Group #43251099 — subgroups

Subgroups — #2084×C₂₂₃ #43251099

Order 28544.

230 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 108 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×#106×C₂₂₃	C₂	1	2	yes	maximal
2	#389×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#384×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#340×C₂₂₃	C₂	1	2	yes	maximal
5	#336×C₂₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₂₂₃	C₂²	1	2	yes	—
7	C₂×C₄⋊C₄×C₂₂₃	—	2	1	no	—
8	C₂×C₄⋊C₄×C₂₂₃	—	2	1	no	—
9	C₂×C₄⋊C₄×C₂₂₃	—	2	1	no	—
10	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
14	#106×C₂₂₃	—	2	4	no	—
15	#95×C₂₂₃	—	2	1	no	—
16	C₂²×C₄×C₂₂₃	—	2	1	no	—
17	C₂²×C₄×C₂₂₃	—	2	1	no	—
18	C₂²×C₄×C₂₂₃	—	2	2	no	—
19	C₂²×C₄×C₂₂₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₂₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₂₂₃	—	2	2	no	—
23	C₂×C₈×C₂₂₃	—	2	2	no	—
24	C₂×C₈×C₂₂₃	Q₈	1	4	yes	—
25	C₄⋊C₄×C₂₂₃	—	2	1	no	—
26	C₄⋊C₄×C₂₂₃	—	2	2	no	—
27	C₄⋊C₄×C₂₂₃	—	2	2	no	—
28	C₄⋊C₄×C₂₂₃	—	2	2	no	—
29	C₄⋊C₄×C₂₂₃	—	4	2	no	—
30	C₄⋊C₄×C₂₂₃	—	4	2	no	—
31	C₄⋊C₄×C₂₂₃	—	4	2	no	—
32	C₄⋊C₄×C₂₂₃	D₄	1	2	yes	—
33	C₄²×C₂₂₃	—	2	2	no	—
34	C₂³×C₂₂₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₂₂₃	—	2	1	no	—
36	C₂×C₄×C₂₂₃	—	2	1	no	—
37	C₂×C₄×C₂₂₃	—	2	1	no	—
38	C₂×C₄×C₂₂₃	—	2	1	no	—
39	C₂×C₄×C₂₂₃	—	2	2	no	—
40	C₂×C₄×C₂₂₃	—	4	1	no	—
41	C₂×C₄×C₂₂₃	—	4	1	no	—
42	C₂×C₄×C₂₂₃	—	4	2	no	—
43	C₂×C₄×C₂₂₃	—	4	1	no	—
44	C₂×C₄×C₂₂₃	—	4	2	no	—
45	C₂×C₄×C₂₂₃	—	4	1	no	—
46	C₂×C₄×C₂₂₃	—	4	2	no	—
47	C₂×C₄×C₂₂₃	—	4	1	no	—
48	C₂×C₄×C₂₂₃	SD₁₆	1	2	yes	—
49	C₂×C₄×C₂₂₃	SD₁₆	1	2	yes	—
50	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₂₃	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₂₂₃	#41	1	2	yes	—
53	C₂×C₄×C₂₂₃	#41	1	1	yes	characteristic
54	C₈×C₂₂₃	—	2	4	no	—
55	C₂²×C₂₂₃	#121	1	1	yes	characteristic
56	C₂²×C₂₂₃	#123	1	1	yes	characteristic
57	C₂²×C₂₂₃	C₄⋊Q₈	1	1	yes	characteristic
58	C₂²×C₂₂₃	C₂×SD₁₆	1	1	yes	characteristic
59	C₂²×C₂₂₃	C₂×SD₁₆	1	1	yes	characteristic
60	C₂²×C₂₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂²×C₂₂₃	#137	1	1	yes	characteristic
62	C₄×C₂₂₃	—	2	2	no	—
63	C₄×C₂₂₃	—	2	2	no	—
64	C₄×C₂₂₃	—	4	2	no	—
65	C₄×C₂₂₃	—	8	2	no	—
66	C₄×C₂₂₃	—	8	1	no	—
67	C₄×C₂₂₃	C₂²⋊Q₈	1	2	yes	—
68	C₄×C₂₂₃	#125	1	2	yes	—
69	C₂×C₂₂₃	#398	1	1	yes	characteristic
70	C₂×C₂₂₃	#460	1	1	yes	characteristic
71	C₂×C₂₂₃	#461	1	1	yes	characteristic
72	C₂×C₂₂₃	#481	1	1	yes	characteristic
73	C₂×C₂₂₃	#483	1	1	yes	characteristic
74	C₂×C₂₂₃	#498	1	1	yes	characteristic
75	C₂×C₂₂₃	#501	1	1	yes	characteristic
76	C₂₂₃	#2084	1	1	yes	characteristic, direct product factor, Sylow
77	#2084	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂×#106	C₂×C₂₂₃	1	2	yes	—
79	#389	C₂×C₂₂₃	1	1	yes	characteristic
80	#384	C₂×C₂₂₃	1	1	yes	characteristic
81	#340	C₂×C₂₂₃	1	2	yes	—
82	#336	C₂×C₂₂₃	1	1	yes	characteristic
83	C₂²×C₈	C₂²×C₂₂₃	1	2	yes	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	2	1	no	—
86	C₂×C₄⋊C₄	—	2	1	no	—
87	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	1	yes	characteristic
88	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	—
89	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	1	yes	characteristic
90	C₂×C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
91	#106	—	2	4	no	—
92	#95	—	2	1	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	D₄×C₂₂₃	1	1	yes	characteristic
97	C₂²×C₄	D₄×C₂₂₃	1	1	yes	characteristic
98	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	Q₈×C₂₂₃	1	4	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄⋊C₄	D₄×C₂₂₃	1	2	yes	—
110	C₄²	—	2	2	no	—
111	C₂³	C₂×D₄×C₂₂₃	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	SD₁₆×C₂₂₃	1	2	yes	—
126	C₂×C₄	SD₁₆×C₂₂₃	1	2	yes	—
127	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
128	C₂×C₄	C₂×Q₈×C₂₂₃	1	2	yes	—
129	C₂×C₄	#41×C₂₂₃	1	2	yes	—
130	C₂×C₄	#41×C₂₂₃	1	1	yes	characteristic
131	C₈	—	2	4	no	—
132	C₂²	#121×C₂₂₃	1	1	yes	characteristic
133	C₂²	#123×C₂₂₃	1	1	yes	characteristic
134	C₂²	C₄⋊Q₈×C₂₂₃	1	1	yes	characteristic
135	C₂²	C₂×SD₁₆×C₂₂₃	1	1	yes	characteristic
136	C₂²	C₂×SD₁₆×C₂₂₃	1	1	yes	characteristic
137	C₂²	AGL(1,ℤ/8)×C₂₂₃	1	1	yes	characteristic
138	C₂²	#137×C₂₂₃	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	4	2	no	—
142	C₄	—	8	2	no	—
143	C₄	—	8	1	no	—
144	C₄	C₂²⋊Q₈×C₂₂₃	1	2	yes	—
145	C₄	#125×C₂₂₃	1	2	yes	—
146	C₂	#398×C₂₂₃	1	1	yes	characteristic
147	C₂	#460×C₂₂₃	1	1	yes	characteristic
148	C₂	#461×C₂₂₃	1	1	yes	characteristic
149	C₂	#481×C₂₂₃	1	1	yes	characteristic
150	C₂	#483×C₂₂₃	1	1	yes	characteristic
151	C₂	#498×C₂₂₃	1	1	yes	characteristic
152	C₂	#501×C₂₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#106×C₂₂₃

C₂

yes

maximal

#389×C₂₂₃

C₂

yes

maximal, characteristic

#384×C₂₂₃

C₂

yes

maximal, characteristic

#340×C₂₂₃

C₂

yes

maximal

#336×C₂₂₃

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₂₃

—

C₂×C₄⋊C₄×C₂₂₃

—

C₂×C₄⋊C₄×C₂₂₃

—

C₂×C₄⋊C₄×C₂₂₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₂₃

C₂²

yes

characteristic

C₂×C₄²×C₂₂₃

C₂²

yes

characteristic

#106×C₂₂₃

—

#95×C₂₂₃

—

C₂²×C₄×C₂₂₃

—

C₂²×C₄×C₂₂₃

—

C₂²×C₄×C₂₂₃

—

C₂²×C₄×C₂₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₂₃

C₂³

yes

characteristic

C₂×C₈×C₂₂₃

—

C₂×C₈×C₂₂₃

—

C₂×C₈×C₂₂₃

Q₈

yes

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

D₄

yes

—

C₄²×C₂₂₃

—

C₂³×C₂₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

SD₁₆

yes

—

C₂×C₄×C₂₂₃

SD₁₆

yes

—

C₂×C₄×C₂₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₂₃

#41

yes

—

C₂×C₄×C₂₂₃

#41

yes

characteristic

C₈×C₂₂₃

—

C₂²×C₂₂₃

yes

C₂²×C₂₂₃

yes

C₂²×C₂₂₃

C₄⋊Q₈

yes

characteristic

C₂²×C₂₂₃

C₂×SD₁₆

yes

characteristic

C₂²×C₂₂₃

C₂×SD₁₆

yes

C₂²×C₂₂₃

yes

C₂²×C₂₂₃

yes

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

C₂²⋊Q₈

yes

—

C₄×C₂₂₃

#125

yes

—

C₂×C₂₂₃

yes

C₂×C₂₂₃

yes

C₂×C₂₂₃

yes

C₂×C₂₂₃

yes

C₂×C₂₂₃

yes

C₂×C₂₂₃

yes

C₂×C₂₂₃

yes

C₂₂₃

yes

characteristic, direct product factor, Sylow

#2084

C₂₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×#106

C₂×C₂₂₃

yes

—

#389

C₂×C₂₂₃

yes

characteristic

#384

C₂×C₂₂₃

yes

characteristic

#340

C₂×C₂₂₃

yes

—

#336

C₂×C₂₂₃

yes

characteristic

C₂²×C₈

C₂²×C₂₂₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₂₂₃

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₂₂₃

yes

—

C₂×C₄⋊C₄

C₂²×C₂₂₃

yes

characteristic

C₂×C₄²

C₂²×C₂₂₃

yes

characteristic

#106

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₂₂₃

yes

characteristic

C₂²×C₄

D₄×C₂₂₃

yes

characteristic

C₂²×C₄

C₂³×C₂₂₃

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

Q₈×C₂₂₃

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

D₄×C₂₂₃

yes

—

110

C₄²

—

111

C₂³

C₂×D₄×C₂₂₃

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

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

SD₁₆×C₂₂₃

yes

—

126

C₂×C₄

SD₁₆×C₂₂₃

yes

—

127

C₂×C₄

C₂×D₄×C₂₂₃

yes

characteristic

128

C₂×C₄

C₂×Q₈×C₂₂₃

yes

—

129

C₂×C₄

#41×C₂₂₃

yes

—

130

C₂×C₄

#41×C₂₂₃

yes

characteristic

131

C₈

—

132

C₂²

#121×C₂₂₃

yes

characteristic

133

C₂²

#123×C₂₂₃

yes

characteristic

134

C₂²

C₄⋊Q₈×C₂₂₃

yes

characteristic

135

C₂²

C₂×SD₁₆×C₂₂₃

yes

characteristic

136

C₂²

C₂×SD₁₆×C₂₂₃

yes

characteristic

137

C₂²

AGL(1,ℤ/8)×C₂₂₃

yes

characteristic

138

C₂²

#137×C₂₂₃

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

C₂²⋊Q₈×C₂₂₃

yes

—

145

C₄

#125×C₂₂₃

yes

—

146

C₂

#398×C₂₂₃

yes

characteristic

147

C₂

#460×C₂₂₃

yes

characteristic

148

C₂

#461×C₂₂₃

yes

characteristic

149

C₂

#481×C₂₂₃

yes

characteristic

150

C₂

#483×C₂₂₃

yes

characteristic

151

C₂

#498×C₂₂₃

yes

characteristic

152

C₂

#501×C₂₂₃

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₂₂₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₂₃ ◁ C₂×C₄²×C₂₂₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₂₂₃

C₂×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₂₂₃

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₂₃ ◁ C₂×C₄²×C₂₂₃ ◁ G

N	H ≅ G/N	type
76. C₂₂₃	#2084	direct (N × H)
77. #2084	C₂₂₃	direct (N × H)

H ≅ G/N

type

76. C₂₂₃

#2084

direct (N × H)

77. #2084

C₂₂₃

direct (N × H)

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

subgroup	index
C₂×#106×C₂₂₃	2
#389×C₂₂₃	2
#384×C₂₂₃	2
#340×C₂₂₃	2
#336×C₂₂₃	2
#2084	223

subgroup

index

C₂×#106×C₂₂₃

#389×C₂₂₃

#384×C₂₂₃

#340×C₂₂₃

#336×C₂₂₃

#2084

223

Subgroups — #2084×C₂₂₃ #43251099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2084×C223 #43251099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2084×C₂₂₃ #43251099