Group #14528913 — subgroups

Subgroups — #14528913

Order 2176.

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

normal subgroups only

This group has 216 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×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
2	#11762674	C₂	1	2	yes	maximal
3	#11762572	C₂	1	1	yes	maximal, characteristic
4	#11762571	C₂	1	2	yes	maximal
5	C₁₇×#340	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
9	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
10	C₂²×#3622	C₂²	1	1	yes	characteristic
11	#10587502	—	2	2	no	—
12	#10587502	—	2	2	no	—
13	C₈×Dic₁₇	C₄	1	4	yes	complemented
14	C₂²×C₄×C₁₇	—	2	2	no	—
15	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
16	C₂²×Dic₁₇	—	2	2	no	—
17	C₂²×Dic₁₇	D₄	1	1	yes	characteristic
18	C₂²×Dic₁₇	D₄	1	1	yes	characteristic
19	C₂×C₈×C₁₇	—	2	2	no	—
20	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
21	C₄⋊C₄×C₁₇	—	2	2	no	—
22	C₄⋊C₄×C₁₇	—	2	2	no	—
23	C₄⋊Dic₁₇	—	2	2	no	—
24	C₄⋊Dic₁₇	—	2	2	no	—
25	C₄×Dic₁₇	C₂×C₄	1	2	yes	—
26	C₄×Dic₁₇	Q₈	1	2	yes	—
27	C₂×#3622	—	2	2	no	—
28	C₂×#3622	C₂×C₄	1	2	yes	—
29	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₇	—	4	2	no	—
31	C₂×C₄×C₁₇	—	4	2	no	—
32	C₂×C₄×C₁₇	—	4	2	no	—
33	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₁₇	#41	1	2	yes	—
36	C₂×C₄×C₁₇	#41	1	2	yes	—
37	C₂×Dic₁₇	—	2	1	no	—
38	C₂×Dic₁₇	—	2	2	no	—
39	C₂×Dic₁₇	—	2	1	no	—
40	C₂×Dic₁₇	—	4	2	no	—
41	C₂×Dic₁₇	—	4	2	no	—
42	C₂×Dic₁₇	—	4	2	no	—
43	C₂×Dic₁₇	C₄⋊C₄	1	2	yes	—
44	C₂×Dic₁₇	C₄⋊C₄	1	2	yes	—
45	C₈×C₁₇	—	2	2	no	—
46	#3622	—	2	2	no	—
47	#1835	—	17	1	no	maximal, Sylow
48	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₇	#124	1	2	yes	—
50	C₂²×C₁₇	#135	1	1	yes	characteristic
51	C₂²×C₁₇	#135	1	2	yes	—
52	C₂²×C₁₇	#135	1	1	yes	characteristic
53	C₄×C₁₇	—	8	2	no	—
54	C₄×C₁₇	#117	1	1	yes	characteristic
55	C₄×C₁₇	#117	1	1	yes	characteristic
56	C₄×C₁₇	#125	1	2	yes	—
57	Dic₁₇	—	2	2	no	—
58	Dic₁₇	—	2	2	no	—
59	Dic₁₇	—	8	2	no	—
60	C₂×C₄×C₈	—	17	1	no	—
61	#382	—	17	2	no	—
62	#340	—	17	1	no	—
63	#340	—	17	2	no	—
64	#340	D₁₇	1	1	yes	characteristic
65	C₂×C₁₇	#382	1	1	yes	characteristic
66	C₂×C₁₇	#427	1	1	yes	characteristic
67	C₂×C₁₇	#427	1	1	yes	characteristic
68	C₂×C₁₇	#488	1	2	yes	—
69	C₂×C₁₇	#488	1	2	yes	—
70	C₂²×C₈	—	17	1	no	—
71	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
72	C₂×C₄⋊C₄	—	17	2	no	—
73	C₂×C₄⋊C₄	C₂×D₁₇	1	2	yes	—
74	C₂×C₄²	—	17	1	no	—
75	C₄×C₈	—	17	4	no	—
76	#95	—	34	2	no	—
77	#95	—	34	2	no	—
78	C₁₇	#1835	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	17	1	no	—
81	C₂²×C₄	—	17	1	no	—
82	C₂²×C₄	—	34	2	no	—
83	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	17	2	no	—
86	C₂×C₈	—	34	2	no	—
87	C₂×C₈	C₄×D₁₇	1	2	yes	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	34	2	no	—
91	C₄⋊C₄	—	34	2	no	—
92	C₄²	—	17	2	no	—
93	C₄²	—	17	2	no	—
94	C₂³	D₄×D₁₇	1	1	yes	characteristic
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	17	2	no	—
99	C₂×C₄	—	17	2	no	—
100	C₂×C₄	—	34	1	no	—
101	C₂×C₄	—	34	2	no	—
102	C₂×C₄	—	34	1	no	—
103	C₂×C₄	—	68	2	no	—
104	C₂×C₄	—	68	2	no	—
105	C₂×C₄	—	68	2	no	—
106	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
107	C₂×C₄	#63252	1	2	yes	—
108	C₂×C₄	#63254	1	2	yes	—
109	C₂×C₄	Q₈×D₁₇	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	—	34	2	no	—
112	C₂²	#10587571	1	2	yes	—
113	C₂²	C₄⋊C₄×D₁₇	1	1	yes	characteristic
114	C₂²	#10587598	1	1	yes	characteristic
115	C₂²	#10587602	1	2	yes	—
116	C₂²	#10587605	1	1	yes	characteristic
117	C₄	—	8	2	no	—
118	C₄	—	34	2	no	—
119	C₄	—	34	2	no	—
120	C₄	—	136	2	no	—
121	C₄	#10587558	1	1	yes	characteristic
122	C₄	#10587563	1	1	yes	characteristic
123	C₄	#10587575	1	2	yes	—
124	C₂	#11762673	1	1	yes	characteristic
125	C₂	#11762844	1	1	yes	characteristic
126	C₂	#11762846	1	1	yes	characteristic
127	C₂	#11763162	1	2	yes	—
128	C₂	#11763165	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₁₇

C₂

yes

maximal, characteristic

#11762674

C₂

yes

maximal

#11762572

C₂

yes

maximal, characteristic

#11762571

C₂

yes

maximal

C₁₇×#340

C₂

yes

maximal, characteristic

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇

C₂²

yes

—

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×C₄×Dic₁₇

C₂²

yes

characteristic

C₂²×#3622

C₂²

yes

characteristic

#10587502

—

#10587502

—

C₈×Dic₁₇

C₄

yes

complemented

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

D₄

yes

characteristic

C₂²×Dic₁₇

D₄

yes

characteristic

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄×Dic₁₇

C₂×C₄

yes

—

C₄×Dic₁₇

Q₈

yes

—

C₂×#3622

—

C₂×#3622

C₂×C₄

yes

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

#41

yes

—

C₂×C₄×C₁₇

#41

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

C₄⋊C₄

yes

—

C₂×Dic₁₇

C₄⋊C₄

yes

—

C₈×C₁₇

—

#3622

—

#1835

—

maximal, Sylow

C₂²×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₇

#124

yes

—

C₂²×C₁₇

#135

yes

characteristic

C₂²×C₁₇

#135

yes

—

C₂²×C₁₇

#135

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

C₂×C₄×C₈

—

—

—

—

D₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

—

C₂×C₁₇

#488

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₁₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

—

C₂×C₄²

—

C₄×C₈

—

#95

—

#95

—

C₁₇

#1835

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₁₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₂³

D₄×D₁₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

107

C₂×C₄

#63252

yes

—

108

C₂×C₄

#63254

yes

—

109

C₂×C₄

Q₈×D₁₇

yes

characteristic

110

C₈

—

111

C₈

—

112

C₂²

#10587571

yes

—

113

C₂²

C₄⋊C₄×D₁₇

yes

characteristic

114

C₂²

#10587598

yes

characteristic

115

C₂²

#10587602

yes

—

116

C₂²

#10587605

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

136

—

121

C₄

#10587558

yes

characteristic

122

C₄

#10587563

yes

characteristic

123

C₄

#10587575

yes

—

124

C₂

#11762673

yes

characteristic

125

C₂

#11762844

yes

characteristic

126

C₂

#11762846

yes

characteristic

127

C₂

#11763162

yes

—

128

C₂

#11763165

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#340	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₄
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	#340	D₁₇
hypocenter	C₁₇	#1835
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₁₇

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#340

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₄

cosocle

C₂²×C₄×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
13. C₈×Dic₁₇	C₄	semidirect
78. C₁₇	#1835	semidirect

H ≅ G/N

type

13. C₈×Dic₁₇

C₄

semidirect

78. C₁₇

#1835

semidirect

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

subgroup	index
C₂×C₈×Dic₁₇	2
#11762674	2
#11762572	2
#11762571	2
C₁₇×#340	2
#1835	17

subgroup

index

C₂×C₈×Dic₁₇

C₁₇×#340

Subgroups — #14528913

Conjugacy classes of subgroups

Complements

Maximal subgroups