Group #21423615 — subgroups

Subgroups — #21423615

Order 7552.

152 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 55 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	#16837807	C₂	1	1	yes	maximal, characteristic
2	#16837613	C₂	1	2	yes	maximal
3	#16837532	C₂	1	1	yes	maximal, characteristic
4	#16837149	C₂	1	2	yes	maximal
5	C₅₉×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₃₆	—	2	2	no	—
7	#14273727	C₂²	1	2	yes	—
8	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
9	#14273661	—	2	1	no	—
10	#14273661	—	2	1	no	—
11	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
12	Dic₅₉⋊C₈	—	2	1	no	—
13	#14273652	—	2	2	no	—
14	#14273637	—	2	1	no	—
15	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
16	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₁₈	—	2	2	no	—
18	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
19	C₂×C₈×C₅₉	—	2	2	no	—
20	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
22	C₄²×C₅₉	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₅₉	—	2	2	no	—
24	C₄⋊Dic₅₉	—	2	2	no	—
25	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
26	Dic₅₉⋊C₄	—	2	1	no	—
27	C₄×Dic₅₉	—	2	1	no	—
28	C₂×C₅₉⋊C₈	—	2	1	no	—
29	Dic₂₃₆	—	4	4	no	—
30	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
33	C₂×Dic₅₉	—	2	1	no	—
34	C₂×Dic₅₉	—	2	1	no	—
35	C₂×Dic₅₉	—	4	2	no	—
36	Dic₁₁₈	—	2	2	no	—
37	Dic₁₁₈	—	4	2	no	—
38	Dic₁₁₈	—	4	2	no	—
39	C₈×C₅₉	—	2	1	no	—
40	C₈×C₅₉	—	4	2	no	—
41	C₈×C₅₉	Q₁₆	1	2	yes	complemented
42	C₅₉⋊C₈	—	4	1	no	—
43	C₂²×C₅₉	#121	1	1	yes	characteristic
44	C₄×C₅₉	—	2	1	no	—
45	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
46	C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
47	C₄×C₅₉	#137	1	1	yes	characteristic
48	Dic₅₉	—	4	1	no	—
49	Dic₅₉	—	4	1	no	—
50	Dic₅₉	—	8	2	no	—
51	#1694	—	59	1	no	maximal, Sylow
52	C₂×C₅₉	#462	1	1	yes	characteristic
53	C₂×C₅₉	#469	1	1	yes	characteristic
54	C₂×C₅₉	#473	1	1	yes	characteristic
55	#500	—	59	1	no	—
56	#462	—	59	2	no	—
57	#446	—	59	1	no	—
58	#334	D₅₉	1	1	yes	characteristic
59	#332	—	59	2	no	—
60	C₅₉	#1694	1	1	yes	characteristic, complemented, Sylow
61	C₂×Q₁₆	—	118	2	no	—
62	C₄⋊Q₈	—	59	2	no	—
63	C₄×Q₈	—	59	1	no	—
64	#107	—	118	1	no	—
65	#107	—	118	1	no	—
66	C₄⋊C₈	—	59	1	no	—
67	C₄⋊C₈	—	118	1	no	—
68	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
69	#103	—	118	2	no	—
70	#97	—	118	1	no	—
71	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
72	C₂×Q₈	—	59	1	no	—
73	C₂×Q₈	—	118	2	no	—
74	Q₁₆	—	236	4	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	118	1	no	—
77	C₂×C₈	D₂₃₆	1	1	yes	characteristic
78	C₂×C₈	D₂₃₆	1	1	yes	characteristic
79	C₄⋊C₄	—	59	1	no	—
80	C₄⋊C₄	—	118	1	no	—
81	C₄⋊C₄	—	118	2	no	—
82	C₄⋊C₄	—	118	2	no	—
83	C₄²	—	118	1	no	—
84	C₄²	C₂²×D₅₉	1	1	yes	characteristic
85	Q₈	—	118	2	no	—
86	Q₈	—	236	2	no	—
87	Q₈	—	236	2	no	—
88	C₂×C₄	—	118	1	no	—
89	C₂×C₄	—	118	1	no	—
90	C₂×C₄	—	236	2	no	—
91	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
92	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
93	C₂×C₄	#11745689	1	1	yes	characteristic
94	C₈	—	2	1	no	—
95	C₈	—	4	2	no	—
96	C₈	—	236	1	no	—
97	C₈	Dic₂₃₆	1	2	yes	complemented
98	C₂²	#14273702	1	1	yes	characteristic
99	C₄	—	2	1	no	—
100	C₄	—	472	2	no	—
101	C₄	—	236	1	no	—
102	C₄	—	236	1	no	—
103	C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
104	C₄	#14273754	1	2	yes	—
105	C₄	#14273760	1	1	yes	characteristic
106	C₂	#16837613	1	1	yes	characteristic
107	C₂	#16837656	1	1	yes	characteristic
108	C₂	#16837678	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837807

C₂

yes

maximal, characteristic

#16837613

C₂

yes

maximal

#16837532

C₂

yes

maximal, characteristic

#16837149

C₂

yes

maximal

C₅₉×#334

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

—

#14273727

C₂²

yes

—

C₄×Dic₁₁₈

C₂²

yes

characteristic

#14273661

—

#14273661

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

Dic₅₉⋊C₈

—

#14273652

—

#14273637

—

C₄⋊C₈×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

D₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

characteristic

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

Dic₂₃₆

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

Q₁₆

yes

complemented

C₅₉⋊C₈

—

C₂²×C₅₉

#121

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

C₂×Q₁₆

yes

characteristic

C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₄×C₅₉

#137

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

—

D₅₉

yes

characteristic

#332

—

C₅₉

#1694

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

118

—

C₄⋊Q₈

—

C₄×Q₈

—

#107

—

118

—

#107

—

118

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄⋊C₈

C₂×D₅₉

yes

—

#103

—

118

—

#97

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

118

—

Q₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

D₂₃₆

yes

characteristic

C₂×C₈

D₂₃₆

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄²

—

118

—

C₄²

C₂²×D₅₉

yes

characteristic

Q₈

—

118

—

Q₈

—

236

—

Q₈

—

236

—

C₂×C₄

—

118

—

C₂×C₄

—

118

—

C₂×C₄

—

236

—

C₂×C₄

C₂×D₂₃₆

yes

characteristic

C₂×C₄

D₄×D₅₉

yes

characteristic

C₂×C₄

#11745689

yes

characteristic

C₈

—

C₈

—

C₈

—

236

—

C₈

Dic₂₃₆

yes

complemented

C₂²

#14273702

yes

characteristic

C₄

—

100

C₄

—

472

—

101

C₄

—

236

—

102

C₄

—

236

—

103

C₄

C₂×Dic₂₃₆

yes

characteristic

104

C₄

#14273754

yes

—

105

C₄

#14273760

yes

characteristic

106

C₂

#16837613

yes

characteristic

107

C₂

#16837656

yes

characteristic

108

C₂

#16837678

yes

characteristic

	subgroup	G/H
center	C₂²	#14273702
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#334	C₂
socle	C₂²×C₅₉	#121
hypercenter	#334	D₅₉
hypocenter	C₅₉	#1694
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#14273702

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#334

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
41. C₈×C₅₉	Q₁₆	semidirect
60. C₅₉	#1694	semidirect
97. C₈	Dic₂₃₆	semidirect

H ≅ G/N

type

41. C₈×C₅₉

Q₁₆

semidirect

60. C₅₉

#1694

semidirect

97. C₈

Dic₂₃₆

semidirect

Subgroups — #21423615

Conjugacy classes of subgroups

Complements