Group #72014 — subgroups

Subgroups — #72014

Order 384.

252 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 105 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×#131	C₂	1	1	yes	maximal, characteristic
2	#5237	C₂	1	2	yes	maximal
3	C₂×#819	C₂	1	2	yes	maximal, complemented
4	C₃⋊#338	C₂	1	2	yes	maximal, complemented
5	#1813	—	3	1	no	maximal, Sylow
6	C₂×C₃×#41	C₂²	1	1	yes	characteristic
7	C₃×#131	—	2	4	no	—
8	C₂×C₃×M₁₆	C₄	1	2	yes	—
9	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₈	C₄	1	2	yes	—
11	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
12	C₂×#208	C₂²	1	4	yes	—
13	#836	—	2	2	no	—
14	#833	—	2	2	no	—
15	#819	—	2	4	no	—
16	C₂×#131	D₃	1	1	yes	characteristic
17	C₂×#108	—	3	2	no	—
18	#413	—	3	2	no	—
19	#338	—	3	2	no	—
20	C₃×#41	—	2	2	no	—
21	C₃×#41	—	2	2	no	—
22	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
23	C₂×C₃×D₄	D₄	1	1	yes	characteristic
24	C₂×C₃×D₄	Q₈	1	2	yes	—
25	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
26	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
27	C₃×M₁₆	—	2	2	no	—
28	C₃×M₁₆	—	2	4	no	—
29	C₂×C₃×C₈	—	2	4	no	—
30	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₈	D₄	1	4	yes	—
34	#208	—	2	4	no	—
35	#208	—	2	4	no	—
36	C₂×C₃⋊C₈	—	2	4	no	—
37	C₂×#41	C₂×D₃	1	1	yes	characteristic
38	#131	—	2	4	no	—
39	C₂×M₁₆	—	3	4	no	—
40	C₂×M₁₆	Dic₃	1	2	yes	—
41	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
42	C₂²×C₈	Dic₃	1	2	yes	—
43	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
44	#108	—	6	4	no	—
45	#101	—	6	2	no	—
46	#100	—	6	2	no	—
47	C₂³×C₃	C₄⋊C₄	1	2	yes	—
48	C₂³×C₃	C₂×Q₈	1	1	yes	characteristic
49	C₃×Q₈	—	2	2	no	—
50	C₃×D₄	—	2	4	no	—
51	C₃×D₄	—	2	2	no	—
52	C₂×C₃×C₄	—	2	2	no	—
53	C₂×C₃×C₄	—	2	2	no	—
54	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
55	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
58	C₂×C₃×C₄	#41	1	2	yes	—
59	C₃×C₈	—	2	2	no	—
60	C₃×C₈	C₂²⋊C₄	1	2	yes	—
61	C₃×C₈	C₂²⋊C₄	1	2	yes	—
62	C₃⋊C₈	—	4	4	no	—
63	#41	—	2	2	no	—
64	#41	—	2	2	no	—
65	C₂×Q₈	D₁₂	1	1	yes	characteristic
66	C₂×D₄	Dic₆	1	2	yes	—
67	C₂×D₄	D₁₂	1	1	yes	characteristic
68	C₂²×C₄	C₂×Dic₃	1	2	yes	—
69	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
70	M₁₆	—	2	2	no	—
71	M₁₆	—	2	4	no	—
72	M₁₆	—	6	4	no	—
73	M₁₆	—	6	4	no	—
74	C₂×C₈	—	2	4	no	—
75	C₂×C₈	—	6	4	no	—
76	C₂×C₈	C₂×Dic₃	1	2	yes	—
77	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
78	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
79	C₂×C₈	#67	1	4	yes	—
80	C₂²×C₃	—	2	2	no	—
81	C₂²×C₃	—	4	1	no	—
82	C₂²×C₃	—	4	2	no	—
83	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₃	C₂²⋊Q₈	1	2	yes	—
85	C₃×C₄	—	2	2	no	—
86	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₃×C₄	#117	1	1	yes	characteristic
88	C₃×C₄	#121	1	2	yes	—
89	C₂³	C₄⋊Dic₃	1	2	yes	—
90	C₂³	C₂×Dic₆	1	1	yes	characteristic
91	Q₈	—	2	2	no	—
92	D₄	—	2	4	no	—
93	D₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	C₄⋊Dic₃	1	2	yes	—
97	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
98	C₂×C₄	#235	1	2	yes	—
99	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
100	C₂×C₄	C₂×#67	1	2	yes	—
101	C₈	—	2	2	no	—
102	C₈	—	12	4	no	—
103	C₈	#217	1	2	yes	—
104	C₈	#217	1	2	yes	—
105	C₂×C₃	—	2	1	no	—
106	C₂×C₃	—	2	2	no	—
107	C₂×C₃	—	4	2	no	—
108	C₂×C₃	#380	1	1	yes	characteristic
109	C₂²	—	2	2	no	—
110	C₂²	—	4	1	no	—
111	C₂²	—	4	2	no	—
112	C₂²	#924	1	2	yes	—
113	C₂²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	#926	1	1	yes	characteristic
116	C₄	#930	1	2	yes	—
117	C₄	C₂×#217	1	1	yes	characteristic
118	C₃	#1813	1	1	yes	characteristic, complemented, Sylow
119	C₂	—	2	1	no	—
120	C₂	—	2	2	no	—
121	C₂	—	4	2	no	—
122	C₂	#5212	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×#131

C₂

yes

maximal, characteristic

#5237

C₂

yes

maximal

C₂×#819

C₂

yes

maximal, complemented

C₃⋊#338

C₂

yes

maximal, complemented

#1813

—

maximal, Sylow

C₂×C₃×#41

C₂²

yes

characteristic

C₃×#131

—

C₂×C₃×M₁₆

C₄

yes

—

C₂×C₃×M₁₆

C₂²

yes

characteristic

C₂²×C₃×C₈

C₄

yes

—

C₂²×C₃×C₈

C₂²

yes

characteristic

C₂×#208

C₂²

yes

—

—

—

—

D₃

yes

—

—

—

C₃×#41

—

C₃×#41

—

C₂×C₃×Q₈

D₄

yes

characteristic

C₂×C₃×D₄

D₄

yes

characteristic

C₂×C₃×D₄

Q₈

yes

—

C₂²×C₃×C₄

C₂×C₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₄

yes

—

C₂×C₃×C₈

C₂×C₄

yes

characteristic

C₂×C₃×C₈

C₂×C₄

yes

characteristic

C₂×C₃×C₈

D₄

yes

—

#208

—

#208

—

C₂×C₃⋊C₈

—

C₂×#41

C₂×D₃

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

Dic₃

yes

—

C₂×M₁₆

C₂×D₃

yes

characteristic

C₂²×C₈

Dic₃

yes

—

C₂²×C₈

C₂×D₃

yes

—

—

—

C₂³×C₃

C₄⋊C₄

yes

—

C₂³×C₃

C₂×Q₈

yes

characteristic

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄⋊C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

#41

yes

—

C₃×C₈

—

C₃×C₈

C₂²⋊C₄

yes

—

C₃×C₈

C₂²⋊C₄

yes

—

C₃⋊C₈

—

#41

—

#41

—

C₂×Q₈

D₁₂

yes

characteristic

C₂×D₄

Dic₆

yes

—

C₂×D₄

D₁₂

yes

characteristic

C₂²×C₄

C₂×Dic₃

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₃

yes

—

C₂×C₈

C₂×Dic₃

yes

characteristic

C₂×C₈

C₂×Dic₃

yes

characteristic

C₂×C₈

#67

yes

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃

C₂²⋊Q₈

yes

—

C₃×C₄

—

C₃×C₄

C₂×C₂²⋊C₄

yes

C₃×C₄

yes

C₃×C₄

yes

—

C₂³

C₄⋊Dic₃

yes

—

C₂³

C₂×Dic₆

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₃

yes

—

C₂×C₄

C₂×D₁₂

yes

characteristic

C₂×C₄

#235

yes

—

C₂×C₄

C₂²×Dic₃

yes

characteristic

100

C₂×C₄

C₂×#67

yes

—

101

C₈

—

102

C₈

—

103

C₈

#217

yes

—

104

C₈

#217

yes

—

105

C₂×C₃

—

106

C₂×C₃

—

107

C₂×C₃

—

108

C₂×C₃

#380

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

#924

yes

—

113

C₂²

C₂×C₄⋊Dic₃

yes

characteristic

114

C₄

—

115

C₄

#926

yes

characteristic

116

C₄

#930

yes

—

117

C₄

C₂×#217

yes

characteristic

118

C₃

#1813

yes

characteristic, complemented, Sylow

119

C₂

—

120

C₂

—

121

C₂

—

122

C₂

#5212

yes

characteristic

	subgroup	G/H
center	C₄	#926
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂×C₃×#131	C₂
socle	C₂×C₃	#380
prime radical	C₂×C₃×#41	C₂²
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×#131	D₃
hypocenter	C₃	#1813
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

subgroup

G/H

center

C₄

#926

commutator

C₂×C₃×C₄

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₂×C₃×#131

C₂

socle

C₂×C₃

#380

prime radical

C₂×C₃×#41

C₂²

cosocle

C₂²×C₃×C₄

C₂³

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

N	H ≅ G/N	type
3. C₂×#819	C₂	semidirect
4. C₃⋊#338	C₂	semidirect
118. C₃	#1813	semidirect

H ≅ G/N

type

3. C₂×#819

C₂

semidirect

4. C₃⋊#338

C₂

semidirect

118. C₃

#1813

semidirect

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

subgroup	index
C₂×C₃×#131	2
#5237	2
C₂×#819	2
C₃⋊#338	2
#1813	3

subgroup

index

C₂×C₃×#131

Subgroups — #72014

Conjugacy classes of subgroups

Complements

Maximal subgroups