Group #21422871 — subgroups

Subgroups — #21422871

Order 7552.

168 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 97 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	#16837426	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#16837155	C₂	1	4	yes	maximal, complemented
4	C₅₉×#431	C₂	1	1	yes	maximal, characteristic
5	C₂×#11745664	—	2	1	no	—
6	C₂²×C₅₉⋊C₈	—	2	1	no	—
7	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
8	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
9	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
10	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₅₉	C₄	1	2	yes	—
13	C₂²⋊C₈×C₅₉	C₄	1	2	yes	—
14	C₄×C₈×C₅₉	C₂²	1	2	yes	—
15	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₉	Q₈	1	1	yes	characteristic
18	C₂×C₈×C₅₉	C₂×C₄	1	4	yes	—
19	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₅₉	D₄	1	1	yes	characteristic
21	C₄²×C₅₉	Q₈	1	1	yes	characteristic
22	C₄²×C₅₉	C₂³	1	1	yes	characteristic
23	#11745664	—	4	2	no	—
24	C₂×C₅₉⋊C₈	—	2	2	no	—
25	C₂×C₅₉⋊C₈	—	2	2	no	—
26	C₂×C₅₉⋊C₈	—	4	2	no	—
27	C₂³×C₅₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₅₉	—	2	2	no	—
29	C₂×C₄×C₅₉	—	4	1	no	—
30	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₉	D₈	1	2	yes	—
34	C₂×C₄×C₅₉	Q₁₆	1	2	yes	—
35	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
38	C₈×C₅₉	—	2	4	no	—
39	C₅₉⋊C₈	—	4	2	no	—
40	C₅₉⋊C₈	—	4	2	no	—
41	C₂²×C₅₉	—	2	1	no	—
42	C₂²×C₅₉	—	2	1	no	—
43	C₂²×C₅₉	#107	1	2	yes	—
44	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₅₉	—	2	1	no	—
46	C₄×C₅₉	—	2	1	no	—
47	C₄×C₅₉	—	2	1	no	—
48	C₄×C₅₉	#107	1	2	yes	—
49	C₄×C₅₉	#131	1	2	yes	—
50	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
51	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
52	#1571	—	59	1	no	maximal, Sylow
53	C₂×C₅₉	—	2	2	no	—
54	C₂×C₅₉	#424	1	1	yes	characteristic
55	C₂×C₅₉	C₂×#107	1	1	yes	characteristic
56	C₂×C₅₉	#430	1	1	yes	characteristic
57	#431	D₅₉	1	1	yes	characteristic
58	#423	—	59	1	no	—
59	C₂×C₄⋊C₈	—	59	1	no	—
60	#335	—	59	4	no	—
61	C₅₉	#1571	1	1	yes	characteristic, complemented, Sylow
62	C₂×M₁₆	—	118	1	no	—
63	C₂²×C₈	—	118	1	no	—
64	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
65	C₄⋊C₈	—	59	2	no	—
66	C₄⋊C₈	—	59	2	no	—
67	C₄⋊C₈	—	118	1	no	—
68	C₄⋊C₈	—	118	1	no	—
69	C₄⋊C₈	Dic₅₉	1	2	yes	—
70	C₂²⋊C₈	Dic₅₉	1	2	yes	—
71	C₄×C₈	C₂×D₅₉	1	2	yes	—
72	C₂²×C₄	Dic₁₁₈	1	1	yes	characteristic
73	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
74	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
75	M₁₆	—	236	2	no	—
76	C₂×C₈	—	118	2	no	—
77	C₂×C₈	—	118	2	no	—
78	C₂×C₈	—	236	2	no	—
79	C₂×C₈	C₂×Dic₅₉	1	4	yes	—
80	C₄²	Dic₁₁₈	1	1	yes	characteristic
81	C₄²	D₂₃₆	1	1	yes	characteristic
82	C₄²	C₂×Dic₅₉	1	1	yes	characteristic
83	C₄²	C₂²×D₅₉	1	1	yes	characteristic
84	C₂³	C₄⋊Dic₅₉	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	D₄₇₂	1	2	yes	—
88	C₂×C₄	Dic₂₃₆	1	2	yes	—
89	C₂×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
90	C₂×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
91	C₂×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
92	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
93	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
94	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
95	C₈	—	2	4	no	—
96	C₈	—	236	2	no	—
97	C₈	—	236	2	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#14273661	1	2	yes	—
101	C₂²	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	#14273661	1	2	yes	—
106	C₄	#14273734	1	2	yes	—
107	C₄	C₂×D₄₇₂	1	1	yes	characteristic
108	C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#16837429	1	1	yes	characteristic
111	C₂	C₂×#14273661	1	1	yes	characteristic
112	C₂	#16837451	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837426

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₅₉⋊C₈)

C₂

yes

maximal, characteristic

#16837155

C₂

yes

maximal, complemented

C₅₉×#431

C₂

yes

maximal, characteristic

C₂×#11745664

—

C₂²×C₅₉⋊C₈

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₄

yes

—

C₂²⋊C₈×C₅₉

C₄

yes

—

C₄×C₈×C₅₉

C₂²

yes

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

Q₈

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

Q₈

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

D₈

yes

—

C₂×C₄×C₅₉

Q₁₆

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#107

yes

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#107

yes

—

C₄×C₅₉

#131

yes

—

C₄×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

C₂×Q₁₆

yes

characteristic

#1571

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

yes

characteristic

#423

—

C₂×C₄⋊C₈

—

#335

—

C₅₉

#1571

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

118

—

C₂²×C₈

—

118

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄⋊C₈

—

118

—

C₄⋊C₈

Dic₅₉

yes

—

C₂²⋊C₈

Dic₅₉

yes

—

C₄×C₈

C₂×D₅₉

yes

—

C₂²×C₄

Dic₁₁₈

yes

characteristic

C₂²×C₄

D₂₃₆

yes

characteristic

C₂²×C₄

C₂×Dic₅₉

yes

characteristic

M₁₆

—

236

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

—

236

—

C₂×C₈

C₂×Dic₅₉

yes

—

C₄²

Dic₁₁₈

yes

characteristic

C₄²

D₂₃₆

yes

characteristic

C₄²

C₂×Dic₅₉

yes

characteristic

C₄²

C₂²×D₅₉

yes

characteristic

C₂³

C₄⋊Dic₅₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₄₇₂

yes

—

C₂×C₄

Dic₂₃₆

yes

—

C₂×C₄

C₄⋊Dic₅₉

yes

characteristic

C₂×C₄

C₄⋊Dic₅₉

yes

characteristic

C₂×C₄

C₄⋊Dic₅₉

yes

characteristic

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

C₂×C₄

C₂×D₂₃₆

yes

characteristic

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

C₈

—

C₈

—

236

—

C₈

—

236

—

C₂²

—

C₂²

—

100

C₂²

#14273661

yes

—

101

C₂²

C₂×C₄⋊Dic₅₉

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#14273661

yes

—

106

C₄

#14273734

yes

—

107

C₄

C₂×D₄₇₂

yes

characteristic

108

C₄

C₂×Dic₂₃₆

yes

characteristic

109

C₂

—

110

C₂

#16837429

yes

characteristic

111

C₂

C₂×#14273661

yes

characteristic

112

C₂

#16837451

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#431	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₄
hypercenter	#431	D₅₉
hypocenter	C₅₉	#1571
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#431

C₂

socle

C₂²×C₅₉

C₂×C₄⋊C₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #16837155	C₂	semidirect
61. C₅₉	#1571	semidirect

H ≅ G/N

type

3. #16837155

C₂

semidirect

61. C₅₉

#1571

semidirect

Subgroups — #21422871

Conjugacy classes of subgroups

Complements