Group #21424834 — subgroups

Subgroups — #21424834

Order 7552.

240 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 109 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×#14273661	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273659	C₂	1	1	yes	maximal, characteristic
3	#16837300	C₂	1	2	yes	maximal
4	#16837165	C₂	1	2	yes	maximal
5	C₂×#97×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
8	C₂×Dic₅₉⋊C₄	—	2	2	no	—
9	#14273661	C₄	1	4	yes	complemented
10	#14273659	—	2	2	no	—
11	C₅₉⋊#95	—	2	2	no	—
12	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
15	#97×C₅₉	—	2	2	no	—
16	C₂²×C₄×C₅₉	D₄	1	2	yes	—
17	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅₉	—	2	2	no	—
19	C₂²×Dic₅₉	—	2	2	no	—
20	C₂×C₈×C₅₉	—	2	1	no	—
21	C₂×C₈×C₅₉	—	2	1	no	—
22	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₅₉	D₄	1	2	yes	—
24	C₂×C₈×C₅₉	Q₈	1	2	yes	—
25	C₂×C₈×C₅₉	Q₈	1	2	yes	—
26	C₄²×C₅₉	—	2	1	no	—
27	C₄²×C₅₉	—	2	1	no	—
28	C₄⋊Dic₅₉	—	2	2	no	—
29	C₄⋊Dic₅₉	—	2	2	no	—
30	C₄⋊Dic₅₉	—	2	2	no	—
31	C₄⋊Dic₅₉	C₂×C₄	1	4	yes	—
32	Dic₅₉⋊C₄	—	4	4	no	—
33	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₅₉	—	2	2	no	—
35	C₂×C₄×C₅₉	—	2	2	no	—
36	C₂×C₄×C₅₉	—	2	2	no	—
37	C₂×C₄×C₅₉	C₂²×C₄	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	#41	1	1	yes	characteristic
42	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
43	C₂×Dic₅₉	—	2	4	no	—
44	C₂×Dic₅₉	—	4	2	no	—
45	C₂×Dic₅₉	—	4	2	no	—
46	C₂×Dic₅₉	—	4	2	no	—
47	C₂×Dic₅₉	—	4	2	no	—
48	C₂×Dic₅₉	—	4	2	no	—
49	C₈×C₅₉	—	2	2	no	—
50	C₈×C₅₉	C₄⋊C₄	1	4	yes	complemented
51	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
52	C₂²×C₅₉	#121	1	1	yes	characteristic
53	C₂²×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
54	C₂²×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
55	C₂²×C₅₉	#137	1	2	yes	—
56	C₄×C₅₉	—	4	2	no	—
57	C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
59	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
60	C₄×C₅₉	#125	1	1	yes	characteristic
61	Dic₅₉	—	4	4	no	—
62	Dic₅₉	—	8	2	no	—
63	#1943	—	59	1	no	maximal, Sylow
64	C₂×C₅₉	#389	1	1	yes	characteristic
65	C₂×C₅₉	#441	1	1	yes	characteristic
66	C₂×C₅₉	#442	1	1	yes	characteristic
67	C₂×C₅₉	#468	1	2	yes	—
68	C₂×C₅₉	#501	1	2	yes	—
69	C₂×#107	—	59	1	no	—
70	C₂×#106	—	59	1	no	—
71	C₂×#97	D₅₉	1	1	yes	characteristic
72	#389	—	59	2	no	—
73	#340	—	59	2	no	—
74	C₅₉	#1943	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
76	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
77	C₂×C₄⋊C₄	—	59	2	no	—
78	C₂×C₄⋊C₄	—	59	2	no	—
79	C₂×C₄⋊C₄	—	118	2	no	—
80	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
81	#107	—	59	4	no	—
82	#106	—	118	2	no	—
83	#97	—	2	2	no	—
84	#95	—	118	2	no	—
85	C₂²×C₄	—	118	2	no	—
86	C₂²×C₄	—	118	2	no	—
87	C₂²×C₄	D₂₃₆	1	2	yes	—
88	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	Dic₁₁₈	1	2	yes	—
92	C₂×C₈	Dic₁₁₈	1	2	yes	—
93	C₂×C₈	C₄×D₅₉	1	2	yes	—
94	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
95	C₄⋊C₄	—	59	4	no	—
96	C₄⋊C₄	—	118	2	no	—
97	C₄⋊C₄	—	118	2	no	—
98	C₄⋊C₄	—	118	2	no	—
99	C₄⋊C₄	—	236	4	no	—
100	C₄²	—	2	1	no	—
101	C₄²	—	2	1	no	—
102	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	118	4	no	—
107	C₂×C₄	—	236	2	no	—
108	C₂×C₄	—	236	2	no	—
109	C₂×C₄	—	236	2	no	—
110	C₂×C₄	—	236	2	no	—
111	C₂×C₄	—	236	2	no	—
112	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
113	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
114	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
115	C₂×C₄	#11745685	1	1	yes	characteristic
116	C₂×C₄	#11745685	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	Dic₅₉⋊C₄	1	4	yes	complemented
120	C₂²	C₄×D₂₃₆	1	1	yes	characteristic
121	C₂²	#14273701	1	1	yes	characteristic
122	C₂²	#14273727	1	1	yes	characteristic
123	C₂²	#14273755	1	2	yes	—
124	C₂²	#14273760	1	2	yes	—
125	C₄	—	4	2	no	—
126	C₄	—	472	2	no	—
127	C₄	—	236	4	no	—
128	C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
129	C₄	C₄×Dic₁₁₈	1	1	yes	characteristic
130	C₄	#14273706	1	1	yes	characteristic
131	C₄	#14273719	1	1	yes	characteristic
132	C₂	#16837300	1	1	yes	characteristic
133	C₂	#16837510	1	1	yes	characteristic
134	C₂	#16837515	1	1	yes	characteristic
135	C₂	#16837648	1	2	yes	—
136	C₂	#16837811	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273661

C₂

yes

maximal, characteristic

C₂×#14273659

C₂

yes

maximal, characteristic

#16837300

C₂

yes

maximal

#16837165

C₂

yes

maximal

C₂×#97×C₅₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×Dic₅₉⋊C₄

—

#14273661

C₄

yes

complemented

#14273659

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

characteristic

#97×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

Q₈

yes

—

C₂×C₈×C₅₉

Q₈

yes

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

Dic₅₉⋊C₄

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₄⋊C₄

yes

complemented

C₂²×C₅₉

C₄×D₄

yes

characteristic

C₂²×C₅₉

#121

yes

characteristic

C₂²×C₅₉

C₄⋊Q₈

yes

characteristic

C₂²×C₅₉

AGL(1,ℤ/8)

yes

—

C₂²×C₅₉

#137

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₉

C₄×Q₈

yes

characteristic

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

#125

yes

characteristic

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#501

yes

—

C₂×#107

—

C₂×#106

—

C₂×#97

D₅₉

yes

characteristic

#389

—

#340

—

C₅₉

#1943

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#107

—

#106

—

118

—

#97

—

#95

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

D₂₃₆

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₁₈

yes

—

C₂×C₈

Dic₁₁₈

yes

—

C₂×C₈

C₄×D₅₉

yes

—

C₂×C₈

C₂²⋊D₅₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

100

C₄²

—

101

C₄²

—

102

C₂³

C₂×D₂₃₆

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

118

—

107

C₂×C₄

—

236

—

108

C₂×C₄

—

236

—

109

C₂×C₄

—

236

—

110

C₂×C₄

—

236

—

111

C₂×C₄

—

236

—

112

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

113

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

114

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

115

C₂×C₄

#11745685

yes

characteristic

116

C₂×C₄

#11745685

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

118

C₈

—

119

C₈

Dic₅₉⋊C₄

yes

complemented

120

C₂²

C₄×D₂₃₆

yes

characteristic

121

C₂²

#14273701

yes

characteristic

122

C₂²

#14273727

yes

characteristic

123

C₂²

#14273755

yes

—

124

C₂²

#14273760

yes

—

125

C₄

—

126

C₄

—

472

—

127

C₄

—

236

—

128

C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

129

C₄

C₄×Dic₁₁₈

yes

characteristic

130

C₄

#14273706

yes

characteristic

131

C₄

#14273719

yes

characteristic

132

C₂

#16837300

yes

characteristic

133

C₂

#16837510

yes

characteristic

134

C₂

#16837515

yes

characteristic

135

C₂

#16837648

yes

—

136

C₂

#16837811

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#97×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×#97	D₅₉
hypocenter	C₅₉	#1943
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#97×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

hypercenter

C₂×#97

D₅₉

hypocenter

C₅₉

#1943

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
9. #14273661	C₄	semidirect
50. C₈×C₅₉	C₄⋊C₄	semidirect
74. C₅₉	#1943	semidirect
119. C₈	Dic₅₉⋊C₄	semidirect

H ≅ G/N

type

9. #14273661

C₄

semidirect

50. C₈×C₅₉

C₄⋊C₄

semidirect

74. C₅₉

#1943

semidirect

119. C₈

Dic₅₉⋊C₄

semidirect

Subgroups — #21424834

Conjugacy classes of subgroups

Complements