Group #16830006 — subgroups

Subgroups — #16830006

Order 3712.

356 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 173 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	#14271720	C₂	1	2	yes	maximal
2	#14271568	C₂	1	2	yes	maximal, complemented
3	#14271565	C₂	1	4	yes	maximal, complemented
4	#14271561	C₂	1	2	yes	maximal, complemented
5	#14271201	C₂	1	4	yes	maximal
6	C₂×C₄×C₈×C₂₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₈×C₂₉	C₂²	1	2	yes	—
8	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
9	#11745288	C₂²	1	4	yes	—
10	#11745286	—	2	4	no	—
11	#11745232	C₂²	1	4	yes	—
12	#11745231	C₂²	1	4	yes	—
13	C₄×Dic₅₈	—	2	4	no	—
14	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₂₉	C₂²	1	2	yes	—
16	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
17	#11745166	C₂²	1	8	yes	—
18	#11745165	C₂²	1	8	yes	—
19	C₂²×C₄×C₂₉	D₄	1	2	yes	—
20	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
21	C₂×Dic₅₈	—	2	4	no	—
22	C₂×C₈×C₂₉	Q₈	1	8	yes	complemented
23	C₂×C₈×C₂₉	C₂³	1	2	yes	—
24	C₂×C₈×C₂₉	C₂³	1	2	yes	—
25	C₄²×C₂₉	D₄	1	2	yes	—
26	C₄²×C₂₉	C₂³	1	1	yes	characteristic
27	C₄²×C₂₉	C₂³	1	1	yes	characteristic
28	C₂²⋊Dic₂₉	—	2	4	no	—
29	C₄⋊Dic₂₉	—	2	4	no	—
30	C₄⋊Dic₂₉	C₂³	1	8	yes	—
31	#90779	—	2	8	no	—
32	C₄×Dic₂₉	—	2	4	no	—
33	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₉	—	2	4	no	—
35	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₂₉	C₂×Q₈	1	4	yes	—
39	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
40	C₂×Dic₂₉	—	2	8	no	—
41	Dic₅₈	—	4	8	no	—
42	C₈×C₂₉	C₂×Q₈	1	4	yes	—
43	C₈×C₂₉	C₂×Q₈	1	4	yes	—
44	#3160	—	29	1	no	maximal, Sylow
45	C₂²×C₂₉	—	2	2	no	—
46	C₂²×C₂₉	C₄⋊Q₈	1	2	yes	—
47	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
48	C₄×C₂₉	—	2	2	no	—
49	C₄×C₂₉	C₄⋊Q₈	1	2	yes	—
50	C₄×C₂₉	#135	1	4	yes	—
51	C₄×C₂₉	C₂²×Q₈	1	2	yes	—
52	Dic₂₉	—	4	8	no	—
53	#533	—	29	2	no	—
54	#500	—	29	2	no	—
55	#499	—	29	4	no	—
56	#498	—	29	2	no	—
57	#427	—	29	4	no	—
58	C₂×C₄×C₈	D₂₉	1	1	yes	characteristic
59	C₂×C₂₉	—	2	2	no	—
60	C₂×C₂₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₂₉	C₂×#135	1	2	yes	—
62	C₂²×C₈	C₂×D₂₉	1	2	yes	—
63	C₄⋊Q₈	—	29	4	no	—
64	#125	—	29	4	no	—
65	C₂²⋊Q₈	—	58	4	no	—
66	C₄×Q₈	—	58	4	no	—
67	#117	—	29	4	no	—
68	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
69	#107	—	29	8	no	—
70	#106	—	29	8	no	—
71	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
72	C₄×C₈	C₂×D₂₉	1	2	yes	—
73	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
74	C₂₉	#3160	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₈	—	58	4	no	—
76	C₂²×C₄	D₁₁₆	1	2	yes	—
77	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
78	C₂×C₈	Dic₅₈	1	8	yes	complemented
79	C₂×C₈	C₂²×D₂₉	1	2	yes	—
80	C₂×C₈	C₂²×D₂₉	1	2	yes	—
81	C₄⋊C₄	—	29	8	no	—
82	C₄⋊C₄	—	58	8	no	—
83	C₄⋊C₄	—	58	4	no	—
84	C₂²⋊C₄	—	58	4	no	—
85	C₄²	—	58	4	no	—
86	C₄²	D₁₁₆	1	2	yes	—
87	C₄²	C₂²×D₂₉	1	1	yes	characteristic
88	C₄²	C₂²×D₂₉	1	1	yes	characteristic
89	C₂³	C₂×D₁₁₆	1	1	yes	characteristic
90	Q₈	—	116	8	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	58	8	no	—
93	C₂×C₄	C₂×Dic₅₈	1	4	yes	—
94	C₂×C₄	C₂×D₁₁₆	1	2	yes	—
95	C₂×C₄	C₂×D₁₁₆	1	2	yes	—
96	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
97	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
98	C₈	C₂×Dic₅₈	1	4	yes	—
99	C₈	C₂×Dic₅₈	1	4	yes	—
100	C₂²	—	2	2	no	—
101	C₂²	#11745231	1	2	yes	—
102	C₂²	C₂²×D₁₁₆	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	—	116	8	no	—
105	C₄	#11745231	1	2	yes	—
106	C₄	#11745266	1	4	yes	—
107	C₄	C₂²×Dic₅₈	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₂×#11745231	1	1	yes	characteristic
110	C₂	C₂×#11745266	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14271720

C₂

yes

maximal

#14271568

C₂

yes

maximal, complemented

#14271565

C₂

yes

maximal, complemented

#14271561

C₂

yes

maximal, complemented

#14271201

C₂

yes

maximal

C₂×C₄×C₈×C₂₉

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₉

C₂²

yes

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

#11745288

C₂²

yes

—

#11745286

—

#11745232

C₂²

yes

—

#11745231

C₂²

yes

—

C₄×Dic₅₈

—

C₄×C₈×C₂₉

C₂²

yes

characteristic

C₄×C₈×C₂₉

C₂²

yes

—

C₄×C₈×C₂₉

C₂²

yes

characteristic

#11745166

C₂²

yes

—

#11745165

C₂²

yes

—

C₂²×C₄×C₂₉

D₄

yes

—

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂×Dic₅₈

—

C₂×C₈×C₂₉

Q₈

yes

complemented

C₂×C₈×C₂₉

C₂³

yes

—

C₂×C₈×C₂₉

C₂³

yes

—

C₄²×C₂₉

D₄

yes

—

C₄²×C₂₉

C₂³

yes

characteristic

C₄²×C₂₉

C₂³

yes

characteristic

C₂²⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

C₂³

yes

—

#90779

—

C₄×Dic₂₉

—

C₂³×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×Q₈

yes

—

C₂×C₄×C₂₉

C₂⁴

yes

characteristic

C₂×Dic₂₉

—

Dic₅₈

—

C₈×C₂₉

C₂×Q₈

yes

—

C₈×C₂₉

C₂×Q₈

yes

—

#3160

—

maximal, Sylow

C₂²×C₂₉

—

C₂²×C₂₉

C₄⋊Q₈

yes

—

C₂²×C₂₉

C₂²×D₄

yes

characteristic

C₄×C₂₉

—

C₄×C₂₉

C₄⋊Q₈

yes

—

C₄×C₂₉

#135

yes

—

C₄×C₂₉

C₂²×Q₈

yes

—

Dic₂₉

—

—

—

—

—

—

C₂×C₄×C₈

D₂₉

yes

characteristic

C₂×C₂₉

—

C₂×C₂₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₉

C₂×#135

yes

—

C₂²×C₈

C₂×D₂₉

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₂₉

yes

characteristic

#107

—

#106

—

C₄×C₈

C₂×D₂₉

yes

characteristic

C₄×C₈

C₂×D₂₉

yes

—

C₄×C₈

C₂×D₂₉

yes

characteristic

C₂₉

#3160

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

D₁₁₆

yes

—

C₂²×C₄

C₂²×D₂₉

yes

characteristic

C₂×C₈

Dic₅₈

yes

complemented

C₂×C₈

C₂²×D₂₉

yes

—

C₂×C₈

C₂²×D₂₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

D₁₁₆

yes

—

C₄²

C₂²×D₂₉

yes

characteristic

C₄²

C₂²×D₂₉

yes

characteristic

C₂³

C₂×D₁₁₆

yes

characteristic

Q₈

—

116

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×Dic₅₈

yes

—

C₂×C₄

C₂×D₁₁₆

yes

—

C₂×C₄

C₂×D₁₁₆

yes

—

C₂×C₄

C₂×D₁₁₆

yes

characteristic

C₂×C₄

C₂³×D₂₉

yes

characteristic

C₈

C₂×Dic₅₈

yes

—

C₈

C₂×Dic₅₈

yes

—

100

C₂²

—

101

C₂²

#11745231

yes

—

102

C₂²

C₂²×D₁₁₆

yes

characteristic

103

C₄

—

104

C₄

—

116

—

105

C₄

#11745231

yes

—

106

C₄

#11745266

yes

—

107

C₄

C₂²×Dic₅₈

yes

—

108

C₂

—

109

C₂

C₂×#11745231

yes

characteristic

110

C₂

C₂×#11745266

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₁₆
commutator, cosocle	C₂×C₄×C₂₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₂×C₄×C₈×C₂₉	C₂
socle	C₂²×C₂₉	C₂²×D₄
prime radical	C₂³×C₂₉	C₂×D₄
hypercenter	C₂×C₄×C₈	D₂₉
hypocenter	C₂₉	#3160
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₁₁₆

commutator, cosocle

C₂×C₄×C₂₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₉

Fitting

C₂×C₄×C₈×C₂₉

C₂

socle

C₂²×C₂₉

C₂²×D₄

prime radical

C₂³×C₂₉

C₂×D₄

hypercenter

C₂×C₄×C₈

D₂₉

hypocenter

C₂₉

#3160

LCS

G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #14271568	C₂	semidirect
3. #14271565	C₂	semidirect
4. #14271561	C₂	semidirect
22. C₂×C₈×C₂₉	Q₈	semidirect
74. C₂₉	#3160	semidirect
78. C₂×C₈	Dic₅₈	semidirect

H ≅ G/N

type

2. #14271568

C₂

semidirect

3. #14271565

C₂

semidirect

4. #14271561

C₂

semidirect

22. C₂×C₈×C₂₉

Q₈

semidirect

74. C₂₉

#3160

semidirect

78. C₂×C₈

Dic₅₈

semidirect

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

subgroup	index
#14271720	2
#14271568	2
#14271565	2
#14271561	2
#14271201	2
C₂×C₄×C₈×C₂₉	2
#3160	29

subgroup

index

C₂×C₄×C₈×C₂₉

Subgroups — #16830006

Conjugacy classes of subgroups

Complements

Maximal subgroups