Group #14528688 — subgroups

Subgroups — #108×Dic₁₇ #14528688

Order 2176.

228 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 121 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×#10587490	C₂	1	1	yes	maximal, characteristic
2	M₁₆×Dic₁₇	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
4	#11762573	C₂	1	2	yes	maximal
5	C₂×C₁₇×#108	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₁₇	C₂²	1	2	yes	—
7	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
9	C₂×#63223	C₂²	1	2	yes	—
10	C₂²×#3622	C₂²	1	1	yes	characteristic
11	C₁₇×#108	C₄	1	4	yes	complemented
12	#10587490	C₄	1	4	yes	complemented
13	#10587486	—	2	2	no	—
14	C₈×Dic₁₇	—	2	2	no	—
15	C₈×Dic₁₇	C₄	1	4	yes	—
16	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
17	C₂²×Dic₁₇	Q₈	1	2	yes	—
18	M₁₆×C₁₇	—	2	2	no	—
19	M₁₆×C₁₇	C₂×C₄	1	4	yes	—
20	C₂×C₈×C₁₇	—	2	2	no	—
21	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
24	C₄×Dic₁₇	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₇	D₄	1	1	yes	characteristic
26	C₄×Dic₁₇	D₄	1	1	yes	characteristic
27	#63223	—	2	2	no	—
28	#63223	C₂×C₄	1	4	yes	—
29	C₂×#3622	—	2	2	no	—
30	C₂×#3622	—	2	1	no	—
31	C₂×#3622	—	2	1	no	—
32	C₂×#3622	C₂×C₄	1	2	yes	—
33	C₂³×C₁₇	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
39	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
40	C₂×Dic₁₇	—	2	4	no	—
41	C₂×Dic₁₇	C₄⋊C₄	1	2	yes	—
42	C₂×Dic₁₇	C₄⋊C₄	1	2	yes	—
43	C₈×C₁₇	—	2	4	no	—
44	C₈×C₁₇	C₄²	1	4	yes	—
45	#3622	—	2	2	no	—
46	#3622	—	2	4	no	—
47	C₄×#108	—	17	1	no	maximal, Sylow
48	C₂²×C₁₇	—	2	2	no	—
49	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₁₇	C₄×Q₈	1	1	yes	characteristic
51	C₂²×C₁₇	C₄×Q₈	1	1	yes	characteristic
52	C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
53	C₄×C₁₇	#117	1	1	yes	characteristic
54	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
55	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
56	Dic₁₇	—	2	2	no	—
57	Dic₁₇	#108	1	4	yes	direct product factor
58	C₂×#108	—	17	1	no	—
59	C₂×#108	D₁₇	1	1	yes	characteristic
60	C₄×M₁₆	—	17	2	no	—
61	C₂×C₄×C₈	—	17	1	no	—
62	#341	—	17	2	no	—
63	C₂×C₁₇	—	2	1	no	—
64	C₂×C₁₇	—	2	1	no	—
65	C₂×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₁₇	C₂×#108	1	2	yes	—
67	C₂×M₁₆	—	17	2	no	—
68	C₂×M₁₆	C₂×D₁₇	1	2	yes	—
69	C₂²×C₈	—	17	1	no	—
70	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
71	C₂×C₄²	—	17	1	no	—
72	#108	—	17	4	no	—
73	#108	Dic₁₇	1	4	yes	direct product factor
74	#97	—	34	2	no	—
75	C₄×C₈	—	17	4	no	—
76	C₄×C₈	—	34	2	no	—
77	C₁₇	C₄×#108	1	1	yes	characteristic, complemented, Sylow
78	C₂²×C₄	—	17	2	no	—
79	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
80	M₁₆	—	2	2	no	—
81	M₁₆	—	17	4	no	—
82	M₁₆	—	34	2	no	—
83	M₁₆	C₂×Dic₁₇	1	4	yes	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	17	2	no	—
86	C₂×C₈	—	34	2	no	—
87	C₂×C₈	—	34	1	no	—
88	C₂×C₈	—	34	1	no	—
89	C₂×C₈	C₄×D₁₇	1	2	yes	—
90	C₂×C₈	C₄×D₁₇	1	2	yes	—
91	C₂×C₈	C₂×Dic₁₇	1	2	yes	—
92	C₄²	—	17	2	no	—
93	C₄²	—	17	1	no	—
94	C₄²	—	17	1	no	—
95	C₂³	Q₈×D₁₇	1	1	yes	characteristic
96	C₂×C₄	—	17	2	no	—
97	C₂×C₄	—	17	2	no	—
98	C₂×C₄	—	34	4	no	—
99	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
100	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
101	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
102	C₂×C₄	#63254	1	1	yes	characteristic
103	C₂×C₄	#63256	1	1	yes	characteristic
104	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
105	C₈	—	2	4	no	—
106	C₈	—	34	2	no	—
107	C₈	—	34	4	no	—
108	C₈	C₄×Dic₁₇	1	4	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	#10587573	1	1	yes	characteristic
111	C₂²	C₄⋊C₄×D₁₇	1	1	yes	characteristic
112	C₂²	Q₈×Dic₁₇	1	1	yes	characteristic
113	C₄	—	17	4	no	—
114	C₄	—	34	2	no	—
115	C₄	#10587578	1	1	yes	characteristic
116	C₄	#10587579	1	1	yes	characteristic
117	C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
118	C₄	D₄×Dic₁₇	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	C₄⋊C₄×Dic₁₇	1	1	yes	characteristic
122	C₂	#108×D₁₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587490

C₂

yes

maximal, characteristic

M₁₆×Dic₁₇

C₂

yes

maximal

C₂×C₈×Dic₁₇

C₂

yes

maximal, characteristic

#11762573

C₂

yes

maximal

C₂×C₁₇×#108

C₂

yes

maximal, characteristic

C₂×M₁₆×C₁₇

C₂²

yes

—

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₂×C₄×Dic₁₇

C₂²

yes

characteristic

C₂×#63223

C₂²

yes

—

C₂²×#3622

C₂²

yes

characteristic

C₁₇×#108

C₄

yes

complemented

#10587490

C₄

yes

complemented

#10587486

—

C₈×Dic₁₇

—

C₈×Dic₁₇

C₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

Q₈

yes

—

M₁₆×C₁₇

—

M₁₆×C₁₇

C₂×C₄

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×C₄

yes

—

C₂×C₈×C₁₇

C₂×C₄

yes

—

C₂×C₈×C₁₇

C₂×C₄

yes

—

C₄×Dic₁₇

C₂×C₄

yes

—

C₄×Dic₁₇

D₄

yes

characteristic

C₄×Dic₁₇

D₄

yes

characteristic

#63223

—

#63223

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂³×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

—

C₂×Dic₁₇

C₄⋊C₄

yes

—

C₂×Dic₁₇

C₄⋊C₄

yes

—

C₈×C₁₇

—

C₈×C₁₇

C₄²

yes

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₇

C₄×Q₈

yes

characteristic

C₂²×C₁₇

C₄×Q₈

yes

characteristic

C₄×C₁₇

C₂×C₄²

yes

characteristic

C₄×C₁₇

#117

yes

characteristic

C₄×C₁₇

C₄×D₄

yes

characteristic

C₄×C₁₇

C₄×D₄

yes

characteristic

Dic₁₇

—

Dic₁₇

#108

yes

direct product factor

C₂×#108

—

C₂×#108

D₁₇

yes

characteristic

C₄×M₁₆

—

C₂×C₄×C₈

—

#341

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₇

C₂×#108

yes

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₇

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₁₇

yes

characteristic

C₂×C₄²

—

#108

—

#108

Dic₁₇

yes

direct product factor

#97

—

C₄×C₈

—

C₄×C₈

—

C₁₇

C₄×#108

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

C₂×Dic₁₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₁₇

yes

—

C₂×C₈

C₄×D₁₇

yes

—

C₂×C₈

C₂×Dic₁₇

yes

—

C₄²

—

C₄²

—

C₄²

—

C₂³

Q₈×D₁₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

100

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

101

C₂×C₄

D₄×D₁₇

yes

characteristic

102

C₂×C₄

#63254

yes

characteristic

103

C₂×C₄

#63256

yes

characteristic

104

C₂×C₄

C₂²×Dic₁₇

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

C₄×Dic₁₇

yes

—

109

C₂²

—

110

C₂²

#10587573

yes

characteristic

111

C₂²

C₄⋊C₄×D₁₇

yes

characteristic

112

C₂²

Q₈×Dic₁₇

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

#10587578

yes

characteristic

116

C₄

#10587579

yes

characteristic

117

C₄

C₂×C₄×Dic₁₇

yes

characteristic

118

C₄

D₄×Dic₁₇

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

C₄⋊C₄×Dic₁₇

yes

characteristic

122

C₂

#108×D₁₇

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₇
commutator	C₄×C₁₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂×C₁₇×#108	C₂
socle	C₂²×C₁₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	C₂×#108	D₁₇
hypocenter	C₁₇	C₄×#108
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₁₇

commutator

C₄×C₁₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₂×C₁₇×#108

C₂

socle

C₂²×C₁₇

C₂×C₄⋊C₄

prime radical

C₂³×C₁₇

C₂×Q₈

cosocle

C₂²×C₄×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
11. C₁₇×#108	C₄	semidirect
12. #10587490	C₄	semidirect
57. Dic₁₇	#108	direct (N × H)
73. #108	Dic₁₇	direct (N × H)
77. C₁₇	C₄×#108	semidirect

H ≅ G/N

type

11. C₁₇×#108

C₄

semidirect

12. #10587490

C₄

semidirect

57. Dic₁₇

#108

direct (N × H)

73. #108

Dic₁₇

direct (N × H)

77. C₁₇

C₄×#108

semidirect

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

subgroup	index
C₂×#10587490	2
M₁₆×Dic₁₇	2
C₂×C₈×Dic₁₇	2
#11762573	2
C₂×C₁₇×#108	2
C₄×#108	17

subgroup

index

C₂×#10587490

M₁₆×Dic₁₇

C₂×C₈×Dic₁₇

#11762573

C₂×C₁₇×#108

C₄×#108

Subgroups — #108×Dic₁₇ #14528688

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #108×Dic17 #14528688

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #108×Dic₁₇ #14528688