Group #21424251 — subgroups

Subgroups — #21424251

Order 7552.

280 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14273659	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273658	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic
4	#16837271	C₂	1	2	yes	maximal
5	C₂×#106×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
8	C₂×Dic₅₉⋊C₄	—	2	4	no	—
9	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
10	#14273659	—	2	2	no	—
11	#14273658	—	2	4	no	—
12	C₈×Dic₅₉	C₄	1	4	yes	complemented
13	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
15	#106×C₅₉	—	2	2	no	—
16	C₂²×C₄×C₅₉	—	2	2	no	—
17	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅₉	—	2	2	no	—
19	C₂²×Dic₅₉	D₄	1	2	yes	—
20	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₅₉	D₄	1	2	yes	—
22	C₂×C₈×C₅₉	Q₈	1	2	yes	—
23	C₄⋊C₄×C₅₉	—	2	2	no	—
24	C₄⋊C₄×C₅₉	—	2	2	no	—
25	C₄⋊Dic₅₉	—	2	2	no	—
26	C₄⋊Dic₅₉	—	2	2	no	—
27	Dic₅₉⋊C₄	—	4	8	no	—
28	C₄×Dic₅₉	C₂×C₄	1	2	yes	—
29	C₄×Dic₅₉	Q₈	1	2	yes	—
30	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
31	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
32	C₂×C₅₉⋊C₈	Q₈	1	2	yes	—
33	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₅₉	—	4	2	no	—
35	C₂×C₄×C₅₉	—	4	4	no	—
36	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×Dic₅₉	—	4	2	no	—
43	C₂×Dic₅₉	—	4	4	no	—
44	C₂×Dic₅₉	C₄⋊C₄	1	4	yes	—
45	C₂×Dic₅₉	SD₁₆	1	8	yes	—
46	C₈×C₅₉	C₄⋊C₄	1	4	yes	complemented
47	C₅₉⋊C₈	C₄⋊C₄	1	4	yes	complemented
48	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₅₉	#127	1	1	yes	characteristic
50	C₂²×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₅₉	C₂×SD₁₆	1	4	yes	—
52	C₄×C₅₉	—	8	2	no	—
53	C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
56	C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
57	Dic₅₉	—	8	2	no	—
58	Dic₅₉	#106	1	8	yes	complemented
59	#1844	—	59	1	no	maximal, Sylow
60	C₂×C₅₉	#384	1	1	yes	characteristic
61	C₂×C₅₉	C₂×#106	1	2	yes	—
62	C₂×C₅₉	#492	1	2	yes	—
63	C₂×C₅₉	#498	1	2	yes	—
64	C₂×#106	—	59	1	no	—
65	C₂×#106	—	59	2	no	—
66	C₂×#106	D₅₉	1	1	yes	characteristic
67	C₂×C₄×C₈	—	59	1	no	—
68	#384	—	59	2	no	—
69	C₅₉	#1844	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₈	—	59	1	no	—
71	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
72	C₂×C₄⋊C₄	—	59	2	no	—
73	C₂×C₄⋊C₄	—	118	4	no	—
74	C₂×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
75	C₂×C₄²	—	59	1	no	—
76	#106	—	2	2	no	—
77	#106	—	118	4	no	—
78	#106	—	118	2	no	—
79	C₄×C₈	—	59	4	no	—
80	C₂²×C₄	—	2	2	no	—
81	C₂²×C₄	—	59	2	no	—
82	C₂²×C₄	—	118	2	no	—
83	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
84	C₂×C₈	—	59	2	no	—
85	C₂×C₈	—	59	2	no	—
86	C₂×C₈	—	59	2	no	—
87	C₂×C₈	Dic₁₁₈	1	2	yes	—
88	C₂×C₈	C₄×D₅₉	1	2	yes	—
89	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	118	2	no	—
93	C₄⋊C₄	—	118	2	no	—
94	C₄⋊C₄	—	236	8	no	—
95	C₄²	—	59	2	no	—
96	C₄²	—	59	2	no	—
97	C₂³	D₄×D₅₉	1	1	yes	characteristic
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	4	4	no	—
100	C₂×C₄	—	59	4	no	—
101	C₂×C₄	—	59	8	no	—
102	C₂×C₄	—	236	2	no	—
103	C₂×C₄	—	236	4	no	—
104	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
106	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
107	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
108	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
109	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
110	C₈	—	59	4	no	—
111	C₈	Dic₅₉⋊C₄	1	4	yes	complemented
112	C₂²	C₄⋊C₄×D₅₉	1	1	yes	characteristic
113	C₂²	#14273723	1	1	yes	characteristic
114	C₂²	#14273726	1	1	yes	characteristic
115	C₂²	SD₁₆×D₅₉	1	4	yes	—
116	C₄	—	8	2	no	—
117	C₄	—	59	8	no	—
118	C₄	—	472	2	no	—
119	C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
120	C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
121	C₄	#14273725	1	1	yes	characteristic
122	C₄	#14273726	1	1	yes	characteristic
123	C₂	#16837271	1	1	yes	characteristic
124	C₂	#106×D₅₉	1	2	yes	—
125	C₂	#16837773	1	2	yes	—
126	C₂	#16837799	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273659

C₂

yes

maximal, characteristic

C₂×#14273658

C₂

yes

maximal

C₂×C₈×Dic₅₉

C₂

yes

maximal, characteristic

#16837271

C₂

yes

maximal

C₂×#106×C₅₉

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×Dic₅₉⋊C₄

—

C₂×C₄×Dic₅₉

C₂²

yes

characteristic

#14273659

—

#14273658

—

C₈×Dic₅₉

C₄

yes

complemented

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

—

#106×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

D₄

yes

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

Q₈

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

C₂×C₄

yes

—

C₄×Dic₅₉

Q₈

yes

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂×C₅₉⋊C₈

Q₈

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₄⋊C₄

yes

—

C₂×Dic₅₉

SD₁₆

yes

—

C₈×C₅₉

C₄⋊C₄

yes

complemented

C₅₉⋊C₈

C₄⋊C₄

yes

complemented

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₉

#127

yes

characteristic

C₂²×C₅₉

C₄⋊Q₈

yes

characteristic

C₂²×C₅₉

C₂×SD₁₆

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₉

C₄⋊Q₈

yes

characteristic

C₄×C₅₉

C₄⋊Q₈

yes

characteristic

Dic₅₉

—

Dic₅₉

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#492

yes

—

C₂×C₅₉

#498

yes

—

C₂×#106

—

C₂×#106

—

C₂×#106

D₅₉

yes

characteristic

C₂×C₄×C₈

—

#384

—

C₅₉

#1844

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

—

C₂×C₄²

—

#106

—

#106

—

118

—

#106

—

118

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₁₈

yes

—

C₂×C₈

C₄×D₅₉

yes

—

C₂×C₈

C₂²⋊D₅₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

C₄²

—

C₄²

—

C₂³

D₄×D₅₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

236

—

103

C₂×C₄

—

236

—

104

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

105

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

106

C₂×C₄

D₄×D₅₉

yes

characteristic

107

C₂×C₄

Q₈×D₅₉

yes

characteristic

108

C₂×C₄

Q₈×D₅₉

yes

characteristic

109

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

110

C₈

—

111

C₈

Dic₅₉⋊C₄

yes

complemented

112

C₂²

C₄⋊C₄×D₅₉

yes

characteristic

113

C₂²

#14273723

yes

characteristic

114

C₂²

#14273726

yes

characteristic

115

C₂²

SD₁₆×D₅₉

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

472

—

119

C₄

C₄⋊C₄×D₅₉

yes

characteristic

120

C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

121

C₄

#14273725

yes

characteristic

122

C₄

#14273726

yes

characteristic

123

C₂

#16837271

yes

characteristic

124

C₂

#106×D₅₉

yes

—

125

C₂

#16837773

yes

—

126

C₂

#16837799

yes

—

	subgroup	G/H
center	C₂³	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#106×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×#106	D₅₉
hypocenter	C₅₉	#1844
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#106×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
12. C₈×Dic₅₉	C₄	semidirect
46. C₈×C₅₉	C₄⋊C₄	semidirect
47. C₅₉⋊C₈	C₄⋊C₄	semidirect
58. Dic₅₉	#106	semidirect
69. C₅₉	#1844	semidirect
111. C₈	Dic₅₉⋊C₄	semidirect

H ≅ G/N

type

12. C₈×Dic₅₉

C₄

semidirect

46. C₈×C₅₉

C₄⋊C₄

semidirect

47. C₅₉⋊C₈

C₄⋊C₄

semidirect

58. Dic₅₉

#106

semidirect

69. C₅₉

#1844

semidirect

111. C₈

Dic₅₉⋊C₄

semidirect

Subgroups — #21424251

Conjugacy classes of subgroups

Complements