Group #72513 — subgroups

Subgroups — C₃×#1798 #72513

Order 384.

312 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 136 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×C₃×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₂²⋊C₈	C₂	1	2	yes	maximal
4	C₃×#336	C₂	1	2	yes	maximal, complemented
5	#1798	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₈	C₂²	1	4	yes	—
8	C₃×#117	—	2	4	no	—
9	C₂×C₃×C₄⋊C₄	C₄	1	2	yes	—
10	C₂×C₃×C₂²⋊C₄	C₄	1	2	yes	—
11	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
13	C₃×C₄⋊C₈	—	2	4	no	—
14	C₃×C₂²⋊C₈	—	2	4	no	—
15	C₂×#117	C₂×C₃	1	1	yes	characteristic
16	C₂×C₄⋊C₈	C₂×C₃	1	2	yes	complemented
17	C₂×C₂²⋊C₈	C₂×C₃	1	2	yes	—
18	#336	C₂×C₃	1	2	yes	complemented
19	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃×C₄	—	2	2	no	—
21	C₂²×C₃×C₄	—	2	2	no	—
22	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
23	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄	D₄	1	2	yes	—
26	C₂²×C₃×C₄	Q₈	1	2	yes	—
27	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
28	C₂×C₃×C₈	—	2	4	no	—
29	C₂×C₃×C₈	—	2	4	no	—
30	C₂×C₃×C₈	—	2	4	no	—
31	C₃×C₄⋊C₄	—	2	4	no	—
32	C₃×C₂²⋊C₄	—	2	4	no	—
33	C₃×C₄²	—	2	2	no	—
34	C₃×C₄²	D₄	1	4	yes	—
35	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
36	C₂²×C₈	C₂²×C₃	1	4	yes	—
37	#117	—	2	4	no	—
38	C₂×C₄⋊C₄	C₃×C₄	1	2	yes	—
39	C₂×C₂²⋊C₄	C₃×C₄	1	2	yes	—
40	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
41	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
42	C₄⋊C₈	—	2	4	no	—
43	C₂²⋊C₈	—	2	4	no	—
44	C₂³×C₃	—	2	1	no	—
45	C₂³×C₃	—	2	1	no	—
46	C₂³×C₃	—	2	4	no	—
47	C₂³×C₃	C₄⋊C₄	1	2	yes	—
48	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₃×C₄	—	2	1	no	—
50	C₂×C₃×C₄	—	2	1	no	—
51	C₂×C₃×C₄	—	2	4	no	—
52	C₂×C₃×C₄	—	2	4	no	—
53	C₂×C₃×C₄	—	2	4	no	—
54	C₂×C₃×C₄	—	2	2	no	—
55	C₂×C₃×C₄	—	2	4	no	—
56	C₂×C₃×C₄	C₂²⋊C₄	1	4	yes	—
57	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
58	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
60	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₃×C₄	#41	1	2	yes	—
62	C₃×C₈	—	4	4	no	—
63	C₂⁴	C₂×C₃×C₄	1	1	yes	characteristic
64	C₂²×C₄	—	2	2	no	—
65	C₂²×C₄	—	2	2	no	—
66	C₂²×C₄	C₂×C₃×C₄	1	2	yes	—
67	C₂²×C₄	C₂×C₃×C₄	1	2	yes	—
68	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
69	C₂²×C₄	C₃×D₄	1	2	yes	—
70	C₂²×C₄	C₃×Q₈	1	2	yes	—
71	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
72	C₂×C₈	—	2	4	no	—
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	4	no	—
75	C₄⋊C₄	—	2	4	no	—
76	C₂²⋊C₄	—	2	4	no	—
77	C₄²	—	2	2	no	—
78	C₄²	C₃×D₄	1	4	yes	—
79	C₂²×C₃	—	2	2	no	—
80	C₂²×C₃	—	2	4	no	—
81	C₂²×C₃	—	4	2	no	—
82	C₂²×C₃	—	4	2	no	—
83	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₃	#117	1	1	yes	characteristic
86	C₂²×C₃	#131	1	4	yes	—
87	C₃×C₄	—	2	2	no	—
88	C₃×C₄	—	2	4	no	—
89	C₃×C₄	—	4	2	no	—
90	C₃×C₄	#121	1	2	yes	—
91	C₃×C₄	C₂²⋊Q₈	1	2	yes	—
92	C₂³	—	2	1	no	—
93	C₂³	—	2	1	no	—
94	C₂³	—	2	4	no	—
95	C₂³	C₃×C₄⋊C₄	1	2	yes	—
96	C₂³	C₂²×C₃×C₄	1	1	yes	characteristic
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	C₃×C₂²⋊C₄	1	4	yes	—
105	C₂×C₄	C₃×C₄⋊C₄	1	2	yes	—
106	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
107	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
108	C₂×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
109	C₂×C₄	C₃×#41	1	2	yes	—
110	C₈	—	4	4	no	—
111	C₂×C₃	—	4	2	no	—
112	C₂×C₃	#380	1	1	yes	characteristic
113	C₂×C₃	#408	1	2	yes	—
114	C₂×C₃	#424	1	2	yes	—
115	C₂×C₃	#433	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	4	no	—
118	C₂²	—	4	2	no	—
119	C₂²	—	4	2	no	—
120	C₂²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
121	C₂²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
122	C₂²	C₃×#117	1	1	yes	characteristic
123	C₂²	C₃×#131	1	4	yes	—
124	C₄	—	2	2	no	—
125	C₄	—	2	4	no	—
126	C₄	—	4	2	no	—
127	C₄	C₃×#121	1	2	yes	—
128	C₄	C₃×C₂²⋊Q₈	1	2	yes	—
129	C₃	#1798	1	1	yes	characteristic, direct product factor, Sylow
130	C₂	—	4	2	no	—
131	C₂	C₃×#380	1	1	yes	characteristic
132	C₂	C₃×#408	1	2	yes	—
133	C₂	C₃×#424	1	2	yes	—
134	C₂	C₃×#433	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×#117

C₂

yes

maximal, characteristic

C₂×C₃×C₄⋊C₈

C₂

yes

maximal, complemented

C₂×C₃×C₂²⋊C₈

C₂

yes

maximal

C₃×#336

C₂

yes

maximal, complemented

#1798

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃×C₄

C₂²

yes

characteristic

C₂²×C₃×C₈

C₂²

yes

—

C₃×#117

—

C₂×C₃×C₄⋊C₄

C₄

yes

—

C₂×C₃×C₂²⋊C₄

C₄

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₃×C₄²

C₂²

yes

characteristic

C₃×C₄⋊C₈

—

C₃×C₂²⋊C₈

—

C₂×#117

C₂×C₃

yes

characteristic

C₂×C₄⋊C₈

C₂×C₃

yes

complemented

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

C₂×C₄

yes

—

C₂²×C₃×C₄

C₂×C₄

yes

—

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

Q₈

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₃×C₄²

D₄

yes

—

C₂³×C₄

C₂²×C₃

yes

characteristic

C₂²×C₈

C₂²×C₃

yes

—

#117

—

C₂×C₄⋊C₄

C₃×C₄

yes

—

C₂×C₂²⋊C₄

C₃×C₄

yes

—

C₂×C₄²

C₂²×C₃

yes

characteristic

C₂×C₄²

C₂²×C₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

C₄⋊C₄

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

C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₄⋊C₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

characteristic

C₂×C₃×C₄

#41

yes

—

C₃×C₈

—

C₂⁴

C₂×C₃×C₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₃×C₄

yes

—

C₂²×C₄

C₂×C₃×C₄

yes

—

C₂²×C₄

C₂×C₃×C₄

yes

characteristic

C₂²×C₄

C₃×D₄

yes

—

C₂²×C₄

C₃×Q₈

yes

—

C₂²×C₄

C₂³×C₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₃×D₄

yes

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃

C₂×C₄⋊C₄

yes

C₂²×C₃

yes

C₂²×C₃

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#121

yes

—

C₃×C₄

C₂²⋊Q₈

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₃×C₄⋊C₄

yes

—

C₂³

C₂²×C₃×C₄

yes

characteristic

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

C₃×C₄⋊C₄

yes

—

106

C₂×C₄

C₂×C₃×D₄

yes

characteristic

107

C₂×C₄

C₂×C₃×D₄

yes

—

108

C₂×C₄

C₂×C₃×Q₈

yes

characteristic

109

C₂×C₄

C₃×#41

yes

—

110

C₈

—

111

C₂×C₃

—

112

C₂×C₃

#380

yes

characteristic

113

C₂×C₃

#408

yes

—

114

C₂×C₃

#424

yes

—

115

C₂×C₃

#433

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×C₃×C₂²⋊C₄

yes

characteristic

121

C₂²

C₂×C₃×C₄⋊C₄

yes

characteristic

122

C₂²

C₃×#117

yes

characteristic

123

C₂²

C₃×#131

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

C₃×#121

yes

—

128

C₄

C₃×C₂²⋊Q₈

yes

—

129

C₃

#1798

yes

characteristic, direct product factor, Sylow

130

C₂

—

131

C₂

C₃×#380

yes

characteristic

132

C₂

C₃×#408

yes

—

133

C₂

C₃×#424

yes

—

134

C₂

C₃×#433

yes

—

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

subgroup

G/H

center

C₂²×C₃×C₄

C₂³

commutator

C₂³

C₂²×C₃×C₄

Frattini, cosocle

C₂²×C₄

C₂³×C₃

socle

C₂³×C₃

C₂²×C₄

prime radical

C₂⁴×C₃

C₂×C₄

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂²×C₃×C₄ ◁ G

N	H ≅ G/N	type
2. C₂×C₃×C₄⋊C₈	C₂	semidirect
4. C₃×#336	C₂	semidirect
5. #1798	C₃	direct (N × H)
16. C₂×C₄⋊C₈	C₂×C₃	semidirect
18. #336	C₂×C₃	semidirect
129. C₃	#1798	direct (N × H)

H ≅ G/N

type

2. C₂×C₃×C₄⋊C₈

C₂

semidirect

4. C₃×#336

C₂

semidirect

5. #1798

C₃

direct (N × H)

16. C₂×C₄⋊C₈

C₂×C₃

semidirect

18. #336

C₂×C₃

semidirect

129. C₃

#1798

direct (N × H)

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

subgroup	index
C₂×C₃×#117	2
C₂×C₃×C₄⋊C₈	2
C₂×C₃×C₂²⋊C₈	2
C₃×#336	2
#1798	3

subgroup

index

C₂×C₃×#117

C₂×C₃×C₄⋊C₈

C₂×C₃×C₂²⋊C₈

C₃×#336

#1798

Subgroups — C₃×#1798 #72513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#1798 #72513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#1798 #72513