Group #71309 — subgroups

Subgroups — #71309

Order 384.

152 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 55 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×#494	C₂	1	1	yes	maximal, characteristic
2	#5094	C₂	1	2	yes	maximal
3	#4679	C₂	1	1	yes	maximal, characteristic
4	#4547	C₂	1	2	yes	maximal
5	#4457	C₂	1	1	yes	maximal, characteristic
6	#1690	—	3	1	no	maximal, Sylow
7	C₂×C₃×Q₁₆	—	2	1	no	—
8	C₂×C₃×Q₁₆	—	2	1	no	—
9	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₄×Dic₆	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	#815	—	2	1	no	—
13	Dic₃⋊C₈	—	2	1	no	—
14	#810	—	2	2	no	—
15	#808	—	2	2	no	—
16	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
17	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	#494	D₃	1	1	yes	characteristic
19	#475	—	3	2	no	—
20	#446	—	3	1	no	—
21	#334	—	3	1	no	—
22	#332	—	3	2	no	—
23	C₂×C₃×Q₈	—	2	2	no	—
24	C₂×Dic₆	D₄	1	1	yes	characteristic
25	C₃×Q₁₆	—	4	2	no	—
26	C₃×Q₁₆	—	4	2	no	—
27	C₂×C₃×C₈	D₄	1	1	yes	characteristic
28	C₂×C₃×C₈	D₄	1	1	yes	characteristic
29	C₃×C₄⋊C₄	—	2	2	no	—
30	C₃×C₄⋊C₄	—	2	2	no	—
31	C₃×C₄²	C₂³	1	1	yes	characteristic
32	C₄⋊Dic₃	D₄	1	1	yes	characteristic
33	Dic₃⋊C₄	—	2	1	no	—
34	C₄×Dic₃	—	2	1	no	—
35	C₂×C₃⋊C₈	—	2	2	no	—
36	C₂×C₃⋊C₈	—	2	1	no	—
37	C₂×Q₁₆	—	2	1	no	—
38	C₂×Q₁₆	—	2	1	no	—
39	C₄⋊Q₈	C₂×D₃	1	2	yes	—
40	C₄×Q₈	—	3	1	no	—
41	#106	—	6	2	no	—
42	C₄⋊C₈	—	3	1	no	—
43	C₄⋊C₈	—	3	2	no	—
44	C₄⋊C₈	—	6	1	no	—
45	#103	—	6	2	no	—
46	#97	—	6	1	no	—
47	C₄×C₈	C₂×D₃	1	1	yes	characteristic
48	C₃×Q₈	—	4	2	no	—
49	C₃×Q₈	—	4	2	no	—
50	C₂×C₃×C₄	—	4	2	no	—
51	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
52	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃×C₄	#41	1	1	yes	characteristic
54	C₂×Dic₃	—	2	1	no	—
55	C₂×Dic₃	—	2	1	no	—
56	Dic₆	—	2	2	no	—
57	C₃×C₈	—	2	1	no	—
58	C₃×C₈	SD₁₆	1	2	yes	—
59	C₃⋊C₈	—	4	1	no	—
60	C₃⋊C₈	—	4	2	no	—
61	C₂×Q₈	—	2	2	no	—
62	C₂×Q₈	—	3	1	no	—
63	Q₁₆	—	4	2	no	—
64	Q₁₆	—	4	2	no	—
65	C₂×C₈	—	6	2	no	—
66	C₂×C₈	—	6	1	no	—
67	C₂×C₈	#67	1	1	yes	characteristic
68	C₂×C₈	#67	1	1	yes	characteristic
69	C₄⋊C₄	—	2	2	no	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	—	3	1	no	—
72	C₄⋊C₄	—	6	1	no	—
73	C₄²	—	6	1	no	—
74	C₄²	C₂²×D₃	1	1	yes	characteristic
75	C₂²×C₃	#121	1	1	yes	characteristic
76	C₃×C₄	—	2	1	no	—
77	C₃×C₄	—	8	2	no	—
78	C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
79	C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₃×C₄	#137	1	2	yes	—
81	Dic₃	—	4	1	no	—
82	Dic₃	—	4	1	no	—
83	Q₈	—	4	2	no	—
84	Q₈	—	4	2	no	—
85	Q₈	—	6	2	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	—	6	1	no	—
88	C₂×C₄	—	6	1	no	—
89	C₂×C₄	D₃×D₄	1	1	yes	characteristic
90	C₂×C₄	#237	1	1	yes	characteristic
91	C₂×C₄	C₂×#67	1	1	yes	characteristic
92	C₈	—	2	1	no	—
93	C₈	—	12	2	no	—
94	C₈	—	12	1	no	—
95	C₈	#214	1	2	yes	—
96	C₂×C₃	#460	1	1	yes	characteristic
97	C₂×C₃	#470	1	1	yes	characteristic
98	C₂×C₃	#473	1	1	yes	characteristic
99	C₂²	#938	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	8	2	no	—
102	C₄	—	12	1	no	—
103	C₄	—	12	1	no	—
104	C₄	#918	1	2	yes	—
105	C₄	#932	1	1	yes	characteristic
106	C₄	C₂×#214	1	1	yes	characteristic
107	C₃	#1690	1	1	yes	characteristic, complemented, Sylow
108	C₂	#5075	1	1	yes	characteristic
109	C₂	#5189	1	1	yes	characteristic
110	C₂	#5192	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#494

C₂

yes

maximal, characteristic

#5094

C₂

yes

maximal

#4679

C₂

yes

maximal, characteristic

#4547

C₂

yes

maximal

#4457

C₂

yes

maximal, characteristic

#1690

—

maximal, Sylow

C₂×C₃×Q₁₆

—

C₂×C₃×Q₁₆

—

C₃×C₄⋊Q₈

C₂²

yes

—

C₄×Dic₆

C₂²

yes

characteristic

C₃×C₄×C₈

C₂²

yes

characteristic

#815

—

Dic₃⋊C₈

—

#810

—

#808

—

C₄⋊(C₃⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₃⋊C₈)

C₂²

yes

—

#494

D₃

yes

—

—

—

—

C₂×C₃×Q₈

—

C₂×Dic₆

D₄

yes

characteristic

C₃×Q₁₆

—

C₃×Q₁₆

—

C₂×C₃×C₈

D₄

yes

characteristic

C₂×C₃×C₈

D₄

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂³

yes

characteristic

C₄⋊Dic₃

D₄

yes

characteristic

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

C₂×D₃

yes

—

C₄×Q₈

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#97

—

C₄×C₈

C₂×D₃

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

#41

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

C₃×C₈

—

C₃×C₈

SD₁₆

yes

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂²×D₃

yes

characteristic

C₂²×C₃

#121

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂×SD₁₆

yes

C₃×C₄

yes

C₃×C₄

yes

—

Dic₃

—

Dic₃

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₃×D₄

yes

characteristic

C₂×C₄

#237

yes

characteristic

C₂×C₄

C₂×#67

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

#214

yes

—

C₂×C₃

yes

C₂×C₃

yes

C₂×C₃

yes

C₂²

yes

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

#918

yes

—

105

C₄

#932

yes

characteristic

106

C₄

C₂×#214

yes

characteristic

107

C₃

#1690

yes

characteristic, complemented, Sylow

108

C₂

#5075

yes

characteristic

109

C₂

#5189

yes

characteristic

110

C₂

#5192

yes

characteristic

	subgroup	G/H
center	C₂²	#938
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#494	C₂
socle, prime radical	C₂²×C₃	#121
hypercenter	#494	D₃
hypocenter	C₃	#1690
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#938

commutator, cosocle

C₃×C₄²

C₂³

Frattini

C₄²

C₂²×D₃

C₂

C₂²×C₃

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
107. C₃	#1690	semidirect

H ≅ G/N

type

107. C₃

#1690

semidirect

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

subgroup	index
C₃×#494	2
#5094	2
#4679	2
#4547	2
#4457	2
#1690	3

subgroup

index

Subgroups — #71309

Conjugacy classes of subgroups

Complements

Maximal subgroups