Group #21426246 — subgroups

Subgroups — #21426246

Order 7552.

130 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16837806	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	#16837228	C₂	1	1	yes	maximal, characteristic
4	#16837227	C₂	1	1	yes	maximal, characteristic
5	Dic₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	#16837206	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#358	C₂	1	1	yes	maximal, characteristic
8	#14273726	—	2	1	no	—
9	Q₈×Dic₅₉	—	2	1	no	—
10	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	#14273661	C₂²	1	1	yes	characteristic
12	#14273661	C₂²	1	1	yes	characteristic
13	#14273660	—	2	1	no	—
14	#14273651	—	2	1	no	—
15	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
18	#107×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₅₉	—	2	1	no	—
20	C₂×Dic₁₁₈	—	4	1	no	—
21	Q₁₆×C₅₉	—	2	1	no	—
22	Q₁₆×C₅₉	Q₈	1	2	yes	complemented
23	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₅₉	—	2	1	no	—
25	C₄⋊Dic₅₉	—	2	1	no	—
26	C₄⋊Dic₅₉	—	2	1	no	—
27	C₄⋊Dic₅₉	—	4	1	no	—
28	Dic₅₉⋊C₄	—	4	1	no	—
29	C₄×Dic₅₉	—	4	1	no	—
30	C₄×Dic₅₉	D₄	1	1	yes	characteristic
31	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
32	C₁₆×C₅₉	—	2	1	no	—
33	C₅₉⋊C₁₆	—	2	1	no	—
34	Q₈×C₅₉	—	2	2	no	—
35	Q₈×C₅₉	—	4	1	no	—
36	C₂×C₄×C₅₉	—	4	1	no	—
37	C₂×C₄×C₅₉	—	4	1	no	—
38	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₅₉	—	4	1	no	—
40	C₂×Dic₅₉	—	4	1	no	—
41	C₂×Dic₅₉	D₈	1	1	yes	characteristic
42	C₂×Dic₅₉	D₈	1	1	yes	characteristic
43	Dic₁₁₈	—	8	2	no	—
44	C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₅₉	#41	1	1	yes	characteristic
46	C₅₉⋊C₈	—	2	1	no	—
47	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
48	C₄×C₅₉	—	4	2	no	—
49	C₄×C₅₉	—	8	1	no	—
50	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
51	C₄×C₅₉	#137	1	1	yes	characteristic
52	Dic₅₉	—	2	1	no	—
53	Dic₅₉	—	8	1	no	—
54	Dic₅₉	—	8	1	no	—
55	Dic₅₉	Q₃₂	1	2	yes	—
56	#2227	—	59	1	no	maximal, Sylow
57	C₂×C₅₉	#474	1	1	yes	characteristic
58	C₂×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
59	C₂×C₅₉	#509	1	1	yes	characteristic
60	#500	—	59	1	no	—
61	C₄×Q₁₆	—	59	1	no	—
62	#366	—	59	1	no	—
63	#366	—	59	1	no	—
64	C₄⋊C₁₆	—	59	1	no	—
65	#358	—	59	1	no	—
66	#358	D₅₉	1	1	yes	characteristic
67	C₅₉	#2227	1	1	yes	characteristic, complemented, Sylow
68	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
69	C₄⋊Q₈	—	118	1	no	—
70	C₄×Q₈	—	118	1	no	—
71	C₂×C₁₆	—	59	1	no	—
72	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
73	#107	—	59	1	no	—
74	#107	—	59	1	no	—
75	#107	—	118	1	no	—
76	#107	C₂×D₅₉	1	1	yes	characteristic
77	#103	—	118	1	no	—
78	C₄×C₈	—	59	1	no	—
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	236	1	no	—
81	Q₁₆	—	2	1	no	—
82	Q₁₆	Dic₁₁₈	1	2	yes	complemented
83	C₂×C₈	—	59	1	no	—
84	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
85	C₄⋊C₄	—	2	1	no	—
86	C₄⋊C₄	—	118	1	no	—
87	C₄⋊C₄	—	118	1	no	—
88	C₄⋊C₄	—	236	1	no	—
89	C₄⋊C₄	—	236	1	no	—
90	C₄²	—	59	1	no	—
91	C₄²	—	236	1	no	—
92	C₁₆	—	2	1	no	—
93	C₁₆	—	118	1	no	—
94	Q₈	—	2	2	no	—
95	Q₈	—	4	1	no	—
96	Q₈	—	472	2	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	59	1	no	—
100	C₂×C₄	—	59	1	no	—
101	C₂×C₄	—	236	1	no	—
102	C₂×C₄	—	236	1	no	—
103	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
104	C₈	—	118	1	no	—
105	C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
106	C₈	#11745687	1	1	yes	characteristic
107	C₂²	D₈×D₅₉	1	1	yes	characteristic
108	C₄	—	4	2	no	—
109	C₄	—	8	1	no	—
110	C₄	—	59	2	no	—
111	C₄	—	118	1	no	—
112	C₄	—	472	1	no	—
113	C₄	—	472	1	no	—
114	C₄	#14273704	1	1	yes	characteristic
115	C₄	#14273756	1	1	yes	characteristic
116	C₂	#16837680	1	1	yes	characteristic
117	C₂	Q₃₂×D₅₉	1	1	yes	characteristic
118	C₂	#16837839	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837806

C₂

yes

maximal, characteristic

Q₁₆×Dic₅₉

C₂

yes

maximal, characteristic

#16837228

C₂

yes

maximal, characteristic

#16837227

C₂

yes

maximal, characteristic

Dic₅₉⋊C₁₆

C₂

yes

maximal, characteristic

#16837206

C₂

yes

maximal, characteristic

C₅₉×#358

C₂

yes

maximal, characteristic

#14273726

—

Q₈×Dic₅₉

—

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273660

—

#14273651

—

C₈×Dic₅₉

C₂²

yes

characteristic

C₂×Q₁₆×C₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

#107×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Dic₁₁₈

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

Q₈

yes

complemented

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

D₈

yes

characteristic

C₂×Dic₅₉

D₈

yes

characteristic

Dic₁₁₈

—

C₈×C₅₉

C₂×Q₈

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

#137

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

Q₃₂

yes

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₂×Q₃₂

yes

C₂×C₅₉

yes

—

C₄×Q₁₆

—

#366

—

#366

—

C₄⋊C₁₆

—

#358

—

#358

D₅₉

yes

characteristic

C₅₉

#2227

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×D₅₉

yes

characteristic

C₄⋊Q₈

—

118

—

C₄×Q₈

—

118

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₉

yes

—

—

—

118

—

#107

C₂×D₅₉

yes

characteristic

#103

—

118

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

236

—

Q₁₆

—

Q₁₆

Dic₁₁₈

yes

complemented

C₂×C₈

—

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

C₄⋊C₄

—

236

—

C₄²

—

C₄²

—

236

—

C₁₆

—

C₁₆

—

118

—

Q₈

—

Q₈

—

Q₈

—

472

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

236

—

102

C₂×C₄

—

236

—

103

C₂×C₄

D₄×D₅₉

yes

characteristic

104

C₈

—

118

—

105

C₈

C₂×Dic₁₁₈

yes

characteristic

106

C₈

#11745687

yes

characteristic

107

C₂²

D₈×D₅₉

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

118

—

112

C₄

—

472

—

113

C₄

—

472

—

114

C₄

#14273704

yes

characteristic

115

C₄

#14273756

yes

characteristic

116

C₂

#16837680

yes

characteristic

117

C₂

Q₃₂×D₅₉

yes

characteristic

118

C₂

#16837839

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#358	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	#358	D₅₉
hypocenter	C₅₉	#2227
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₅₉

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#358

C₂

socle

C₂²×C₅₉

C₂×D₈

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
22. Q₁₆×C₅₉	Q₈	semidirect
67. C₅₉	#2227	semidirect
82. Q₁₆	Dic₁₁₈	semidirect

H ≅ G/N

type

22. Q₁₆×C₅₉

Q₈

semidirect

67. C₅₉

#2227

semidirect

82. Q₁₆

Dic₁₁₈

semidirect

Subgroups — #21426246

Conjugacy classes of subgroups

Complements