Group #21424874 — subgroups

Subgroups — #21424874

Order 7552.

196 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 89 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×#14273663	C₂	1	1	yes	maximal, characteristic
2	#16837442	C₂	1	2	yes	maximal
3	#16837357	C₂	1	1	yes	maximal, characteristic
4	#16837171	C₂	1	2	yes	maximal
5	C₂×#106×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745664	C₂²	1	2	yes	—
7	C₂×#11745664	C₂²	1	1	yes	characteristic
8	#14273686	C₂²	1	1	yes	characteristic
9	#14273663	C₄	1	4	yes	complemented
10	#14273660	—	2	2	no	—
11	#14273658	—	2	2	no	—
12	#14273637	—	2	1	no	—
13	C₈×Dic₅₉	—	2	1	no	—
14	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
16	#106×C₅₉	—	2	1	no	—
17	#106×C₅₉	—	2	1	no	—
18	C₂²×C₄×C₅₉	—	2	2	no	—
19	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
20	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
22	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
23	C₂×C₈×C₅₉	Q₈	1	2	yes	—
24	C₄⋊C₄×C₅₉	—	2	2	no	—
25	C₄⋊C₄×C₅₉	—	2	2	no	—
26	C₂²⋊Dic₅₉	—	2	1	no	—
27	C₄⋊Dic₅₉	D₄	1	2	yes	—
28	C₄×Dic₅₉	—	2	1	no	—
29	#11745664	—	2	2	no	—
30	#11745664	—	2	1	no	—
31	#11745664	C₂×C₄	1	4	yes	—
32	#11745664	Q₈	1	2	yes	—
33	C₂×C₅₉⋊C₈	—	2	1	no	—
34	C₂×C₅₉⋊C₈	—	2	2	no	—
35	C₂³×C₅₉	#41	1	1	yes	characteristic
36	C₂×C₄×C₅₉	—	4	2	no	—
37	C₂×C₄×C₅₉	—	8	2	no	—
38	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
44	C₂×Dic₅₉	—	2	2	no	—
45	C₈×C₅₉	C₄⋊C₄	1	2	yes	—
46	C₈×C₅₉	C₄⋊C₄	1	2	yes	—
47	C₅₉⋊C₈	—	2	4	no	—
48	C₅₉⋊C₈	—	2	2	no	—
49	C₂²×C₅₉	—	2	2	no	—
50	C₂²×C₅₉	C₄×Q₈	1	1	yes	characteristic
51	C₂²×C₅₉	#121	1	1	yes	characteristic
52	C₂²×C₅₉	#125	1	1	yes	characteristic
53	C₄×C₅₉	—	8	2	no	—
54	C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
56	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
57	C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
58	Dic₅₉	—	4	2	no	—
59	#1951	—	59	1	no	maximal, Sylow
60	C₂×C₅₉	—	2	1	no	—
61	C₂×C₅₉	—	2	1	no	—
62	C₂×C₅₉	#389	1	1	yes	characteristic
63	C₂×C₅₉	#471	1	2	yes	—
64	C₂×#108	—	59	1	no	—
65	#428	—	59	2	no	—
66	C₂×#106	D₅₉	1	1	yes	characteristic
67	#405	—	59	1	no	—
68	#343	—	59	2	no	—
69	C₅₉	#1951	1	1	yes	characteristic, complemented, Sylow
70	C₂×M₁₆	—	59	1	no	—
71	C₂×M₁₆	—	59	2	no	—
72	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
73	#117	—	59	1	no	—
74	C₂×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
75	#108	—	59	4	no	—
76	#107	—	118	2	no	—
77	#106	—	2	1	no	—
78	#106	—	2	1	no	—
79	#106	—	118	2	no	—
80	#97	—	118	1	no	—
81	C₄×C₈	—	118	1	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
84	M₁₆	—	59	4	no	—
85	M₁₆	—	59	2	no	—
86	M₁₆	—	118	2	no	—
87	M₁₆	—	118	1	no	—
88	C₂×C₈	—	118	2	no	—
89	C₂×C₈	—	118	1	no	—
90	C₂×C₈	Dic₁₁₈	1	2	yes	—
91	C₂×C₈	C₄×D₅₉	1	2	yes	—
92	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
93	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	59	2	no	—
97	C₂²⋊C₄	—	118	1	no	—
98	C₄²	—	118	1	no	—
99	C₂³	#11745687	1	1	yes	characteristic
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	8	2	no	—
102	C₂×C₄	—	118	2	no	—
103	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
104	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
105	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
107	C₂×C₄	#11745689	1	1	yes	characteristic
108	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
109	C₈	—	118	4	no	—
110	C₈	—	118	2	no	—
111	C₈	Dic₅₉⋊C₄	1	2	yes	—
112	C₈	Dic₅₉⋊C₄	1	2	yes	—
113	C₂²	—	2	2	no	—
114	C₂²	#14273692	1	1	yes	characteristic
115	C₂²	#14273698	1	1	yes	characteristic
116	C₂²	#14273718	1	1	yes	characteristic
117	C₄	—	8	2	no	—
118	C₄	—	236	2	no	—
119	C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
120	C₄	#14273689	1	1	yes	characteristic
121	C₄	#14273703	1	1	yes	characteristic
122	C₄	#14273726	1	1	yes	characteristic
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	#16837298	1	1	yes	characteristic
126	C₂	#16837662	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273663

C₂

yes

maximal, characteristic

#16837442

C₂

yes

maximal

#16837357

C₂

yes

maximal, characteristic

#16837171

C₂

yes

maximal

C₂×#106×C₅₉

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

—

C₂×#11745664

C₂²

yes

characteristic

#14273686

C₂²

yes

characteristic

#14273663

C₄

yes

—

—

—

C₈×Dic₅₉

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

—

#106×C₅₉

—

#106×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

Q₈

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

—

C₄×Dic₅₉

—

#11745664

—

#11745664

—

#11745664

C₂×C₄

yes

—

#11745664

Q₈

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#41

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

#41

yes

characteristic

C₂×Dic₅₉

—

C₈×C₅₉

C₄⋊C₄

yes

—

C₈×C₅₉

C₄⋊C₄

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

C₄×Q₈

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

C₄⋊Q₈

yes

characteristic

Dic₅₉

—

#1951

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#389

yes

characteristic

C₂×C₅₉

#471

yes

—

C₂×#108

—

#428

—

C₂×#106

D₅₉

yes

characteristic

#405

—

#343

—

C₅₉

#1951

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

#117

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

—

#108

—

#107

—

118

—

#106

—

#106

—

#106

—

118

—

#97

—

118

—

C₄×C₈

—

118

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

118

—

M₁₆

—

118

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

Dic₁₁₈

yes

—

C₂×C₈

C₄×D₅₉

yes

—

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

118

—

C₄²

—

118

—

C₂³

#11745687

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

118

—

103

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

104

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

105

C₂×C₄

D₄×D₅₉

yes

characteristic

106

C₂×C₄

Q₈×D₅₉

yes

characteristic

107

C₂×C₄

#11745689

yes

characteristic

108

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

109

C₈

—

118

—

110

C₈

—

118

—

111

C₈

Dic₅₉⋊C₄

yes

—

112

C₈

Dic₅₉⋊C₄

yes

—

113

C₂²

—

114

C₂²

#14273692

yes

characteristic

115

C₂²

#14273698

yes

characteristic

116

C₂²

#14273718

yes

characteristic

117

C₄

—

118

C₄

—

236

—

119

C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

120

C₄

#14273689

yes

characteristic

121

C₄

#14273703

yes

characteristic

122

C₄

#14273726

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#16837298

yes

characteristic

126

C₂

#16837662

yes

—

	subgroup	G/H
center	C₂²	#14273698
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#106×C₅₉	C₂
socle	C₂²×C₅₉	#121
hypercenter	C₂×#106	D₅₉
hypocenter	C₅₉	#1951
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂²

#14273698

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#106×C₅₉

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
9. #14273663	C₄	semidirect
69. C₅₉	#1951	semidirect

H ≅ G/N

type

9. #14273663

C₄

semidirect

69. C₅₉

#1951

semidirect

Subgroups — #21424874

Conjugacy classes of subgroups

Complements