Group #21426227 — subgroups

Subgroups — #21426227

Order 7552.

144 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 45 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×#14273672	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837818	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837678	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837673	C₂	1	1	yes	maximal, characteristic
5	#16837217	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837214	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#368	C₂	1	1	yes	maximal, characteristic, complemented
8	#14273760	—	2	1	no	—
9	#14273755	—	2	1	no	—
10	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
11	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
12	#14273733	C₂²	1	1	yes	characteristic
13	#14273672	—	2	1	no	—
14	#14273672	—	2	1	no	—
15	#14273667	C₂²	1	1	yes	characteristic
16	#14273666	—	2	1	no	—
17	C₁₆×D₅₉	—	2	1	no	—
18	#14273645	—	2	1	no	—
19	#14273643	—	2	1	no	—
20	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
21	#108×C₅₉	C₂²	1	1	yes	characteristic
22	#108×C₅₉	C₂²	1	1	yes	characteristic
23	#11745685	D₄	1	1	yes	characteristic
24	C₂×D₂₃₆	—	2	1	no	—
25	C₂×Dic₁₁₈	—	2	1	no	—
26	M₁₆×C₅₉	—	2	1	no	—
27	M₁₆×C₅₉	—	2	1	no	—
28	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	#11745664	D₄	1	1	yes	characteristic
30	Dic₂₃₆	—	2	1	no	—
31	Dic₂₃₆	—	2	1	no	—
32	Dic₂₃₆	—	4	1	no	—
33	D₄₇₂	—	2	1	no	—
34	D₄₇₂	—	2	1	no	—
35	D₄₇₂	—	4	1	no	—
36	#11745660	—	4	1	no	—
37	#11745660	—	4	1	no	—
38	#11745659	—	2	1	no	—
39	C₈×D₅₉	—	2	1	no	—
40	C₁₆×C₅₉	D₄	1	1	yes	characteristic
41	C₁₆×C₅₉	D₄	1	1	yes	characteristic
42	C₅₉⋊C₁₆	—	2	1	no	—
43	C₂²×D₅₉	—	4	1	no	—
44	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂²⋊D₅₉	—	2	1	no	—
46	C₂×Dic₅₉	—	4	1	no	—
47	D₂₃₆	—	4	1	no	—
48	D₂₃₆	—	4	1	no	—
49	D₂₃₆	D₈	1	1	yes	characteristic
50	C₄×D₅₉	—	2	1	no	—
51	Dic₁₁₈	—	4	1	no	—
52	Dic₁₁₈	—	4	1	no	—
53	Dic₁₁₈	D₈	1	1	yes	characteristic
54	C₈×C₅₉	—	4	1	no	—
55	C₈×C₅₉	—	4	1	no	—
56	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₈×C₅₉	#41	1	1	yes	characteristic
58	C₅₉⋊C₈	—	2	1	no	—
59	C₂²×C₅₉	#135	1	1	yes	characteristic
60	C₂×D₅₉	—	2	1	no	—
61	C₂×D₅₉	—	8	1	no	—
62	C₂×D₅₉	—	16	1	no	—
63	C₄×C₅₉	#121	1	1	yes	characteristic
64	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
65	Dic₅₉	—	2	1	no	—
66	Dic₅₉	—	8	1	no	—
67	#2223	—	59	1	no	maximal, Sylow
68	C₂×C₅₉	—	2	1	no	—
69	C₂×C₅₉	#466	1	1	yes	characteristic
70	D₅₉	—	4	1	no	—
71	D₅₉	—	16	1	no	—
72	C₂×SD₃₂	—	59	1	no	—
73	#504	—	59	1	no	—
74	#473	—	59	1	no	—
75	#472	—	59	1	no	—
76	#368	D₅₉	1	1	yes	characteristic, complemented
77	#362	—	59	1	no	—
78	#361	—	59	1	no	—
79	C₅₉	#2223	1	1	yes	characteristic, complemented, Sylow
80	#137	—	118	1	no	—
81	AGL(1,ℤ/8)	—	118	1	no	—
82	C₂×Q₁₆	—	59	1	no	—
83	C₂×D₈	—	59	1	no	—
84	#131	—	59	1	no	—
85	SD₃₂	—	118	1	no	—
86	SD₃₂	—	118	1	no	—
87	M₃₂	—	59	1	no	—
88	M₃₂	—	118	1	no	—
89	C₂×C₁₆	—	118	1	no	—
90	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
91	#108	C₂×D₅₉	1	1	yes	characteristic
92	#108	C₂×D₅₉	1	1	yes	characteristic
93	#101	—	118	1	no	—
94	#100	—	118	1	no	—
95	#41	—	59	1	no	—
96	C₂×Q₈	—	118	1	no	—
97	C₂×D₄	—	118	1	no	—
98	Q₁₆	—	118	1	no	—
99	Q₁₆	—	118	1	no	—
100	Q₁₆	—	236	1	no	—
101	SD₁₆	—	236	1	no	—
102	SD₁₆	—	236	1	no	—
103	D₈	—	118	1	no	—
104	D₈	—	118	1	no	—
105	D₈	—	236	1	no	—
106	M₁₆	—	2	1	no	—
107	M₁₆	—	2	1	no	—
108	M₁₆	—	59	1	no	—
109	M₁₆	—	118	1	no	—
110	C₂×C₈	—	118	1	no	—
111	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
112	C₁₆	—	118	1	no	—
113	C₁₆	D₂₃₆	1	1	yes	characteristic
114	C₁₆	D₂₃₆	1	1	yes	characteristic
115	C₂³	—	236	1	no	—
116	Q₈	—	59	1	no	—
117	Q₈	—	236	1	no	—
118	Q₈	—	236	1	no	—
119	D₄	—	59	1	no	—
120	D₄	—	118	1	no	—
121	D₄	—	236	1	no	—
122	D₄	—	236	1	no	—
123	C₂×C₄	—	118	1	no	—
124	C₂×C₄	—	236	1	no	—
125	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
126	C₈	—	4	1	no	—
127	C₈	—	4	1	no	—
128	C₈	—	118	1	no	—
129	C₈	C₂×D₂₃₆	1	1	yes	characteristic
130	C₈	#11745689	1	1	yes	characteristic
131	C₂²	—	118	1	no	—
132	C₂²	—	944	1	no	—
133	C₂²	—	472	1	no	—
134	C₂²	#14273750	1	1	yes	characteristic
135	C₄	—	118	1	no	—
136	C₄	—	472	1	no	—
137	C₄	#14273702	1	1	yes	characteristic
138	C₄	D₈×D₅₉	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	944	1	no	—
141	C₂	—	236	1	no	—
142	C₂	#16837639	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273672

C₂

yes

maximal, characteristic, complemented

#16837818

C₂

yes

maximal, characteristic, complemented

#16837678

C₂

yes

maximal, characteristic, complemented

#16837673

C₂

yes

maximal, characteristic

#16837217

C₂

yes

maximal, characteristic, complemented

#16837214

C₂

yes

maximal, characteristic, complemented

C₅₉×#368

C₂

yes

maximal, characteristic, complemented

#14273760

—

#14273755

—

C₂×Dic₂₃₆

C₂²

yes

characteristic

C₂×D₄₇₂

C₂²

yes

characteristic

#14273733

C₂²

yes

—

—

C₂²

yes

characteristic

#14273666

—

C₁₆×D₅₉

—

#14273645

—

#14273643

—

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

#108×C₅₉

C₂²

yes

characteristic

#108×C₅₉

C₂²

yes

characteristic

#11745685

D₄

yes

characteristic

C₂×D₂₃₆

—

C₂×Dic₁₁₈

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

#11745664

D₄

yes

characteristic

Dic₂₃₆

—

Dic₂₃₆

—

Dic₂₃₆

—

D₄₇₂

—

D₄₇₂

—

D₄₇₂

—

#11745660

—

#11745660

—

#11745659

—

C₈×D₅₉

—

C₁₆×C₅₉

D₄

yes

characteristic

C₁₆×C₅₉

D₄

yes

characteristic

C₅₉⋊C₁₆

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

D₈

yes

characteristic

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

D₈

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

—

C₂²×C₅₉

#135

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

#121

yes

characteristic

C₄×C₅₉

C₂×D₈

yes

characteristic

Dic₅₉

—

Dic₅₉

—

#2223

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#466

yes

characteristic

D₅₉

—

D₅₉

—

C₂×SD₃₂

—

—

—

—

D₅₉

yes

characteristic, complemented

#362

—

#361

—

C₅₉

#2223

yes

characteristic, complemented, Sylow

#137

—

118

—

AGL(1,ℤ/8)

—

118

—

C₂×Q₁₆

—

C₂×D₈

—

#131

—

SD₃₂

—

118

—

SD₃₂

—

118

—

M₃₂

—

M₃₂

—

118

—

C₂×C₁₆

—

118

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

#108

C₂×D₅₉

yes

characteristic

#108

C₂×D₅₉

yes

characteristic

#101

—

118

—

#100

—

118

—

#41

—

C₂×Q₈

—

118

—

C₂×D₄

—

118

—

Q₁₆

—

118

—

Q₁₆

—

118

—

100

Q₁₆

—

236

—

101

SD₁₆

—

236

—

102

SD₁₆

—

236

—

103

D₈

—

118

—

104

D₈

—

118

—

105

D₈

—

236

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

118

—

110

C₂×C₈

—

118

—

111

C₂×C₈

C₂²×D₅₉

yes

characteristic

112

C₁₆

—

118

—

113

C₁₆

D₂₃₆

yes

characteristic

114

C₁₆

D₂₃₆

yes

characteristic

115

C₂³

—

236

—

116

Q₈

—

117

Q₈

—

236

—

118

Q₈

—

236

—

119

D₄

—

120

D₄

—

118

—

121

D₄

—

236

—

122

D₄

—

236

—

123

C₂×C₄

—

118

—

124

C₂×C₄

—

236

—

125

C₂×C₄

D₄×D₅₉

yes

characteristic

126

C₈

—

127

C₈

—

128

C₈

—

118

—

129

C₈

C₂×D₂₃₆

yes

characteristic

130

C₈

#11745689

yes

characteristic

131

C₂²

—

118

—

132

C₂²

—

944

—

133

C₂²

—

472

—

134

C₂²

#14273750

yes

characteristic

135

C₄

—

118

—

136

C₄

—

472

—

137

C₄

#14273702

yes

characteristic

138

C₄

D₈×D₅₉

yes

characteristic

139

C₂

—

140

C₂

—

944

—

141

C₂

—

236

—

142

C₂

#16837639

yes

characteristic

	subgroup	G/H
center	C₂	#16837639
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#368	C₂
socle	C₂×C₅₉	#466
hypercenter	#368	D₅₉
hypocenter	C₅₉	#2223
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

subgroup

G/H

center

C₂

#16837639

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#368

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

N	H ≅ G/N	type
1. C₂×#14273672	C₂	semidirect
2. #16837818	C₂	semidirect
3. #16837678	C₂	semidirect
5. #16837217	C₂	semidirect
6. #16837214	C₂	semidirect
7. C₅₉×#368	C₂	semidirect
76. #368	D₅₉	semidirect
79. C₅₉	#2223	semidirect

H ≅ G/N

type

1. C₂×#14273672

C₂

semidirect

2. #16837818

C₂

semidirect

3. #16837678

C₂

semidirect

5. #16837217

C₂

semidirect

6. #16837214

C₂

semidirect

7. C₅₉×#368

C₂

semidirect

76. #368

D₅₉

semidirect

79. C₅₉

#2223

semidirect

Subgroups — #21426227

Conjugacy classes of subgroups

Complements