Group #71277 — subgroups

Subgroups — #71277

Order 384.

220 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 65 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×#493	C₂	1	1	yes	maximal, characteristic
2	#5076	C₂	1	2	yes	maximal, complemented
3	#5070	C₂	1	2	yes	maximal, complemented
4	#4680	C₂	1	1	yes	maximal, characteristic, complemented
5	#4455	C₂	1	1	yes	maximal, characteristic, complemented
6	#1715	—	3	1	no	maximal, Sylow
7	C₂×C₃×D₈	—	2	2	no	—
8	C₃×#127	C₂²	1	1	yes	characteristic
9	C₃×#127	C₂²	1	1	yes	characteristic
10	C₂×#214	—	2	4	no	—
11	#869	C₂²	1	2	yes	—
12	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	#832	—	2	4	no	—
14	#817	—	2	2	no	—
15	C₄×C₃⋊C₈	C₂²	1	2	yes	—
16	#498	—	3	1	no	—
17	#493	D₃	1	1	yes	characteristic
18	#492	—	3	2	no	—
19	#486	—	3	2	no	—
20	#322	—	3	1	no	—
21	C₂×C₃×D₄	—	2	2	no	—
22	C₂×C₃×D₄	—	2	2	no	—
23	C₂×C₃×D₄	—	2	1	no	—
24	C₂×C₃×D₄	—	2	1	no	—
25	C₂×Dic₆	—	2	2	no	—
26	C₃×D₈	—	4	4	no	—
27	C₂×C₃×C₈	D₄	1	2	yes	—
28	C₃×C₄²	C₂³	1	1	yes	characteristic
29	#214	—	4	8	no	—
30	C₄⋊Dic₃	—	2	2	no	—
31	C₄⋊Dic₃	—	2	2	no	—
32	C₂×C₃⋊C₈	D₄	1	4	yes	—
33	C₂×SD₁₆	—	6	4	no	—
34	C₂×D₈	—	2	2	no	—
35	C₄⋊Q₈	—	3	2	no	—
36	#127	C₂×D₃	1	1	yes	characteristic
37	#127	C₂×D₃	1	1	yes	characteristic
38	#106	—	6	2	no	—
39	#102	—	6	4	no	—
40	C₄×C₈	—	3	2	no	—
41	C₄×C₈	C₂×D₃	1	1	yes	characteristic
42	C₂³×C₃	—	4	1	no	—
43	C₂³×C₃	—	4	1	no	—
44	C₃×D₄	—	4	2	no	—
45	C₃×D₄	—	4	2	no	—
46	C₃×D₄	—	4	4	no	—
47	C₃×D₄	—	4	4	no	—
48	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
49	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₃	—	4	2	no	—
51	Dic₆	—	4	4	no	—
52	C₃×C₈	SD₁₆	1	4	yes	complemented
53	C₃⋊C₈	—	2	4	no	—
54	C₂×Q₈	—	6	2	no	—
55	C₂×D₄	—	2	2	no	—
56	C₂×D₄	—	2	2	no	—
57	C₂×D₄	—	2	1	no	—
58	C₂×D₄	—	2	1	no	—
59	SD₁₆	—	12	8	no	—
60	D₈	—	4	4	no	—
61	C₂×C₈	—	3	4	no	—
62	C₂×C₈	#67	1	2	yes	—
63	C₄⋊C₄	—	6	2	no	—
64	C₄⋊C₄	—	6	2	no	—
65	C₄²	C₂²×D₃	1	1	yes	characteristic
66	C₂²×C₃	—	8	1	no	—
67	C₂²×C₃	—	8	1	no	—
68	C₂²×C₃	—	8	2	no	—
69	C₂²×C₃	—	8	2	no	—
70	C₂²×C₃	#127	1	1	yes	characteristic
71	C₃×C₄	C₂×SD₁₆	1	2	yes	—
72	C₃×C₄	AGL(1,ℤ/8)	1	4	yes	—
73	Dic₃	—	8	2	no	—
74	C₂³	—	4	1	no	—
75	C₂³	—	4	1	no	—
76	Q₈	—	12	4	no	—
77	D₄	—	4	4	no	—
78	D₄	—	4	4	no	—
79	D₄	—	4	2	no	—
80	D₄	—	4	2	no	—
81	C₂×C₄	—	12	2	no	—
82	C₂×C₄	D₃×D₄	1	2	yes	—
83	C₂×C₄	C₂×#67	1	1	yes	characteristic
84	C₈	—	6	4	no	—
85	C₈	#214	1	4	yes	complemented
86	C₂×C₃	—	16	1	no	—
87	C₂×C₃	—	16	1	no	—
88	C₂×C₃	#492	1	1	yes	characteristic
89	C₂×C₃	#496	1	2	yes	—
90	C₂²	—	8	2	no	—
91	C₂²	—	8	2	no	—
92	C₂²	—	8	1	no	—
93	C₂²	—	8	1	no	—
94	C₂²	#940	1	1	yes	characteristic
95	C₄	—	24	2	no	—
96	C₄	#911	1	4	yes	—
97	C₄	C₂×#214	1	2	yes	—
98	C₃	#1715	1	1	yes	characteristic, complemented, Sylow
99	C₂	—	16	1	no	—
100	C₂	—	16	1	no	—
101	C₂	#5076	1	1	yes	characteristic
102	C₂	#5154	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#493

C₂

yes

maximal, characteristic

#5076

C₂

yes

maximal, complemented

#5070

C₂

yes

maximal, complemented

#4680

C₂

yes

maximal, characteristic, complemented

#4455

C₂

yes

maximal, characteristic, complemented

#1715

—

maximal, Sylow

C₂×C₃×D₈

—

C₃×#127

C₂²

yes

characteristic

C₃×#127

C₂²

yes

characteristic

C₂×#214

—

#869

C₂²

yes

—

C₃×C₄×C₈

C₂²

yes

characteristic

#832

—

#817

—

C₄×C₃⋊C₈

C₂²

yes

—

#498

—

#493

D₃

yes

—

—

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×Dic₆

—

C₃×D₈

—

C₂×C₃×C₈

D₄

yes

—

C₃×C₄²

C₂³

yes

characteristic

#214

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

D₄

yes

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

#127

C₂×D₃

yes

characteristic

#127

C₂×D₃

yes

characteristic

#106

—

#102

—

C₄×C₈

—

C₄×C₈

C₂×D₃

yes

characteristic

C₂³×C₃

—

C₂³×C₃

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×Dic₃

—

Dic₆

—

C₃×C₈

SD₁₆

yes

complemented

C₃⋊C₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

D₈

—

C₂×C₈

—

C₂×C₈

#67

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₃

yes

characteristic

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

#127

yes

characteristic

C₃×C₄

C₂×SD₁₆

yes

—

C₃×C₄

AGL(1,ℤ/8)

yes

—

Dic₃

—

C₂³

—

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

D₃×D₄

yes

—

C₂×C₄

C₂×#67

yes

characteristic

C₈

—

C₈

#214

yes

complemented

C₂×C₃

—

C₂×C₃

—

C₂×C₃

#492

yes

characteristic

C₂×C₃

#496

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#940

yes

characteristic

C₄

—

C₄

#911

yes

—

C₄

C₂×#214

yes

—

C₃

#1715

yes

characteristic, complemented, Sylow

C₂

—

100

C₂

—

101

C₂

#5076

yes

characteristic

102

C₂

#5154

yes

—

	subgroup	G/H
center	C₂²	#940
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting, prime radical	C₃×#493	C₂
socle	C₂²×C₃	#127
hypercenter	#493	D₃
hypocenter	C₃	#1715
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#940

commutator, cosocle

C₃×C₄²

C₂³

Frattini

C₄²

C₂²×D₃

Fitting, prime radical

C₂

C₂²×C₃

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
2. #5076	C₂	semidirect
3. #5070	C₂	semidirect
4. #4680	C₂	semidirect
5. #4455	C₂	semidirect
52. C₃×C₈	SD₁₆	semidirect
85. C₈	#214	semidirect
98. C₃	#1715	semidirect

H ≅ G/N

type

2. #5076

C₂

semidirect

3. #5070

C₂

semidirect

4. #4680

C₂

semidirect

5. #4455

C₂

semidirect

52. C₃×C₈

SD₁₆

semidirect

85. C₈

#214

semidirect

98. C₃

#1715

semidirect

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

subgroup	index
C₃×#493	2
#5076	2
#5070	2
#4680	2
#4455	2
#1715	3

subgroup

index

Subgroups — #71277

Conjugacy classes of subgroups

Complements

Maximal subgroups