Group #70543 — subgroups

Subgroups — #70543

Order 384.

256 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 69 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×Dic₁₂	C₂	1	1	yes	maximal, characteristic
2	C₃×#424	C₂	1	1	yes	maximal, characteristic
3	C₂×#824	C₂	1	2	yes	maximal
4	#5004	C₂	1	2	yes	maximal
5	#4552	C₂	1	1	yes	maximal, characteristic
6	#2047	—	3	1	no	maximal, Sylow
7	C₂²×Dic₆	C₂²	1	2	yes	—
8	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
10	C₃×#117	C₂²	1	1	yes	characteristic
11	C₂×#208	C₂²	1	2	yes	—
12	C₂×Dic₁₂	—	2	4	no	—
13	C₂×Dic₁₂	—	2	2	no	—
14	C₃×C₄⋊C₈	—	2	2	no	—
15	#824	—	2	4	no	—
16	#810	—	2	4	no	—
17	C₂²×Q₁₆	—	3	1	no	—
18	#424	D₃	1	1	yes	characteristic
19	#419	—	3	2	no	—
20	C₂×#101	—	3	2	no	—
21	#341	—	3	1	no	—
22	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
23	C₂²×Dic₃	—	2	2	no	—
24	C₂×Dic₆	—	2	2	no	—
25	C₂×Dic₆	—	2	2	no	—
26	C₂×Dic₆	—	2	2	no	—
27	C₂×Dic₆	—	2	2	no	—
28	C₂×Dic₆	—	2	2	no	—
29	C₂×Dic₆	D₄	1	4	yes	—
30	C₃×M₁₆	—	2	2	no	—
31	C₂×C₃×C₈	—	2	2	no	—
32	C₂×C₃×C₈	—	2	1	no	—
33	C₂×C₃×C₈	D₄	1	2	yes	—
34	C₃×C₄⋊C₄	D₄	1	2	yes	—
35	C₃×C₂²⋊C₄	—	2	1	no	—
36	C₃×C₄²	—	2	1	no	—
37	#208	—	2	2	no	—
38	#208	—	2	2	no	—
39	C₂×C₃⋊C₈	—	2	2	no	—
40	Dic₁₂	—	2	8	no	—
41	C₂²×Q₈	—	3	2	no	—
42	C₂×Q₁₆	—	6	4	no	—
43	C₂×Q₁₆	—	6	2	no	—
44	C₂×M₁₆	—	3	2	no	—
45	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
46	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
47	#117	C₂×D₃	1	1	yes	characteristic
48	C₄⋊C₈	—	2	2	no	—
49	#103	—	6	4	no	—
50	#101	—	6	4	no	—
51	C₂³×C₃	#41	1	1	yes	characteristic
52	C₂×C₃×C₄	—	2	2	no	—
53	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
54	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
55	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
56	C₂×C₃×C₄	#41	1	2	yes	—
57	C₂×Dic₃	—	2	4	no	—
58	C₂×Dic₃	—	4	2	no	—
59	C₂×Dic₃	—	4	2	no	—
60	Dic₆	—	2	4	no	—
61	Dic₆	—	2	4	no	—
62	Dic₆	—	4	2	no	—
63	Dic₆	—	4	2	no	—
64	C₃×C₈	—	2	2	no	—
65	C₃×C₈	—	4	1	no	—
66	C₃⋊C₈	—	4	2	no	—
67	C₂×Q₈	—	3	4	no	—
68	C₂×Q₈	—	6	2	no	—
69	C₂×Q₈	—	6	2	no	—
70	C₂×Q₈	—	6	2	no	—
71	C₂×Q₈	—	6	2	no	—
72	C₂×Q₈	—	6	2	no	—
73	C₂²×C₄	—	6	2	no	—
74	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
75	Q₁₆	—	6	8	no	—
76	M₁₆	—	2	2	no	—
77	M₁₆	—	6	2	no	—
78	M₁₆	—	6	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	6	2	no	—
82	C₂×C₈	#67	1	2	yes	—
83	C₄⋊C₄	D₁₂	1	2	yes	—
84	C₂²⋊C₄	—	2	1	no	—
85	C₄²	—	2	1	no	—
86	C₂²×C₃	—	2	2	no	—
87	C₂²×C₃	#121	1	1	yes	characteristic
88	C₂²×C₃	#123	1	2	yes	—
89	C₃×C₄	—	4	2	no	—
90	C₃×C₄	#120	1	1	yes	characteristic
91	C₃×C₄	#121	1	2	yes	—
92	C₃×C₄	#124	1	1	yes	characteristic
93	Dic₃	—	4	4	no	—
94	C₂³	#235	1	1	yes	characteristic
95	Q₈	—	6	4	no	—
96	Q₈	—	6	4	no	—
97	Q₈	—	12	2	no	—
98	Q₈	—	12	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	6	4	no	—
101	C₂×C₄	—	12	2	no	—
102	C₂×C₄	—	12	2	no	—
103	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
104	C₂×C₄	D₃×D₄	1	2	yes	—
105	C₂×C₄	#239	1	2	yes	—
106	C₂×C₄	C₂×#67	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	4	1	no	—
109	C₈	—	12	2	no	—
110	C₂×C₃	—	2	2	no	—
111	C₂×C₃	#394	1	1	yes	characteristic
112	C₂×C₃	#473	1	2	yes	—
113	C₂²	—	2	2	no	—
114	C₂²	#894	1	2	yes	—
115	C₂²	#930	1	1	yes	characteristic
116	C₄	—	4	2	no	—
117	C₄	—	12	4	no	—
118	C₄	#895	1	2	yes	—
119	C₄	#937	1	1	yes	characteristic
120	C₄	#947	1	1	yes	characteristic
121	C₃	#2047	1	1	yes	characteristic, complemented, Sylow
122	C₂	—	2	2	no	—
123	C₂	#4991	1	1	yes	characteristic
124	C₂	#5143	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₂

C₂

yes

maximal, characteristic

C₃×#424

C₂

yes

maximal, characteristic

C₂×#824

C₂

yes

maximal

#5004

C₂

yes

maximal

#4552

C₂

yes

maximal, characteristic

#2047

—

maximal, Sylow

C₂²×Dic₆

C₂²

yes

—

C₂×C₃×M₁₆

C₂²

yes

characteristic

C₂²×C₃×C₈

C₂²

yes

characteristic

C₃×#117

C₂²

yes

characteristic

C₂×#208

C₂²

yes

—

C₂×Dic₁₂

—

C₂×Dic₁₂

—

C₃×C₄⋊C₈

—

#824

—

#810

—

C₂²×Q₁₆

—

#424

D₃

yes

—

—

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

D₄

yes

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

D₄

yes

—

C₃×C₄⋊C₄

D₄

yes

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

#208

—

#208

—

C₂×C₃⋊C₈

—

Dic₁₂

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃

yes

characteristic

C₂²×C₈

C₂×D₃

yes

characteristic

#117

C₂×D₃

yes

characteristic

C₄⋊C₈

—

#103

—

#101

—

C₂³×C₃

#41

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

characteristic

Q₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#67

yes

—

C₄⋊C₄

D₁₂

yes

—

C₂²⋊C₄

—

C₄²

—

C₂²×C₃

—

C₂²×C₃

#121

yes

characteristic

C₂²×C₃

#123

yes

—

C₃×C₄

—

C₃×C₄

#120

yes

characteristic

C₃×C₄

#121

yes

—

C₃×C₄

#124

yes

characteristic

Dic₃

—

C₂³

#235

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂×D₁₂

yes

characteristic

104

C₂×C₄

D₃×D₄

yes

—

105

C₂×C₄

#239

yes

—

106

C₂×C₄

C₂×#67

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

110

C₂×C₃

—

111

C₂×C₃

#394

yes

characteristic

112

C₂×C₃

#473

yes

—

113

C₂²

—

114

C₂²

#894

yes

—

115

C₂²

#930

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

#895

yes

—

119

C₄

#937

yes

characteristic

120

C₄

#947

yes

characteristic

121

C₃

#2047

yes

characteristic, complemented, Sylow

122

C₂

—

123

C₂

#4991

yes

characteristic

124

C₂

#5143

yes

—

	subgroup	G/H
center	C₂²	#930
commutator, cosocle	C₂²×C₃×C₄	C₂³
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#424	C₂
socle	C₂²×C₃	#121
prime radical	C₂³×C₃	#41
hypercenter	#424	D₃
hypocenter	C₃	#2047
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#930

commutator, cosocle

C₂²×C₃×C₄

C₂³

Frattini

C₂²×C₄

C₂²×D₃

C₂

C₂²×C₃

C₂³×C₃

D₃

C₃

G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
121. C₃	#2047	semidirect

H ≅ G/N

type

121. C₃

#2047

semidirect

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

subgroup	index
C₂²×Dic₁₂	2
C₃×#424	2
C₂×#824	2
#5004	2
#4552	2
#2047	3

subgroup

index

C₂²×Dic₁₂

Subgroups — #70543

Conjugacy classes of subgroups

Complements

Maximal subgroups