Group #17948834 — subgroups

Subgroups — #17948834

Order 4352.

190 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 66 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	#14544346	C₂	1	2	yes	maximal
2	C₂×#11762627	C₂	1	1	yes	maximal, characteristic
3	#11763830	—	2	2	no	—
4	#11763672	—	2	2	no	—
5	#11763671	C₄	1	4	yes	—
6	C₂²×#63214	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
8	#11762627	—	2	2	no	—
9	C₂²×C₁₆×C₁₇	C₄	1	1	yes	characteristic
10	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
12	C₂²×#3622	Q₈	1	1	yes	characteristic
13	C₂²⋊#3624	—	4	2	no	—
14	#10587542	—	4	2	no	—
15	C₄⋊#3624	—	2	4	no	—
16	C₂×C₁₆×C₁₇	—	2	1	no	—
17	C₂×C₁₆×C₁₇	—	4	1	no	—
18	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
19	C₈×Dic₁₇	D₄	1	2	yes	—
20	C₂×#63214	—	2	1	no	—
21	C₂×#63214	—	4	1	no	—
22	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
23	C₂²×Dic₁₇	D₈	1	1	yes	characteristic
24	C₂²×Dic₁₇	Q₁₆	1	1	yes	characteristic
25	C₂×#3624	—	4	4	no	—
26	C₂×C₈×C₁₇	—	2	1	no	—
27	C₂×C₈×C₁₇	—	2	1	no	—
28	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
29	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₄×Dic₁₇	C₂²⋊C₄	1	2	yes	—
31	C₄×Dic₁₇	SD₁₆	1	2	yes	—
32	C₂×#3622	—	4	1	no	—
33	C₂×#3622	C₄⋊C₄	1	2	yes	—
34	C₁₆×C₁₇	—	4	1	no	—
35	#63214	—	4	1	no	—
36	#7242	—	17	1	no	maximal, Sylow
37	C₂³×C₁₇	#107	1	1	yes	characteristic
38	C₂×C₄×C₁₇	—	2	1	no	—
39	C₂×C₄×C₁₇	—	2	1	no	—
40	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
41	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
42	C₂×Dic₁₇	—	2	1	no	—
43	C₂×Dic₁₇	—	2	1	no	—
44	C₂×Dic₁₇	—	4	1	no	—
45	C₂×Dic₁₇	#102	1	2	yes	—
46	C₂×Dic₁₇	#103	1	2	yes	—
47	#3624	—	8	4	no	—
48	C₈×C₁₇	—	2	1	no	—
49	C₈×C₁₇	#100	1	1	yes	characteristic
50	C₈×C₁₇	#101	1	1	yes	characteristic
51	#3622	—	2	2	no	—
52	C₂×C₄⋊C₁₆	—	17	1	no	—
53	#1568	—	17	2	no	—
54	C₂²×C₁₇	—	2	1	no	—
55	C₂²×C₁₇	—	2	1	no	—
56	C₂²×C₁₇	#340	1	1	yes	characteristic
57	C₂²×C₁₇	#368	1	1	yes	characteristic
58	C₂²×C₁₇	#368	1	1	yes	characteristic
59	C₄×C₁₇	—	2	1	no	—
60	C₄×C₁₇	#338	1	1	yes	characteristic
61	C₄×C₁₇	#343	1	1	yes	characteristic
62	Dic₁₇	—	2	2	no	—
63	Dic₁₇	—	2	2	no	—
64	C₂²×C₁₆	—	17	1	no	—
65	C₂²×C₁₆	#595	1	1	yes	characteristic
66	#432	—	34	2	no	—
67	C₂×C₄×C₈	—	17	1	no	—
68	C₄⋊C₁₆	—	34	2	no	—
69	#335	—	34	2	no	—
70	#334	—	17	4	no	—
71	C₂×C₁₇	—	2	1	no	—
72	C₂×C₁₇	—	2	1	no	—
73	C₂×C₁₇	#1292	1	1	yes	characteristic
74	C₂×C₁₇	#1381	1	1	yes	characteristic
75	C₂×C₁₇	#1384	1	1	yes	characteristic
76	C₂²×C₈	—	17	1	no	—
77	C₂²×C₈	C₂×#595	1	1	yes	characteristic
78	C₂×C₄²	—	17	1	no	—
79	C₂×C₁₆	—	2	1	no	—
80	C₂×C₁₆	—	4	1	no	—
81	C₂×C₁₆	—	34	1	no	—
82	C₂×C₁₆	—	68	1	no	—
83	C₄⋊C₈	—	34	4	no	—
84	C₂²⋊C₈	—	68	2	no	—
85	#97	—	68	2	no	—
86	C₄×C₈	—	17	2	no	—
87	C₄×C₈	—	17	2	no	—
88	C₁₇	#7242	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₄	—	17	1	no	—
90	C₂²×C₄	—	17	1	no	—
91	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	17	2	no	—
95	C₂×C₈	—	68	1	no	—
96	C₂×C₈	—	68	4	no	—
97	C₂×C₈	C₄×#595	1	1	yes	characteristic
98	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
99	C₄²	—	17	2	no	—
100	C₄²	—	17	2	no	—
101	C₁₆	—	4	1	no	—
102	C₁₆	—	68	1	no	—
103	C₂³	#10587534	1	1	yes	characteristic
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	17	2	no	—
107	C₂×C₄	—	17	2	no	—
108	C₂×C₄	—	34	1	no	—
109	C₂×C₄	—	34	1	no	—
110	C₂×C₄	—	68	1	no	—
111	C₂×C₄	#10587533	1	1	yes	characteristic
112	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
113	C₈	—	2	1	no	—
114	C₈	—	34	2	no	—
115	C₈	—	136	4	no	—
116	C₈	#10587545	1	1	yes	characteristic
117	C₈	#10587553	1	1	yes	characteristic
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	#11763683	1	1	yes	characteristic
121	C₂²	#11763737	1	1	yes	characteristic
122	C₂²	#11763738	1	1	yes	characteristic
123	C₄	—	2	1	no	—
124	C₄	—	34	2	no	—
125	C₄	—	34	2	no	—
126	C₄	#11763678	1	1	yes	characteristic
127	C₄	#11763689	1	1	yes	characteristic
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#14543654	1	1	yes	characteristic
131	C₂	#14543844	1	1	yes	characteristic
132	C₂	#14543852	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14544346

C₂

yes

maximal

C₂×#11762627

C₂

yes

maximal, characteristic

#11763830

—

#11763672

—

#11763671

C₄

yes

—

C₂²×#63214

C₄

yes

characteristic

C₂×C₈×Dic₁₇

C₂²

yes

characteristic

#11762627

—

C₂²×C₁₆×C₁₇

C₄

yes

characteristic

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₇

D₄

yes

characteristic

C₂²×#3622

Q₈

yes

characteristic

C₂²⋊#3624

—

#10587542

—

C₄⋊#3624

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

—

C₈×Dic₁₇

C₂×C₄

yes

—

C₈×Dic₁₇

D₄

yes

—

C₂×#63214

—

C₂×#63214

—

C₂²×C₄×C₁₇

C₄⋊C₄

yes

characteristic

C₂²×Dic₁₇

D₈

yes

characteristic

C₂²×Dic₁₇

Q₁₆

yes

characteristic

C₂×#3624

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

C₂²⋊C₄

yes

—

C₄×Dic₁₇

SD₁₆

yes

—

C₂×#3622

—

C₂×#3622

C₄⋊C₄

yes

—

C₁₆×C₁₇

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

#102

yes

—

C₂×Dic₁₇

#103

yes

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

yes

C₈×C₁₇

yes

—

C₂×C₄⋊C₁₆

—

#1568

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

C₂²×C₁₆

—

C₂²×C₁₆

#595

yes

characteristic

#432

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

#335

—

#334

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×C₈

—

C₂²×C₈

C₂×#595

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₁₇

#7242

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄⋊#595

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×#595

yes

characteristic

C₂×C₈

C₂²⋊#595

yes

characteristic

C₄²

—

100

C₄²

—

101

C₁₆

—

102

C₁₆

—

103

C₂³

#10587534

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

#10587533

yes

characteristic

112

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

136

—

116

C₈

#10587545

yes

characteristic

117

C₈

#10587553

yes

characteristic

118

C₂²

—

119

C₂²

—

120

C₂²

#11763683

yes

characteristic

121

C₂²

#11763737

yes

characteristic

122

C₂²

#11763738

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#11763678

yes

characteristic

127

C₄

#11763689

yes

characteristic

128

C₂

—

129

C₂

—

130

C₂

#14543654

yes

characteristic

131

C₂

#14543844

yes

characteristic

132

C₂

#14543852

yes

characteristic

	subgroup	G/H
center	C₂²	#11763683
commutator	C₂×C₈×C₁₇	C₄²
Frattini	C₂²×C₈	C₂×#595
Fitting	C₂²×C₁₆×C₁₇	C₄
socle	C₂²×C₁₇	#340
prime radical	C₂³×C₁₇	#107
cosocle	C₂×C₈×Dic₁₇	C₂²
hypercenter	C₂²×C₁₆	#595
hypocenter	C₁₇	#7242
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂²

#11763683

commutator

C₂×C₈×C₁₇

C₄²

Frattini

C₂²×C₈

C₂×#595

Fitting

C₂²×C₁₆×C₁₇

C₄

C₂²×C₁₇

C₂³×C₁₇

C₂×C₈×Dic₁₇

C₂²

C₂²×C₁₆

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
88. C₁₇	#7242	semidirect

H ≅ G/N

type

88. C₁₇

#7242

semidirect

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

subgroup	index
#14544346	2
C₂×#11762627	2
#7242	17

subgroup

index

#14544346

C₂×#11762627

#7242

Subgroups — #17948834

Conjugacy classes of subgroups

Complements

Maximal subgroups