Group #72532 — subgroups

Subgroups — C₃×#1817 #72532

Order 384.

380 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 160 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	C₂²×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₂²⋊C₈	C₂	1	2	yes	maximal
3	C₃×#336	C₂	1	4	yes	maximal, complemented
4	#1817	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂³×C₃×C₄	C₄	1	1	yes	characteristic
6	C₂³×C₃×C₄	C₄	1	1	yes	characteristic
7	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₈	C₂²	1	4	yes	—
9	C₂×C₃×C₄²	—	2	4	no	—
10	C₂×C₃×C₄²	C₄	1	2	yes	—
11	C₂×C₃×C₄²	C₂²	1	2	yes	—
12	C₃×C₂²⋊C₈	—	2	4	no	—
13	C₂²×C₄²	C₂×C₃	1	1	yes	characteristic
14	C₂×C₂²⋊C₈	C₂×C₃	1	2	yes	—
15	#336	C₂×C₃	1	4	yes	complemented
16	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₃×C₄	—	2	2	no	—
18	C₂²×C₃×C₄	—	2	2	no	—
19	C₂²×C₃×C₄	—	2	2	no	—
20	C₂²×C₃×C₄	—	2	4	no	—
21	C₂²×C₃×C₄	—	2	4	no	—
22	C₂²×C₃×C₄	C₈	1	4	yes	complemented
23	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
26	C₂²×C₃×C₄	D₄	1	4	yes	—
27	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
28	C₂×C₃×C₈	—	2	4	no	—
29	C₂×C₃×C₈	—	2	8	no	—
30	C₃×C₄²	—	2	4	no	—
31	C₃×C₄²	—	2	4	no	—
32	C₂³×C₄	C₃×C₄	1	1	yes	characteristic
33	C₂³×C₄	C₃×C₄	1	1	yes	characteristic
34	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
35	C₂²×C₈	C₂²×C₃	1	4	yes	—
36	C₂×C₄²	—	2	4	no	—
37	C₂×C₄²	C₃×C₄	1	2	yes	—
38	C₂×C₄²	C₂²×C₃	1	2	yes	—
39	C₂²⋊C₈	—	2	4	no	—
40	C₂³×C₃	—	2	2	no	—
41	C₂³×C₃	—	2	2	no	—
42	C₂³×C₃	C₂²⋊C₄	1	2	yes	—
43	C₂³×C₃	C₂×C₈	1	2	yes	—
44	C₂³×C₃	M₁₆	1	2	yes	—
45	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₃×C₄	—	2	8	no	—
47	C₂×C₃×C₄	—	2	8	no	—
48	C₂×C₃×C₄	—	2	2	no	—
49	C₂×C₃×C₄	—	2	2	no	—
50	C₂×C₃×C₄	—	2	2	no	—
51	C₂×C₃×C₄	—	2	2	no	—
52	C₂×C₃×C₄	—	2	2	no	—
53	C₂×C₃×C₄	—	2	8	no	—
54	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
55	C₂×C₃×C₄	C₂×C₈	1	4	yes	—
56	C₂×C₃×C₄	M₁₆	1	4	yes	—
57	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
58	C₂×C₃×C₄	#41	1	4	yes	—
59	C₃×C₈	—	4	4	no	—
60	C₂⁴	C₂×C₃×C₄	1	1	yes	characteristic
61	C₂²×C₄	—	2	2	no	—
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	—	2	2	no	—
64	C₂²×C₄	—	2	4	no	—
65	C₂²×C₄	—	2	4	no	—
66	C₂²×C₄	C₃×C₈	1	4	yes	complemented
67	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
68	C₂²×C₄	C₂×C₃×C₄	1	2	yes	—
69	C₂²×C₄	C₂×C₃×C₄	1	2	yes	—
70	C₂²×C₄	C₃×D₄	1	4	yes	—
71	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
72	C₂×C₈	—	2	4	no	—
73	C₂×C₈	—	2	8	no	—
74	C₄²	—	2	4	no	—
75	C₄²	—	2	4	no	—
76	C₂²×C₃	—	2	2	no	—
77	C₂²×C₃	—	2	2	no	—
78	C₂²×C₃	—	2	4	no	—
79	C₂²×C₃	—	2	4	no	—
80	C₂²×C₃	C₂²⋊C₈	1	4	yes	—
81	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₃	#117	1	2	yes	—
83	C₂²×C₃	C₂²×C₈	1	1	yes	characteristic
84	C₂²×C₃	C₂×M₁₆	1	1	yes	characteristic
85	C₂²×C₃	C₂×M₁₆	1	2	yes	—
86	C₃×C₄	—	2	2	no	—
87	C₃×C₄	—	2	4	no	—
88	C₃×C₄	—	2	4	no	—
89	C₃×C₄	#123	1	4	yes	—
90	C₂³	—	2	2	no	—
91	C₂³	—	2	2	no	—
92	C₂³	C₃×C₂²⋊C₄	1	2	yes	—
93	C₂³	C₂×C₃×C₈	1	2	yes	—
94	C₂³	C₃×M₁₆	1	2	yes	—
95	C₂³	C₂²×C₃×C₄	1	1	yes	characteristic
96	C₂×C₄	—	2	8	no	—
97	C₂×C₄	—	2	8	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	8	no	—
104	C₂×C₄	C₃×C₂²⋊C₄	1	2	yes	—
105	C₂×C₄	C₂×C₃×C₈	1	4	yes	—
106	C₂×C₄	C₃×M₁₆	1	4	yes	—
107	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
108	C₂×C₄	C₃×#41	1	4	yes	—
109	C₈	—	4	4	no	—
110	C₂×C₃	—	2	4	no	—
111	C₂×C₃	#381	1	1	yes	characteristic
112	C₂×C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
113	C₂×C₃	#407	1	1	yes	characteristic
114	C₂×C₃	#431	1	2	yes	—
115	C₂×C₃	#432	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	4	no	—
119	C₂²	—	2	4	no	—
120	C₂²	C₃×C₂²⋊C₈	1	4	yes	—
121	C₂²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
122	C₂²	C₃×#117	1	2	yes	—
123	C₂²	C₂²×C₃×C₈	1	1	yes	characteristic
124	C₂²	C₂×C₃×M₁₆	1	1	yes	characteristic
125	C₂²	C₂×C₃×M₁₆	1	2	yes	—
126	C₄	—	2	2	no	—
127	C₄	—	2	4	no	—
128	C₄	—	2	4	no	—
129	C₄	C₃×#123	1	4	yes	—
130	C₃	#1817	1	1	yes	characteristic, direct product factor, Sylow
131	C₂	—	2	4	no	—
132	C₂	C₃×#381	1	1	yes	characteristic
133	C₂	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
134	C₂	C₃×#407	1	1	yes	characteristic
135	C₂	C₃×#431	1	2	yes	—
136	C₂	C₃×#432	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₃×C₄²

C₂

yes

maximal, characteristic

C₂×C₃×C₂²⋊C₈

C₂

yes

maximal

C₃×#336

C₂

yes

maximal, complemented

#1817

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃×C₄

C₄

yes

characteristic

C₂³×C₃×C₄

C₄

yes

characteristic

C₂³×C₃×C₄

C₂²

yes

characteristic

C₂²×C₃×C₈

C₂²

yes

—

C₂×C₃×C₄²

—

C₂×C₃×C₄²

C₄

yes

—

C₂×C₃×C₄²

C₂²

yes

—

C₃×C₂²⋊C₈

—

C₂²×C₄²

C₂×C₃

yes

characteristic

C₂×C₂²⋊C₈

C₂×C₃

yes

—

#336

C₂×C₃

yes

complemented

C₂⁴×C₃

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₈

yes

complemented

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

C₂×C₄

yes

—

C₂²×C₃×C₄

C₂×C₄

yes

—

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

—

C₃×C₄²

—

C₂³×C₄

C₃×C₄

yes

characteristic

C₂³×C₄

C₃×C₄

yes

characteristic

C₂³×C₄

C₂²×C₃

yes

characteristic

C₂²×C₈

C₂²×C₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₃×C₄

yes

—

C₂×C₄²

C₂²×C₃

yes

—

C₂²⋊C₈

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

C₂²⋊C₄

yes

—

C₂³×C₃

C₂×C₈

yes

—

C₂³×C₃

M₁₆

yes

—

C₂³×C₃

C₂²×C₄

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

C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₂×C₈

yes

—

C₂×C₃×C₄

M₁₆

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

#41

yes

—

C₃×C₈

—

C₂⁴

C₂×C₃×C₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₃×C₈

yes

complemented

C₂²×C₄

C₂×C₃×C₄

yes

characteristic

C₂²×C₄

C₂×C₃×C₄

yes

—

C₂²×C₄

C₂×C₃×C₄

yes

—

C₂²×C₄

C₃×D₄

yes

—

C₂²×C₄

C₂³×C₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂²⋊C₈

yes

—

C₂²×C₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃

#117

yes

—

C₂²×C₃

C₂²×C₈

yes

characteristic

C₂²×C₃

C₂×M₁₆

yes

characteristic

C₂²×C₃

C₂×M₁₆

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#123

yes

—

C₂³

—

C₂³

—

C₂³

C₃×C₂²⋊C₄

yes

—

C₂³

C₂×C₃×C₈

yes

—

C₂³

C₃×M₁₆

yes

—

C₂³

C₂²×C₃×C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₃×C₂²⋊C₄

yes

—

105

C₂×C₄

C₂×C₃×C₈

yes

—

106

C₂×C₄

C₃×M₁₆

yes

—

107

C₂×C₄

C₂×C₃×D₄

yes

—

108

C₂×C₄

C₃×#41

yes

—

109

C₈

—

110

C₂×C₃

—

111

C₂×C₃

#381

yes

characteristic

112

C₂×C₃

C₂×C₂²⋊C₈

yes

characteristic

113

C₂×C₃

#407

yes

characteristic

114

C₂×C₃

#431

yes

—

115

C₂×C₃

#432

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₃×C₂²⋊C₈

yes

—

121

C₂²

C₂×C₃×C₂²⋊C₄

yes

characteristic

122

C₂²

C₃×#117

yes

—

123

C₂²

C₂²×C₃×C₈

yes

characteristic

124

C₂²

C₂×C₃×M₁₆

yes

characteristic

125

C₂²

C₂×C₃×M₁₆

yes

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

C₃×#123

yes

—

130

C₃

#1817

yes

characteristic, direct product factor, Sylow

131

C₂

—

132

C₂

C₃×#381

yes

characteristic

133

C₂

C₂×C₃×C₂²⋊C₈

yes

characteristic

134

C₂

C₃×#407

yes

characteristic

135

C₂

C₃×#431

yes

—

136

C₂

C₃×#432

yes

—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂³
commutator	C₂²	C₂²×C₃×C₈
Frattini, cosocle	C₂²×C₄	C₂³×C₃
socle	C₂³×C₃	C₂²×C₄
prime radical	C₂⁴×C₃	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

subgroup

G/H

center

C₂²×C₃×C₄

C₂³

commutator

C₂²

C₂²×C₃×C₈

Frattini, cosocle

C₂²×C₄

C₂³×C₃

socle

C₂³×C₃

C₂²×C₄

prime radical

C₂⁴×C₃

C₂×C₄

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₃×C₄ ◁ G

N	H ≅ G/N	type
3. C₃×#336	C₂	semidirect
4. #1817	C₃	direct (N × H)
15. #336	C₂×C₃	semidirect
22. C₂²×C₃×C₄	C₈	semidirect
66. C₂²×C₄	C₃×C₈	semidirect
130. C₃	#1817	direct (N × H)

H ≅ G/N

type

3. C₃×#336

C₂

semidirect

4. #1817

C₃

direct (N × H)

15. #336

C₂×C₃

semidirect

22. C₂²×C₃×C₄

C₈

semidirect

66. C₂²×C₄

C₃×C₈

semidirect

130. C₃

#1817

direct (N × H)

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

subgroup	index
C₂²×C₃×C₄²	2
C₂×C₃×C₂²⋊C₈	2
C₃×#336	2
#1817	3

subgroup

index

C₂²×C₃×C₄²

C₂×C₃×C₂²⋊C₈

C₃×#336

#1817

Subgroups — C₃×#1817 #72532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#1817 #72532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#1817 #72532