Group #70749 — subgroups

Subgroups — #70749

Order 384.

168 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 97 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×#431	C₂	1	1	yes	maximal, characteristic
2	#4924	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#4457	C₂	1	4	yes	maximal, complemented
5	#1570	—	3	1	no	maximal, Sylow
6	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
7	C₂×#208	—	2	1	no	—
8	C₂²×C₃⋊C₈	—	2	1	no	—
9	C₃×C₄⋊C₈	C₄	1	2	yes	—
10	C₃×C₂²⋊C₈	C₄	1	2	yes	—
11	C₃×C₄×C₈	C₂²	1	2	yes	—
12	C₄⋊(C₃⋊C₈)	—	2	1	no	—
13	C₄⋊(C₃⋊C₈)	—	2	1	no	—
14	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
15	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
16	#431	D₃	1	1	yes	characteristic
17	#423	—	3	1	no	—
18	C₂×C₄⋊C₈	—	3	1	no	—
19	#334	—	3	4	no	—
20	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
22	C₂²×C₃×C₄	Q₈	1	1	yes	characteristic
23	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
24	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
25	C₃×C₄²	D₄	1	1	yes	characteristic
26	C₃×C₄²	Q₈	1	1	yes	characteristic
27	C₃×C₄²	C₂³	1	1	yes	characteristic
28	#208	—	4	2	no	—
29	C₂×C₃⋊C₈	—	2	2	no	—
30	C₂×C₃⋊C₈	—	2	2	no	—
31	C₂×C₃⋊C₈	—	4	2	no	—
32	C₂×M₁₆	—	6	1	no	—
33	C₂²×C₈	—	6	1	no	—
34	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
35	C₄⋊C₈	—	3	2	no	—
36	C₄⋊C₈	—	3	2	no	—
37	C₄⋊C₈	—	6	1	no	—
38	C₄⋊C₈	—	6	1	no	—
39	C₄⋊C₈	Dic₃	1	2	yes	—
40	C₂²⋊C₈	Dic₃	1	2	yes	—
41	C₄×C₈	C₂×D₃	1	2	yes	—
42	C₂³×C₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₃×C₄	—	2	2	no	—
44	C₂×C₃×C₄	—	4	1	no	—
45	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₃×C₄	SD₁₆	1	2	yes	—
49	C₂×C₃×C₄	SD₁₆	1	2	yes	—
50	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
52	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
53	C₃×C₈	—	2	4	no	—
54	C₃⋊C₈	—	4	2	no	—
55	C₃⋊C₈	—	4	2	no	—
56	C₂²×C₄	Dic₆	1	1	yes	characteristic
57	C₂²×C₄	D₁₂	1	1	yes	characteristic
58	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
59	M₁₆	—	12	2	no	—
60	C₂×C₈	—	6	2	no	—
61	C₂×C₈	—	6	2	no	—
62	C₂×C₈	—	12	2	no	—
63	C₂×C₈	C₂×Dic₃	1	4	yes	—
64	C₄²	Dic₆	1	1	yes	characteristic
65	C₄²	D₁₂	1	1	yes	characteristic
66	C₄²	C₂×Dic₃	1	1	yes	characteristic
67	C₄²	C₂²×D₃	1	1	yes	characteristic
68	C₂²×C₃	—	2	1	no	—
69	C₂²×C₃	—	2	1	no	—
70	C₂²×C₃	#106	1	2	yes	—
71	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₃×C₄	—	2	1	no	—
73	C₃×C₄	—	2	1	no	—
74	C₃×C₄	—	2	1	no	—
75	C₃×C₄	#106	1	2	yes	—
76	C₃×C₄	#131	1	2	yes	—
77	C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
78	C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
79	C₂³	C₄⋊Dic₃	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	#204	1	2	yes	—
83	C₂×C₄	#204	1	2	yes	—
84	C₂×C₄	C₄⋊Dic₃	1	1	yes	characteristic
85	C₂×C₄	C₄⋊Dic₃	1	1	yes	characteristic
86	C₂×C₄	C₄⋊Dic₃	1	1	yes	characteristic
87	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
88	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
89	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
90	C₈	—	2	4	no	—
91	C₈	—	12	2	no	—
92	C₈	—	12	2	no	—
93	C₂×C₃	—	2	2	no	—
94	C₂×C₃	#424	1	1	yes	characteristic
95	C₂×C₃	C₂×#106	1	1	yes	characteristic
96	C₂×C₃	#430	1	1	yes	characteristic
97	C₂²	—	2	1	no	—
98	C₂²	—	2	1	no	—
99	C₂²	#817	1	2	yes	—
100	C₂²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	#817	1	2	yes	—
105	C₄	C₂×#204	1	1	yes	characteristic
106	C₄	C₂×#204	1	1	yes	characteristic
107	C₄	#948	1	2	yes	—
108	C₃	#1570	1	1	yes	characteristic, complemented, Sylow
109	C₂	—	2	2	no	—
110	C₂	#4999	1	1	yes	characteristic
111	C₂	C₂×#817	1	1	yes	characteristic
112	C₂	#5125	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#431

C₂

yes

maximal, characteristic

#4924

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₃⋊C₈)

C₂

yes

maximal, characteristic

#4457

C₂

yes

maximal, complemented

#1570

—

maximal, Sylow

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×#208

—

C₂²×C₃⋊C₈

—

C₃×C₄⋊C₈

C₄

yes

—

C₃×C₂²⋊C₈

C₄

yes

—

C₃×C₄×C₈

C₂²

yes

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

C₂²

yes

—

C₄⋊(C₃⋊C₈)

C₂²

yes

—

#431

D₃

yes

characteristic

#423

—

C₂×C₄⋊C₈

—

#334

—

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

D₄

yes

characteristic

C₂²×C₃×C₄

Q₈

yes

characteristic

C₂×C₃×C₈

C₂×C₄

yes

—

C₃×C₄²

C₂×C₄

yes

characteristic

C₃×C₄²

D₄

yes

characteristic

C₃×C₄²

Q₈

yes

characteristic

C₃×C₄²

C₂³

yes

characteristic

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

Dic₃

yes

—

C₂²⋊C₈

Dic₃

yes

—

C₄×C₈

C₂×D₃

yes

—

C₂³×C₃

C₄⋊C₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃×C₄

SD₁₆

yes

—

C₂×C₃×C₄

SD₁₆

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

characteristic

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂²×C₄

Dic₆

yes

characteristic

C₂²×C₄

D₁₂

yes

characteristic

C₂²×C₄

C₂×Dic₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₃

yes

—

C₄²

Dic₆

yes

characteristic

C₄²

D₁₂

yes

characteristic

C₄²

C₂×Dic₃

yes

characteristic

C₄²

C₂²×D₃

yes

characteristic

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

#106

yes

—

C₂²×C₃

C₂×C₄⋊C₄

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#106

yes

—

C₃×C₄

#131

yes

—

C₃×C₄

C₂×SD₁₆

yes

characteristic

C₃×C₄

C₂×SD₁₆

yes

characteristic

C₂³

C₄⋊Dic₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#204

yes

—

C₂×C₄

#204

yes

—

C₂×C₄

C₄⋊Dic₃

yes

characteristic

C₂×C₄

C₄⋊Dic₃

yes

characteristic

C₂×C₄

C₄⋊Dic₃

yes

characteristic

C₂×C₄

C₂×Dic₆

yes

characteristic

C₂×C₄

C₂×D₁₂

yes

characteristic

C₂×C₄

C₂²×Dic₃

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂×C₃

—

C₂×C₃

yes

C₂×C₃

yes

C₂×C₃

yes

C₂²

—

C₂²

—

C₂²

#817

yes

—

100

C₂²

C₂×C₄⋊Dic₃

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

#817

yes

—

105

C₄

C₂×#204

yes

characteristic

106

C₄

C₂×#204

yes

characteristic

107

C₄

#948

yes

—

108

C₃

#1570

yes

characteristic, complemented, Sylow

109

C₂

—

110

C₂

#4999

yes

characteristic

111

C₂

C₂×#817

yes

characteristic

112

C₂

#5125

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#431	C₂
socle	C₂²×C₃	C₂×C₄⋊C₄
prime radical	C₂³×C₃	C₄⋊C₄
cosocle	C₃×C₄²	C₂³
hypercenter	#431	D₃
hypocenter	C₃	#1570
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₁₂

commutator

C₂×C₃×C₄

C₂²×C₄

Frattini

C₄²

C₂²×D₃

Fitting

C₃×#431

C₂

socle

C₂²×C₃

C₂×C₄⋊C₄

prime radical

C₂³×C₃

C₄⋊C₄

cosocle

C₃×C₄²

C₂³

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
4. #4457	C₂	semidirect
108. C₃	#1570	semidirect

H ≅ G/N

type

4. #4457

C₂

semidirect

108. C₃

#1570

semidirect

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

subgroup	index
C₃×#431	2
#4924	2
C₂×C₄⋊(C₃⋊C₈)	2
#4457	2
#1570	3

subgroup

index

C₂×C₄⋊(C₃⋊C₈)

Subgroups — #70749

Conjugacy classes of subgroups

Complements

Maximal subgroups