Group #71834 — subgroups

Subgroups — #71834

Order 384.

210 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 85 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×#533	C₂	1	1	yes	maximal, characteristic
2	#4924	C₂	1	2	yes	maximal
3	#4547	C₂	1	4	yes	maximal, complemented
4	#1561	—	3	1	no	maximal, Sylow
5	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
6	C₃×#125	C₄	1	2	yes	—
7	C₃×C₂²⋊Q₈	—	2	2	no	—
8	C₃×C₄×Q₈	—	2	2	no	—
9	C₃×#117	C₄	1	2	yes	—
10	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
11	C₂×#208	—	2	2	no	—
12	C₄⋊(C₃⋊C₈)	—	2	2	no	—
13	C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	—
14	#533	D₃	1	1	yes	characteristic
15	#423	—	3	2	no	—
16	#332	—	3	4	no	—
17	C₂×C₃×Q₈	—	2	2	no	—
18	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃×C₄	D₄	1	2	yes	—
20	C₃×C₄⋊C₄	—	2	4	no	—
21	C₃×C₄⋊C₄	—	2	2	no	—
22	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
23	C₃×C₂²⋊C₄	—	2	2	no	—
24	C₃×C₄²	—	2	2	no	—
25	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
26	C₃×C₄²	D₄	1	2	yes	—
27	C₃×C₄²	C₂³	1	1	yes	characteristic
28	#208	—	4	4	no	—
29	C₂×C₃⋊C₈	—	2	4	no	—
30	C₂×M₁₆	—	6	2	no	—
31	C₄⋊Q₈	C₂×D₃	1	2	yes	—
32	#125	Dic₃	1	2	yes	—
33	C₂²⋊Q₈	—	2	2	no	—
34	C₄×Q₈	—	2	2	no	—
35	#117	Dic₃	1	2	yes	—
36	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
37	C₄⋊C₈	—	3	4	no	—
38	C₄⋊C₈	—	6	2	no	—
39	C₂³×C₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₃×Q₈	—	4	2	no	—
41	C₃×Q₈	—	4	2	no	—
42	C₂×C₃×C₄	—	2	2	no	—
43	C₂×C₃×C₄	—	2	2	no	—
44	C₂×C₃×C₄	—	2	2	no	—
45	C₂×C₃×C₄	—	2	4	no	—
46	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
47	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
50	C₃⋊C₈	—	4	4	no	—
51	C₂×Q₈	—	2	2	no	—
52	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
53	C₂²×C₄	#67	1	2	yes	—
54	M₁₆	—	12	4	no	—
55	C₂×C₈	—	6	4	no	—
56	C₄⋊C₄	—	2	4	no	—
57	C₄⋊C₄	—	2	2	no	—
58	C₄⋊C₄	C₂×Dic₃	1	4	yes	—
59	C₂²⋊C₄	—	2	2	no	—
60	C₄²	—	2	2	no	—
61	C₄²	C₂×Dic₃	1	1	yes	characteristic
62	C₄²	#67	1	2	yes	—
63	C₄²	C₂²×D₃	1	1	yes	characteristic
64	C₂²×C₃	—	2	1	no	—
65	C₂²×C₃	—	2	2	no	—
66	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₃×C₄	—	2	1	no	—
68	C₃×C₄	—	2	2	no	—
69	C₃×C₄	—	4	4	no	—
70	C₃×C₄	#101	1	2	yes	—
71	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
72	C₃×C₄	#137	1	2	yes	—
73	C₂³	#217	1	1	yes	characteristic
74	Q₈	—	4	2	no	—
75	Q₈	—	4	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	4	no	—
80	C₂×C₄	#217	1	2	yes	—
81	C₂×C₄	#217	1	1	yes	characteristic
82	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
83	C₂×C₄	C₂×#67	1	2	yes	—
84	C₈	—	12	4	no	—
85	C₂×C₃	—	4	1	no	—
86	C₂×C₃	C₂×#101	1	1	yes	characteristic
87	C₂×C₃	#417	1	2	yes	—
88	C₂²	—	2	1	no	—
89	C₂²	—	2	2	no	—
90	C₂²	C₂×#217	1	1	yes	characteristic
91	C₄	—	2	1	no	—
92	C₄	—	2	2	no	—
93	C₄	—	4	4	no	—
94	C₄	#836	1	2	yes	—
95	C₄	#949	1	2	yes	—
96	C₄	#951	1	2	yes	—
97	C₃	#1561	1	1	yes	characteristic, complemented, Sylow
98	C₂	—	4	1	no	—
99	C₂	C₂×#836	1	1	yes	characteristic
100	C₂	#5232	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#533

C₂

yes

maximal, characteristic

#4924

C₂

yes

maximal

#4547

C₂

yes

maximal, complemented

#1561

—

maximal, Sylow

C₃×C₄⋊Q₈

C₂²

yes

—

C₃×#125

C₄

yes

—

C₃×C₂²⋊Q₈

—

C₃×C₄×Q₈

—

C₃×#117

C₄

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×#208

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

C₂²

yes

—

#533

D₃

yes

characteristic

#423

—

#332

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

D₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₂×C₄

yes

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×C₄

yes

characteristic

C₃×C₄²

D₄

yes

—

C₃×C₄²

C₂³

yes

characteristic

#208

—

C₂×C₃⋊C₈

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₃

yes

—

#125

Dic₃

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

Dic₃

yes

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂³×C₃

C₂²⋊C₄

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

—

C₃⋊C₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₃

yes

characteristic

C₂²×C₄

#67

yes

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×Dic₃

yes

characteristic

C₄²

#67

yes

—

C₄²

C₂²×D₃

yes

characteristic

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂×C₂²⋊C₄

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#101

yes

—

C₃×C₄

AGL(1,ℤ/8)

yes

—

C₃×C₄

#137

yes

—

C₂³

#217

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#217

yes

—

C₂×C₄

#217

yes

characteristic

C₂×C₄

C₂²×Dic₃

yes

characteristic

C₂×C₄

C₂×#67

yes

—

C₈

—

C₂×C₃

—

C₂×C₃

C₂×#101

yes

characteristic

C₂×C₃

#417

yes

—

C₂²

—

C₂²

—

C₂²

C₂×#217

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#836

yes

—

C₄

#949

yes

—

C₄

#951

yes

—

C₃

#1561

yes

characteristic, complemented, Sylow

C₂

—

C₂

C₂×#836

yes

characteristic

100

C₂

#5232

yes

—

	subgroup	G/H
center	C₂²	C₂×#217
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#533	C₂
socle	C₂²×C₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₃	C₂²⋊C₄
cosocle	C₃×C₄²	C₂³
hypercenter	#533	D₃
hypocenter	C₃	#1561
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂×#217

commutator

C₂×C₃×C₄

C₂²×C₄

Frattini

C₄²

C₂²×D₃

Fitting

C₃×#533

C₂

socle

C₂²×C₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₃

C₂²⋊C₄

cosocle

C₃×C₄²

C₂³

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
3. #4547	C₂	semidirect
97. C₃	#1561	semidirect

H ≅ G/N

type

3. #4547

C₂

semidirect

97. C₃

#1561

semidirect

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

subgroup	index
C₃×#533	2
#4924	2
#4547	2
#1561	3

subgroup

index

Subgroups — #71834

Conjugacy classes of subgroups

Complements

Maximal subgroups