Group #43251114 — subgroups

Subgroups — #2099×C₂₂₃ #43251114

Order 28544.

214 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 100 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	#389×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	#384×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#340×C₂₂₃	C₂	1	2	yes	maximal
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	2	yes	—
11	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
14	#95×C₂₂₃	—	2	1	no	—
15	C₂²×C₄×C₂₂₃	—	2	1	no	—
16	C₂²×C₄×C₂₂₃	—	2	1	no	—
17	C₂²×C₄×C₂₂₃	—	2	2	no	—
18	C₂²×C₄×C₂₂₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₂₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
21	C₂×C₈×C₂₂₃	—	2	2	no	—
22	C₂×C₈×C₂₂₃	—	2	2	no	—
23	C₂×C₈×C₂₂₃	—	2	2	no	—
24	C₄⋊C₄×C₂₂₃	—	2	1	no	—
25	C₄⋊C₄×C₂₂₃	—	2	2	no	—
26	C₄⋊C₄×C₂₂₃	—	2	4	no	—
27	C₄⋊C₄×C₂₂₃	—	4	2	no	—
28	C₄⋊C₄×C₂₂₃	—	4	2	no	—
29	C₄⋊C₄×C₂₂₃	—	4	2	no	—
30	C₄⋊C₄×C₂₂₃	Q₈	1	2	yes	—
31	C₄²×C₂₂₃	—	2	2	no	—
32	C₂³×C₂₂₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₂₂₃	—	2	1	no	—
34	C₂×C₄×C₂₂₃	—	2	1	no	—
35	C₂×C₄×C₂₂₃	—	2	1	no	—
36	C₂×C₄×C₂₂₃	—	2	1	no	—
37	C₂×C₄×C₂₂₃	—	2	2	no	—
38	C₂×C₄×C₂₂₃	—	4	1	no	—
39	C₂×C₄×C₂₂₃	—	4	1	no	—
40	C₂×C₄×C₂₂₃	—	4	1	no	—
41	C₂×C₄×C₂₂₃	—	4	2	no	—
42	C₂×C₄×C₂₂₃	—	4	1	no	—
43	C₂×C₄×C₂₂₃	—	4	2	no	—
44	C₂×C₄×C₂₂₃	—	4	2	no	—
45	C₂×C₄×C₂₂₃	—	4	1	no	—
46	C₂×C₄×C₂₂₃	D₈	1	2	yes	—
47	C₂×C₄×C₂₂₃	SD₁₆	1	2	yes	—
48	C₂×C₄×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×C₂₂₃	#41	1	2	yes	—
50	C₂×C₄×C₂₂₃	#41	1	2	yes	—
51	C₂×C₄×C₂₂₃	#41	1	1	yes	characteristic
52	C₈×C₂₂₃	—	4	2	no	—
53	C₂²×C₂₂₃	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₂₂₃	#123	1	1	yes	characteristic
55	C₂²×C₂₂₃	#124	1	1	yes	characteristic
56	C₂²×C₂₂₃	C₂×D₈	1	1	yes	characteristic
57	C₂²×C₂₂₃	C₂×SD₁₆	1	1	yes	characteristic
58	C₂²×C₂₂₃	#137	1	1	yes	characteristic
59	C₂²×C₂₂₃	#137	1	1	yes	characteristic
60	C₄×C₂₂₃	—	2	2	no	—
61	C₄×C₂₂₃	—	2	2	no	—
62	C₄×C₂₂₃	—	4	2	no	—
63	C₄×C₂₂₃	—	8	1	no	—
64	C₄×C₂₂₃	—	8	2	no	—
65	C₄×C₂₂₃	#125	1	2	yes	—
66	C₄×C₂₂₃	#126	1	2	yes	—
67	C₂×C₂₂₃	#400	1	1	yes	characteristic
68	C₂×C₂₂₃	#474	1	1	yes	characteristic
69	C₂×C₂₂₃	#475	1	1	yes	characteristic
70	C₂×C₂₂₃	#480	1	1	yes	characteristic
71	C₂×C₂₂₃	#483	1	1	yes	characteristic
72	C₂×C₂₂₃	#486	1	1	yes	characteristic
73	C₂×C₂₂₃	#491	1	1	yes	characteristic
74	C₂₂₃	#2099	1	1	yes	characteristic, direct product factor, Sylow
75	#2099	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	#389	C₂×C₂₂₃	1	1	yes	characteristic
77	#384	C₂×C₂₂₃	1	1	yes	characteristic
78	#340	C₂×C₂₂₃	1	2	yes	—
79	#340	C₂×C₂₂₃	1	2	yes	—
80	#336	C₂×C₂₂₃	1	1	yes	characteristic
81	C₂²×C₈	C₂²×C₂₂₃	1	2	yes	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	—
86	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	1	yes	characteristic
87	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	1	yes	characteristic
88	C₂×C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
89	#95	—	2	1	no	—
90	C₂²×C₄	—	2	1	no	—
91	C₂²×C₄	—	2	1	no	—
92	C₂²×C₄	—	2	2	no	—
93	C₂²×C₄	D₄×C₂₂₃	1	1	yes	characteristic
94	C₂²×C₄	D₄×C₂₂₃	1	1	yes	characteristic
95	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	2	4	no	—
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	Q₈×C₂₂₃	1	2	yes	—
106	C₄²	—	2	2	no	—
107	C₂³	C₂×D₄×C₂₂₃	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	D₈×C₂₂₃	1	2	yes	—
122	C₂×C₄	SD₁₆×C₂₂₃	1	2	yes	—
123	C₂×C₄	C₂×Q₈×C₂₂₃	1	1	yes	characteristic
124	C₂×C₄	#41×C₂₂₃	1	2	yes	—
125	C₂×C₄	#41×C₂₂₃	1	2	yes	—
126	C₂×C₄	#41×C₂₂₃	1	1	yes	characteristic
127	C₈	—	4	2	no	—
128	C₂²	C₂²⋊Q₈×C₂₂₃	1	1	yes	characteristic
129	C₂²	#123×C₂₂₃	1	1	yes	characteristic
130	C₂²	#124×C₂₂₃	1	1	yes	characteristic
131	C₂²	C₂×D₈×C₂₂₃	1	1	yes	characteristic
132	C₂²	C₂×SD₁₆×C₂₂₃	1	1	yes	characteristic
133	C₂²	#137×C₂₂₃	1	1	yes	characteristic
134	C₂²	#137×C₂₂₃	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	4	2	no	—
138	C₄	—	8	1	no	—
139	C₄	—	8	2	no	—
140	C₄	#125×C₂₂₃	1	2	yes	—
141	C₄	#126×C₂₂₃	1	2	yes	—
142	C₂	#400×C₂₂₃	1	1	yes	characteristic
143	C₂	#474×C₂₂₃	1	1	yes	characteristic
144	C₂	#475×C₂₂₃	1	1	yes	characteristic
145	C₂	#480×C₂₂₃	1	1	yes	characteristic
146	C₂	#483×C₂₂₃	1	1	yes	characteristic
147	C₂	#486×C₂₂₃	1	1	yes	characteristic
148	C₂	#491×C₂₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#389×C₂₂₃

C₂

yes

maximal, characteristic

#384×C₂₂₃

C₂

yes

maximal, characteristic

#340×C₂₂₃

C₂

yes

maximal

#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

—

C₂×C₄⋊C₄×C₂₂₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₂₃

C₂²

yes

characteristic

C₂×C₄²×C₂₂₃

C₂²

yes

characteristic

#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

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

Q₈

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

D₈

yes

—

C₂×C₄×C₂₂₃

SD₁₆

yes

—

C₂×C₄×C₂₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₂₃

#41

yes

—

C₂×C₄×C₂₂₃

#41

yes

—

C₂×C₄×C₂₂₃

#41

yes

characteristic

C₈×C₂₂₃

—

C₂²×C₂₂₃

C₂²⋊Q₈

yes

C₂²×C₂₂₃

yes

C₂²×C₂₂₃

yes

C₂²×C₂₂₃

C₂×D₈

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

#125

yes

—

C₄×C₂₂₃

#126

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

#2099

C₂₂₃

yes

maximal, characteristic, direct product factor, Sylow

#389

C₂×C₂₂₃

yes

characteristic

#384

C₂×C₂₂₃

yes

characteristic

#340

C₂×C₂₂₃

yes

—

#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

—

C₂×C₄⋊C₄

C₂²×C₂₂₃

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₂₂₃

yes

characteristic

C₂×C₄²

C₂²×C₂₂₃

yes

characteristic

#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

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

Q₈×C₂₂₃

yes

—

106

C₄²

—

107

C₂³

C₂×D₄×C₂₂₃

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

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

D₈×C₂₂₃

yes

—

122

C₂×C₄

SD₁₆×C₂₂₃

yes

—

123

C₂×C₄

C₂×Q₈×C₂₂₃

yes

characteristic

124

C₂×C₄

#41×C₂₂₃

yes

—

125

C₂×C₄

#41×C₂₂₃

yes

—

126

C₂×C₄

#41×C₂₂₃

yes

characteristic

127

C₈

—

128

C₂²

C₂²⋊Q₈×C₂₂₃

yes

characteristic

129

C₂²

#123×C₂₂₃

yes

characteristic

130

C₂²

#124×C₂₂₃

yes

characteristic

131

C₂²

C₂×D₈×C₂₂₃

yes

characteristic

132

C₂²

C₂×SD₁₆×C₂₂₃

yes

characteristic

133

C₂²

#137×C₂₂₃

yes

characteristic

134

C₂²

#137×C₂₂₃

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#125×C₂₂₃

yes

—

141

C₄

#126×C₂₂₃

yes

—

142

C₂

#400×C₂₂₃

yes

characteristic

143

C₂

#474×C₂₂₃

yes

characteristic

144

C₂

#475×C₂₂₃

yes

characteristic

145

C₂

#480×C₂₂₃

yes

characteristic

146

C₂

#483×C₂₂₃

yes

characteristic

147

C₂

#486×C₂₂₃

yes

characteristic

148

C₂

#491×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
74. C₂₂₃	#2099	direct (N × H)
75. #2099	C₂₂₃	direct (N × H)

H ≅ G/N

type

74. C₂₂₃

#2099

direct (N × H)

75. #2099

C₂₂₃

direct (N × H)

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

subgroup	index
#389×C₂₂₃	2
#384×C₂₂₃	2
#340×C₂₂₃	2
#336×C₂₂₃	2
#2099	223

subgroup

index

#389×C₂₂₃

#384×C₂₂₃

#340×C₂₂₃

#336×C₂₂₃

#2099

223

Subgroups — #2099×C₂₂₃ #43251114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2099×C223 #43251114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2099×C₂₂₃ #43251114