Group #10647902 — subgroups

Subgroups — C₃×#29295 #10647902

Order 768.

630 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 270 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	C₃×#2330	C₂	1	2	yes	maximal
2	C₃×#2088	C₂	1	4	yes	maximal
3	C₃×#1989	C₂	1	4	yes	maximal
4	C₃×#1948	C₂	1	4	yes	maximal
5	C₃×#1843	C₂	1	1	yes	maximal, characteristic
6	#29295	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×#125	C₂²	1	2	yes	—
8	C₂×C₃×#106	C₂²	1	4	yes	—
9	C₂×C₃×C₄⋊C₈	C₂²	1	4	yes	—
10	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₃×#389	—	2	4	no	—
12	C₃×#389	C₂²	1	4	yes	—
13	C₃×#387	—	2	4	no	—
14	C₃×#384	C₂²	1	2	yes	—
15	C₃×C₄×C₄⋊C₄	C₂²	1	4	yes	—
16	C₃×#376	C₂²	1	1	yes	characteristic
17	C₃×#340	C₂²	1	4	yes	—
18	C₃×#340	C₂²	1	8	yes	—
19	#2330	C₂×C₃	1	2	yes	—
20	#2088	C₂×C₃	1	4	yes	—
21	#1989	C₂×C₃	1	4	yes	—
22	#1948	C₂×C₃	1	4	yes	—
23	#1843	C₂×C₃	1	1	yes	characteristic
24	C₂²×C₃×C₈	C₂³	1	4	yes	—
25	C₃×#125	—	2	4	no	—
26	C₃×#125	Q₈	1	8	yes	—
27	C₂×C₃×C₄⋊C₄	—	2	4	no	—
28	C₂×C₃×C₄⋊C₄	—	2	4	no	—
29	C₂×C₃×C₄⋊C₄	—	2	4	no	—
30	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
31	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
32	C₂×C₃×C₄²	—	2	4	no	—
33	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
34	C₂×C₃×C₄²	C₂³	1	2	yes	—
35	C₃×#106	—	4	4	no	—
36	C₃×C₄⋊C₈	—	2	8	no	—
37	C₃×C₄⋊C₈	—	4	2	no	—
38	C₃×#95	—	2	4	no	—
39	C₃×#95	—	2	4	no	—
40	C₃×#95	D₄	1	4	yes	—
41	C₂×#125	C₂²×C₃	1	2	yes	—
42	C₂×#106	C₂²×C₃	1	4	yes	—
43	C₂×C₄⋊C₈	C₂²×C₃	1	4	yes	—
44	C₂×C₄⋊C₈	C₂²×C₃	1	2	yes	—
45	#389	—	2	4	no	—
46	#389	C₂²×C₃	1	4	yes	—
47	#387	—	2	4	no	—
48	#384	C₂²×C₃	1	2	yes	—
49	C₄×C₄⋊C₄	C₂²×C₃	1	4	yes	—
50	#376	C₂²×C₃	1	1	yes	characteristic
51	#340	C₂²×C₃	1	8	yes	—
52	#340	C₂²×C₃	1	4	yes	—
53	C₂²×C₃×C₄	—	2	4	no	—
54	C₂²×C₃×C₄	—	2	4	no	—
55	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
56	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
57	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
58	C₂×C₃×C₈	—	2	8	no	—
59	C₂×C₃×C₈	—	2	4	no	—
60	C₃×C₄⋊C₄	—	2	16	no	—
61	C₃×C₄⋊C₄	—	2	4	no	—
62	C₃×C₄⋊C₄	—	2	4	no	—
63	C₃×C₄⋊C₄	—	2	4	no	—
64	C₃×C₄⋊C₄	—	8	4	no	—
65	C₃×C₄⋊C₄	—	8	4	no	—
66	C₃×C₄⋊C₄	C₂×Q₈	1	8	yes	—
67	C₃×C₄²	—	2	1	no	—
68	C₃×C₄²	—	2	1	no	—
69	C₃×C₄²	—	8	4	no	—
70	C₃×C₄²	C₂×Q₈	1	4	yes	—
71	C₃×C₄²	#41	1	4	yes	—
72	C₂²×C₈	C₂³×C₃	1	4	yes	—
73	#125	—	2	4	no	—
74	#125	C₃×Q₈	1	8	yes	—
75	C₂×C₄⋊C₄	—	2	4	no	—
76	C₂×C₄⋊C₄	—	2	4	no	—
77	C₂×C₄⋊C₄	—	2	4	no	—
78	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
79	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
80	C₂×C₄²	—	2	4	no	—
81	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
82	C₂×C₄²	C₂³×C₃	1	2	yes	—
83	#106	—	4	4	no	—
84	C₄⋊C₈	—	2	8	no	—
85	C₄⋊C₈	—	4	2	no	—
86	#95	—	2	4	no	—
87	#95	—	2	4	no	—
88	#95	C₃×D₄	1	4	yes	—
89	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₃×C₄	—	2	4	no	—
91	C₂×C₃×C₄	—	2	4	no	—
92	C₂×C₃×C₄	—	2	8	no	—
93	C₂×C₃×C₄	—	2	2	no	—
94	C₂×C₃×C₄	—	2	4	no	—
95	C₂×C₃×C₄	—	4	4	no	—
96	C₂×C₃×C₄	—	4	4	no	—
97	C₂×C₃×C₄	—	4	4	no	—
98	C₂×C₃×C₄	—	4	4	no	—
99	C₂×C₃×C₄	—	4	4	no	—
100	C₂×C₃×C₄	C₂²⋊Q₈	1	8	yes	—
101	C₂×C₃×C₄	C₂²×Q₈	1	2	yes	—
102	C₂×C₃×C₄	C₂×#41	1	2	yes	—
103	C₂×C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
104	C₂×C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
105	C₃×C₈	—	4	4	no	—
106	C₂²×C₄	—	2	4	no	—
107	C₂²×C₄	—	2	4	no	—
108	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
109	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
110	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
111	C₂×C₈	—	2	8	no	—
112	C₂×C₈	—	2	4	no	—
113	C₄⋊C₄	—	2	16	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	8	4	no	—
118	C₄⋊C₄	—	8	4	no	—
119	C₄⋊C₄	C₂×C₃×Q₈	1	8	yes	—
120	C₄²	—	2	1	no	—
121	C₄²	—	2	1	no	—
122	C₄²	—	8	4	no	—
123	C₄²	C₂×C₃×Q₈	1	4	yes	—
124	C₄²	C₃×#41	1	4	yes	—
125	C₂²×C₃	C₂×C₂²⋊Q₈	1	2	yes	—
126	C₂²×C₃	#537	1	1	yes	characteristic
127	C₂²×C₃	#577	1	4	yes	—
128	C₃×C₄	—	2	8	no	—
129	C₃×C₄	—	4	8	no	—
130	C₃×C₄	—	4	2	no	—
131	C₃×C₄	—	8	4	no	—
132	C₃×C₄	#557	1	2	yes	—
133	C₃×C₄	Q₈²	1	1	yes	characteristic
134	C₃×C₄	#559	1	1	yes	characteristic
135	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	4	4	no	—
142	C₂×C₄	—	4	4	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	—	4	4	no	—
145	C₂×C₄	—	4	4	no	—
146	C₂×C₄	C₃×C₂²⋊Q₈	1	8	yes	—
147	C₂×C₄	C₂²×C₃×Q₈	1	2	yes	—
148	C₂×C₄	C₂×C₃×#41	1	2	yes	—
149	C₂×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
150	C₂×C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
151	C₈	—	4	4	no	—
152	C₂×C₃	#2582	1	1	yes	characteristic
153	C₂×C₃	#3084	1	4	yes	—
154	C₂×C₃	#3260	1	2	yes	—
155	C₂²	C₂×C₃×C₂²⋊Q₈	1	2	yes	—
156	C₂²	C₃×#537	1	1	yes	characteristic
157	C₂²	C₃×#577	1	4	yes	—
158	C₄	—	2	8	no	—
159	C₄	—	4	2	no	—
160	C₄	—	4	8	no	—
161	C₄	—	8	4	no	—
162	C₄	C₃×#557	1	2	yes	—
163	C₄	C₃×Q₈²	1	1	yes	characteristic
164	C₄	C₃×#559	1	1	yes	characteristic
165	C₃	#29295	1	1	yes	characteristic, direct product factor, Sylow
166	C₂	C₃×#2582	1	1	yes	characteristic
167	C₂	C₃×#3084	1	4	yes	—
168	C₂	C₃×#3260	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#2330

C₂

yes

maximal

C₃×#2088

C₂

yes

maximal

C₃×#1989

C₂

yes

maximal

C₃×#1948

C₂

yes

maximal

C₃×#1843

C₂

yes

maximal, characteristic

#29295

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₃×#125

C₂²

yes

—

C₂×C₃×#106

C₂²

yes

—

C₂×C₃×C₄⋊C₈

C₂²

yes

—

C₂×C₃×C₄⋊C₈

C₂²

yes

—

C₃×#389

—

C₃×#389

C₂²

yes

—

C₃×#387

—

C₃×#384

C₂²

yes

—

C₃×C₄×C₄⋊C₄

C₂²

yes

—

C₃×#376

C₂²

yes

characteristic

C₃×#340

C₂²

yes

—

C₃×#340

C₂²

yes

—

#2330

C₂×C₃

yes

—

#2088

C₂×C₃

yes

—

#1989

C₂×C₃

yes

—

#1948

C₂×C₃

yes

—

#1843

C₂×C₃

yes

characteristic

C₂²×C₃×C₈

C₂³

yes

—

C₃×#125

—

C₃×#125

Q₈

yes

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

C₂³

yes

—

C₂×C₃×C₄⋊C₄

C₂³

yes

—

C₂×C₃×C₄²

—

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂×C₃×C₄²

C₂³

yes

—

C₃×#106

—

C₃×C₄⋊C₈

—

C₃×C₄⋊C₈

—

C₃×#95

—

C₃×#95

—

C₃×#95

D₄

yes

—

C₂×#125

C₂²×C₃

yes

—

C₂×#106

C₂²×C₃

yes

—

C₂×C₄⋊C₈

C₂²×C₃

yes

—

C₂×C₄⋊C₈

C₂²×C₃

yes

—

#389

—

#389

C₂²×C₃

yes

—

#387

—

#384

C₂²×C₃

yes

—

C₄×C₄⋊C₄

C₂²×C₃

yes

—

#376

C₂²×C₃

yes

characteristic

#340

C₂²×C₃

yes

—

#340

C₂²×C₃

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×D₄

yes

—

C₂²×C₃×C₄

C₂×D₄

yes

—

C₂²×C₃×C₄

C₂⁴

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₂×Q₈

yes

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₂×Q₈

yes

—

C₃×C₄²

#41

yes

—

C₂²×C₈

C₂³×C₃

yes

—

#125

—

#125

C₃×Q₈

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂³×C₃

yes

—

C₂×C₄⋊C₄

C₂³×C₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³×C₃

yes

characteristic

C₂×C₄²

C₂³×C₃

yes

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

#95

—

#95

—

#95

C₃×D₄

yes

—

C₂³×C₃

C₂²×D₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

100

C₂×C₃×C₄

C₂²⋊Q₈

yes

—

101

C₂×C₃×C₄

C₂²×Q₈

yes

—

102

C₂×C₃×C₄

C₂×#41

yes

—

103

C₂×C₃×C₄

H(5,𝔽₂)

yes

characteristic

104

C₂×C₃×C₄

2₋¹⁺⁴

yes

characteristic

105

C₃×C₈

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₂×C₃×D₄

yes

—

109

C₂²×C₄

C₂×C₃×D₄

yes

—

110

C₂²×C₄

C₂⁴×C₃

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

C₂×C₃×Q₈

yes

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₄²

C₂×C₃×Q₈

yes

—

124

C₄²

C₃×#41

yes

—

125

C₂²×C₃

C₂×C₂²⋊Q₈

yes

—

126

C₂²×C₃

#537

yes

characteristic

127

C₂²×C₃

#577

yes

—

128

C₃×C₄

—

129

C₃×C₄

—

130

C₃×C₄

—

131

C₃×C₄

—

132

C₃×C₄

#557

yes

—

133

C₃×C₄

Q₈²

yes

characteristic

134

C₃×C₄

#559

yes

characteristic

135

C₂³

C₂²×C₃×D₄

yes

characteristic

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₃×C₂²⋊Q₈

yes

—

147

C₂×C₄

C₂²×C₃×Q₈

yes

—

148

C₂×C₄

C₂×C₃×#41

yes

—

149

C₂×C₄

C₃×H(5,𝔽₂)

yes

characteristic

150

C₂×C₄

C₃×2₋¹⁺⁴

yes

characteristic

151

C₈

—

152

C₂×C₃

#2582

yes

characteristic

153

C₂×C₃

#3084

yes

—

154

C₂×C₃

#3260

yes

—

155

C₂²

C₂×C₃×C₂²⋊Q₈

yes

—

156

C₂²

C₃×#537

yes

characteristic

157

C₂²

C₃×#577

yes

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

C₃×#557

yes

—

163

C₄

C₃×Q₈²

yes

characteristic

164

C₄

C₃×#559

yes

characteristic

165

C₃

#29295

yes

characteristic, direct product factor, Sylow

166

C₂

C₃×#2582

yes

characteristic

167

C₂

C₃×#3084

yes

—

168

C₂

C₃×#3260

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×#376 ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₃

C₂²×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴×C₃

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₃ ◁ C₃×#376 ◁ G

N	H ≅ G/N	type
6. #29295	C₃	direct (N × H)
165. C₃	#29295	direct (N × H)

H ≅ G/N

type

6. #29295

C₃

direct (N × H)

165. C₃

#29295

direct (N × H)

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

subgroup	index
C₃×#2330	2
C₃×#2088	2
C₃×#1989	2
C₃×#1948	2
C₃×#1843	2
#29295	3

subgroup

index

Subgroups — C₃×#29295 #10647902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#29295 #10647902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#29295 #10647902