Group #16821213 — subgroups

Subgroups — #16821213

Order 3712.

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

normal subgroups only

This group has 220 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	C₂²×Dic₁₁₆	C₂	1	1	yes	maximal, characteristic
2	#14271201	C₂	1	1	yes	maximal, characteristic
3	#14270968	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₉×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₈	—	2	1	no	—
6	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
7	#11745288	—	2	1	no	—
8	C₂×Dic₁₁₆	—	2	1	no	—
9	C₂×Dic₁₁₆	—	2	2	no	—
10	C₂×Dic₁₁₆	C₄	1	4	yes	complemented
11	C₂×Dic₁₁₆	C₂²	1	2	yes	—
12	C₂₉×M₃₂	—	2	2	no	—
13	C₂×C₁₆×C₂₉	C₂²	1	2	yes	—
14	#11745166	C₂²	1	2	yes	—
15	#11745165	—	2	1	no	—
16	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
17	C₂²×Dic₂₉	—	4	1	no	—
18	C₂×Dic₅₈	—	2	4	no	—
19	C₂×Dic₅₈	—	2	2	no	—
20	C₂×Dic₅₈	—	4	1	no	—
21	C₂×Dic₅₈	—	4	2	no	—
22	C₂×C₈×C₂₉	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₂₉	D₄	1	1	yes	characteristic
24	C₂×C₈×C₂₉	D₄	1	2	yes	—
25	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
26	C₂²⋊Dic₂₉	—	4	1	no	—
27	C₄⋊Dic₂₉	—	2	2	no	—
28	C₄×Dic₂₉	—	4	1	no	—
29	Dic₁₁₆	—	2	4	no	—
30	Dic₁₁₆	—	2	2	no	—
31	Dic₁₁₆	C₂×C₄	1	4	yes	—
32	C₁₆×C₂₉	—	2	2	no	—
33	C₂³×C₂₉	D₈	1	1	yes	characteristic
34	C₂×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₂₉	D₈	1	1	yes	characteristic
36	C₂×C₄×C₂₉	SD₁₆	1	2	yes	—
37	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₂₉	—	4	4	no	—
39	C₂×Dic₂₉	—	4	2	no	—
40	C₂×Dic₂₉	—	4	2	no	—
41	Dic₅₈	—	2	4	no	—
42	Dic₅₈	—	4	4	no	—
43	Dic₅₈	—	4	2	no	—
44	C₈×C₂₉	C₂²⋊C₄	1	2	yes	—
45	C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
46	C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
47	#2141	—	29	1	no	maximal, Sylow
48	C₂²×C₂₉	—	2	2	no	—
49	C₂²×C₂₉	#102	1	2	yes	—
50	C₂²×C₂₉	C₂×D₈	1	1	yes	characteristic
51	C₄×C₂₉	#102	1	2	yes	—
52	C₄×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
54	Dic₂₉	—	4	4	no	—
55	Dic₂₉	—	8	2	no	—
56	C₂²×Q₁₆	—	29	1	no	—
57	C₂×M₃₂	D₂₉	1	1	yes	characteristic
58	#427	—	29	1	no	—
59	#358	—	29	4	no	—
60	C₂×C₂₉	—	2	2	no	—
61	C₂×C₂₉	C₂×#102	1	1	yes	characteristic
62	C₂×C₂₉	#510	1	2	yes	—
63	C₂²×Q₈	—	58	1	no	—
64	C₂×Q₁₆	—	29	4	no	—
65	C₂×Q₁₆	—	29	2	no	—
66	C₂×Q₁₆	—	58	1	no	—
67	C₂×Q₁₆	—	58	2	no	—
68	C₂²×C₈	C₂×D₂₉	1	1	yes	characteristic
69	#117	—	58	1	no	—
70	M₃₂	—	2	2	no	—
71	C₂×C₁₆	C₂×D₂₉	1	2	yes	—
72	#107	—	29	2	no	—
73	#106	—	58	1	no	—
74	C₂₉	#2141	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₈	—	58	4	no	—
76	C₂×Q₈	—	58	2	no	—
77	C₂×Q₈	—	116	2	no	—
78	C₂×Q₈	—	116	1	no	—
79	C₂²×C₄	—	116	1	no	—
80	C₂²×C₄	D₁₁₆	1	1	yes	characteristic
81	Q₁₆	—	29	4	no	—
82	Q₁₆	—	58	4	no	—
83	Q₁₆	—	58	2	no	—
84	C₂×C₈	C₄×D₂₉	1	2	yes	—
85	C₂×C₈	D₁₁₆	1	1	yes	characteristic
86	C₂×C₈	C₂²⋊D₂₉	1	2	yes	—
87	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
88	C₄⋊C₄	—	58	2	no	—
89	C₂²⋊C₄	—	116	1	no	—
90	C₄²	—	116	1	no	—
91	C₁₆	—	2	2	no	—
92	C₂³	D₂₃₂	1	1	yes	characteristic
93	Q₈	—	58	4	no	—
94	Q₈	—	116	4	no	—
95	Q₈	—	116	2	no	—
96	C₂×C₄	—	116	4	no	—
97	C₂×C₄	—	116	2	no	—
98	C₂×C₄	—	116	2	no	—
99	C₂×C₄	#90773	1	2	yes	—
100	C₂×C₄	D₂₃₂	1	1	yes	characteristic
101	C₂×C₄	#90781	1	2	yes	—
102	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
103	C₈	#90781	1	2	yes	—
104	C₈	C₂×C₄×D₂₉	1	1	yes	characteristic
105	C₈	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
106	C₂²	—	2	2	no	—
107	C₂²	#11745169	1	2	yes	—
108	C₂²	C₂×D₂₃₂	1	1	yes	characteristic
109	C₄	—	116	4	no	—
110	C₄	—	232	2	no	—
111	C₄	#11745169	1	2	yes	—
112	C₄	C₂×#90773	1	1	yes	characteristic
113	C₄	C₂×#90781	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	C₂×#11745169	1	1	yes	characteristic
116	C₂	#14271606	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₁₆

C₂

yes

maximal, characteristic

#14271201

C₂

yes

maximal, characteristic

#14270968

C₂

yes

maximal, complemented

C₂×C₂₉×M₃₂

C₂

yes

maximal, characteristic

C₂²×Dic₅₈

—

C₂²×C₈×C₂₉

C₂²

yes

characteristic

#11745288

—

C₂×Dic₁₁₆

—

C₂×Dic₁₁₆

—

C₂×Dic₁₁₆

C₄

yes

complemented

C₂×Dic₁₁₆

C₂²

yes

—

C₂₉×M₃₂

—

C₂×C₁₆×C₂₉

C₂²

yes

—

#11745166

C₂²

yes

—

#11745165

—

C₂²×C₄×C₂₉

D₄

yes

characteristic

C₂²×Dic₂₉

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×C₈×C₂₉

C₂×C₄

yes

—

C₂×C₈×C₂₉

D₄

yes

characteristic

C₂×C₈×C₂₉

D₄

yes

—

C₂×C₈×C₂₉

C₂³

yes

characteristic

C₂²⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄×Dic₂₉

—

Dic₁₁₆

—

Dic₁₁₆

—

Dic₁₁₆

C₂×C₄

yes

—

C₁₆×C₂₉

—

C₂³×C₂₉

D₈

yes

characteristic

C₂×C₄×C₂₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉

D₈

yes

characteristic

C₂×C₄×C₂₉

SD₁₆

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

Dic₅₈

—

Dic₅₈

—

Dic₅₈

—

C₈×C₂₉

C₂²⋊C₄

yes

—

C₈×C₂₉

C₂²×C₄

yes

characteristic

C₈×C₂₉

C₂×D₄

yes

characteristic

#2141

—

maximal, Sylow

C₂²×C₂₉

—

C₂²×C₂₉

#102

yes

—

C₂²×C₂₉

C₂×D₈

yes

characteristic

C₄×C₂₉

#102

yes

—

C₄×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₉

C₂×SD₁₆

yes

characteristic

Dic₂₉

—

Dic₂₉

—

C₂²×Q₁₆

—

C₂×M₃₂

D₂₉

yes

characteristic

#427

—

#358

—

C₂×C₂₉

—

C₂×C₂₉

C₂×#102

yes

characteristic

C₂×C₂₉

#510

yes

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

C₂×D₂₉

yes

characteristic

#117

—

M₃₂

—

C₂×C₁₆

C₂×D₂₉

yes

—

#107

—

#106

—

C₂₉

#2141

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

116

—

C₂×Q₈

—

116

—

C₂²×C₄

—

116

—

C₂²×C₄

D₁₁₆

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

C₄×D₂₉

yes

—

C₂×C₈

D₁₁₆

yes

characteristic

C₂×C₈

C₂²⋊D₂₉

yes

—

C₂×C₈

C₂²×D₂₉

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

116

—

C₄²

—

116

—

C₁₆

—

C₂³

D₂₃₂

yes

characteristic

Q₈

—

Q₈

—

116

—

Q₈

—

116

—

C₂×C₄

—

116

—

C₂×C₄

—

116

—

C₂×C₄

—

116

—

C₂×C₄

#90773

yes

—

100

C₂×C₄

D₂₃₂

yes

characteristic

101

C₂×C₄

#90781

yes

—

102

C₂×C₄

C₂×D₁₁₆

yes

characteristic

103

C₈

#90781

yes

—

104

C₈

C₂×C₄×D₂₉

yes

characteristic

105

C₈

C₂×C₂²⋊D₂₉

yes

characteristic

106

C₂²

—

107

C₂²

#11745169

yes

—

108

C₂²

C₂×D₂₃₂

yes

characteristic

109

C₄

—

116

—

110

C₄

—

232

—

111

C₄

#11745169

yes

—

112

C₄

C₂×#90773

yes

characteristic

113

C₄

C₂×#90781

yes

characteristic

114

C₂

—

115

C₂

C₂×#11745169

yes

characteristic

116

C₂

#14271606

yes

—

	subgroup	G/H
center	C₂²	C₂×D₂₃₂
commutator	C₈×C₂₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₂₉
Fitting	C₂×C₂₉×M₃₂	C₂
socle	C₂²×C₂₉	C₂×D₈
prime radical	C₂³×C₂₉	D₈
cosocle	C₂×C₈×C₂₉	C₂³
hypercenter	C₂×M₃₂	D₂₉
hypocenter	C₂₉	#2141
LCS	G ▷ C₈×C₂₉ ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂²

C₂×D₂₃₂

commutator

C₈×C₂₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₂₉

Fitting

C₂×C₂₉×M₃₂

C₂

socle

C₂²×C₂₉

C₂×D₈

prime radical

C₂³×C₂₉

D₈

cosocle

C₂×C₈×C₂₉

C₂³

hypercenter

C₂×M₃₂

D₂₉

hypocenter

C₂₉

#2141

LCS

G ▷ C₈×C₂₉ ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #14270968	C₂	semidirect
10. C₂×Dic₁₁₆	C₄	semidirect
74. C₂₉	#2141	semidirect

H ≅ G/N

type

3. #14270968

C₂

semidirect

10. C₂×Dic₁₁₆

C₄

semidirect

74. C₂₉

#2141

semidirect

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

subgroup	index
C₂²×Dic₁₁₆	2
#14271201	2
#14270968	2
C₂×C₂₉×M₃₂	2
#2141	29

subgroup

index

C₂²×Dic₁₁₆

#14271201

#14270968

C₂×C₂₉×M₃₂

#2141

Subgroups — #16821213

Conjugacy classes of subgroups

Complements

Maximal subgroups