Group #16821543 — subgroups

Subgroups — #16821543

Order 3712.

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×Dic₂₃₂	C₂	1	2	yes	maximal
2	#14271543	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₁₁₆	C₂	1	1	yes	maximal, characteristic
4	#14270968	C₂	1	2	yes	maximal
5	C₂₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₁₆	—	2	1	no	—
7	C₂×Dic₁₁₆	C₂²	1	2	yes	—
8	C₂×Dic₁₁₆	C₂²	1	1	yes	characteristic
9	#11745231	—	2	1	no	—
10	C₄×Dic₅₈	—	2	1	no	—
11	C₂×C₁₆×C₂₉	C₂²	1	2	yes	—
12	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
13	#11745166	C₂²	1	1	yes	characteristic
14	#11745164	—	2	1	no	—
15	Dic₂₃₂	—	2	4	no	—
16	C₂×Dic₅₈	—	2	1	no	—
17	C₂×Dic₅₈	—	2	2	no	—
18	C₂×C₈×C₂₉	D₄	1	1	yes	characteristic
19	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
20	C₄²×C₂₉	D₄	1	1	yes	characteristic
21	C₄⋊Dic₂₉	—	2	1	no	—
22	C₄⋊Dic₂₉	—	4	1	no	—
23	C₄⋊Dic₂₉	—	4	1	no	—
24	#90779	—	4	1	no	—
25	C₄×Dic₂₉	—	4	1	no	—
26	Dic₁₁₆	—	2	2	no	—
27	Dic₁₁₆	—	2	1	no	—
28	Dic₁₁₆	—	2	2	no	—
29	Dic₁₁₆	—	4	2	no	—
30	Dic₁₁₆	D₄	1	2	yes	—
31	C₁₆×C₂₉	—	2	2	no	—
32	C₂×C₄×C₂₉	D₈	1	1	yes	characteristic
33	C₂×C₄×C₂₉	D₈	1	1	yes	characteristic
34	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₂₉	—	4	1	no	—
36	C₂×Dic₂₉	—	4	1	no	—
37	C₂×Dic₂₉	—	4	2	no	—
38	Dic₅₈	—	2	2	no	—
39	Dic₅₈	—	4	1	no	—
40	Dic₅₈	—	4	2	no	—
41	Dic₅₈	—	4	2	no	—
42	C₈×C₂₉	—	2	1	no	—
43	C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
44	C₈×C₂₉	#41	1	1	yes	characteristic
45	#2209	—	29	1	no	maximal, Sylow
46	C₂²×C₂₉	C₂×D₈	1	1	yes	characteristic
47	C₄×C₂₉	—	2	1	no	—
48	C₄×C₂₉	Q₃₂	1	2	yes	complemented
49	C₄×C₂₉	#121	1	1	yes	characteristic
50	C₄×C₂₉	AGL(1,ℤ/8)	1	1	yes	characteristic
51	Dic₂₉	—	4	2	no	—
52	Dic₂₉	—	8	2	no	—
53	Dic₂₉	—	8	1	no	—
54	C₂×Q₃₂	—	29	2	no	—
55	#494	—	29	1	no	—
56	C₄×Q₁₆	—	29	1	no	—
57	C₄⋊C₁₆	D₂₉	1	1	yes	characteristic
58	#358	—	29	2	no	—
59	C₂×C₂₉	#459	1	1	yes	characteristic
60	C₂×C₂₉	C₂×Q₃₂	1	1	yes	characteristic
61	C₂×C₂₉	#510	1	1	yes	characteristic
62	C₂×Q₁₆	—	29	1	no	—
63	C₂×Q₁₆	—	29	2	no	—
64	C₂×Q₁₆	—	58	1	no	—
65	C₄⋊Q₈	—	58	1	no	—
66	C₄×Q₈	—	58	1	no	—
67	Q₃₂	—	58	4	no	—
68	C₂×C₁₆	C₂×D₂₉	1	2	yes	—
69	#107	—	29	1	no	—
70	#103	—	58	1	no	—
71	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
72	C₂₉	#2209	1	1	yes	characteristic, complemented, Sylow
73	C₂×Q₈	—	58	1	no	—
74	C₂×Q₈	—	58	2	no	—
75	Q₁₆	—	29	2	no	—
76	Q₁₆	—	58	2	no	—
77	Q₁₆	—	58	1	no	—
78	Q₁₆	—	58	2	no	—
79	Q₁₆	—	116	2	no	—
80	C₂×C₈	D₁₁₆	1	1	yes	characteristic
81	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
82	C₄⋊C₄	—	58	1	no	—
83	C₄⋊C₄	—	116	1	no	—
84	C₄⋊C₄	—	116	1	no	—
85	C₄⋊C₄	—	116	1	no	—
86	C₄²	—	116	1	no	—
87	C₄²	D₁₁₆	1	1	yes	characteristic
88	C₁₆	—	2	2	no	—
89	Q₈	—	58	2	no	—
90	Q₈	—	116	1	no	—
91	Q₈	—	116	2	no	—
92	Q₈	—	116	2	no	—
93	C₂×C₄	—	116	1	no	—
94	C₂×C₄	—	116	1	no	—
95	C₂×C₄	—	116	2	no	—
96	C₂×C₄	D₂₃₂	1	1	yes	characteristic
97	C₂×C₄	D₂₃₂	1	1	yes	characteristic
98	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	D₄×D₂₉	1	1	yes	characteristic
101	C₈	#90809	1	1	yes	characteristic
102	C₂²	C₂×D₂₃₂	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	116	2	no	—
105	C₄	—	232	1	no	—
106	C₄	—	232	2	no	—
107	C₄	Dic₂₃₂	1	2	yes	complemented
108	C₄	#11745257	1	1	yes	characteristic
109	C₄	#11745270	1	1	yes	characteristic
110	C₂	#14271359	1	1	yes	characteristic
111	C₂	C₂×Dic₂₃₂	1	1	yes	characteristic
112	C₂	#14271606	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₂₃₂

C₂

yes

maximal

#14271543

C₂

yes

maximal, characteristic

C₄×Dic₁₁₆

C₂

yes

maximal, characteristic

#14270968

C₂

yes

maximal

C₂₉×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂×Dic₁₁₆

—

C₂×Dic₁₁₆

C₂²

yes

—

C₂×Dic₁₁₆

C₂²

yes

characteristic

#11745231

—

C₄×Dic₅₈

—

C₂×C₁₆×C₂₉

C₂²

yes

—

C₄×C₈×C₂₉

C₂²

yes

characteristic

#11745166

C₂²

yes

characteristic

#11745164

—

Dic₂₃₂

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×C₈×C₂₉

D₄

yes

characteristic

C₂×C₈×C₂₉

C₂³

yes

characteristic

C₄²×C₂₉

D₄

yes

characteristic

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

#90779

—

C₄×Dic₂₉

—

Dic₁₁₆

—

Dic₁₁₆

—

Dic₁₁₆

—

Dic₁₁₆

—

Dic₁₁₆

D₄

yes

—

C₁₆×C₂₉

—

C₂×C₄×C₂₉

D₈

yes

characteristic

C₂×C₄×C₂₉

D₈

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

Dic₅₈

—

Dic₅₈

—

Dic₅₈

—

Dic₅₈

—

C₈×C₂₉

—

C₈×C₂₉

C₂×D₄

yes

C₈×C₂₉

yes

—

C₂²×C₂₉

C₂×D₈

yes

characteristic

C₄×C₂₉

—

C₄×C₂₉

Q₃₂

yes

C₄×C₂₉

yes

C₄×C₂₉

yes

Dic₂₉

—

Dic₂₉

—

Dic₂₉

—

C₂×Q₃₂

—

#494

—

C₄×Q₁₆

—

C₄⋊C₁₆

D₂₉

yes

characteristic

#358

—

C₂×C₂₉

#459

yes

characteristic

C₂×C₂₉

C₂×Q₃₂

yes

characteristic

C₂×C₂₉

#510

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

C₂×D₂₉

yes

—

#107

—

#103

—

C₄×C₈

C₂×D₂₉

yes

characteristic

C₂₉

#2209

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

116

—

C₂×C₈

D₁₁₆

yes

characteristic

C₂×C₈

C₂²×D₂₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

116

—

C₄⋊C₄

—

116

—

C₄⋊C₄

—

116

—

C₄²

—

116

—

C₄²

D₁₁₆

yes

characteristic

C₁₆

—

Q₈

—

Q₈

—

116

—

Q₈

—

116

—

Q₈

—

116

—

C₂×C₄

—

116

—

C₂×C₄

—

116

—

C₂×C₄

—

116

—

C₂×C₄

D₂₃₂

yes

characteristic

C₂×C₄

D₂₃₂

yes

characteristic

C₂×C₄

C₂×D₁₁₆

yes

characteristic

C₈

—

100

C₈

D₄×D₂₉

yes

characteristic

101

C₈

#90809

yes

characteristic

102

C₂²

C₂×D₂₃₂

yes

characteristic

103

C₄

—

104

C₄

—

116

—

105

C₄

—

232

—

106

C₄

—

232

—

107

C₄

Dic₂₃₂

yes

complemented

108

C₄

#11745257

yes

characteristic

109

C₄

#11745270

yes

characteristic

110

C₂

#14271359

yes

characteristic

111

C₂

C₂×Dic₂₃₂

yes

characteristic

112

C₂

#14271606

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₂₃₂
commutator, cosocle	C₂×C₈×C₂₉	C₂³
Frattini	C₂×C₈	C₂²×D₂₉
Fitting	C₂₉×C₄⋊C₁₆	C₂
socle, prime radical	C₂²×C₂₉	C₂×D₈
hypercenter	C₄⋊C₁₆	D₂₉
hypocenter	C₂₉	#2209
LCS	G ▷ C₂×C₈×C₂₉ ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₂₃₂

commutator, cosocle

C₂×C₈×C₂₉

C₂³

Frattini

C₂×C₈

C₂²×D₂₉

Fitting

C₂₉×C₄⋊C₁₆

C₂

socle, prime radical

C₂²×C₂₉

C₂×D₈

hypercenter

C₄⋊C₁₆

D₂₉

hypocenter

C₂₉

#2209

LCS

G ▷ C₂×C₈×C₂₉ ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
48. C₄×C₂₉	Q₃₂	semidirect
72. C₂₉	#2209	semidirect
107. C₄	Dic₂₃₂	semidirect

H ≅ G/N

type

48. C₄×C₂₉

Q₃₂

semidirect

72. C₂₉

#2209

semidirect

107. C₄

Dic₂₃₂

semidirect

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

subgroup	index
C₂×Dic₂₃₂	2
#14271543	2
C₄×Dic₁₁₆	2
#14270968	2
C₂₉×C₄⋊C₁₆	2
#2209	29

subgroup

index

C₂×Dic₂₃₂

#14271543

C₄×Dic₁₁₆

#14270968

C₂₉×C₄⋊C₁₆

#2209

Subgroups — #16821543

Conjugacy classes of subgroups

Complements

Maximal subgroups