Group #21432883 — subgroups

Subgroups — C₂×#16837509 #21432883

Order 7552.

480 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 237 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×Q₈×Dic₅₉	C₂	1	2	yes	maximal
2	#16837509	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14273659	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273651	C₂	1	2	yes	maximal
5	C₂×#14273637	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
8	Q₈×Dic₅₉	—	2	4	no	—
9	Q₈×Dic₅₉	C₂²	1	8	yes	—
10	C₂×C₄⋊Dic₅₉	—	2	2	no	—
11	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
12	C₂×C₄×Dic₅₉	—	2	2	no	—
13	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
14	#14273659	C₂²	1	4	yes	—
15	#14273651	C₂²	1	8	yes	—
16	#14273637	C₂²	1	4	yes	—
17	C₂²×Q₈×C₅₉	C₂²	1	2	yes	—
18	C₂×Q₁₆×C₅₉	C₄	1	8	yes	complemented
19	C₂×Q₁₆×C₅₉	C₂²	1	4	yes	—
20	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₅₉	—	2	2	no	—
22	C₂×Q₈×C₅₉	—	2	4	no	—
23	C₂×Q₈×C₅₉	C₂×C₄	1	8	yes	—
24	C₂×Q₈×C₅₉	C₂³	1	4	yes	—
25	C₂²×C₄×C₅₉	—	2	2	no	—
26	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
27	C₂²×Dic₅₉	—	2	2	no	—
28	C₂²×Dic₅₉	D₄	1	2	yes	—
29	Q₁₆×C₅₉	C₂×C₄	1	16	yes	complemented
30	C₂×C₈×C₅₉	C₂×C₄	1	4	yes	—
31	C₂×C₈×C₅₉	C₂³	1	2	yes	—
32	C₄⋊Dic₅₉	—	2	2	no	—
33	C₄⋊Dic₅₉	—	2	8	no	—
34	C₄⋊Dic₅₉	C₂³	1	4	yes	—
35	C₄×Dic₅₉	—	2	8	no	—
36	C₄×Dic₅₉	D₄	1	2	yes	—
37	C₄×Dic₅₉	C₂³	1	2	yes	—
38	C₂×C₅₉⋊C₈	—	2	2	no	—
39	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
40	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
41	Q₈×C₅₉	—	2	8	no	—
42	Q₈×C₅₉	—	2	4	no	—
43	Q₈×C₅₉	C₂²×C₄	1	8	yes	—
44	C₂×C₄×C₅₉	—	2	8	no	—
45	C₂×C₄×C₅₉	—	2	4	no	—
46	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₉	#41	1	2	yes	—
49	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₅₉	—	2	4	no	—
51	C₂×Dic₅₉	—	2	4	no	—
52	C₂×Dic₅₉	—	4	4	no	—
53	C₂×Dic₅₉	C₂×D₄	1	4	yes	—
54	C₈×C₅₉	C₂²×C₄	1	4	yes	—
55	C₅₉⋊C₈	—	2	2	no	—
56	C₂²×C₅₉	C₄×D₄	1	2	yes	—
57	C₂²×C₅₉	#137	1	4	yes	—
58	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₅₉	—	2	8	no	—
60	C₄×C₅₉	C₄×D₄	1	2	yes	—
61	C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
62	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
63	Dic₅₉	—	2	4	no	—
64	Dic₅₉	—	4	4	no	—
65	C₂×#441	—	59	1	no	maximal, Sylow
66	C₂×C₅₉	#441	1	4	yes	complemented
67	C₂×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
68	C₂×C₅₉	C₂×#137	1	2	yes	—
69	C₂²×Q₁₆	D₅₉	1	1	yes	characteristic
70	C₂×C₄×Q₈	—	59	2	no	—
71	#441	—	59	8	no	—
72	C₂×#106	—	59	1	no	—
73	C₂×#103	—	59	2	no	—
74	C₂×#97	—	59	1	no	—
75	C₅₉	C₂×#441	1	1	yes	characteristic, complemented, Sylow
76	C₂²×Q₈	C₂×D₅₉	1	2	yes	—
77	C₂×Q₁₆	Dic₅₉	1	8	yes	complemented
78	C₂×Q₁₆	C₂×D₅₉	1	4	yes	—
79	C₂²×C₈	—	59	1	no	—
80	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
81	C₄×Q₈	—	59	8	no	—
82	C₄×Q₈	—	118	4	no	—
83	C₂×C₄⋊C₄	—	59	2	no	—
84	C₂×C₄⋊C₄	—	118	2	no	—
85	C₂×C₄²	—	59	1	no	—
86	C₂×C₄²	—	118	2	no	—
87	#106	—	59	4	no	—
88	#103	—	59	8	no	—
89	#97	—	59	4	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	2	4	no	—
92	C₂×Q₈	C₂×Dic₅₉	1	8	yes	—
93	C₂×Q₈	C₂²×D₅₉	1	4	yes	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	59	2	no	—
96	C₂²×C₄	—	118	2	no	—
97	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
98	Q₁₆	C₂×Dic₅₉	1	16	yes	complemented
99	C₂×C₈	—	59	2	no	—
100	C₂×C₈	—	118	2	no	—
101	C₂×C₈	C₂×Dic₅₉	1	4	yes	—
102	C₂×C₈	C₂²×D₅₉	1	2	yes	—
103	C₄⋊C₄	—	59	4	no	—
104	C₄⋊C₄	—	118	8	no	—
105	C₄⋊C₄	—	118	2	no	—
106	C₄²	—	59	2	no	—
107	C₄²	—	59	2	no	—
108	C₄²	—	118	8	no	—
109	C₂³	D₄×D₅₉	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	Q₈	—	2	4	no	—
112	Q₈	C₂²×Dic₅₉	1	8	yes	—
113	C₂×C₄	—	2	8	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	59	4	no	—
116	C₂×C₄	—	118	4	no	—
117	C₂×C₄	—	118	4	no	—
118	C₂×C₄	—	236	4	no	—
119	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
120	C₂×C₄	#11745687	1	2	yes	—
121	C₂×C₄	C₂²×Dic₅₉	1	2	yes	—
122	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
123	C₈	—	118	2	no	—
124	C₈	C₂²×Dic₅₉	1	4	yes	—
125	C₂²	D₄×Dic₅₉	1	2	yes	—
126	C₂²	#14273759	1	4	yes	—
127	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
128	C₄	—	2	8	no	—
129	C₄	—	118	4	no	—
130	C₄	—	236	4	no	—
131	C₄	D₄×Dic₅₉	1	2	yes	—
132	C₄	C₂³×Dic₅₉	1	1	yes	characteristic
133	C₄	C₂×#11745687	1	1	yes	characteristic
134	C₂	#16837509	1	4	yes	direct product factor
135	C₂	C₂×D₄×Dic₅₉	1	1	yes	characteristic
136	C₂	C₂×#14273759	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₅₉

C₂

yes

maximal

#16837509

C₂

yes

maximal, direct product factor

C₂×#14273659

C₂

yes

maximal, characteristic

C₂×#14273651

C₂

yes

maximal

C₂×#14273637

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₅₉

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

Q₈×Dic₅₉

—

Q₈×Dic₅₉

C₂²

yes

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄×Dic₅₉

—

C₂×C₄×Dic₅₉

C₂²

yes

characteristic

#14273659

C₂²

yes

—

#14273651

C₂²

yes

—

#14273637

C₂²

yes

—

C₂²×Q₈×C₅₉

C₂²

yes

—

C₂×Q₁₆×C₅₉

C₄

yes

complemented

C₂×Q₁₆×C₅₉

C₂²

yes

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

C₂×C₄

yes

—

C₂×Q₈×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

D₄

yes

—

Q₁₆×C₅₉

C₂×C₄

yes

complemented

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂³

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

C₄×Dic₅₉

—

C₄×Dic₅₉

D₄

yes

—

C₄×Dic₅₉

C₂³

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂×D₄

yes

—

C₈×C₅₉

C₂²×C₄

yes

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₄×D₄

yes

—

C₂²×C₅₉

#137

yes

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

C₄×D₄

yes

—

C₄×C₅₉

C₂³×C₄

yes

characteristic

C₄×C₅₉

C₂×#41

yes

characteristic

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₂×C₄×D₄

yes

characteristic

C₂×C₅₉

C₂×#137

yes

—

C₂²×Q₁₆

D₅₉

yes

characteristic

C₂×C₄×Q₈

—

#441

—

C₂×#106

—

C₂×#103

—

C₂×#97

—

C₅₉

C₂×#441

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₉

yes

—

C₂×Q₁₆

Dic₅₉

yes

complemented

C₂×Q₁₆

C₂×D₅₉

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

118

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄²

—

C₂×C₄²

—

118

—

#106

—

#103

—

#97

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₅₉

yes

—

C₂×Q₈

C₂²×D₅₉

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

Q₁₆

C₂×Dic₅₉

yes

complemented

C₂×C₈

—

100

C₂×C₈

—

118

—

101

C₂×C₈

C₂×Dic₅₉

yes

—

102

C₂×C₈

C₂²×D₅₉

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

118

—

105

C₄⋊C₄

—

118

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

118

—

109

C₂³

D₄×D₅₉

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

C₂²×Dic₅₉

yes

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

118

—

117

C₂×C₄

—

118

—

118

C₂×C₄

—

236

—

119

C₂×C₄

D₄×D₅₉

yes

characteristic

120

C₂×C₄

#11745687

yes

—

121

C₂×C₄

C₂²×Dic₅₉

yes

—

122

C₂×C₄

C₂³×D₅₉

yes

characteristic

123

C₈

—

118

—

124

C₈

C₂²×Dic₅₉

yes

—

125

C₂²

D₄×Dic₅₉

yes

—

126

C₂²

#14273759

yes

—

127

C₂²

C₂×D₄×D₅₉

yes

characteristic

128

C₄

—

129

C₄

—

118

—

130

C₄

—

236

—

131

C₄

D₄×Dic₅₉

yes

—

132

C₄

C₂³×Dic₅₉

yes

characteristic

133

C₄

C₂×#11745687

yes

characteristic

134

C₂

#16837509

yes

direct product factor

135

C₂

C₂×D₄×Dic₅₉

yes

characteristic

136

C₂

C₂×#14273759

yes

—

	subgroup	G/H
center	C₂³	D₄×D₅₉
commutator	C₄×C₅₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₂²×Q₁₆×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂²×Q₁₆	D₅₉
hypocenter	C₅₉	C₂×#441
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂³

D₄×D₅₉

commutator

C₄×C₅₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₂²×Q₁₆×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

hypercenter

C₂²×Q₁₆

D₅₉

hypocenter

C₅₉

C₂×#441

LCS

G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
2. #16837509	C₂	direct (N × H)
18. C₂×Q₁₆×C₅₉	C₄	semidirect
29. Q₁₆×C₅₉	C₂×C₄	semidirect
66. C₂×C₅₉	#441	semidirect
75. C₅₉	C₂×#441	semidirect
77. C₂×Q₁₆	Dic₅₉	semidirect
98. Q₁₆	C₂×Dic₅₉	semidirect
134. C₂	#16837509	direct (N × H)

H ≅ G/N

type

2. #16837509

C₂

direct (N × H)

18. C₂×Q₁₆×C₅₉

C₄

semidirect

29. Q₁₆×C₅₉

C₂×C₄

semidirect

66. C₂×C₅₉

#441

semidirect

75. C₅₉

C₂×#441

semidirect

77. C₂×Q₁₆

Dic₅₉

semidirect

98. Q₁₆

C₂×Dic₅₉

semidirect

134. C₂

#16837509

direct (N × H)

Subgroups — C₂×#16837509 #21432883

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16837509 #21432883

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16837509 #21432883