Group #21423478 — subgroups

Subgroups — #21423478

Order 7552.

150 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16837800	C₂	1	1	yes	maximal, characteristic
2	#16837697	C₂	1	1	yes	maximal, characteristic
3	#16837612	C₂	1	1	yes	maximal, characteristic
4	#16837153	C₂	1	1	yes	maximal, characteristic
5	#16837150	C₂	1	1	yes	maximal, characteristic
6	#16837147	C₂	1	1	yes	maximal, characteristic
7	C₈×Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#11745672	—	2	1	no	—
9	#14273727	C₂²	1	1	yes	characteristic
10	#14273727	C₂²	1	1	yes	characteristic
11	#14273660	—	2	1	no	—
12	#14273659	—	2	1	no	—
13	#14273659	—	2	1	no	—
14	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
16	#14273651	—	2	1	no	—
17	#14273650	—	2	1	no	—
18	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₅₉	—	2	1	no	—
20	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₅₉	—	2	1	no	—
22	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
24	C₂×Dic₁₁₈	—	2	1	no	—
25	C₂×Dic₁₁₈	—	2	1	no	—
26	C₂×C₈×C₅₉	—	2	1	no	—
27	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
28	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
29	C₄⋊C₄×C₅₉	—	2	1	no	—
30	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
31	C₄²×C₅₉	—	2	1	no	—
32	C₄²×C₅₉	C₂³	1	1	yes	characteristic
33	#11745672	—	4	2	no	—
34	C₄⋊Dic₅₉	—	2	1	no	—
35	C₄⋊Dic₅₉	—	2	1	no	—
36	C₄⋊Dic₅₉	—	2	1	no	—
37	C₄⋊Dic₅₉	—	2	1	no	—
38	C₂×C₅₉⋊C₈	—	2	1	no	—
39	C₂×C₅₉⋊C₈	—	2	1	no	—
40	Q₈×C₅₉	—	2	1	no	—
41	Q₈×C₅₉	SD₁₆	1	2	yes	—
42	C₂×C₄×C₅₉	—	2	1	no	—
43	C₂×C₄×C₅₉	—	2	1	no	—
44	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
47	C₂×Dic₅₉	—	4	1	no	—
48	C₂×Dic₅₉	—	4	1	no	—
49	Dic₁₁₈	—	4	1	no	—
50	Dic₁₁₈	—	4	1	no	—
51	Dic₁₁₈	—	8	1	no	—
52	C₈×C₅₉	—	2	1	no	—
53	C₈×C₅₉	—	4	1	no	—
54	C₈×C₅₉	Q₁₆	1	2	yes	complemented
55	C₅₉⋊C₈	—	4	1	no	—
56	C₅₉⋊C₈	—	4	1	no	—
57	C₂²×C₅₉	#121	1	1	yes	characteristic
58	C₄×C₅₉	—	2	2	no	—
59	C₄×C₅₉	—	2	1	no	—
60	C₄×C₅₉	—	4	1	no	—
61	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
63	C₄×C₅₉	#135	1	1	yes	characteristic
64	C₄×C₅₉	#137	1	1	yes	characteristic
65	Dic₅₉	—	8	1	no	—
66	Dic₅₉	—	8	1	no	—
67	#1672	—	59	1	no	maximal, Sylow
68	C₂×C₅₉	#462	1	1	yes	characteristic
69	C₂×C₅₉	#465	1	1	yes	characteristic
70	C₂×C₅₉	#471	1	1	yes	characteristic
71	#498	—	59	1	no	—
72	#477	—	59	1	no	—
73	#462	—	59	1	no	—
74	C₈×Q₈	D₅₉	1	1	yes	characteristic
75	#334	—	59	1	no	—
76	#333	—	59	1	no	—
77	#332	—	59	1	no	—
78	C₅₉	#1672	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₁₆	—	118	1	no	—
80	C₄⋊Q₈	—	59	1	no	—
81	C₄⋊Q₈	—	59	1	no	—
82	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
83	#107	—	118	1	no	—
84	#106	—	118	1	no	—
85	#106	—	118	1	no	—
86	C₄⋊C₈	—	2	1	no	—
87	C₄⋊C₈	—	59	1	no	—
88	C₄⋊C₈	—	59	1	no	—
89	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
90	#103	—	118	1	no	—
91	#103	—	118	1	no	—
92	C₄×C₈	—	2	1	no	—
93	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
94	C₂×Q₈	—	118	1	no	—
95	C₂×Q₈	—	118	1	no	—
96	C₂×Q₈	D₂₃₆	1	1	yes	characteristic
97	Q₁₆	—	236	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	118	1	no	—
100	C₂×C₈	—	118	1	no	—
101	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
102	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	118	1	no	—
105	C₄⋊C₄	—	118	1	no	—
106	C₄⋊C₄	—	118	1	no	—
107	C₄⋊C₄	—	118	1	no	—
108	C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	C₂²×D₅₉	1	1	yes	characteristic
111	Q₈	—	2	1	no	—
112	Q₈	—	472	1	no	—
113	Q₈	—	236	1	no	—
114	Q₈	—	236	1	no	—
115	Q₈	#11745660	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	236	1	no	—
119	C₂×C₄	—	236	1	no	—
120	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
121	C₂×C₄	#11745685	1	1	yes	characteristic
122	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	236	1	no	—
126	C₈	—	236	1	no	—
127	C₈	#11745672	1	2	yes	complemented
128	C₂²	#14273701	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	2	no	—
131	C₄	—	4	1	no	—
132	C₄	—	472	1	no	—
133	C₄	—	472	1	no	—
134	C₄	C₂×#11745660	1	1	yes	characteristic
135	C₄	C₂×#11745672	1	1	yes	characteristic
136	C₄	#14273749	1	1	yes	characteristic
137	C₄	#14273757	1	1	yes	characteristic
138	C₂	#16837612	1	1	yes	characteristic
139	C₂	#16837631	1	1	yes	characteristic
140	C₂	#16837665	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837800

C₂

yes

maximal, characteristic

#16837697

C₂

yes

maximal, characteristic

#16837612

C₂

yes

maximal, characteristic

#16837153

C₂

yes

maximal, characteristic

#16837150

C₂

yes

maximal, characteristic

#16837147

C₂

yes

maximal, characteristic

C₈×Q₈×C₅₉

C₂

yes

maximal, characteristic

C₂×#11745672

—

#14273727

C₂²

yes

characteristic

#14273727

C₂²

yes

—

—

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

#14273651

—

#14273650

—

C₄×Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

#11745672

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

Q₈×C₅₉

—

Q₈×C₅₉

SD₁₆

yes

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

Q₁₆

yes

complemented

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

#121

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₄×C₅₉

C₂×Q₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

—

C₈×Q₈

D₅₉

yes

—

—

—

C₅₉

#1672

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

118

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

C₂×D₅₉

yes

characteristic

#107

—

118

—

#106

—

118

—

#106

—

118

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₉

yes

characteristic

#103

—

118

—

#103

—

118

—

C₄×C₈

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

118

—

C₂×Q₈

—

118

—

C₂×Q₈

D₂₃₆

yes

characteristic

Q₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

118

—

100

C₂×C₈

—

118

—

101

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

102

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

118

—

105

C₄⋊C₄

—

118

—

106

C₄⋊C₄

—

118

—

107

C₄⋊C₄

—

118

—

108

C₄⋊C₄

D₂₃₆

yes

characteristic

109

C₄²

—

110

C₄²

C₂²×D₅₉

yes

characteristic

111

Q₈

—

112

Q₈

—

472

—

113

Q₈

—

236

—

114

Q₈

—

236

—

115

Q₈

#11745660

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

236

—

119

C₂×C₄

—

236

—

120

C₂×C₄

C₂×D₂₃₆

yes

characteristic

121

C₂×C₄

#11745685

yes

characteristic

122

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

236

—

126

C₈

—

236

—

127

C₈

#11745672

yes

complemented

128

C₂²

#14273701

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

472

—

133

C₄

—

472

—

134

C₄

C₂×#11745660

yes

characteristic

135

C₄

C₂×#11745672

yes

characteristic

136

C₄

#14273749

yes

characteristic

137

C₄

#14273757

yes

characteristic

138

C₂

#16837612

yes

characteristic

139

C₂

#16837631

yes

characteristic

140

C₂

#16837665

yes

characteristic

	subgroup	G/H
center	C₂²	#14273701
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₈×Q₈×C₅₉	C₂
socle	C₂²×C₅₉	#121
hypercenter	C₈×Q₈	D₅₉
hypocenter	C₅₉	#1672
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#14273701

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₈×Q₈×C₅₉

C₂

socle

C₂²×C₅₉

#121

hypercenter

C₈×Q₈

D₅₉

hypocenter

C₅₉

#1672

LCS

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
54. C₈×C₅₉	Q₁₆	semidirect
78. C₅₉	#1672	semidirect
127. C₈	#11745672	semidirect

H ≅ G/N

type

54. C₈×C₅₉

Q₁₆

semidirect

78. C₅₉

#1672

semidirect

127. C₈

#11745672

semidirect

Subgroups — #21423478

Conjugacy classes of subgroups

Complements