Group #72226216 — subgroups

Subgroups — #72226216

Order 3072.

966 conjugacy classes of subgroups; only the 447 normal subgroups are shown.

normal subgroups only

This group has 966 conjugacy classes of subgroups — too many to draw. Only its 447 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×#122211	C₂	1	1	yes	maximal, characteristic
2	#72225825	C₂	1	3	yes	maximal
3	#72225813	C₂	1	3	yes	maximal
4	#72214029	—	3	1	no	maximal, Sylow
5	C₄×#4512	C₄	1	6	yes	—
6	#10704187	C₄	1	6	yes	—
7	#10704180	C₂²	1	3	yes	—
8	#10684562	C₂²	1	1	yes	characteristic
9	#10681803	C₄	1	12	yes	complemented
10	C₃×C₈×M₃₂	C₂²	1	3	yes	—
11	C₃×#7052	C₄	1	3	yes	complemented
12	C₃×C₁₆²	C₄	1	1	yes	characteristic, complemented
13	#126544	—	3	3	no	—
14	#122220	—	3	3	no	—
15	#122211	D₃	1	1	yes	characteristic
16	C₃×C₄×M₃₂	C₂×C₄	1	6	yes	—
17	C₂×C₃×C₈²	C₂³	1	1	yes	characteristic
18	C₂×#4512	—	2	12	no	—
19	C₄×#815	C₂×C₄	1	3	yes	—
20	C₄×#812	C₂×C₄	1	6	yes	—
21	C₂×#4455	C₂×C₄	1	3	yes	—
22	C₃×#1311	C₂×C₄	1	6	yes	—
23	C₃×C₈×C₁₆	C₂×C₄	1	3	yes	—
24	C₃×C₈×C₁₆	C₂×C₄	1	3	yes	—
25	#67511	—	2	12	no	—
26	#67438	C₂×C₄	1	12	yes	—
27	C₄×#347	—	3	6	no	—
28	#11404	—	3	6	no	—
29	#11402	—	3	3	no	—
30	C₈×M₃₂	C₂×D₃	1	3	yes	—
31	#7846	—	3	1	no	—
32	#7053	—	3	12	no	—
33	#7052	Dic₃	1	3	yes	complemented
34	C₁₆²	Dic₃	1	1	yes	characteristic, complemented
35	C₂×C₃×M₃₂	M₁₆	1	12	yes	—
36	C₂×C₃×C₄×C₈	C₂²×C₄	1	3	yes	—
37	C₂×#815	—	2	6	no	—
38	C₂×#812	—	2	6	no	—
39	C₄²×Dic₃	C₄²	1	1	yes	characteristic
40	C₂×C₄×C₃⋊C₈	C₄²	1	3	yes	—
41	C₃×#346	M₁₆	1	12	yes	—
42	C₃×C₄×C₁₆	C₄²	1	6	yes	—
43	C₃×C₄×C₁₆	C₄²	1	6	yes	—
44	C₃×C₈²	C₂²×C₄	1	3	yes	—
45	C₃×C₈²	C₂²×C₄	1	1	yes	characteristic
46	#4512	—	4	24	no	—
47	#4461	—	2	6	no	—
48	C₄×C₃⋊C₁₆	C₄²	1	12	yes	—
49	#4455	—	2	6	no	—
50	C₂×#347	—	6	12	no	—
51	C₄×M₃₂	—	3	6	no	—
52	C₄×M₃₂	C₄×D₃	1	6	yes	—
53	C₄×#97	—	3	3	no	—
54	C₂×#322	—	3	3	no	—
55	C₂×C₈²	C₂²×D₃	1	1	yes	characteristic
56	#1313	—	6	12	no	—
57	#1312	—	3	12	no	—
58	#1311	C₄×D₃	1	6	yes	—
59	C₈×C₁₆	C₂×Dic₃	1	3	yes	—
60	C₈×C₁₆	C₂×Dic₃	1	3	yes	—
61	C₂²×C₃×C₈	C₂×M₁₆	1	6	yes	—
62	C₂×C₃×C₄²	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×Dic₃	—	2	3	no	—
64	C₂²×C₃⋊C₈	—	2	3	no	—
65	C₃×M₃₂	—	2	24	no	—
66	C₂×C₃×C₁₆	#97	1	12	yes	—
67	C₂×C₃×C₁₆	#97	1	12	yes	—
68	C₃×C₄×C₈	C₂×C₄²	1	3	yes	—
69	C₃×C₄×C₈	C₂×C₄²	1	3	yes	—
70	C₃×C₄×C₈	C₂×M₁₆	1	6	yes	—
71	#815	—	4	12	no	—
72	#812	—	4	12	no	—
73	C₂×C₃⋊C₁₆	—	2	12	no	—
74	C₄×C₃⋊C₈	—	2	6	no	—
75	C₂×M₃₂	—	6	6	no	—
76	C₂×M₃₂	ΓL(1,𝔽₂₅)	1	12	yes	—
77	C₂×#97	—	6	6	no	—
78	C₂×C₄×C₈	—	3	3	no	—
79	C₂×C₄×C₈	C₂×C₄×D₃	1	3	yes	—
80	C₄³	—	3	1	no	—
81	#347	—	12	24	no	—
82	#346	—	6	6	no	—
83	#346	ΓL(1,𝔽₂₅)	1	12	yes	—
84	C₄×C₁₆	—	3	12	no	—
85	C₄×C₁₆	C₄×Dic₃	1	6	yes	—
86	C₄×C₁₆	C₄×Dic₃	1	6	yes	—
87	#322	—	6	6	no	—
88	C₈²	C₂×C₄×D₃	1	3	yes	—
89	C₈²	C₂²×Dic₃	1	1	yes	characteristic
90	C₂²×C₃×C₄	C₄×M₁₆	1	3	yes	—
91	C₂²×Dic₃	—	4	1	no	—
92	C₂×C₃×C₈	—	2	12	no	—
93	C₂×C₃×C₈	C₂×#97	1	6	yes	—
94	C₂×C₃×C₈	C₄×M₁₆	1	6	yes	—
95	C₃×C₄²	C₄³	1	1	yes	characteristic
96	C₃×C₄²	C₄×M₁₆	1	3	yes	—
97	C₄×Dic₃	—	4	6	no	—
98	C₂×C₃⋊C₈	—	4	6	no	—
99	C₂×C₃⋊C₈	—	4	3	no	—
100	C₃×C₁₆	—	2	12	no	—
101	C₃×C₁₆	#347	1	24	yes	complemented
102	C₃⋊C₁₆	—	4	12	no	—
103	C₂²×C₈	—	6	3	no	—
104	C₂²×C₈	C₂×ΓL(1,𝔽₂₅)	1	6	yes	—
105	C₂×C₄²	—	6	3	no	—
106	C₂×C₄²	C₄²×D₃	1	1	yes	characteristic
107	M₃₂	—	2	24	no	—
108	M₃₂	—	12	12	no	—
109	C₂×C₁₆	—	6	12	no	—
110	C₂×C₁₆	#815	1	12	yes	—
111	C₂×C₁₆	#815	1	12	yes	—
112	#97	—	12	12	no	—
113	C₄×C₈	—	6	6	no	—
114	C₄×C₈	C₄²×D₃	1	3	yes	—
115	C₄×C₈	C₂×ΓL(1,𝔽₂₅)	1	6	yes	—
116	C₄×C₈	C₂×C₄×Dic₃	1	3	yes	—
117	C₂³×C₃	#1455	1	1	yes	characteristic
118	C₂×C₃×C₄	—	2	6	no	—
119	C₂×C₃×C₄	#1455	1	3	yes	—
120	C₂×C₃×C₄	C₄×#97	1	3	yes	—
121	C₂×Dic₃	—	8	3	no	—
122	C₃×C₈	—	2	6	no	—
123	C₃×C₈	C₂×#347	1	12	yes	—
124	C₃⋊C₈	—	8	3	no	—
125	C₂²×C₄	—	12	1	no	—
126	C₂²×C₄	C₄×ΓL(1,𝔽₂₅)	1	3	yes	—
127	C₂×C₈	—	2	12	no	—
128	C₂×C₈	—	12	6	no	—
129	C₂×C₈	—	12	3	no	—
130	C₂×C₈	C₄×ΓL(1,𝔽₂₅)	1	6	yes	—
131	C₂×C₈	C₂×#815	1	6	yes	—
132	C₄²	—	12	6	no	—
133	C₄²	C₄×ΓL(1,𝔽₂₅)	1	3	yes	—
134	C₄²	C₄²×Dic₃	1	1	yes	characteristic
135	C₁₆	—	2	12	no	—
136	C₁₆	—	12	12	no	—
137	C₁₆	#4512	1	24	yes	complemented
138	C₂²×C₃	—	2	3	no	—
139	C₂²×C₃	#7846	1	1	yes	characteristic
140	C₃×C₄	—	2	3	no	—
141	C₃×C₄	C₄×#347	1	6	yes	—
142	Dic₃	—	16	1	no	—
143	C₂³	#68050	1	1	yes	characteristic
144	C₂×C₄	—	2	6	no	—
145	C₂×C₄	—	24	3	no	—
146	C₂×C₄	#68050	1	3	yes	—
147	C₂×C₄	C₄×#815	1	3	yes	—
148	C₈	—	2	6	no	—
149	C₈	—	24	3	no	—
150	C₈	C₂×#4512	1	12	yes	—
151	C₂×C₃	—	4	1	no	—
152	C₂×C₃	#126544	1	3	yes	—
153	C₂²	—	2	3	no	—
154	C₂²	#10684562	1	1	yes	characteristic
155	C₄	—	2	3	no	—
156	C₄	—	48	1	no	—
157	C₄	C₄×#4512	1	6	yes	—
158	C₃	#72214029	1	1	yes	characteristic, complemented, Sylow
159	C₂	—	4	1	no	—
160	C₂	#72225813	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#122211

C₂

yes

maximal, characteristic

#72225825

C₂

yes

maximal

#72225813

C₂

yes

—

C₄

yes

—

#10704187

C₄

yes

—

#10704180

C₂²

yes

—

#10684562

C₂²

yes

characteristic

#10681803

C₄

yes

complemented

C₃×C₈×M₃₂

C₂²

yes

—

C₃×#7052

C₄

yes

complemented

C₃×C₁₆²

C₄

yes

characteristic, complemented

#126544

—

#122220

—

#122211

D₃

yes

characteristic

C₃×C₄×M₃₂

C₂×C₄

yes

—

C₂×C₃×C₈²

C₂³

yes

characteristic

C₂×#4512

—

C₄×#815

C₂×C₄

yes

—

C₄×#812

C₂×C₄

yes

—

C₂×#4455

C₂×C₄

yes

—

C₃×#1311

C₂×C₄

yes

—

C₃×C₈×C₁₆

C₂×C₄

yes

—

C₃×C₈×C₁₆

C₂×C₄

yes

—

#67511

—

#67438

C₂×C₄

yes

—

C₄×#347

—

#11404

—

#11402

—

C₈×M₃₂

C₂×D₃

yes

—

#7846

—

#7053

—

#7052

Dic₃

yes

complemented

C₁₆²

Dic₃

yes

characteristic, complemented

C₂×C₃×M₃₂

M₁₆

yes

—

C₂×C₃×C₄×C₈

C₂²×C₄

yes

—

C₂×#815

—

C₂×#812

—

C₄²×Dic₃

C₄²

yes

characteristic

C₂×C₄×C₃⋊C₈

C₄²

yes

—

C₃×#346

M₁₆

yes

—

C₃×C₄×C₁₆

C₄²

yes

—

C₃×C₄×C₁₆

C₄²

yes

—

C₃×C₈²

C₂²×C₄

yes

—

C₃×C₈²

C₂²×C₄

yes

characteristic

#4512

—

#4461

—

C₄×C₃⋊C₁₆

C₄²

yes

—

#4455

—

C₂×#347

—

C₄×M₃₂

—

C₄×M₃₂

C₄×D₃

yes

—

C₄×#97

—

C₂×#322

—

C₂×C₈²

C₂²×D₃

yes

—

—

C₄×D₃

yes

—

C₈×C₁₆

C₂×Dic₃

yes

—

C₈×C₁₆

C₂×Dic₃

yes

—

C₂²×C₃×C₈

C₂×M₁₆

yes

—

C₂×C₃×C₄²

C₂×C₄²

yes

characteristic

C₂×C₄×Dic₃

—

C₂²×C₃⋊C₈

—

C₃×M₃₂

—

C₂×C₃×C₁₆

#97

yes

—

C₂×C₃×C₁₆

#97

yes

—

C₃×C₄×C₈

C₂×C₄²

yes

—

C₃×C₄×C₈

C₂×C₄²

yes

—

C₃×C₄×C₈

C₂×M₁₆

yes

—

#815

—

#812

—

C₂×C₃⋊C₁₆

—

C₄×C₃⋊C₈

—

C₂×M₃₂

—

C₂×M₃₂

ΓL(1,𝔽₂₅)

yes

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×C₄×D₃

yes

—

C₄³

—

#347

—

#346

—

#346

ΓL(1,𝔽₂₅)

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₄×Dic₃

yes

—

C₄×C₁₆

C₄×Dic₃

yes

—

#322

—

C₈²

C₂×C₄×D₃

yes

—

C₈²

C₂²×Dic₃

yes

characteristic

C₂²×C₃×C₄

C₄×M₁₆

yes

—

C₂²×Dic₃

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×#97

yes

—

C₂×C₃×C₈

C₄×M₁₆

yes

—

C₃×C₄²

C₄³

yes

characteristic

C₃×C₄²

C₄×M₁₆

yes

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

100

C₃×C₁₆

—

101

C₃×C₁₆

#347

yes

complemented

102

C₃⋊C₁₆

—

103

C₂²×C₈

—

104

C₂²×C₈

C₂×ΓL(1,𝔽₂₅)

yes

—

105

C₂×C₄²

—

106

C₂×C₄²

C₄²×D₃

yes

characteristic

107

M₃₂

—

108

M₃₂

—

109

C₂×C₁₆

—

110

C₂×C₁₆

#815

yes

—

111

C₂×C₁₆

#815

yes

—

112

#97

—

113

C₄×C₈

—

114

C₄×C₈

C₄²×D₃

yes

—

115

C₄×C₈

C₂×ΓL(1,𝔽₂₅)

yes

—

116

C₄×C₈

C₂×C₄×Dic₃

yes

—

117

C₂³×C₃

#1455

yes

characteristic

118

C₂×C₃×C₄

—

119

C₂×C₃×C₄

#1455

yes

—

120

C₂×C₃×C₄

C₄×#97

yes

—

121

C₂×Dic₃

—

122

C₃×C₈

—

123

C₃×C₈

C₂×#347

yes

—

124

C₃⋊C₈

—

125

C₂²×C₄

—

126

C₂²×C₄

C₄×ΓL(1,𝔽₂₅)

yes

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

C₄×ΓL(1,𝔽₂₅)

yes

—

131

C₂×C₈

C₂×#815

yes

—

132

C₄²

—

133

C₄²

C₄×ΓL(1,𝔽₂₅)

yes

—

134

C₄²

C₄²×Dic₃

yes

characteristic

135

C₁₆

—

136

C₁₆

—

137

C₁₆

#4512

yes

complemented

138

C₂²×C₃

—

139

C₂²×C₃

#7846

yes

characteristic

140

C₃×C₄

—

141

C₃×C₄

C₄×#347

yes

—

142

Dic₃

—

143

C₂³

#68050

yes

characteristic

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

#68050

yes

—

147

C₂×C₄

C₄×#815

yes

—

148

C₈

—

149

C₈

—

150

C₈

C₂×#4512

yes

—

151

C₂×C₃

—

152

C₂×C₃

#126544

yes

—

153

C₂²

—

154

C₂²

#10684562

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

C₄×#4512

yes

—

158

C₃

#72214029

yes

characteristic, complemented, Sylow

159

C₂

—

160

C₂

#72225813

yes

—

	subgroup	G/H
center	C₄²	C₄²×Dic₃
commutator	C₃×C₄²	C₄³
Frattini	C₂×C₈²	C₂²×D₃
Fitting	C₃×#122211	C₂
socle	C₂²×C₃	#7846
prime radical	C₂³×C₃	#1455
cosocle	C₂×C₃×C₈²	C₂³
hypercenter	#122211	D₃
hypocenter	C₃	#72214029
LCS	G ▷ C₃×C₄² ▷ C₃
UCS	C₁ ◁ C₄² ◁ #122211

subgroup

G/H

center

C₄²

C₄²×Dic₃

commutator

C₃×C₄²

C₄³

Frattini

C₂×C₈²

C₂²×D₃

C₂

C₂²×C₃

C₂³×C₃

C₂×C₃×C₈²

C₂³

D₃

C₃

G ▷ C₃×C₄² ▷ C₃

C₁ ◁ C₄² ◁ #122211

N	H ≅ G/N	type
9. #10681803	C₄	semidirect
11. C₃×#7052	C₄	semidirect
12. C₃×C₁₆²	C₄	semidirect
33. #7052	Dic₃	semidirect
34. C₁₆²	Dic₃	semidirect
101. C₃×C₁₆	#347	semidirect
137. C₁₆	#4512	semidirect
158. C₃	#72214029	semidirect

H ≅ G/N

type

9. #10681803

C₄

semidirect

11. C₃×#7052

C₄

semidirect

12. C₃×C₁₆²

C₄

semidirect

33. #7052

Dic₃

semidirect

34. C₁₆²

Dic₃

semidirect

101. C₃×C₁₆

#347

semidirect

137. C₁₆

#4512

semidirect

158. C₃

#72214029

semidirect

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

subgroup	index
C₃×#122211	2
#72225825	2
#72225813	2
#72214029	3

subgroup

index

Subgroups — #72226216

Conjugacy classes of subgroups

Complements

Maximal subgroups