Group #16847762 — subgroups

Subgroups — #16847762

Order 3968.

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

normal subgroups only

This group has 154 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	C₂×Dic₂₄₈	C₂	1	2	yes	maximal
2	#14521429	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₁₂₄	C₂	1	1	yes	maximal, characteristic
4	#14520854	C₂	1	2	yes	maximal
5	C₃₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₂₄	—	2	1	no	—
7	C₂×Dic₁₂₄	C₂²	1	2	yes	—
8	C₂×Dic₁₂₄	C₂²	1	1	yes	characteristic
9	#11758446	—	2	1	no	—
10	C₄×Dic₆₂	—	2	1	no	—
11	C₂×C₁₆×C₃₁	C₂²	1	2	yes	—
12	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	#11758406	C₂²	1	1	yes	characteristic
14	#11758404	—	2	1	no	—
15	Dic₂₄₈	—	2	4	no	—
16	C₂×Dic₆₂	—	2	1	no	—
17	C₂×Dic₆₂	—	2	2	no	—
18	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
19	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
20	C₄²×C₃₁	D₄	1	1	yes	characteristic
21	C₄⋊Dic₃₁	—	2	1	no	—
22	C₄⋊Dic₃₁	—	4	1	no	—
23	C₄⋊Dic₃₁	—	4	1	no	—
24	Dic₃₁⋊C₄	—	4	1	no	—
25	C₄×Dic₃₁	—	4	1	no	—
26	Dic₁₂₄	—	2	2	no	—
27	Dic₁₂₄	—	2	1	no	—
28	Dic₁₂₄	—	2	2	no	—
29	Dic₁₂₄	—	4	2	no	—
30	Dic₁₂₄	D₄	1	2	yes	—
31	C₁₆×C₃₁	—	2	2	no	—
32	C₂×C₄×C₃₁	D₈	1	1	yes	characteristic
33	C₂×C₄×C₃₁	D₈	1	1	yes	characteristic
34	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₃₁	—	4	1	no	—
36	C₂×Dic₃₁	—	4	1	no	—
37	C₂×Dic₃₁	—	4	2	no	—
38	Dic₆₂	—	2	2	no	—
39	Dic₆₂	—	4	1	no	—
40	Dic₆₂	—	4	2	no	—
41	Dic₆₂	—	4	2	no	—
42	C₈×C₃₁	—	2	1	no	—
43	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
44	C₈×C₃₁	#41	1	1	yes	characteristic
45	#2209	—	31	1	no	maximal, Sylow
46	C₂²×C₃₁	C₂×D₈	1	1	yes	characteristic
47	C₄×C₃₁	—	2	1	no	—
48	C₄×C₃₁	Q₃₂	1	2	yes	complemented
49	C₄×C₃₁	#121	1	1	yes	characteristic
50	C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
51	Dic₃₁	—	4	2	no	—
52	Dic₃₁	—	8	2	no	—
53	Dic₃₁	—	8	1	no	—
54	C₂×Q₃₂	—	31	2	no	—
55	#494	—	31	1	no	—
56	C₄×Q₁₆	—	31	1	no	—
57	C₄⋊C₁₆	D₃₁	1	1	yes	characteristic
58	#358	—	31	2	no	—
59	C₂×C₃₁	#459	1	1	yes	characteristic
60	C₂×C₃₁	C₂×Q₃₂	1	1	yes	characteristic
61	C₂×C₃₁	#510	1	1	yes	characteristic
62	C₂×Q₁₆	—	31	1	no	—
63	C₂×Q₁₆	—	31	2	no	—
64	C₂×Q₁₆	—	62	1	no	—
65	C₄⋊Q₈	—	62	1	no	—
66	C₄×Q₈	—	62	1	no	—
67	Q₃₂	—	62	4	no	—
68	C₂×C₁₆	C₂×D₃₁	1	2	yes	—
69	#107	—	31	1	no	—
70	#103	—	62	1	no	—
71	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
72	C₃₁	#2209	1	1	yes	characteristic, complemented, Sylow
73	C₂×Q₈	—	62	1	no	—
74	C₂×Q₈	—	62	2	no	—
75	Q₁₆	—	31	2	no	—
76	Q₁₆	—	62	2	no	—
77	Q₁₆	—	62	1	no	—
78	Q₁₆	—	62	2	no	—
79	Q₁₆	—	124	2	no	—
80	C₂×C₈	D₁₂₄	1	1	yes	characteristic
81	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
82	C₄⋊C₄	—	62	1	no	—
83	C₄⋊C₄	—	124	1	no	—
84	C₄⋊C₄	—	124	1	no	—
85	C₄⋊C₄	—	124	1	no	—
86	C₄²	—	124	1	no	—
87	C₄²	D₁₂₄	1	1	yes	characteristic
88	C₁₆	—	2	2	no	—
89	Q₈	—	62	2	no	—
90	Q₈	—	124	1	no	—
91	Q₈	—	124	2	no	—
92	Q₈	—	124	2	no	—
93	C₂×C₄	—	124	1	no	—
94	C₂×C₄	—	124	1	no	—
95	C₂×C₄	—	124	2	no	—
96	C₂×C₄	D₂₄₈	1	1	yes	characteristic
97	C₂×C₄	D₂₄₈	1	1	yes	characteristic
98	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	D₄×D₃₁	1	1	yes	characteristic
101	C₈	#92499	1	1	yes	characteristic
102	C₂²	C₂×D₂₄₈	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	124	2	no	—
105	C₄	—	248	1	no	—
106	C₄	—	248	2	no	—
107	C₄	Dic₂₄₈	1	2	yes	complemented
108	C₄	#11758472	1	1	yes	characteristic
109	C₄	#11758485	1	1	yes	characteristic
110	C₂	#14521245	1	1	yes	characteristic
111	C₂	C₂×Dic₂₄₈	1	1	yes	characteristic
112	C₂	#14521492	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₂₄₈

C₂

yes

maximal

#14521429

C₂

yes

maximal, characteristic

C₄×Dic₁₂₄

C₂

yes

maximal, characteristic

#14520854

C₂

yes

maximal

C₃₁×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂×Dic₁₂₄

—

C₂×Dic₁₂₄

C₂²

yes

—

C₂×Dic₁₂₄

C₂²

yes

characteristic

#11758446

—

C₄×Dic₆₂

—

C₂×C₁₆×C₃₁

C₂²

yes

—

C₄×C₈×C₃₁

C₂²

yes

characteristic

#11758406

C₂²

yes

characteristic

#11758404

—

Dic₂₄₈

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×C₈×C₃₁

D₄

yes

characteristic

C₂×C₈×C₃₁

C₂³

yes

characteristic

C₄²×C₃₁

D₄

yes

characteristic

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

Dic₁₂₄

—

Dic₁₂₄

—

Dic₁₂₄

—

Dic₁₂₄

—

Dic₁₂₄

D₄

yes

—

C₁₆×C₃₁

—

C₂×C₄×C₃₁

D₈

yes

characteristic

C₂×C₄×C₃₁

D₈

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

—

C₈×C₃₁

—

C₈×C₃₁

C₂×D₄

yes

C₈×C₃₁

yes

—

C₂²×C₃₁

C₂×D₈

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

Q₃₂

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

C₂×Q₃₂

—

#494

—

C₄×Q₁₆

—

C₄⋊C₁₆

D₃₁

yes

characteristic

#358

—

C₂×C₃₁

#459

yes

characteristic

C₂×C₃₁

C₂×Q₃₂

yes

characteristic

C₂×C₃₁

#510

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

C₂×D₃₁

yes

—

#107

—

#103

—

C₄×C₈

C₂×D₃₁

yes

characteristic

C₃₁

#2209

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

124

—

C₂×C₈

D₁₂₄

yes

characteristic

C₂×C₈

C₂²×D₃₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

124

—

C₄⋊C₄

—

124

—

C₄⋊C₄

—

124

—

C₄²

—

124

—

C₄²

D₁₂₄

yes

characteristic

C₁₆

—

Q₈

—

Q₈

—

124

—

Q₈

—

124

—

Q₈

—

124

—

C₂×C₄

—

124

—

C₂×C₄

—

124

—

C₂×C₄

—

124

—

C₂×C₄

D₂₄₈

yes

characteristic

C₂×C₄

D₂₄₈

yes

characteristic

C₂×C₄

C₂×D₁₂₄

yes

characteristic

C₈

—

100

C₈

D₄×D₃₁

yes

characteristic

101

C₈

#92499

yes

characteristic

102

C₂²

C₂×D₂₄₈

yes

characteristic

103

C₄

—

104

C₄

—

124

—

105

C₄

—

248

—

106

C₄

—

248

—

107

C₄

Dic₂₄₈

yes

complemented

108

C₄

#11758472

yes

characteristic

109

C₄

#11758485

yes

characteristic

110

C₂

#14521245

yes

characteristic

111

C₂

C₂×Dic₂₄₈

yes

characteristic

112

C₂

#14521492

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₂₄₈
commutator, cosocle	C₂×C₈×C₃₁	C₂³
Frattini	C₂×C₈	C₂²×D₃₁
Fitting	C₃₁×C₄⋊C₁₆	C₂
socle, prime radical	C₂²×C₃₁	C₂×D₈
hypercenter	C₄⋊C₁₆	D₃₁
hypocenter	C₃₁	#2209
LCS	G ▷ C₂×C₈×C₃₁ ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₂₄₈

commutator, cosocle

C₂×C₈×C₃₁

C₂³

Frattini

C₂×C₈

C₂²×D₃₁

Fitting

C₃₁×C₄⋊C₁₆

C₂

socle, prime radical

C₂²×C₃₁

C₂×D₈

hypercenter

C₄⋊C₁₆

D₃₁

hypocenter

C₃₁

#2209

LCS

G ▷ C₂×C₈×C₃₁ ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
48. C₄×C₃₁	Q₃₂	semidirect
72. C₃₁	#2209	semidirect
107. C₄	Dic₂₄₈	semidirect

H ≅ G/N

type

48. C₄×C₃₁

Q₃₂

semidirect

72. C₃₁

#2209

semidirect

107. C₄

Dic₂₄₈

semidirect

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

subgroup	index
C₂×Dic₂₄₈	2
#14521429	2
C₄×Dic₁₂₄	2
#14520854	2
C₃₁×C₄⋊C₁₆	2
#2209	31

subgroup

index

C₂×Dic₂₄₈

#14521429

C₄×Dic₁₂₄

#14520854

C₃₁×C₄⋊C₁₆

#2209

Subgroups — #16847762

Conjugacy classes of subgroups

Complements

Maximal subgroups