Group #21423466 — subgroups

Subgroups — #21423466

Order 7552.

156 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16837782	C₂	1	1	yes	maximal, characteristic
2	#16837687	C₂	1	2	yes	maximal
3	#16837155	C₂	1	1	yes	maximal, characteristic
4	#16837149	C₂	1	2	yes	maximal
5	C₈×Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₃₆	—	2	1	no	—
7	C₂×Dic₂₃₆	—	2	1	no	—
8	#14273727	C₂²	1	2	yes	—
9	#14273658	—	2	2	no	—
10	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
11	#14273651	—	2	2	no	—
12	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₅₉	—	2	1	no	—
14	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₅₉	—	2	1	no	—
16	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
18	C₂×Dic₁₁₈	—	2	2	no	—
19	C₂×C₈×C₅₉	—	2	1	no	—
20	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
22	C₄⋊C₄×C₅₉	—	2	1	no	—
23	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
24	C₄²×C₅₉	—	2	1	no	—
25	C₄²×C₅₉	C₂³	1	1	yes	characteristic
26	C₄⋊Dic₅₉	—	2	2	no	—
27	C₄⋊Dic₅₉	—	2	2	no	—
28	C₂×C₅₉⋊C₈	—	2	2	no	—
29	Dic₂₃₆	—	4	2	no	—
30	Dic₂₃₆	—	4	2	no	—
31	Q₈×C₅₉	—	2	1	no	—
32	Q₈×C₅₉	Q₁₆	1	2	yes	complemented
33	C₂×C₄×C₅₉	—	2	1	no	—
34	C₂×C₄×C₅₉	—	2	1	no	—
35	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
38	C₂×Dic₅₉	—	4	2	no	—
39	Dic₁₁₈	—	4	2	no	—
40	Dic₁₁₈	—	4	2	no	—
41	C₈×C₅₉	—	2	1	no	—
42	C₈×C₅₉	—	4	1	no	—
43	C₈×C₅₉	SD₁₆	1	2	yes	—
44	C₅₉⋊C₈	—	4	2	no	—
45	C₂²×C₅₉	#121	1	1	yes	characteristic
46	C₄×C₅₉	—	2	2	no	—
47	C₄×C₅₉	—	2	1	no	—
48	C₄×C₅₉	—	4	1	no	—
49	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
50	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
51	C₄×C₅₉	#135	1	1	yes	characteristic
52	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
53	Dic₅₉	—	8	2	no	—
54	#1670	—	59	1	no	maximal, Sylow
55	C₂×C₅₉	#460	1	1	yes	characteristic
56	C₂×C₅₉	#467	1	1	yes	characteristic
57	C₂×C₅₉	#473	1	1	yes	characteristic
58	#494	—	59	1	no	—
59	#475	—	59	2	no	—
60	C₈×Q₈	D₅₉	1	1	yes	characteristic
61	#335	—	59	1	no	—
62	#332	—	59	2	no	—
63	C₅₉	#1670	1	1	yes	characteristic, complemented, Sylow
64	C₂×Q₁₆	—	118	1	no	—
65	C₂×Q₁₆	—	118	1	no	—
66	C₄⋊Q₈	—	59	2	no	—
67	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
68	#106	—	118	2	no	—
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	—	59	2	no	—
71	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
72	#103	—	118	2	no	—
73	C₄×C₈	—	2	1	no	—
74	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
75	C₂×Q₈	—	118	2	no	—
76	C₂×Q₈	D₂₃₆	1	1	yes	characteristic
77	Q₁₆	—	236	2	no	—
78	Q₁₆	—	236	2	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	118	2	no	—
81	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
82	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
83	C₄⋊C₄	—	2	1	no	—
84	C₄⋊C₄	—	118	2	no	—
85	C₄⋊C₄	—	118	2	no	—
86	C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
87	C₄²	—	2	1	no	—
88	C₄²	C₂²×D₅₉	1	1	yes	characteristic
89	Q₈	—	2	1	no	—
90	Q₈	—	236	2	no	—
91	Q₈	—	236	2	no	—
92	Q₈	Dic₂₃₆	1	2	yes	complemented
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	236	2	no	—
96	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
97	C₂×C₄	#11745685	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	—	4	1	no	—
101	C₈	—	236	2	no	—
102	C₈	#11745671	1	2	yes	—
103	C₂²	#14273701	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	4	1	no	—
107	C₄	—	472	2	no	—
108	C₄	C₂×#11745671	1	1	yes	characteristic
109	C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
110	C₄	#14273749	1	1	yes	characteristic
111	C₄	#14273752	1	1	yes	characteristic
112	C₂	#16837602	1	1	yes	characteristic
113	C₂	#16837643	1	1	yes	characteristic
114	C₂	#16837677	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837782

C₂

yes

maximal, characteristic

#16837687

C₂

yes

maximal

#16837155

C₂

yes

maximal, characteristic

#16837149

C₂

yes

maximal

C₈×Q₈×C₅₉

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

#14273727

C₂²

yes

—

#14273658

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

#14273651

—

C₄×Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

Dic₂₃₆

—

Dic₂₃₆

—

Q₈×C₅₉

—

Q₈×C₅₉

Q₁₆

yes

complemented

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

SD₁₆

yes

—

C₅₉⋊C₈

—

C₂²×C₅₉

#121

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₄×C₅₉

C₂×Q₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

C₈×Q₈

D₅₉

yes

characteristic

#335

—

#332

—

C₅₉

#1670

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

118

—

C₂×Q₁₆

—

118

—

C₄⋊Q₈

—

C₄×Q₈

C₂×D₅₉

yes

characteristic

#106

—

118

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₉

yes

characteristic

#103

—

118

—

C₄×C₈

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

118

—

C₂×Q₈

D₂₃₆

yes

characteristic

Q₁₆

—

236

—

Q₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

D₂₃₆

yes

characteristic

C₄²

—

C₄²

C₂²×D₅₉

yes

characteristic

Q₈

—

Q₈

—

236

—

Q₈

—

236

—

Q₈

Dic₂₃₆

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

236

—

C₂×C₄

C₂×D₂₃₆

yes

characteristic

C₂×C₄

#11745685

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

236

—

102

C₈

#11745671

yes

—

103

C₂²

#14273701

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

472

—

108

C₄

C₂×#11745671

yes

characteristic

109

C₄

C₂×Dic₂₃₆

yes

characteristic

110

C₄

#14273749

yes

characteristic

111

C₄

#14273752

yes

characteristic

112

C₂

#16837602

yes

characteristic

113

C₂

#16837643

yes

characteristic

114

C₂

#16837677

yes

characteristic

	subgroup	G/H
center	C₂²	#14273701
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₈×Q₈×C₅₉	C₂
socle	C₂²×C₅₉	#121
hypercenter	C₈×Q₈	D₅₉
hypocenter	C₅₉	#1670
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#14273701

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₈×Q₈×C₅₉

C₂

socle

C₂²×C₅₉

#121

hypercenter

C₈×Q₈

D₅₉

hypocenter

C₅₉

#1670

LCS

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
32. Q₈×C₅₉	Q₁₆	semidirect
63. C₅₉	#1670	semidirect
92. Q₈	Dic₂₃₆	semidirect

H ≅ G/N

type

32. Q₈×C₅₉

Q₁₆

semidirect

63. C₅₉

#1670

semidirect

92. Q₈

Dic₂₃₆

semidirect

Subgroups — #21423466

Conjugacy classes of subgroups

Complements