Group #21424230 — subgroups

Subgroups — #21424230

Order 7552.

224 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 109 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	#16837272	C₂	1	1	yes	maximal, characteristic
2	#16837268	C₂	1	1	yes	maximal, characteristic
3	#16837165	C₂	1	4	yes	maximal
4	C₂×#97×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₅₉	—	2	2	no	—
6	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
8	C₅₉⋊#95	—	2	2	no	—
9	C₂²×C₈×C₅₉	C₂²	1	2	yes	—
10	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
11	#97×C₅₉	C₄	1	4	yes	complemented
12	C₂²×C₄×C₅₉	D₄	1	2	yes	—
13	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
14	C₂²×Dic₅₉	—	2	2	no	—
15	C₂²×Dic₅₉	—	2	2	no	—
16	C₂×C₈×C₅₉	—	2	2	no	—
17	C₂×C₈×C₅₉	—	2	2	no	—
18	C₂×C₈×C₅₉	C₂×C₄	1	4	yes	—
19	C₄²×C₅₉	C₂×C₄	1	2	yes	—
20	C₄²×C₅₉	Q₈	1	2	yes	—
21	C₄⋊Dic₅₉	—	2	4	no	—
22	C₄⋊Dic₅₉	—	2	4	no	—
23	C₄⋊Dic₅₉	—	4	4	no	—
24	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
25	C₂×C₄×C₅₉	—	2	2	no	—
26	C₂×C₄×C₅₉	—	2	2	no	—
27	C₂×C₄×C₅₉	C₄⋊C₄	1	4	yes	—
28	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₄×C₅₉	#41	1	2	yes	—
31	C₂×C₄×C₅₉	#41	1	2	yes	—
32	C₂×Dic₅₉	—	4	2	no	—
33	C₂×Dic₅₉	—	4	2	no	—
34	C₂×Dic₅₉	—	4	4	no	—
35	C₂×Dic₅₉	—	4	2	no	—
36	C₂×Dic₅₉	—	4	2	no	—
37	C₈×C₅₉	—	2	4	no	—
38	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₅₉	#124	1	1	yes	characteristic
40	C₂²×C₅₉	#124	1	1	yes	characteristic
41	C₂²×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
42	C₂²×C₅₉	#137	1	2	yes	—
43	C₄×C₅₉	—	2	4	no	—
44	C₄×C₅₉	#117	1	2	yes	—
45	C₄×C₅₉	#125	1	2	yes	—
46	Dic₅₉	—	8	2	no	—
47	Dic₅₉	—	8	2	no	—
48	#1836	—	59	1	no	maximal, Sylow
49	C₂×C₅₉	#382	1	1	yes	characteristic
50	C₂×C₅₉	#428	1	2	yes	—
51	C₂×C₅₉	#489	1	2	yes	—
52	C₂×C₅₉	#490	1	1	yes	characteristic
53	C₂×C₅₉	#491	1	1	yes	characteristic
54	C₂×#97	D₅₉	1	1	yes	characteristic
55	#384	—	59	1	no	—
56	#382	—	59	1	no	—
57	#340	—	59	4	no	—
58	C₅₉	#1836	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₈	C₂×D₅₉	1	2	yes	—
60	C₂×C₄⋊C₄	—	59	2	no	—
61	C₂×C₄⋊C₄	—	59	2	no	—
62	C₂×C₄⋊C₄	—	118	2	no	—
63	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
64	#97	Dic₅₉	1	4	yes	complemented
65	#95	—	118	2	no	—
66	C₂²×C₄	—	118	2	no	—
67	C₂²×C₄	—	118	2	no	—
68	C₂²×C₄	D₂₃₆	1	2	yes	—
69	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	C₂×Dic₅₉	1	4	yes	—
73	C₄⋊C₄	—	118	4	no	—
74	C₄⋊C₄	—	118	4	no	—
75	C₄⋊C₄	—	236	4	no	—
76	C₄²	Dic₁₁₈	1	2	yes	—
77	C₄²	C₂×Dic₅₉	1	2	yes	—
78	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	236	2	no	—
82	C₂×C₄	—	236	2	no	—
83	C₂×C₄	—	236	4	no	—
84	C₂×C₄	—	236	2	no	—
85	C₂×C₄	—	236	2	no	—
86	C₂×C₄	C₄⋊Dic₅₉	1	4	yes	—
87	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
88	C₂×C₄	#11745685	1	2	yes	—
89	C₂×C₄	#11745685	1	2	yes	—
90	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
91	C₈	—	2	4	no	—
92	C₂²	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
93	C₂²	#14273715	1	1	yes	characteristic
94	C₂²	#14273715	1	1	yes	characteristic
95	C₂²	#14273755	1	2	yes	—
96	C₂²	#14273760	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	472	2	no	—
99	C₄	—	472	2	no	—
100	C₄	#14273686	1	2	yes	—
101	C₄	#14273719	1	2	yes	—
102	C₂	#16837268	1	1	yes	characteristic
103	C₂	#16837444	1	2	yes	—
104	C₂	#16837762	1	2	yes	—
105	C₂	#16837767	1	1	yes	characteristic
106	C₂	#16837771	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837272

C₂

yes

maximal, characteristic

#16837268

C₂

yes

maximal, characteristic

#16837165

C₂

yes

maximal

C₂×#97×C₅₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

#97×C₅₉

C₄

yes

complemented

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

Q₈

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#137

yes

—

C₄×C₅₉

—

C₄×C₅₉

#117

yes

—

C₄×C₅₉

#125

yes

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#489

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×#97

D₅₉

yes

—

—

—

C₅₉

#1836

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×D₅₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#97

Dic₅₉

yes

complemented

#95

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

D₂₃₆

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₅₉

yes

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

C₄²

Dic₁₁₈

yes

—

C₄²

C₂×Dic₅₉

yes

—

C₂³

C₂×D₂₃₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

C₄⋊Dic₅₉

yes

—

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

C₂×C₄

#11745685

yes

—

C₂×C₄

#11745685

yes

—

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

C₈

—

C₂²

C₂×C₄⋊Dic₅₉

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#14273760

yes

—

C₄

—

C₄

—

472

—

C₄

—

472

—

100

C₄

#14273686

yes

—

101

C₄

#14273719

yes

—

102

C₂

#16837268

yes

characteristic

103

C₂

#16837444

yes

—

104

C₂

#16837762

yes

—

105

C₂

#16837767

yes

characteristic

106

C₂

#16837771

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#97×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×#97	D₅₉
hypocenter	C₅₉	#1836
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#97×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

hypercenter

C₂×#97

D₅₉

hypocenter

C₅₉

#1836

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
11. #97×C₅₉	C₄	semidirect
58. C₅₉	#1836	semidirect
64. #97	Dic₅₉	semidirect

H ≅ G/N

type

11. #97×C₅₉

C₄

semidirect

58. C₅₉

#1836

semidirect

64. #97

Dic₅₉

semidirect

Subgroups — #21424230

Conjugacy classes of subgroups

Complements