555 — subgroups

Subgroups — #1888×C₂₄₁ #44503555

Order 30848.

344 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 112 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×#124×C₂₄₁	C₂	1	1	yes	maximal, characteristic
2	#519×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#418×C₂₄₁	C₂	1	2	yes	maximal
4	C₂×#101×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#339×C₂₄₁	C₂	1	2	yes	maximal, complemented
6	C₂²×Q₈×C₂₄₁	C₂²	1	1	yes	characteristic
7	C₂²×D₄×C₂₄₁	C₂²	1	1	yes	characteristic
8	C₂×M₁₆×C₂₄₁	C₂²	1	2	yes	—
9	#124×C₂₄₁	—	2	2	no	—
10	#124×C₂₄₁	C₄	1	4	yes	complemented
11	C₄×Q₈×C₂₄₁	—	2	4	no	—
12	#117×C₂₄₁	—	2	2	no	—
13	#117×C₂₄₁	C₂²	1	2	yes	—
14	C₂×C₂²⋊C₄×C₂₄₁	—	2	2	no	—
15	C₂×C₄²×C₂₄₁	C₂²	1	1	yes	characteristic
16	#102×C₂₄₁	—	2	4	no	—
17	#101×C₂₄₁	—	2	2	no	—
18	C₂⁴×C₂₄₁	—	2	1	no	—
19	C₂×Q₈×C₂₄₁	—	2	2	no	—
20	C₂×Q₈×C₂₄₁	—	2	2	no	—
21	C₂×Q₈×C₂₄₁	C₂×C₄	1	2	yes	—
22	C₂×Q₈×C₂₄₁	D₄	1	2	yes	—
23	C₂×D₄×C₂₄₁	—	2	2	no	—
24	C₂×D₄×C₂₄₁	—	2	1	no	—
25	C₂×D₄×C₂₄₁	—	2	2	no	—
26	C₂×D₄×C₂₄₁	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₂₄₁	—	2	1	no	—
28	C₂²×C₄×C₂₄₁	D₄	1	2	yes	—
29	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
30	M₁₆×C₂₄₁	—	2	2	no	—
31	M₁₆×C₂₄₁	—	2	2	no	—
32	C₂×C₈×C₂₄₁	—	2	2	no	—
33	C₄⋊C₄×C₂₄₁	—	2	4	no	—
34	C₄⋊C₄×C₂₄₁	D₄	1	4	yes	—
35	C₂²⋊C₄×C₂₄₁	—	2	2	no	—
36	C₂²⋊C₄×C₂₄₁	—	2	8	no	—
37	C₄²×C₂₄₁	—	2	4	no	—
38	C₄²×C₂₄₁	—	2	1	no	—
39	C₄²×C₂₄₁	—	2	2	no	—
40	C₄²×C₂₄₁	C₂×C₄	1	2	yes	—
41	C₂³×C₂₄₁	—	2	2	no	—
42	C₂³×C₂₄₁	—	4	4	no	—
43	C₂³×C₂₄₁	—	4	1	no	—
44	C₂³×C₂₄₁	—	4	1	no	—
45	C₂³×C₂₄₁	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₂₄₁	—	2	4	no	—
47	Q₈×C₂₄₁	—	2	4	no	—
48	D₄×C₂₄₁	—	2	4	no	—
49	D₄×C₂₄₁	—	4	2	no	—
50	C₂×C₄×C₂₄₁	—	2	2	no	—
51	C₂×C₄×C₂₄₁	—	2	2	no	—
52	C₂×C₄×C₂₄₁	—	2	2	no	—
53	C₂×C₄×C₂₄₁	—	2	4	no	—
54	C₂×C₄×C₂₄₁	—	4	2	no	—
55	C₂×C₄×C₂₄₁	C₂²⋊C₄	1	4	yes	—
56	C₂×C₄×C₂₄₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₂₄₁	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₂₄₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₂₄₁	#41	1	2	yes	—
60	C₈×C₂₄₁	—	4	2	no	—
61	C₂²×C₂₄₁	—	2	2	no	—
62	C₂²×C₂₄₁	—	4	4	no	—
63	C₂²×C₂₄₁	—	4	2	no	—
64	C₂²×C₂₄₁	—	8	2	no	—
65	C₂²×C₂₄₁	—	8	2	no	—
66	C₂²×C₂₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₂₄₁	#120	1	1	yes	characteristic
68	C₂²×C₂₄₁	#123	1	1	yes	characteristic
69	C₄×C₂₄₁	—	2	4	no	—
70	C₄×C₂₄₁	—	4	2	no	—
71	C₄×C₂₄₁	—	4	4	no	—
72	C₄×C₂₄₁	C₄×D₄	1	2	yes	—
73	C₄×C₂₄₁	#121	1	2	yes	—
74	C₂×C₂₄₁	—	2	1	no	—
75	C₂×C₂₄₁	—	2	1	no	—
76	C₂×C₂₄₁	—	8	2	no	—
77	C₂×C₂₄₁	#386	1	1	yes	characteristic
78	C₂×C₂₄₁	#454	1	2	yes	—
79	C₂₄₁	#1888	1	1	yes	characteristic, direct product factor, Sylow
80	#1888	C₂₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂×#124	C₂×C₂₄₁	1	1	yes	characteristic
82	#519	C₂×C₂₄₁	1	1	yes	characteristic, complemented
83	#418	C₂×C₂₄₁	1	2	yes	—
84	C₂×#101	C₂×C₂₄₁	1	1	yes	characteristic, complemented
85	#339	C₂×C₂₄₁	1	2	yes	complemented
86	C₂²×Q₈	C₂²×C₂₄₁	1	1	yes	characteristic
87	C₂²×D₄	C₂²×C₂₄₁	1	1	yes	characteristic
88	C₂×M₁₆	C₂²×C₂₄₁	1	2	yes	—
89	#124	—	2	2	no	—
90	#124	C₄×C₂₄₁	1	4	yes	complemented
91	C₄×Q₈	—	2	4	no	—
92	#117	—	2	2	no	—
93	#117	C₂²×C₂₄₁	1	2	yes	—
94	C₂×C₂²⋊C₄	—	2	2	no	—
95	C₂×C₄²	C₂²×C₂₄₁	1	1	yes	characteristic
96	#102	—	2	4	no	—
97	#101	—	2	2	no	—
98	C₂⁴	—	2	1	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	C₂×C₄×C₂₄₁	1	2	yes	—
102	C₂×Q₈	D₄×C₂₄₁	1	2	yes	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	2	2	no	—
106	C₂×D₄	C₂×C₄×C₂₄₁	1	2	yes	—
107	C₂²×C₄	—	2	1	no	—
108	C₂²×C₄	D₄×C₂₄₁	1	2	yes	—
109	C₂²×C₄	C₂³×C₂₄₁	1	1	yes	characteristic
110	M₁₆	—	2	2	no	—
111	M₁₆	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₄⋊C₄	—	2	4	no	—
114	C₄⋊C₄	D₄×C₂₄₁	1	4	yes	—
115	C₂²⋊C₄	—	2	2	no	—
116	C₂²⋊C₄	—	2	8	no	—
117	C₄²	—	2	4	no	—
118	C₄²	—	2	1	no	—
119	C₄²	—	2	2	no	—
120	C₄²	C₂×C₄×C₂₄₁	1	2	yes	—
121	C₂³	—	2	2	no	—
122	C₂³	—	4	4	no	—
123	C₂³	—	4	1	no	—
124	C₂³	—	4	1	no	—
125	C₂³	C₂×D₄×C₂₄₁	1	1	yes	characteristic
126	Q₈	—	2	4	no	—
127	Q₈	—	2	4	no	—
128	D₄	—	2	4	no	—
129	D₄	—	4	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	C₂²⋊C₄×C₂₄₁	1	4	yes	—
136	C₂×C₄	C₂²×C₄×C₂₄₁	1	1	yes	characteristic
137	C₂×C₄	C₂×D₄×C₂₄₁	1	2	yes	—
138	C₂×C₄	C₂×D₄×C₂₄₁	1	1	yes	characteristic
139	C₂×C₄	#41×C₂₄₁	1	2	yes	—
140	C₈	—	4	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	4	4	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	8	2	no	—
145	C₂²	—	8	2	no	—
146	C₂²	C₂×C₂²⋊C₄×C₂₄₁	1	1	yes	characteristic
147	C₂²	#120×C₂₄₁	1	1	yes	characteristic
148	C₂²	#123×C₂₄₁	1	1	yes	characteristic
149	C₄	—	2	4	no	—
150	C₄	—	4	2	no	—
151	C₄	—	4	4	no	—
152	C₄	C₄×D₄×C₂₄₁	1	2	yes	—
153	C₄	#121×C₂₄₁	1	2	yes	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	—	8	2	no	—
157	C₂	#386×C₂₄₁	1	1	yes	characteristic
158	C₂	#454×C₂₄₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#124×C₂₄₁

C₂

yes

maximal, characteristic

#519×C₂₄₁

C₂

yes

maximal, characteristic, complemented

#418×C₂₄₁

C₂

yes

maximal

C₂×#101×C₂₄₁

C₂

yes

maximal, characteristic, complemented

#339×C₂₄₁

C₂

yes

maximal, complemented

C₂²×Q₈×C₂₄₁

C₂²

yes

characteristic

C₂²×D₄×C₂₄₁

C₂²

yes

characteristic

C₂×M₁₆×C₂₄₁

C₂²

yes

—

#124×C₂₄₁

—

#124×C₂₄₁

C₄

yes

complemented

C₄×Q₈×C₂₄₁

—

#117×C₂₄₁

—

#117×C₂₄₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₄₁

—

C₂×C₄²×C₂₄₁

C₂²

yes

characteristic

#102×C₂₄₁

—

#101×C₂₄₁

—

C₂⁴×C₂₄₁

—

C₂×Q₈×C₂₄₁

—

C₂×Q₈×C₂₄₁

—

C₂×Q₈×C₂₄₁

C₂×C₄

yes

—

C₂×Q₈×C₂₄₁

D₄

yes

—

C₂×D₄×C₂₄₁

—

C₂×D₄×C₂₄₁

—

C₂×D₄×C₂₄₁

—

C₂×D₄×C₂₄₁

C₂×C₄

yes

—

C₂²×C₄×C₂₄₁

—

C₂²×C₄×C₂₄₁

D₄

yes

—

C₂²×C₄×C₂₄₁

C₂³

yes

characteristic

M₁₆×C₂₄₁

—

M₁₆×C₂₄₁

—

C₂×C₈×C₂₄₁

—

C₄⋊C₄×C₂₄₁

—

C₄⋊C₄×C₂₄₁

D₄

yes

—

C₂²⋊C₄×C₂₄₁

—

C₂²⋊C₄×C₂₄₁

—

C₄²×C₂₄₁

—

C₄²×C₂₄₁

—

C₄²×C₂₄₁

—

C₄²×C₂₄₁

C₂×C₄

yes

—

C₂³×C₂₄₁

—

C₂³×C₂₄₁

—

C₂³×C₂₄₁

—

C₂³×C₂₄₁

—

C₂³×C₂₄₁

C₂×D₄

yes

characteristic

Q₈×C₂₄₁

—

Q₈×C₂₄₁

—

D₄×C₂₄₁

—

D₄×C₂₄₁

—

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₄₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₄₁

C₂×D₄

yes

—

C₂×C₄×C₂₄₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₄₁

#41

yes

—

C₈×C₂₄₁

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

C₂×C₂²⋊C₄

yes

C₂²×C₂₄₁

yes

C₂²×C₂₄₁

yes

C₄×C₂₄₁

—

C₄×C₂₄₁

—

C₄×C₂₄₁

—

C₄×C₂₄₁

C₄×D₄

yes

—

C₄×C₂₄₁

#121

yes

—

C₂×C₂₄₁

—

C₂×C₂₄₁

—

C₂×C₂₄₁

—

C₂×C₂₄₁

#386

yes

characteristic

C₂×C₂₄₁

#454

yes

—

C₂₄₁

#1888

yes

characteristic, direct product factor, Sylow

#1888

C₂₄₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×#124

C₂×C₂₄₁

yes

characteristic

#519

C₂×C₂₄₁

yes

characteristic, complemented

#418

C₂×C₂₄₁

yes

—

C₂×#101

C₂×C₂₄₁

yes

characteristic, complemented

#339

C₂×C₂₄₁

yes

complemented

C₂²×Q₈

C₂²×C₂₄₁

yes

characteristic

C₂²×D₄

C₂²×C₂₄₁

yes

characteristic

C₂×M₁₆

C₂²×C₂₄₁

yes

—

#124

—

#124

C₄×C₂₄₁

yes

complemented

C₄×Q₈

—

#117

—

#117

C₂²×C₂₄₁

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂²×C₂₄₁

yes

characteristic

#102

—

#101

—

C₂⁴

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

C₂×C₄×C₂₄₁

yes

—

102

C₂×Q₈

D₄×C₂₄₁

yes

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

C₂×C₄×C₂₄₁

yes

—

107

C₂²×C₄

—

108

C₂²×C₄

D₄×C₂₄₁

yes

—

109

C₂²×C₄

C₂³×C₂₄₁

yes

characteristic

110

M₁₆

—

111

M₁₆

—

112

C₂×C₈

—

113

C₄⋊C₄

—

114

C₄⋊C₄

D₄×C₂₄₁

yes

—

115

C₂²⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₂×C₄×C₂₄₁

yes

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

C₂×D₄×C₂₄₁

yes

characteristic

126

Q₈

—

127

Q₈

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂²⋊C₄×C₂₄₁

yes

—

136

C₂×C₄

C₂²×C₄×C₂₄₁

yes

characteristic

137

C₂×C₄

C₂×D₄×C₂₄₁

yes

—

138

C₂×C₄

C₂×D₄×C₂₄₁

yes

characteristic

139

C₂×C₄

#41×C₂₄₁

yes

—

140

C₈

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×C₂²⋊C₄×C₂₄₁

yes

characteristic

147

C₂²

#120×C₂₄₁

yes

characteristic

148

C₂²

#123×C₂₄₁

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

C₄×D₄×C₂₄₁

yes

—

153

C₄

#121×C₂₄₁

yes

—

154

C₂

—

155

C₂

—

156

C₂

—

157

C₂

#386×C₂₄₁

yes

characteristic

158

C₂

#454×C₂₄₁

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₄₁	#120
commutator	C₂×C₄	C₂²×C₄×C₂₄₁
Frattini, cosocle	C₂²×C₄	C₂³×C₂₄₁
prime radical	C₂²×D₄×C₂₄₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₄₁ ◁ C₂²×C₄×C₂₄₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₄₁

#120

commutator

C₂×C₄

C₂²×C₄×C₂₄₁

Frattini, cosocle

C₂²×C₄

C₂³×C₂₄₁

prime radical

C₂²×D₄×C₂₄₁

C₂²

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₂₄₁ ◁ C₂²×C₄×C₂₄₁ ◁ G

N	H ≅ G/N	type
2. #519×C₂₄₁	C₂	semidirect
4. C₂×#101×C₂₄₁	C₂	semidirect
5. #339×C₂₄₁	C₂	semidirect
10. #124×C₂₄₁	C₄	semidirect
79. C₂₄₁	#1888	direct (N × H)
80. #1888	C₂₄₁	direct (N × H)
82. #519	C₂×C₂₄₁	semidirect
84. C₂×#101	C₂×C₂₄₁	semidirect
85. #339	C₂×C₂₄₁	semidirect
90. #124	C₄×C₂₄₁	semidirect

H ≅ G/N

type

2. #519×C₂₄₁

C₂

semidirect

4. C₂×#101×C₂₄₁

C₂

semidirect

5. #339×C₂₄₁

C₂

semidirect

10. #124×C₂₄₁

C₄

semidirect

79. C₂₄₁

#1888

direct (N × H)

80. #1888

C₂₄₁

direct (N × H)

82. #519

C₂×C₂₄₁

semidirect

84. C₂×#101

C₂×C₂₄₁

semidirect

85. #339

C₂×C₂₄₁

semidirect

90. #124

C₄×C₂₄₁

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#124×C₂₄₁	2
#519×C₂₄₁	2
#418×C₂₄₁	2
C₂×#101×C₂₄₁	2
#339×C₂₄₁	2
#1888	241

subgroup

index

C₂×#124×C₂₄₁

#519×C₂₄₁

#418×C₂₄₁

C₂×#101×C₂₄₁

#339×C₂₄₁

#1888

241

Subgroups — #1888×C₂₄₁ #44503555

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1888×C241 #44503555

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1888×C₂₄₁ #44503555