Group #14530277 — subgroups

Subgroups — #14530277

Order 2176.

224 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 89 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×#10587489	C₂	1	1	yes	maximal, characteristic
2	#11762707	C₂	1	2	yes	maximal
3	#11762572	C₂	1	2	yes	maximal
4	#11762572	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
11	C₂×#63225	—	2	2	no	—
12	C₁₇×C₄⋊C₈	—	2	2	no	—
13	#10587502	—	2	2	no	—
14	#10587489	—	2	2	no	—
15	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
18	C₂²×Dic₁₇	—	2	2	no	—
19	C₂²×Dic₁₇	—	2	2	no	—
20	C₂×C₈×C₁₇	—	2	2	no	—
21	C₂×C₈×C₁₇	—	2	1	no	—
22	C₂×C₈×C₁₇	—	2	2	no	—
23	C₂×C₈×C₁₇	D₄	1	2	yes	—
24	C₄²×C₁₇	—	2	1	no	—
25	C₄²×C₁₇	—	2	1	no	—
26	C₄⋊Dic₁₇	—	2	2	no	—
27	C₄⋊Dic₁₇	—	2	2	no	—
28	C₄⋊Dic₁₇	—	2	2	no	—
29	C₄⋊Dic₁₇	Q₈	1	4	yes	—
30	#63225	—	4	4	no	—
31	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₇	—	2	1	no	—
33	C₂×C₄×C₁₇	—	2	1	no	—
34	C₂×C₄×C₁₇	—	2	1	no	—
35	C₂×C₄×C₁₇	—	2	2	no	—
36	C₂×C₄×C₁₇	Q₁₆	1	2	yes	—
37	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
40	C₂×C₄×C₁₇	#41	1	2	yes	—
41	C₂×Dic₁₇	—	2	4	no	—
42	C₂×Dic₁₇	—	4	2	no	—
43	C₂×Dic₁₇	—	4	2	no	—
44	C₂×Dic₁₇	—	4	2	no	—
45	C₂×Dic₁₇	—	4	2	no	—
46	C₂×Dic₁₇	—	4	2	no	—
47	C₈×C₁₇	—	2	2	no	—
48	C₈×C₁₇	—	4	1	no	—
49	#2087	—	17	1	no	maximal, Sylow
50	C₂²×C₁₇	#121	1	1	yes	characteristic
51	C₂²×C₁₇	C₂²⋊Q₈	1	2	yes	—
52	C₂²×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
53	C₂²×C₁₇	#135	1	1	yes	characteristic
54	C₂²×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₁₇	—	2	2	no	—
56	C₄×C₁₇	—	4	1	no	—
57	C₄×C₁₇	#123	1	1	yes	characteristic
58	C₄×C₁₇	#125	1	2	yes	—
59	C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
60	Dic₁₇	—	4	4	no	—
61	Dic₁₇	—	8	2	no	—
62	C₂×#107	—	17	1	no	—
63	C₂×C₄⋊C₈	D₁₇	1	1	yes	characteristic
64	#389	—	17	2	no	—
65	#340	—	17	1	no	—
66	#340	—	17	2	no	—
67	C₂×C₁₇	#398	1	1	yes	characteristic
68	C₂×C₁₇	#467	1	1	yes	characteristic
69	C₂×C₁₇	#469	1	1	yes	characteristic
70	C₂×C₁₇	#477	1	2	yes	—
71	C₂×C₁₇	#478	1	2	yes	—
72	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
73	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
74	C₂×C₄⋊C₄	—	17	2	no	—
75	C₂×C₄⋊C₄	—	17	2	no	—
76	C₂×C₄⋊C₄	—	34	2	no	—
77	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
78	#107	—	34	2	no	—
79	C₄⋊C₈	—	2	2	no	—
80	#95	—	34	2	no	—
81	C₁₇	#2087	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₄	—	34	2	no	—
83	C₂²×C₄	—	34	2	no	—
84	C₂²×C₄	D₆₈	1	1	yes	characteristic
85	C₂²×C₄	D₆₈	1	1	yes	characteristic
86	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	C₂²⋊D₁₇	1	2	yes	—
91	C₄⋊C₄	—	17	4	no	—
92	C₄⋊C₄	—	34	2	no	—
93	C₄⋊C₄	—	34	2	no	—
94	C₄⋊C₄	—	34	2	no	—
95	C₄⋊C₄	—	68	4	no	—
96	C₄²	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₂³	C₂×D₆₈	1	1	yes	characteristic
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	34	4	no	—
104	C₂×C₄	—	68	2	no	—
105	C₂×C₄	—	68	2	no	—
106	C₂×C₄	—	68	2	no	—
107	C₂×C₄	—	68	2	no	—
108	C₂×C₄	—	68	2	no	—
109	C₂×C₄	Dic₆₈	1	2	yes	—
110	C₂×C₄	#63252	1	1	yes	characteristic
111	C₂×C₄	#63254	1	2	yes	—
112	C₂×C₄	Q₈×D₁₇	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	4	1	no	—
116	C₂²	#10587583	1	2	yes	—
117	C₂²	#10587590	1	1	yes	characteristic
118	C₂²	C₂×Dic₆₈	1	1	yes	characteristic
119	C₂²	#10587594	1	2	yes	—
120	C₂²	#10587616	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	4	1	no	—
123	C₄	—	68	4	no	—
124	C₄	—	136	2	no	—
125	C₄	#10587575	1	2	yes	—
126	C₄	#10587621	1	1	yes	characteristic
127	C₄	#10587630	1	1	yes	characteristic
128	C₂	#11762739	1	1	yes	characteristic
129	C₂	#11763050	1	1	yes	characteristic
130	C₂	#11763062	1	1	yes	characteristic
131	C₂	#11763105	1	2	yes	—
132	C₂	#11763110	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587489

C₂

yes

maximal, characteristic

#11762707

C₂

yes

maximal

#11762572

C₂

yes

maximal

#11762572

C₂

yes

maximal, characteristic

C₂×C₁₇×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×#63225

—

C₁₇×C₄⋊C₈

—

#10587502

—

#10587489

—

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₄

yes

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

Q₈

yes

—

#63225

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

Q₁₆

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

—

C₂×C₄×C₁₇

#41

yes

characteristic

C₂×C₄×C₁₇

#41

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

—

C₂²×C₁₇

yes

C₂²×C₁₇

C₂²⋊Q₈

yes

—

C₂²×C₁₇

C₂×Q₁₆

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#123

yes

characteristic

C₄×C₁₇

#125

yes

—

C₄×C₁₇

C₄⋊Q₈

yes

characteristic

Dic₁₇

—

Dic₁₇

—

C₂×#107

—

C₂×C₄⋊C₈

D₁₇

yes

—

—

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

—

C₂×C₁₇

#478

yes

—

C₂²×C₈

C₂×D₁₇

yes

characteristic

C₂²×C₈

C₂×D₁₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

#107

—

C₄⋊C₈

—

#95

—

C₁₇

#2087

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₆₈

yes

characteristic

C₂²×C₄

D₆₈

yes

characteristic

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₁₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₂³

C₂×D₆₈

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

Dic₆₈

yes

—

110

C₂×C₄

#63252

yes

characteristic

111

C₂×C₄

#63254

yes

—

112

C₂×C₄

Q₈×D₁₇

yes

—

113

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

114

C₈

—

115

C₈

—

116

C₂²

#10587583

yes

—

117

C₂²

#10587590

yes

characteristic

118

C₂²

C₂×Dic₆₈

yes

characteristic

119

C₂²

#10587594

yes

—

120

C₂²

#10587616

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

136

—

125

C₄

#10587575

yes

—

126

C₄

#10587621

yes

characteristic

127

C₄

#10587630

yes

characteristic

128

C₂

#11762739

yes

characteristic

129

C₂

#11763050

yes

characteristic

130

C₂

#11763062

yes

characteristic

131

C₂

#11763105

yes

—

132

C₂

#11763110

yes

—

	subgroup	G/H
center	C₂³	C₂×D₆₈
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂×C₁₇×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₁₇
hypocenter	C₁₇	#2087
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₆₈

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₂×C₁₇×C₄⋊C₈

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₁₇

hypocenter

C₁₇

#2087

LCS

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
81. C₁₇	#2087	semidirect

H ≅ G/N

type

81. C₁₇

#2087

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#10587489	2
#11762707	2
#11762572	2
C₂×C₁₇×C₄⋊C₈	2
#2087	17

subgroup

index

C₂×C₁₇×C₄⋊C₈

Subgroups — #14530277

Conjugacy classes of subgroups

Complements

Maximal subgroups