Group #21426030 — subgroups

Subgroups — #21426030

Order 7552.

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

normal subgroups only

This group has 216 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	C₂×#14273673	C₂	1	2	yes	maximal, complemented
2	#16837643	C₂	1	1	yes	maximal, characteristic
3	#16837206	C₂	1	2	yes	maximal, complemented
4	C₂²⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
7	#14273706	—	2	1	no	—
8	#14273673	—	2	4	no	—
9	C₂×C₅₉⋊C₁₆	C₂²	1	2	yes	—
10	#14273661	C₂²	1	1	yes	characteristic
11	#14273652	—	2	1	no	—
12	C₂²×Q₈×C₅₉	—	2	1	no	—
13	C₂×Q₁₆×C₅₉	—	2	4	no	—
14	C₂×Q₁₆×C₅₉	—	2	1	no	—
15	C₂×Q₁₆×C₅₉	C₂²	1	2	yes	—
16	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₉	—	2	4	no	—
18	C₂×Q₈×C₅₉	—	2	2	no	—
19	C₂×Q₈×C₅₉	—	4	2	no	—
20	C₂×Q₈×C₅₉	—	4	1	no	—
21	C₂²×C₄×C₅₉	—	4	1	no	—
22	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
23	C₂×Dic₁₁₈	—	2	1	no	—
24	Q₁₆×C₅₉	—	2	4	no	—
25	Q₁₆×C₅₉	—	2	2	no	—
26	Q₁₆×C₅₉	D₄	1	4	yes	—
27	C₂×C₈×C₅₉	—	2	2	no	—
28	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
29	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
30	C₂²⋊Dic₅₉	—	4	1	no	—
31	C₄⋊Dic₅₉	—	2	1	no	—
32	Dic₅₉⋊C₄	—	4	1	no	—
33	Dic₂₃₆	—	2	2	no	—
34	C₅₉⋊C₁₆	—	2	2	no	—
35	C₂³×C₅₉	D₈	1	1	yes	characteristic
36	Q₈×C₅₉	—	2	4	no	—
37	Q₈×C₅₉	—	4	4	no	—
38	Q₈×C₅₉	—	4	2	no	—
39	C₂×C₄×C₅₉	—	2	1	no	—
40	C₂×C₄×C₅₉	—	2	1	no	—
41	C₂×C₄×C₅₉	—	4	4	no	—
42	C₂×C₄×C₅₉	—	4	2	no	—
43	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
44	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₅₉	—	4	1	no	—
46	C₂×Dic₅₉	—	4	1	no	—
47	Dic₁₁₈	—	4	1	no	—
48	Dic₁₁₈	—	4	1	no	—
49	C₈×C₅₉	—	2	1	no	—
50	C₈×C₅₉	C₂×D₄	1	2	yes	—
51	C₂²×C₅₉	—	2	1	no	—
52	C₂²×C₅₉	—	2	1	no	—
53	C₂²×C₅₉	Q₃₂	1	2	yes	complemented
54	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
55	C₄×C₅₉	—	2	1	no	—
56	C₄×C₅₉	—	4	4	no	—
57	C₄×C₅₉	#120	1	1	yes	characteristic
58	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
59	Dic₅₉	—	8	1	no	—
60	Dic₅₉	—	8	1	no	—
61	#2188	—	59	1	no	maximal, Sylow
62	C₂×C₅₉	—	2	2	no	—
63	C₂×C₅₉	#447	1	1	yes	characteristic
64	C₂×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
65	C₂×C₅₉	#510	1	1	yes	characteristic
66	C₂²×Q₁₆	D₅₉	1	1	yes	characteristic
67	C₂×Q₃₂	—	59	2	no	—
68	#467	—	59	1	no	—
69	#358	—	59	2	no	—
70	C₂²⋊C₁₆	—	59	1	no	—
71	C₅₉	#2188	1	1	yes	characteristic, complemented, Sylow
72	C₂²×Q₈	—	2	1	no	—
73	C₂×Q₁₆	—	2	4	no	—
74	C₂×Q₁₆	—	2	1	no	—
75	C₂×Q₁₆	—	59	1	no	—
76	C₂×Q₁₆	C₂×D₅₉	1	2	yes	—
77	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
78	C₂²⋊Q₈	—	118	1	no	—
79	Q₃₂	—	118	4	no	—
80	C₂×C₁₆	—	59	2	no	—
81	#107	—	59	1	no	—
82	#103	—	118	1	no	—
83	C₂×Q₈	—	2	4	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	—	4	2	no	—
86	C₂×Q₈	—	4	1	no	—
87	C₂×Q₈	—	118	1	no	—
88	C₂²×C₄	—	4	1	no	—
89	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
90	Q₁₆	—	2	4	no	—
91	Q₁₆	—	2	2	no	—
92	Q₁₆	—	118	2	no	—
93	Q₁₆	C₂²⋊D₅₉	1	4	yes	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
96	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
97	C₄⋊C₄	—	118	1	no	—
98	C₄⋊C₄	—	236	1	no	—
99	C₂²⋊C₄	—	236	1	no	—
100	C₁₆	—	118	2	no	—
101	C₂³	#11745669	1	1	yes	characteristic
102	Q₈	—	2	4	no	—
103	Q₈	—	4	4	no	—
104	Q₈	—	4	2	no	—
105	Q₈	—	236	1	no	—
106	Q₈	—	236	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	4	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	236	1	no	—
112	C₂×C₄	—	236	1	no	—
113	C₂×C₄	#11745669	1	1	yes	characteristic
114	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
115	C₈	—	2	1	no	—
116	C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	#14273673	1	2	yes	complemented
120	C₂²	C₂×#11745669	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	4	4	no	—
123	C₄	—	472	1	no	—
124	C₄	—	472	1	no	—
125	C₄	#14273696	1	1	yes	characteristic
126	C₄	#14273753	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	#16837536	1	1	yes	characteristic
129	C₂	C₂×#14273673	1	1	yes	characteristic
130	C₂	#16837843	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273673

C₂

yes

maximal, complemented

#16837643

C₂

yes

maximal, characteristic

#16837206

C₂

yes

maximal, complemented

C₂²⋊(C₅₉⋊C₁₆)

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₅₉

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

C₂²

yes

characteristic

#14273706

—

#14273673

—

C₂×C₅₉⋊C₁₆

C₂²

yes

—

#14273661

C₂²

yes

characteristic

#14273652

—

C₂²×Q₈×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

C₂²

yes

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂×Dic₁₁₈

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

Dic₂₃₆

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

D₈

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂×D₄

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

Q₃₂

yes

complemented

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

#2188

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#447

yes

characteristic

C₂×C₅₉

C₂×Q₃₂

yes

characteristic

C₂×C₅₉

#510

yes

characteristic

C₂²×Q₁₆

D₅₉

yes

characteristic

C₂×Q₃₂

—

#467

—

#358

—

C₂²⋊C₁₆

—

C₅₉

#2188

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₅₉

yes

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₂²⋊Q₈

—

118

—

Q₃₂

—

118

—

C₂×C₁₆

—

#107

—

#103

—

118

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

118

—

Q₁₆

C₂²⋊D₅₉

yes

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

C₂²⋊C₄

—

236

—

100

C₁₆

—

118

—

101

C₂³

#11745669

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

105

Q₈

—

236

—

106

Q₈

—

236

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

236

—

112

C₂×C₄

—

236

—

113

C₂×C₄

#11745669

yes

characteristic

114

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

115

C₈

—

116

C₈

C₂×C₂²⋊D₅₉

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

#14273673

yes

complemented

120

C₂²

C₂×#11745669

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

472

—

124

C₄

—

472

—

125

C₄

#14273696

yes

characteristic

126

C₄

#14273753

yes

characteristic

127

C₂

—

128

C₂

#16837536

yes

characteristic

129

C₂

C₂×#14273673

yes

characteristic

130

C₂

#16837843

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11745669
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₂²×Q₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	C₂²×Q₁₆	D₅₉
hypocenter	C₅₉	#2188
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

C₂×#11745669

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₂²×Q₁₆×C₅₉

C₂

socle

C₂²×C₅₉

C₂×D₈

hypercenter

C₂²×Q₁₆

D₅₉

hypocenter

C₅₉

#2188

LCS

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
1. C₂×#14273673	C₂	semidirect
3. #16837206	C₂	semidirect
53. C₂²×C₅₉	Q₃₂	semidirect
71. C₅₉	#2188	semidirect
119. C₂²	#14273673	semidirect

H ≅ G/N

type

1. C₂×#14273673

C₂

semidirect

3. #16837206

C₂

semidirect

53. C₂²×C₅₉

Q₃₂

semidirect

71. C₅₉

#2188

semidirect

119. C₂²

#14273673

semidirect

Subgroups — #21426030

Conjugacy classes of subgroups

Complements