Group #16817328 — subgroups

Subgroups — #16817328

Order 3712.

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

normal subgroups only

This group has 172 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	C₄×#90777	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#6597	C₂	1	1	yes	maximal, characteristic
3	C₂₉×#431	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
5	C₂×#90777	C₄	1	2	yes	—
6	C₂²×#6597	—	2	1	no	—
7	C₂₉×C₄⋊C₈	C₄	1	2	yes	—
8	C₂₉×C₂²⋊C₈	—	2	1	no	—
9	C₄×C₈×C₂₉	—	2	1	no	—
10	#11745151	—	2	1	no	—
11	C₄×#6597	—	2	1	no	—
12	C₄×#6597	—	2	1	no	—
13	C₄×#6597	C₄	1	2	yes	—
14	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₂₉	Q₈	1	1	yes	characteristic
17	C₂×C₈×C₂₉	—	2	2	no	—
18	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₂₉	D₄	1	1	yes	characteristic
21	C₄²×C₂₉	D₄	1	1	yes	characteristic
22	#90777	—	2	2	no	—
23	#90777	C₈	1	4	yes	complemented
24	C₂×#6597	—	2	4	no	—
25	C₂×#6597	—	2	2	no	—
26	C₂×#6597	—	2	2	no	—
27	C₂³×C₂₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₂₉	—	2	2	no	—
29	C₂×C₄×C₂₉	—	2	2	no	—
30	C₂×C₄×C₂₉	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₉	C₂×C₈	1	2	yes	—
37	C₂×C₄×C₂₉	M₁₆	1	2	yes	—
38	C₈×C₂₉	—	4	2	no	—
39	#6597	—	2	4	no	—
40	#6597	—	2	4	no	—
41	#1278	—	29	1	no	maximal, Sylow
42	C₂²×C₂₉	—	2	1	no	—
43	C₂²×C₂₉	—	2	1	no	—
44	C₂²×C₂₉	#95	1	1	yes	characteristic
45	C₂²×C₂₉	C₄⋊C₈	1	2	yes	—
46	C₄×C₂₉	—	2	1	no	—
47	C₄×C₂₉	—	2	1	no	—
48	C₄×C₂₉	C₄×C₈	1	1	yes	characteristic
49	C₄×C₂₉	#97	1	1	yes	characteristic
50	C₄×C₂₉	C₂²⋊C₈	1	2	yes	—
51	C₄×C₂₉	#104	1	2	yes	—
52	C₄×C₂₉	#108	1	2	yes	—
53	#431	D₂₉	1	1	yes	characteristic
54	C₄×M₁₆	—	29	1	no	—
55	C₂×C₄×C₈	—	29	1	no	—
56	C₂×C₂₉	—	2	2	no	—
57	C₂×C₂₉	#336	1	1	yes	characteristic
58	C₂×C₂₉	#339	1	1	yes	characteristic
59	C₂×C₂₉	#341	1	1	yes	characteristic
60	C₂×M₁₆	—	29	2	no	—
61	C₂²×C₈	—	58	1	no	—
62	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
63	C₄⋊C₈	Dic₂₉	1	2	yes	—
64	C₂²⋊C₈	—	2	1	no	—
65	#97	—	58	1	no	—
66	C₄×C₈	—	2	1	no	—
67	C₄×C₈	—	29	2	no	—
68	C₄×C₈	—	58	1	no	—
69	C₄×C₈	—	58	1	no	—
70	C₂₉	#1278	1	1	yes	characteristic, complemented, Sylow
71	C₂²×C₄	Dic₅₈	1	1	yes	characteristic
72	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
73	C₂²×C₄	C₂²⋊D₂₉	1	1	yes	characteristic
74	M₁₆	—	29	4	no	—
75	M₁₆	—	58	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	58	4	no	—
78	C₂×C₈	—	58	2	no	—
79	C₂×C₈	—	58	2	no	—
80	C₄²	C₄×D₂₉	1	1	yes	characteristic
81	C₄²	D₁₁₆	1	1	yes	characteristic
82	C₄²	C₂×Dic₂₉	1	1	yes	characteristic
83	C₄²	C₂²⋊D₂₉	1	1	yes	characteristic
84	C₂³	#90779	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	C₈×D₂₉	1	2	yes	—
88	C₂×C₄	#90772	1	2	yes	—
89	C₂×C₄	C₄×Dic₂₉	1	1	yes	characteristic
90	C₂×C₄	#90779	1	1	yes	characteristic
91	C₂×C₄	C₄⋊Dic₂₉	1	1	yes	characteristic
92	C₂×C₄	#90781	1	1	yes	characteristic
93	C₂×C₄	#90781	1	1	yes	characteristic
94	C₂×C₄	C₂²⋊Dic₂₉	1	1	yes	characteristic
95	C₈	—	4	2	no	—
96	C₈	—	58	4	no	—
97	C₈	—	58	4	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#11745162	1	2	yes	—
101	C₂²	#11745179	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	C₈×Dic₂₉	1	1	yes	characteristic
105	C₄	#11745163	1	1	yes	characteristic
106	C₄	#11745168	1	2	yes	—
107	C₄	#11745170	1	2	yes	—
108	C₄	#11745185	1	2	yes	—
109	C₂	—	2	2	no	—
110	C₂	#14270917	1	1	yes	characteristic
111	C₂	#14270922	1	1	yes	characteristic
112	C₂	#14270927	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#90777

C₂

yes

maximal, characteristic

C₂×C₄×#6597

C₂

yes

maximal, characteristic

C₂₉×#431

C₂

yes

maximal, characteristic

C₂×C₄²×C₂₉

C₂²

yes

characteristic

C₂×#90777

C₄

yes

—

C₂²×#6597

—

C₂₉×C₄⋊C₈

C₄

yes

—

C₂₉×C₂²⋊C₈

—

C₄×C₈×C₂₉

—

—

—

—

C₄

yes

—

C₂²×C₄×C₂₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₉

D₄

yes

characteristic

C₂²×C₄×C₂₉

Q₈

yes

characteristic

C₂×C₈×C₂₉

—

C₄²×C₂₉

C₂×C₄

yes

characteristic

C₄²×C₂₉

C₂×C₄

yes

characteristic

C₄²×C₂₉

D₄

yes

characteristic

C₄²×C₂₉

D₄

yes

characteristic

#90777

—

#90777

C₈

yes

—

—

—

C₂³×C₂₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₄²

yes

characteristic

C₂×C₄×C₂₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₉

C₂×C₈

yes

—

C₂×C₄×C₂₉

M₁₆

yes

—

C₈×C₂₉

—

—

—

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

#95

yes

characteristic

C₂²×C₂₉

C₄⋊C₈

yes

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

C₄×C₈

yes

characteristic

C₄×C₂₉

#97

yes

characteristic

C₄×C₂₉

C₂²⋊C₈

yes

—

C₄×C₂₉

#104

yes

—

C₄×C₂₉

#108

yes

—

#431

D₂₉

yes

characteristic

C₄×M₁₆

—

C₂×C₄×C₈

—

C₂×C₂₉

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

C₂×D₂₉

yes

characteristic

C₄⋊C₈

Dic₂₉

yes

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂₉

#1278

yes

characteristic, complemented, Sylow

C₂²×C₄

Dic₅₈

yes

characteristic

C₂²×C₄

C₄×D₂₉

yes

characteristic

C₂²×C₄

C₂²⋊D₂₉

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₄×D₂₉

yes

characteristic

C₄²

D₁₁₆

yes

characteristic

C₄²

C₂×Dic₂₉

yes

characteristic

C₄²

C₂²⋊D₂₉

yes

characteristic

C₂³

#90779

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₂₉

yes

—

C₂×C₄

#90772

yes

—

C₂×C₄

C₄×Dic₂₉

yes

characteristic

C₂×C₄

#90779

yes

characteristic

C₂×C₄

C₄⋊Dic₂₉

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₂₉

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

100

C₂²

#11745162

yes

—

101

C₂²

#11745179

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

C₈×Dic₂₉

yes

characteristic

105

C₄

#11745163

yes

characteristic

106

C₄

#11745168

yes

—

107

C₄

#11745170

yes

—

108

C₄

#11745185

yes

—

109

C₂

—

110

C₂

#14270917

yes

characteristic

111

C₂

#14270922

yes

characteristic

112

C₂

#14270927

yes

characteristic

	subgroup	G/H
center	C₄²	C₂²⋊D₂₉
commutator	C₄×C₂₉	C₄×C₈
Frattini	C₂×C₄²	C₂×D₂₉
Fitting	C₂₉×#431	C₂
socle	C₂²×C₂₉	#95
prime radical	C₂³×C₂₉	C₄⋊C₄
cosocle	C₂×C₄²×C₂₉	C₂²
hypercenter	#431	D₂₉
hypocenter	C₂₉	#1278
LCS	G ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₄²

C₂²⋊D₂₉

commutator

C₄×C₂₉

C₄×C₈

Frattini

C₂×C₄²

C₂×D₂₉

Fitting

C₂₉×#431

C₂

socle

C₂²×C₂₉

#95

prime radical

C₂³×C₂₉

C₄⋊C₄

cosocle

C₂×C₄²×C₂₉

C₂²

D₂₉

C₂₉

G ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
23. #90777	C₈	semidirect
70. C₂₉	#1278	semidirect

H ≅ G/N

type

23. #90777

C₈

semidirect

70. C₂₉

#1278

semidirect

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

subgroup	index
C₄×#90777	2
C₂×C₄×#6597	2
C₂₉×#431	2
#1278	29

subgroup

index

C₄×#90777

C₂×C₄×#6597

C₂₉×#431

#1278

Subgroups — #16817328

Conjugacy classes of subgroups

Complements

Maximal subgroups