Group #14529564 — subgroups

Subgroups — #14529564

Order 2176.

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

normal subgroups only

This group has 196 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×#10587490	C₂	1	1	yes	maximal, characteristic
2	#11762852	C₂	1	2	yes	maximal
3	C₂×#10587488	C₂	1	1	yes	maximal, characteristic
4	#11762577	C₂	1	2	yes	maximal
5	C₁₇×#405	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
8	C₁₇×#117	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
10	C₂×#63223	C₂²	1	2	yes	—
11	C₁₇×#97	—	2	1	no	—
12	C₄×C₈×C₁₇	—	2	1	no	—
13	#10587490	C₄	1	4	yes	complemented
14	#10587488	—	2	1	no	—
15	#10587488	—	2	1	no	—
16	#10587479	—	2	2	no	—
17	#10587478	—	2	2	no	—
18	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
19	C₂²×Dic₁₇	—	2	2	no	—
20	M₁₆×C₁₇	—	2	1	no	—
21	M₁₆×C₁₇	Q₈	1	2	yes	—
22	C₂×C₈×C₁₇	—	2	1	no	—
23	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
25	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
26	C₂×C₈×C₁₇	Q₈	1	2	yes	—
27	C₄⋊C₄×C₁₇	D₄	1	2	yes	—
28	C₂²⋊C₄×C₁₇	—	2	1	no	—
29	C₄²×C₁₇	—	2	1	no	—
30	C₄⋊Dic₁₇	—	2	2	no	—
31	C₄⋊Dic₁₇	—	2	2	no	—
32	#63223	—	2	2	no	—
33	#63223	C₂×C₄	1	4	yes	—
34	C₂×#3622	—	2	2	no	—
35	C₂³×C₁₇	#41	1	1	yes	characteristic
36	C₂×C₄×C₁₇	—	2	2	no	—
37	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
43	C₂×Dic₁₇	—	4	2	no	—
44	C₂×Dic₁₇	—	8	2	no	—
45	C₈×C₁₇	—	2	2	no	—
46	C₈×C₁₇	C₄⋊C₄	1	2	yes	—
47	C₈×C₁₇	C₄⋊C₄	1	2	yes	—
48	#3622	—	2	4	no	—
49	#1951	—	17	1	no	maximal, Sylow
50	C₂²×C₁₇	—	2	2	no	—
51	C₂²×C₁₇	C₄×Q₈	1	1	yes	characteristic
52	C₂²×C₁₇	#121	1	1	yes	characteristic
53	C₂²×C₁₇	#125	1	1	yes	characteristic
54	C₄×C₁₇	—	4	2	no	—
55	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
57	C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
58	C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
59	Dic₁₇	—	8	2	no	—
60	C₂×#108	—	17	1	no	—
61	#428	—	17	2	no	—
62	C₂×#106	—	17	1	no	—
63	#405	D₁₇	1	1	yes	characteristic
64	#343	—	17	2	no	—
65	C₂×C₁₇	—	2	1	no	—
66	C₂×C₁₇	—	2	1	no	—
67	C₂×C₁₇	#389	1	1	yes	characteristic
68	C₂×C₁₇	#471	1	2	yes	—
69	C₂×M₁₆	—	17	2	no	—
70	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
71	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
72	#117	C₂×D₁₇	1	1	yes	characteristic
73	C₂×C₄⋊C₄	—	17	2	no	—
74	#108	—	17	4	no	—
75	#107	—	34	2	no	—
76	#106	—	34	1	no	—
77	#106	—	34	1	no	—
78	#106	—	34	2	no	—
79	#97	—	2	1	no	—
80	C₄×C₈	—	2	1	no	—
81	C₁₇	#1951	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₄	—	34	2	no	—
83	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
84	M₁₆	—	2	1	no	—
85	M₁₆	—	17	4	no	—
86	M₁₆	—	34	2	no	—
87	M₁₆	Dic₃₄	1	2	yes	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	34	2	no	—
90	C₂×C₈	Dic₃₄	1	2	yes	—
91	C₂×C₈	C₄×D₁₇	1	2	yes	—
92	C₂×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
93	C₂×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
94	C₄⋊C₄	—	34	2	no	—
95	C₄⋊C₄	—	34	2	no	—
96	C₄⋊C₄	D₆₈	1	2	yes	—
97	C₂²⋊C₄	—	2	1	no	—
98	C₄²	—	2	1	no	—
99	C₂³	#63252	1	1	yes	characteristic
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	68	2	no	—
102	C₂×C₄	—	136	2	no	—
103	C₂×C₄	C₂×Dic₃₄	1	1	yes	characteristic
104	C₂×C₄	C₂×Dic₃₄	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
106	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
107	C₂×C₄	#63252	1	1	yes	characteristic
108	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	34	4	no	—
111	C₈	#63225	1	2	yes	—
112	C₈	#63225	1	2	yes	—
113	C₂²	—	2	2	no	—
114	C₂²	C₄×Dic₃₄	1	1	yes	characteristic
115	C₂²	#10587556	1	1	yes	characteristic
116	C₂²	#10587616	1	1	yes	characteristic
117	C₄	—	4	2	no	—
118	C₄	—	136	2	no	—
119	C₄	#10587555	1	1	yes	characteristic
120	C₄	C₄×D₆₈	1	1	yes	characteristic
121	C₄	C₂×#63225	1	1	yes	characteristic
122	C₄	#10587610	1	1	yes	characteristic
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	#11762707	1	1	yes	characteristic
126	C₂	#11763072	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587490

C₂

yes

maximal, characteristic

#11762852

C₂

yes

maximal

C₂×#10587488

C₂

yes

maximal, characteristic

#11762577

C₂

yes

maximal

C₁₇×#405

C₂

yes

maximal, characteristic

C₂×M₁₆×C₁₇

C₂²

yes

characteristic

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#117

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×#63223

C₂²

yes

—

C₁₇×#97

—

C₄×C₈×C₁₇

—

#10587490

C₄

yes

—

—

—

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

M₁₆×C₁₇

—

M₁₆×C₁₇

Q₈

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×C₄

yes

—

C₂×C₈×C₁₇

D₄

yes

characteristic

C₂×C₈×C₁₇

D₄

yes

characteristic

C₂×C₈×C₁₇

Q₈

yes

—

C₄⋊C₄×C₁₇

D₄

yes

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63223

—

#63223

C₂×C₄

yes

—

C₂×#3622

—

C₂³×C₁₇

#41

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

#41

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₄⋊C₄

yes

—

C₈×C₁₇

C₄⋊C₄

yes

—

#3622

—

#1951

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

C₄×Q₈

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₇

C₄×D₄

yes

characteristic

C₄×C₁₇

C₂²⋊Q₈

yes

characteristic

C₄×C₁₇

C₄⋊Q₈

yes

characteristic

Dic₁₇

—

—

—

—

D₁₇

yes

characteristic

#343

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#389

yes

characteristic

C₂×C₁₇

#471

yes

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₇

yes

characteristic

C₂²×C₈

C₂×D₁₇

yes

characteristic

#117

C₂×D₁₇

yes

characteristic

C₂×C₄⋊C₄

—

—

—

—

—

—

—

C₄×C₈

—

C₁₇

#1951

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

Dic₃₄

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₃₄

yes

—

C₂×C₈

C₄×D₁₇

yes

—

C₂×C₈

C₂²⋊D₁₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₆₈

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#63252

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

136

—

103

C₂×C₄

C₂×Dic₃₄

yes

characteristic

104

C₂×C₄

C₂×Dic₃₄

yes

characteristic

105

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

106

C₂×C₄

C₂×D₆₈

yes

characteristic

107

C₂×C₄

#63252

yes

characteristic

108

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

#63225

yes

—

112

C₈

#63225

yes

—

113

C₂²

—

114

C₂²

C₄×Dic₃₄

yes

characteristic

115

C₂²

#10587556

yes

characteristic

116

C₂²

#10587616

yes

characteristic

117

C₄

—

118

C₄

—

136

—

119

C₄

#10587555

yes

characteristic

120

C₄

C₄×D₆₈

yes

characteristic

121

C₄

C₂×#63225

yes

characteristic

122

C₄

#10587610

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#11762707

yes

characteristic

126

C₂

#11763072

yes

—

	subgroup	G/H
center	C₂²	#10587616
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#405	C₂
socle	C₂²×C₁₇	#121
prime radical	C₂³×C₁₇	#41
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	#405	D₁₇
hypocenter	C₁₇	#1951
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

subgroup

G/H

center

C₂²

#10587616

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#405

C₂

C₂²×C₁₇

C₂³×C₁₇

C₂²×C₄×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

N	H ≅ G/N	type
13. #10587490	C₄	semidirect
81. C₁₇	#1951	semidirect

H ≅ G/N

type

13. #10587490

C₄

semidirect

81. C₁₇

#1951

semidirect

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

subgroup	index
C₂×#10587490	2
#11762852	2
C₂×#10587488	2
#11762577	2
C₁₇×#405	2
#1951	17

subgroup

index

C₁₇×#405

Subgroups — #14529564

Conjugacy classes of subgroups

Complements

Maximal subgroups