Group #16817375 — subgroups

Subgroups — #16817375

Order 3712.

188 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 105 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×C₄⋊#6597	C₂	1	2	yes	maximal
2	C₂₉×#384	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
4	C₂×C₄⋊C₄×C₂₉	C₄	1	2	yes	—
5	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
6	C₂²×#6597	—	2	2	no	—
7	C₄⋊#6597	—	2	2	no	—
8	C₄⋊#6597	C₄	1	4	yes	complemented
9	C₂²×C₄×C₂₉	—	2	2	no	—
10	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₄×C₂₉	D₄	1	2	yes	—
12	C₄⋊C₄×C₂₉	—	4	2	no	—
13	C₄⋊C₄×C₂₉	—	4	2	no	—
14	C₄⋊C₄×C₂₉	—	4	2	no	—
15	C₄²×C₂₉	C₂×C₄	1	2	yes	—
16	C₄²×C₂₉	D₄	1	1	yes	characteristic
17	C₄²×C₂₉	Q₈	1	1	yes	characteristic
18	C₂×#6597	—	2	4	no	—
19	C₂×#6597	—	4	2	no	—
20	C₂×#6597	—	4	2	no	—
21	C₂³×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₉	—	2	1	no	—
23	C₂×C₄×C₂₉	—	2	1	no	—
24	C₂×C₄×C₂₉	—	4	2	no	—
25	C₂×C₄×C₂₉	—	4	4	no	—
26	C₂×C₄×C₂₉	C₄²	1	1	yes	characteristic
27	C₂×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
28	C₂×C₄×C₂₉	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₂₉	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₂₉	D₈	1	2	yes	—
31	C₂×C₄×C₂₉	SD₁₆	1	2	yes	—
32	C₂×C₄×C₂₉	SD₁₆	1	2	yes	—
33	C₂×C₄×C₂₉	Q₁₆	1	2	yes	—
34	#6597	—	4	4	no	—
35	#1296	—	29	1	no	maximal, Sylow
36	C₂²×C₂₉	#95	1	1	yes	characteristic
37	C₂²×C₂₉	#100	1	1	yes	characteristic
38	C₂²×C₂₉	#101	1	1	yes	characteristic
39	C₂²×C₂₉	#102	1	2	yes	—
40	C₂²×C₂₉	#103	1	2	yes	—
41	C₄×C₂₉	—	2	1	no	—
42	C₄×C₂₉	—	2	1	no	—
43	C₄×C₂₉	—	8	2	no	—
44	C₄×C₂₉	#102	1	2	yes	—
45	C₄×C₂₉	#103	1	2	yes	—
46	C₄×C₂₉	#106	1	2	yes	—
47	C₄×C₂₉	#107	1	2	yes	—
48	C₂×C₄⋊C₈	—	29	2	no	—
49	#384	D₂₉	1	1	yes	characteristic
50	C₂×C₂₉	#331	1	1	yes	characteristic
51	C₂×C₂₉	#332	1	2	yes	—
52	C₂×C₂₉	#333	1	1	yes	characteristic
53	C₂×C₂₉	#340	1	2	yes	—
54	C₂×C₂₉	#343	1	1	yes	characteristic
55	C₂²×C₈	—	58	2	no	—
56	C₂×C₄⋊C₄	—	2	2	no	—
57	C₂×C₄⋊C₄	Dic₂₉	1	2	yes	—
58	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
59	C₄⋊C₈	—	29	4	no	—
60	C₄⋊C₈	—	58	2	no	—
61	C₂₉	#1296	1	1	yes	characteristic, complemented, Sylow
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
64	C₂²×C₄	C₂²⋊D₂₉	1	2	yes	—
65	C₂×C₈	—	58	4	no	—
66	C₂×C₈	—	116	2	no	—
67	C₂×C₈	—	116	2	no	—
68	C₄⋊C₄	—	4	2	no	—
69	C₄⋊C₄	—	4	2	no	—
70	C₄⋊C₄	—	4	2	no	—
71	C₄²	Dic₅₈	1	1	yes	characteristic
72	C₄²	C₄×D₂₉	1	2	yes	—
73	C₄²	D₁₁₆	1	1	yes	characteristic
74	C₂³	C₂²⋊Dic₂₉	1	1	yes	characteristic
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	4	4	no	—
78	C₂×C₄	—	4	2	no	—
79	C₂×C₄	C₄×Dic₂₉	1	1	yes	characteristic
80	C₂×C₄	#90779	1	2	yes	—
81	C₂×C₄	C₄⋊Dic₂₉	1	1	yes	characteristic
82	C₂×C₄	#90781	1	2	yes	—
83	C₂×C₄	#90782	1	2	yes	—
84	C₂×C₄	#90783	1	2	yes	—
85	C₂×C₄	#90784	1	2	yes	—
86	C₂×C₄	#90785	1	2	yes	—
87	C₈	—	116	4	no	—
88	C₂²	#11745179	1	1	yes	characteristic
89	C₂²	#11745180	1	2	yes	—
90	C₂²	#11745181	1	1	yes	characteristic
91	C₂²	#11745183	1	2	yes	—
92	C₂²	#11745184	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	1	no	—
95	C₄	—	8	2	no	—
96	C₄	#11745155	1	2	yes	—
97	C₄	#11745156	1	2	yes	—
98	C₄	#11745157	1	2	yes	—
99	C₄	#11745158	1	2	yes	—
100	C₂	#14270907	1	1	yes	characteristic
101	C₂	#14270909	1	2	yes	—
102	C₂	#14270912	1	1	yes	characteristic
103	C₂	#14270925	1	2	yes	—
104	C₂	#14270932	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#6597

C₂

yes

maximal

C₂₉×#384

C₂

yes

maximal, characteristic

C₂×C₄⋊C₄×C₂₉

—

C₂×C₄⋊C₄×C₂₉

C₄

yes

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

C₂²×#6597

—

C₄⋊#6597

—

C₄⋊#6597

C₄

yes

complemented

C₂²×C₄×C₂₉

—

C₂²×C₄×C₂₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₉

D₄

yes

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄²×C₂₉

C₂×C₄

yes

—

C₄²×C₂₉

D₄

yes

characteristic

C₄²×C₂₉

Q₈

yes

—

—

—

C₂³×C₂₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₄²

yes

characteristic

C₂×C₄×C₂₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉

C₄⋊C₄

yes

—

C₂×C₄×C₂₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₉

D₈

yes

—

C₂×C₄×C₂₉

SD₁₆

yes

—

C₂×C₄×C₂₉

SD₁₆

yes

—

C₂×C₄×C₂₉

Q₁₆

yes

—

—

—

C₂²×C₂₉

yes

C₂²×C₂₉

yes

C₂²×C₂₉

yes

C₂²×C₂₉

yes

—

C₂²×C₂₉

#103

yes

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

#102

yes

—

C₄×C₂₉

#103

yes

—

C₄×C₂₉

#106

yes

—

C₄×C₂₉

#107

yes

—

C₂×C₄⋊C₈

—

#384

D₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

—

C₂×C₂₉

#333

yes

characteristic

C₂×C₂₉

#340

yes

—

C₂×C₂₉

#343

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₂₉

yes

—

C₂×C₄²

C₂×D₂₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂₉

#1296

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₂×Dic₂₉

yes

characteristic

C₂²×C₄

C₂²⋊D₂₉

yes

—

C₂×C₈

—

C₂×C₈

—

116

—

C₂×C₈

—

116

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Dic₅₈

yes

characteristic

C₄²

C₄×D₂₉

yes

—

C₄²

D₁₁₆

yes

characteristic

C₂³

C₂²⋊Dic₂₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₂₉

yes

characteristic

C₂×C₄

#90779

yes

—

C₂×C₄

C₄⋊Dic₂₉

yes

characteristic

C₂×C₄

#90781

yes

—

C₂×C₄

#90782

yes

—

C₂×C₄

#90783

yes

—

C₂×C₄

#90784

yes

—

C₂×C₄

#90785

yes

—

C₈

—

116

—

C₂²

#11745179

yes

characteristic

C₂²

#11745180

yes

—

C₂²

#11745181

yes

characteristic

C₂²

#11745183

yes

—

C₂²

#11745184

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#11745155

yes

—

C₄

#11745156

yes

—

C₄

#11745157

yes

—

C₄

#11745158

yes

—

100

C₂

#14270907

yes

characteristic

101

C₂

#14270909

yes

—

102

C₂

#14270912

yes

characteristic

103

C₂

#14270925

yes

—

104

C₂

#14270932

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₂₉
commutator	C₂×C₄×C₂₉	C₄²
Frattini	C₂×C₄²	C₂×D₂₉
Fitting	C₂₉×#384	C₂
socle, prime radical	C₂³×C₂₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₂₉	C₂²
hypercenter	#384	D₂₉
hypocenter	C₂₉	#1296
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂²⋊Dic₂₉

commutator

C₂×C₄×C₂₉

C₄²

Frattini

C₂×C₄²

C₂×D₂₉

Fitting

C₂₉×#384

C₂

socle, prime radical

C₂³×C₂₉

C₂²⋊C₄

cosocle

C₂×C₄²×C₂₉

C₂²

D₂₉

C₂₉

G ▷ C₂×C₄×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
8. C₄⋊#6597	C₄	semidirect
61. C₂₉	#1296	semidirect

H ≅ G/N

type

8. C₄⋊#6597

C₄

semidirect

61. C₂₉

#1296

semidirect

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

subgroup	index
C₂×C₄⋊#6597	2
C₂₉×#384	2
#1296	29

subgroup

index

C₂×C₄⋊#6597

C₂₉×#384

#1296

Subgroups — #16817375

Conjugacy classes of subgroups

Complements

Maximal subgroups