Group #14528528 — subgroups

Subgroups — #14528528

Order 2176.

296 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 149 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	2	yes	maximal
2	C₄²×Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	#11762564	C₂	1	1	yes	maximal, characteristic
4	#11762563	C₂	1	2	yes	maximal
5	C₁₇×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₁₇	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₇	C₄	1	2	yes	—
9	C₂×C₄×Dic₁₇	C₄	1	2	yes	—
10	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
12	C₂²×#3622	C₂²	1	2	yes	—
13	C₈×Dic₁₇	C₄	1	8	yes	complemented
14	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
15	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
16	C₂²×Dic₁₇	C₂×C₄	1	2	yes	—
17	C₂²×Dic₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×Dic₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₁₇	—	2	2	no	—
20	C₂×C₈×C₁₇	—	2	2	no	—
21	C₂×C₈×C₁₇	C₂×C₄	1	4	yes	—
22	C₄²×C₁₇	—	2	2	no	—
23	C₄×Dic₁₇	—	2	1	no	—
24	C₄×Dic₁₇	—	2	1	no	—
25	C₄×Dic₁₇	—	2	4	no	—
26	C₄×Dic₁₇	C₈	1	8	yes	complemented
27	C₄×Dic₁₇	C₂×C₄	1	4	yes	—
28	C₂×#3622	—	2	2	no	—
29	C₂×#3622	—	2	2	no	—
30	C₂×#3622	C₂×C₄	1	4	yes	—
31	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₇	—	2	2	no	—
33	C₂×C₄×C₁₇	—	2	2	no	—
34	C₂×C₄×C₁₇	C₂×C₈	1	4	yes	—
35	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
36	C₂×C₄×C₁₇	#41	1	2	yes	—
37	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
38	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
39	C₂×Dic₁₇	—	2	2	no	—
40	C₂×Dic₁₇	—	2	2	no	—
41	C₂×Dic₁₇	C₄²	1	2	yes	—
42	C₂×Dic₁₇	C₄²	1	2	yes	—
43	C₂×Dic₁₇	C₂×C₈	1	4	yes	—
44	C₂×Dic₁₇	C₂×C₈	1	4	yes	—
45	C₂×Dic₁₇	M₁₆	1	4	yes	—
46	C₈×C₁₇	—	2	4	no	—
47	#3622	—	2	4	no	—
48	#1744	—	17	1	no	maximal, Sylow
49	C₂²×C₁₇	C₂×C₄²	1	1	yes	characteristic
50	C₂²×C₁₇	#117	1	1	yes	characteristic
51	C₂²×C₁₇	#117	1	1	yes	characteristic
52	C₂²×C₁₇	C₂²×C₈	1	2	yes	—
53	C₂²×C₁₇	C₂×M₁₆	1	2	yes	—
54	C₄×C₁₇	—	2	4	no	—
55	C₄×C₁₇	#117	1	2	yes	—
56	C₄×C₁₇	#117	1	2	yes	—
57	Dic₁₇	—	2	4	no	—
58	Dic₁₇	C₄×C₈	1	4	yes	—
59	Dic₁₇	#97	1	4	yes	—
60	C₂×C₄×C₈	—	17	2	no	—
61	C₄³	—	17	1	no	—
62	#336	—	17	2	no	—
63	#336	—	17	1	no	—
64	#336	D₁₇	1	1	yes	characteristic
65	C₂×C₁₇	#376	1	1	yes	characteristic
66	C₂×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
67	C₂×C₁₇	C₂×#97	1	1	yes	characteristic
68	C₂×C₁₇	#431	1	2	yes	—
69	C₂×C₁₇	#431	1	2	yes	—
70	C₂²×C₈	—	17	2	no	—
71	C₂²×C₈	C₂×D₁₇	1	2	yes	—
72	C₂×C₄²	—	17	2	no	—
73	C₂×C₄²	—	17	1	no	—
74	C₂×C₄²	—	17	2	no	—
75	C₂×C₄²	—	17	1	no	—
76	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
77	C₄×C₈	—	17	8	no	—
78	C₁₇	#1744	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₄	—	17	1	no	—
80	C₂²×C₄	—	17	1	no	—
81	C₂²×C₄	—	17	2	no	—
82	C₂²×C₄	C₄×D₁₇	1	2	yes	—
83	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	17	4	no	—
87	C₂×C₈	—	34	2	no	—
88	C₂×C₈	—	34	2	no	—
89	C₂×C₈	C₄×D₁₇	1	4	yes	—
90	C₄²	—	2	2	no	—
91	C₄²	—	17	4	no	—
92	C₄²	—	17	8	no	—
93	C₄²	—	34	4	no	—
94	C₄²	—	34	1	no	—
95	C₄²	—	34	1	no	—
96	C₂³	C₂×C₄×D₁₇	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	17	2	no	—
100	C₂×C₄	—	17	2	no	—
101	C₂×C₄	—	17	4	no	—
102	C₂×C₄	—	17	4	no	—
103	C₂×C₄	—	17	4	no	—
104	C₂×C₄	—	34	2	no	—
105	C₂×C₄	—	34	2	no	—
106	C₂×C₄	C₈×D₁₇	1	4	yes	—
107	C₂×C₄	C₂×C₄×D₁₇	1	2	yes	—
108	C₂×C₄	#63254	1	1	yes	characteristic
109	C₂×C₄	#63254	1	2	yes	—
110	C₂×C₄	#63256	1	1	yes	characteristic
111	C₈	—	2	4	no	—
112	C₈	—	34	4	no	—
113	C₂²	C₄²×D₁₇	1	1	yes	characteristic
114	C₂²	#10587563	1	1	yes	characteristic
115	C₂²	#10587578	1	1	yes	characteristic
116	C₂²	C₂×C₈×D₁₇	1	2	yes	—
117	C₂²	M₁₆×D₁₇	1	2	yes	—
118	C₄	—	2	4	no	—
119	C₄	—	17	4	no	—
120	C₄	—	17	4	no	—
121	C₄	—	34	4	no	—
122	C₄	#10587563	1	2	yes	—
123	C₄	#10587578	1	2	yes	—
124	C₂	#11762654	1	1	yes	characteristic
125	C₂	C₄×C₈×D₁₇	1	1	yes	characteristic
126	C₂	#97×D₁₇	1	1	yes	characteristic
127	C₂	#11762859	1	2	yes	—
128	C₂	#11762861	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₁₇

C₂

yes

maximal

C₄²×Dic₁₇

C₂

yes

maximal, characteristic

#11762564

C₂

yes

maximal, characteristic

#11762563

C₂

yes

maximal

C₁₇×#336

C₂

yes

maximal, characteristic

C₂²×C₈×C₁₇

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄×Dic₁₇

C₄

yes

—

C₂×C₄×Dic₁₇

C₄

yes

—

C₂×C₄×Dic₁₇

C₂²

yes

characteristic

C₂×C₄×Dic₁₇

C₂²

yes

characteristic

C₂²×#3622

C₂²

yes

—

C₈×Dic₁₇

C₄

yes

complemented

C₂²×C₄×C₁₇

C₂×C₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

C₂×C₄

yes

—

C₂²×Dic₁₇

C₂×C₄

yes

characteristic

C₂²×Dic₁₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

C₈

yes

complemented

C₄×Dic₁₇

C₂×C₄

yes

—

C₂×#3622

—

C₂×#3622

—

C₂×#3622

C₂×C₄

yes

—

C₂³×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₈

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

—

C₂×C₄×C₁₇

#41

yes

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

C₄²

yes

—

C₂×Dic₁₇

C₄²

yes

—

C₂×Dic₁₇

C₂×C₈

yes

—

C₂×Dic₁₇

C₂×C₈

yes

—

C₂×Dic₁₇

M₁₆

yes

—

C₈×C₁₇

—

#3622

—

#1744

—

maximal, Sylow

C₂²×C₁₇

C₂×C₄²

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

C₂²×C₈

yes

—

C₂²×C₁₇

C₂×M₁₆

yes

—

C₄×C₁₇

—

C₄×C₁₇

#117

yes

—

C₄×C₁₇

#117

yes

—

Dic₁₇

—

Dic₁₇

C₄×C₈

yes

—

Dic₁₇

#97

yes

—

C₂×C₄×C₈

—

C₄³

—

#336

—

#336

—

#336

D₁₇

yes

characteristic

C₂×C₁₇

#376

yes

characteristic

C₂×C₁₇

C₂×C₄×C₈

yes

characteristic

C₂×C₁₇

C₂×#97

yes

characteristic

C₂×C₁₇

#431

yes

—

C₂×C₁₇

#431

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₁₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

C₄×C₈

—

C₁₇

#1744

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₁₇

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂³

C₂×C₄×D₁₇

yes

characteristic

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

C₈×D₁₇

yes

—

107

C₂×C₄

C₂×C₄×D₁₇

yes

—

108

C₂×C₄

#63254

yes

characteristic

109

C₂×C₄

#63254

yes

—

110

C₂×C₄

#63256

yes

characteristic

111

C₈

—

112

C₈

—

113

C₂²

C₄²×D₁₇

yes

characteristic

114

C₂²

#10587563

yes

characteristic

115

C₂²

#10587578

yes

characteristic

116

C₂²

C₂×C₈×D₁₇

yes

—

117

C₂²

M₁₆×D₁₇

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

#10587563

yes

—

123

C₄

#10587578

yes

—

124

C₂

#11762654

yes

characteristic

125

C₂

C₄×C₈×D₁₇

yes

characteristic

126

C₂

#97×D₁₇

yes

characteristic

127

C₂

#11762859

yes

—

128

C₂

#11762861

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₇
commutator	C₂×C₁₇	C₂×C₄×C₈
Fitting	C₁₇×#336	C₂
socle, prime radical	C₂³×C₁₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	#336	D₁₇
hypocenter	C₁₇	#1744
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₇

commutator

C₂×C₁₇

C₂×C₄×C₈

Fitting

C₁₇×#336

C₂

socle, prime radical

C₂³×C₁₇

C₂²×C₄

cosocle

C₂²×C₄×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₁₇ ▷ C₁₇

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
13. C₈×Dic₁₇	C₄	semidirect
26. C₄×Dic₁₇	C₈	semidirect
78. C₁₇	#1744	semidirect

H ≅ G/N

type

13. C₈×Dic₁₇

C₄

semidirect

26. C₄×Dic₁₇

C₈

semidirect

78. C₁₇

#1744

semidirect

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

subgroup	index
C₂×C₈×Dic₁₇	2
C₄²×Dic₁₇	2
#11762564	2
#11762563	2
C₁₇×#336	2
#1744	17

subgroup

index

C₂×C₈×Dic₁₇

C₄²×Dic₁₇

#11762564

#11762563

C₁₇×#336

#1744

Subgroups — #14528528

Conjugacy classes of subgroups

Complements

Maximal subgroups