Group #14530322 — subgroups

Subgroups — #14530322

Order 2176.

254 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 93 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×#10587478	C₂	1	2	yes	maximal
2	C₂×#10587506	C₂	1	2	yes	maximal
3	#11762679	C₂	1	1	yes	maximal, characteristic
4	#11762564	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#391	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₇	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₇	—	2	2	no	—
10	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
11	C₂²×#3622	C₂²	1	2	yes	—
12	C₁₇×#95	—	2	2	no	—
13	#10587506	—	2	4	no	—
14	#10587478	—	2	4	no	—
15	C₂×Q₈×C₁₇	—	2	1	no	—
16	C₂×Q₈×C₁₇	—	2	2	no	—
17	C₂×Q₈×C₁₇	—	2	2	no	—
18	C₂×Q₈×C₁₇	Q₈	1	2	yes	—
19	C₂²×C₄×C₁₇	—	2	1	no	—
20	C₂²×C₄×C₁₇	—	2	1	no	—
21	C₂²×C₄×C₁₇	D₄	1	2	yes	—
22	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
23	C₂²×Dic₁₇	—	2	2	no	—
24	C₄⋊C₄×C₁₇	—	2	2	no	—
25	C₄²×C₁₇	—	2	2	no	—
26	C₄⋊Dic₁₇	—	2	2	no	—
27	C₄⋊Dic₁₇	—	2	2	no	—
28	C₄⋊Dic₁₇	—	4	4	no	—
29	C₂×#3622	—	2	2	no	—
30	C₂×#3622	—	2	2	no	—
31	C₂×#3622	D₄	1	4	yes	—
32	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
33	Q₈×C₁₇	—	2	4	no	—
34	Q₈×C₁₇	—	4	2	no	—
35	C₂×C₄×C₁₇	—	2	2	no	—
36	C₂×C₄×C₁₇	—	2	2	no	—
37	C₂×C₄×C₁₇	—	2	2	no	—
38	C₂×C₄×C₁₇	—	4	1	no	—
39	C₂×C₄×C₁₇	—	4	1	no	—
40	C₂×C₄×C₁₇	—	4	1	no	—
41	C₂×C₄×C₁₇	—	4	2	no	—
42	C₂×C₄×C₁₇	Q₁₆	1	4	yes	—
43	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
46	C₂×C₄×C₁₇	#41	1	2	yes	—
47	C₂×Dic₁₇	—	4	2	no	—
48	C₂×Dic₁₇	—	4	4	no	—
49	#3622	—	2	4	no	—
50	#2095	—	17	1	no	maximal, Sylow
51	C₂²×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
52	C₂²×C₁₇	#123	1	1	yes	characteristic
53	C₂²×C₁₇	#127	1	1	yes	characteristic
54	C₂²×C₁₇	C₂×Q₁₆	1	2	yes	—
55	C₂²×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
56	C₄×C₁₇	—	2	4	no	—
57	C₄×C₁₇	—	4	2	no	—
58	C₄×C₁₇	—	8	1	no	—
59	C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
60	C₄×C₁₇	#123	1	2	yes	—
61	Dic₁₇	—	8	2	no	—
62	C₂×#107	—	17	2	no	—
63	C₂×#103	—	17	2	no	—
64	#391	D₁₇	1	1	yes	characteristic
65	#384	—	17	1	no	—
66	#336	—	17	1	no	—
67	C₂×C₁₇	#399	1	1	yes	characteristic
68	C₂×C₁₇	#477	1	2	yes	—
69	C₂×C₁₇	#484	1	2	yes	—
70	C₂×C₁₇	#494	1	1	yes	characteristic
71	C₂×C₁₇	#496	1	1	yes	characteristic
72	C₂²×Q₈	C₂×D₁₇	1	1	yes	characteristic
73	C₂²×C₈	—	17	2	no	—
74	C₂×C₄⋊C₄	—	17	2	no	—
75	C₂×C₄⋊C₄	—	34	2	no	—
76	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
77	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
78	#107	—	34	4	no	—
79	#103	—	34	4	no	—
80	#95	—	2	2	no	—
81	C₁₇	#2095	1	1	yes	characteristic, complemented, Sylow
82	C₂×Q₈	—	2	1	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	Dic₃₄	1	2	yes	—
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	34	2	no	—
89	C₂²×C₄	C₂²⋊D₁₇	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
91	C₂×C₈	—	17	4	no	—
92	C₂×C₈	—	34	2	no	—
93	C₂×C₈	—	34	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	34	2	no	—
96	C₄⋊C₄	—	34	2	no	—
97	C₄⋊C₄	—	68	4	no	—
98	C₄²	—	2	2	no	—
99	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
100	Q₈	—	2	4	no	—
101	Q₈	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	68	2	no	—
110	C₂×C₄	—	68	4	no	—
111	C₂×C₄	#63231	1	4	yes	—
112	C₂×C₄	C₂×Dic₃₄	1	1	yes	characteristic
113	C₂×C₄	#63252	1	1	yes	characteristic
114	C₂×C₄	D₄×D₁₇	1	2	yes	—
115	C₂×C₄	#63254	1	2	yes	—
116	C₈	—	34	4	no	—
117	C₂²	#10587610	1	1	yes	characteristic
118	C₂²	#10587621	1	1	yes	characteristic
119	C₂²	#10587626	1	1	yes	characteristic
120	C₂²	C₂×#63231	1	2	yes	—
121	C₂²	#10587635	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	4	2	no	—
124	C₄	—	8	1	no	—
125	C₄	—	136	2	no	—
126	C₄	#10587564	1	2	yes	—
127	C₄	#10587569	1	2	yes	—
128	C₂	#11762745	1	1	yes	characteristic
129	C₂	#11763104	1	2	yes	—
130	C₂	#11763142	1	2	yes	—
131	C₂	#11763188	1	1	yes	characteristic
132	C₂	#11763199	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587478

C₂

yes

maximal

C₂×#10587506

C₂

yes

maximal

#11762679

C₂

yes

maximal, characteristic

#11762564

C₂

yes

maximal, characteristic

C₁₇×#391

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇

C₂²

yes

characteristic

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂²×#3622

C₂²

yes

—

C₁₇×#95

—

#10587506

—

#10587478

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

Q₈

yes

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₂×#3622

—

C₂×#3622

—

C₂×#3622

D₄

yes

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

Q₁₆

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×Q₈

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

#3622

—

#2095

—

maximal, Sylow

C₂²×C₁₇

C₂²⋊Q₈

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

C₂×Q₁₆

yes

—

C₂²×C₁₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂²⋊Q₈

yes

—

C₄×C₁₇

#123

yes

—

Dic₁₇

—

C₂×#107

—

C₂×#103

—

#391

D₁₇

yes

characteristic

#384

—

#336

—

C₂×C₁₇

#399

yes

characteristic

C₂×C₁₇

#477

yes

—

C₂×C₁₇

#484

yes

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×Q₈

C₂×D₁₇

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

characteristic

C₂×C₄²

C₂×D₁₇

yes

—

—

—

C₁₇

#2095

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₃₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₂³

C₂×C₂²⋊D₁₇

yes

characteristic

100

Q₈

—

101

Q₈

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

#63231

yes

—

112

C₂×C₄

C₂×Dic₃₄

yes

characteristic

113

C₂×C₄

#63252

yes

characteristic

114

C₂×C₄

D₄×D₁₇

yes

—

115

C₂×C₄

#63254

yes

—

116

C₈

—

117

C₂²

#10587610

yes

characteristic

118

C₂²

#10587621

yes

characteristic

119

C₂²

#10587626

yes

characteristic

120

C₂²

C₂×#63231

yes

—

121

C₂²

#10587635

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

136

—

126

C₄

#10587564

yes

—

127

C₄

#10587569

yes

—

128

C₂

#11762745

yes

characteristic

129

C₂

#11763104

yes

—

130

C₂

#11763142

yes

—

131

C₂

#11763188

yes

characteristic

132

C₂

#11763199

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₇
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#391	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₄
hypercenter	#391	D₁₇
hypocenter	C₁₇	#2095
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₇

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#391

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₄

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
81. C₁₇	#2095	semidirect

H ≅ G/N

type

81. C₁₇

#2095

semidirect

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

subgroup	index
C₂×#10587478	2
C₂×#10587506	2
#11762679	2
#11762564	2
C₁₇×#391	2
#2095	17

subgroup

index

C₁₇×#391

Subgroups — #14530322

Conjugacy classes of subgroups

Complements

Maximal subgroups