Group #21426133 — subgroups

Subgroups — #21426133

Order 7552.

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

normal subgroups only

This group has 174 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	C₂×#14273671	C₂	1	2	yes	maximal, complemented
2	C₄×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837203	C₂	1	2	yes	maximal, complemented
5	C₅₉×#493	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
7	C₄×Dic₁₁₈	—	2	1	no	—
8	#14273671	—	2	4	no	—
9	C₂×C₅₉⋊C₁₆	C₂²	1	2	yes	—
10	#14273661	C₂²	1	1	yes	characteristic
11	#14273652	—	2	1	no	—
12	C₂×D₈×C₅₉	—	2	1	no	—
13	C₂×D₈×C₅₉	C₂²	1	2	yes	—
14	#127×C₅₉	—	2	1	no	—
15	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×D₄×C₅₉	—	2	2	no	—
17	C₂×D₄×C₅₉	—	4	1	no	—
18	C₂×D₄×C₅₉	—	4	1	no	—
19	C₂×Dic₁₁₈	—	2	1	no	—
20	D₈×C₅₉	—	2	2	no	—
21	D₈×C₅₉	—	2	2	no	—
22	D₈×C₅₉	—	4	2	no	—
23	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
24	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
25	C₄²×C₅₉	D₄	1	1	yes	characteristic
26	C₄⋊Dic₅₉	—	2	1	no	—
27	Dic₅₉⋊C₄	—	4	1	no	—
28	C₄×Dic₅₉	—	4	1	no	—
29	Dic₂₃₆	—	2	1	no	—
30	Dic₂₃₆	D₄	1	2	yes	complemented
31	C₅₉⋊C₁₆	—	2	2	no	—
32	C₂³×C₅₉	—	4	2	no	—
33	D₄×C₅₉	—	4	2	no	—
34	D₄×C₅₉	—	4	2	no	—
35	D₄×C₅₉	—	8	2	no	—
36	D₄×C₅₉	—	8	2	no	—
37	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
38	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
39	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₅₉	—	4	1	no	—
41	C₂×Dic₅₉	—	4	1	no	—
42	Dic₁₁₈	—	2	2	no	—
43	Dic₁₁₈	—	4	1	no	—
44	C₈×C₅₉	—	2	1	no	—
45	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₈×C₅₉	#41	1	1	yes	characteristic
47	C₂²×C₅₉	—	8	2	no	—
48	C₂²×C₅₉	—	8	2	no	—
49	C₂²×C₅₉	—	8	2	no	—
50	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
51	C₄×C₅₉	—	2	1	no	—
52	C₄×C₅₉	SD₃₂	1	2	yes	complemented
53	C₄×C₅₉	#121	1	1	yes	characteristic
54	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
55	Dic₅₉	—	4	2	no	—
56	Dic₅₉	—	8	1	no	—
57	#2207	—	59	1	no	maximal, Sylow
58	C₂×C₅₉	—	16	2	no	—
59	C₂×C₅₉	#459	1	1	yes	characteristic
60	C₂×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
61	C₂×C₅₉	#509	1	1	yes	characteristic
62	C₂×SD₃₂	—	59	2	no	—
63	#493	D₅₉	1	1	yes	characteristic
64	C₄×Q₁₆	—	59	1	no	—
65	C₄⋊C₁₆	—	59	1	no	—
66	#357	—	59	2	no	—
67	C₅₉	#2207	1	1	yes	characteristic, complemented, Sylow
68	C₂×Q₁₆	—	59	1	no	—
69	C₂×D₈	—	2	1	no	—
70	C₂×D₈	C₂×D₅₉	1	2	yes	—
71	#127	—	2	1	no	—
72	C₄×Q₈	—	118	1	no	—
73	SD₃₂	—	118	4	no	—
74	C₂×C₁₆	—	59	2	no	—
75	#107	—	59	1	no	—
76	#103	—	118	1	no	—
77	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
78	C₂×Q₈	—	118	1	no	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	—	4	1	no	—
81	C₂×D₄	—	4	1	no	—
82	Q₁₆	—	59	2	no	—
83	Q₁₆	—	118	1	no	—
84	D₈	—	2	2	no	—
85	D₈	—	2	2	no	—
86	D₈	—	4	2	no	—
87	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
88	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
89	C₄⋊C₄	—	118	1	no	—
90	C₄⋊C₄	—	236	1	no	—
91	C₄²	—	236	1	no	—
92	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
93	C₁₆	—	118	2	no	—
94	C₂³	—	4	2	no	—
95	Q₈	—	118	2	no	—
96	Q₈	—	236	1	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	D₄	—	8	2	no	—
100	D₄	—	8	2	no	—
101	C₂×C₄	—	236	1	no	—
102	C₂×C₄	—	236	1	no	—
103	C₂×C₄	#11745669	1	1	yes	characteristic
104	C₂×C₄	#11745669	1	1	yes	characteristic
105	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
106	C₈	—	2	1	no	—
107	C₈	D₄×D₅₉	1	1	yes	characteristic
108	C₈	#11745687	1	1	yes	characteristic
109	C₂²	—	8	2	no	—
110	C₂²	—	8	2	no	—
111	C₂²	—	8	2	no	—
112	C₂²	C₂×#11745669	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	472	1	no	—
115	C₄	—	236	2	no	—
116	C₄	#14273671	1	2	yes	complemented
117	C₄	#14273698	1	1	yes	characteristic
118	C₄	#14273753	1	1	yes	characteristic
119	C₂	—	16	2	no	—
120	C₂	#16837594	1	1	yes	characteristic
121	C₂	C₂×#14273671	1	1	yes	characteristic
122	C₂	#16837838	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273671

C₂

yes

maximal, complemented

C₄×Dic₂₃₆

C₂

yes

maximal, characteristic, complemented

C₄⋊(C₅₉⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

#16837203

C₂

yes

maximal, complemented

C₅₉×#493

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

C₂²

yes

characteristic

C₄×Dic₁₁₈

—

#14273671

—

C₂×C₅₉⋊C₁₆

C₂²

yes

—

#14273661

C₂²

yes

characteristic

#14273652

—

C₂×D₈×C₅₉

—

C₂×D₈×C₅₉

C₂²

yes

—

#127×C₅₉

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×Dic₁₁₈

—

D₈×C₅₉

—

D₈×C₅₉

—

D₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

Dic₂₃₆

—

Dic₂₃₆

D₄

yes

complemented

C₅₉⋊C₁₆

—

C₂³×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

SD₃₂

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

#2207

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#459

yes

characteristic

C₂×C₅₉

C₂×SD₃₂

yes

characteristic

C₂×C₅₉

#509

yes

characteristic

C₂×SD₃₂

—

#493

D₅₉

yes

characteristic

C₄×Q₁₆

—

C₄⋊C₁₆

—

#357

—

C₅₉

#2207

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

C₂×D₅₉

yes

—

#127

—

C₄×Q₈

—

118

—

SD₃₂

—

118

—

C₂×C₁₆

—

#107

—

#103

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

118

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

118

—

D₈

—

D₈

—

D₈

—

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

C₄²

—

236

—

C₄²

C₂²⋊D₅₉

yes

characteristic

C₁₆

—

118

—

C₂³

—

Q₈

—

118

—

Q₈

—

236

—

D₄

—

D₄

—

D₄

—

100

D₄

—

101

C₂×C₄

—

236

—

102

C₂×C₄

—

236

—

103

C₂×C₄

#11745669

yes

characteristic

104

C₂×C₄

#11745669

yes

characteristic

105

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

106

C₈

—

107

C₈

D₄×D₅₉

yes

characteristic

108

C₈

#11745687

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

C₂×#11745669

yes

characteristic

113

C₄

—

114

C₄

—

472

—

115

C₄

—

236

—

116

C₄

#14273671

yes

complemented

117

C₄

#14273698

yes

characteristic

118

C₄

#14273753

yes

characteristic

119

C₂

—

120

C₂

#16837594

yes

characteristic

121

C₂

C₂×#14273671

yes

characteristic

122

C₂

#16837838

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11745669
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#493	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	#493	D₅₉
hypocenter	C₅₉	#2207
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #493

subgroup

G/H

center

C₂²

C₂×#11745669

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#493

C₂

socle

C₂²×C₅₉

C₂×D₈

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #493

N	H ≅ G/N	type
1. C₂×#14273671	C₂	semidirect
2. C₄×Dic₂₃₆	C₂	semidirect
3. C₄⋊(C₅₉⋊C₁₆)	C₂	semidirect
4. #16837203	C₂	semidirect
30. Dic₂₃₆	D₄	semidirect
52. C₄×C₅₉	SD₃₂	semidirect
67. C₅₉	#2207	semidirect
116. C₄	#14273671	semidirect

H ≅ G/N

type

1. C₂×#14273671

C₂

semidirect

2. C₄×Dic₂₃₆

C₂

semidirect

3. C₄⋊(C₅₉⋊C₁₆)

C₂

semidirect

4. #16837203

C₂

semidirect

30. Dic₂₃₆

D₄

semidirect

52. C₄×C₅₉

SD₃₂

semidirect

67. C₅₉

#2207

semidirect

116. C₄

#14273671

semidirect

Subgroups — #21426133

Conjugacy classes of subgroups

Complements