Group #17949412 — subgroups

Subgroups — #17949412

Order 4352.

166 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 64 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×#11763740	C₂	1	1	yes	maximal, characteristic
2	D₁₇⋊#368	C₂	1	1	yes	maximal, characteristic
3	D₁₇×#368	C₂	1	1	yes	maximal, characteristic
4	D₁₇⋊#108	—	2	1	no	—
5	#11763740	—	2	1	no	—
6	#11763740	C₄	1	2	yes	—
7	C₂×#10587467	—	2	1	no	—
8	#11763738	—	2	1	no	—
9	#11763737	C₄	1	2	yes	—
10	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
11	#108×D₁₇	—	2	1	no	—
12	#11762639	C₄	1	1	yes	characteristic
13	#11762638	—	2	1	no	—
14	C₁₇×#368	C₄	1	1	yes	characteristic
15	D₁₇⋊M₁₆	—	4	1	no	—
16	M₁₆×D₁₇	—	4	1	no	—
17	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
18	#10587536	—	2	2	no	—
19	#10587535	—	4	1	no	—
20	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₁₇×#108	—	2	1	no	—
22	#10587493	—	4	1	no	—
23	#10587490	—	2	1	no	—
24	C₂×#63214	D₄	1	1	yes	characteristic
25	C₁₆×D₁₇	C₂×C₄	1	1	yes	characteristic
26	C₁₆×D₁₇	C₂×C₄	1	1	yes	characteristic
27	C₁₆×D₁₇	D₄	1	1	yes	characteristic
28	C₁₆×D₁₇	Q₈	1	1	yes	characteristic
29	#10587467	—	2	2	no	—
30	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
31	#63247	—	8	1	no	—
32	#63243	—	4	2	no	—
33	#63242	—	8	1	no	—
34	M₁₆×C₁₇	—	4	1	no	—
35	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
36	#63223	—	4	1	no	—
37	C₂×#3622	SD₁₆	1	1	yes	characteristic
38	#63218	—	8	1	no	—
39	C₈×D₁₇	—	8	1	no	—
40	C₈×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₈×D₁₇	C₄⋊C₄	1	1	yes	characteristic
42	C₈×D₁₇	SD₁₆	1	1	yes	characteristic
43	C₈×D₁₇	Q₁₆	1	1	yes	characteristic
44	C₁₆×C₁₇	C₄²	1	1	yes	characteristic
45	C₁₆×C₁₇	C₄⋊C₄	1	1	yes	characteristic
46	#63214	C₂²⋊C₄	1	1	yes	characteristic
47	#63214	C₄⋊C₄	1	1	yes	characteristic
48	#7472	—	17	1	no	maximal, Sylow
49	C₂²×D₁₇	Q₃₂	1	1	yes	characteristic
50	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
51	C₂×Dic₁₇	SD₃₂	1	1	yes	characteristic
52	C₄×D₁₇	#102	1	1	yes	characteristic
53	C₄×D₁₇	#107	1	1	yes	characteristic
54	C₄×D₁₇	D₁₆	1	1	yes	characteristic
55	C₄×D₁₇	SD₃₂	1	1	yes	characteristic
56	#3624	—	8	2	no	—
57	C₈×C₁₇	—	8	1	no	—
58	C₈×C₁₇	#95	1	1	yes	characteristic
59	C₈×C₁₇	#103	1	1	yes	characteristic
60	#3622	—	8	1	no	—
61	#3622	#103	1	1	yes	characteristic
62	#3622	#106	1	1	yes	characteristic
63	C₂²×C₃₂	—	17	1	no	—
64	C₂×#368	—	17	1	no	—
65	C₂×#368	—	17	1	no	—
66	C₂²×C₁₇	#358	1	1	yes	characteristic
67	C₂×D₁₇	—	2	2	no	—
68	C₂×D₁₇	#358	1	1	yes	characteristic
69	C₂×D₁₇	#366	1	1	yes	characteristic
70	C₄×C₁₇	#340	1	1	yes	characteristic
71	C₄×C₁₇	#357	1	1	yes	characteristic
72	Dic₁₇	#357	1	1	yes	characteristic
73	Dic₁₇	#367	1	1	yes	characteristic
74	C₂²×C₁₆	—	17	1	no	—
75	C₂×#108	—	34	1	no	—
76	C₂×#108	—	34	1	no	—
77	C₂×C₃₂	—	17	2	no	—
78	C₂×C₃₂	—	34	1	no	—
79	C₂×C₃₂	—	34	1	no	—
80	#368	—	17	2	no	—
81	#368	—	17	1	no	—
82	#368	—	34	1	no	—
83	#368	—	34	1	no	—
84	#368	#595	1	1	yes	characteristic
85	C₂×C₁₇	—	2	1	no	—
86	C₂×C₁₇	#1380	1	1	yes	characteristic
87	D₁₇	—	2	1	no	—
88	D₁₇	#1425	1	2	yes	—
89	C₂×M₁₆	—	68	1	no	—
90	C₂×M₁₆	—	68	1	no	—
91	C₂²×C₈	—	17	1	no	—
92	C₂×C₁₆	—	17	1	no	—
93	C₂×C₁₆	—	17	1	no	—
94	C₂×C₁₆	—	17	1	no	—
95	C₂×C₁₆	—	17	1	no	—
96	C₂×C₁₆	—	17	1	no	—
97	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
98	#108	—	2	1	no	—
99	#108	—	34	2	no	—
100	#108	—	34	1	no	—
101	#108	—	68	1	no	—
102	#108	—	68	1	no	—
103	C₃₂	—	34	2	no	—
104	C₁₇	#7472	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₄	—	17	1	no	—
106	M₁₆	—	4	1	no	—
107	M₁₆	—	68	2	no	—
108	M₁₆	—	68	1	no	—
109	M₁₆	—	136	1	no	—
110	M₁₆	—	136	1	no	—
111	C₂×C₈	—	17	1	no	—
112	C₂×C₈	—	17	1	no	—
113	C₂×C₈	—	17	1	no	—
114	C₂×C₈	—	17	1	no	—
115	C₂×C₈	—	17	1	no	—
116	C₂×C₈	—	136	1	no	—
117	C₂×C₈	—	136	1	no	—
118	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
119	C₁₆	—	17	1	no	—
120	C₁₆	—	17	1	no	—
121	C₁₆	C₄×#595	1	1	yes	characteristic
122	C₁₆	C₄⋊#595	1	1	yes	characteristic
123	C₂³	—	17	1	no	—
124	C₂×C₄	—	17	1	no	—
125	C₂×C₄	—	17	1	no	—
126	C₂×C₄	—	17	1	no	—
127	C₂×C₄	—	17	1	no	—
128	C₂×C₄	—	17	1	no	—
129	C₂×C₄	#10587547	1	1	yes	characteristic
130	C₈	—	8	1	no	—
131	C₈	—	17	1	no	—
132	C₈	—	17	1	no	—
133	C₈	—	136	2	no	—
134	C₈	—	136	1	no	—
135	C₈	(C₂×#595)⋊C₄	1	1	yes	characteristic
136	C₈	#10587549	1	1	yes	characteristic
137	C₂²	—	17	1	no	—
138	C₂²	—	17	1	no	—
139	C₂²	—	34	2	no	—
140	C₂²	#11763722	1	1	yes	characteristic
141	C₄	—	17	1	no	—
142	C₄	—	17	1	no	—
143	C₄	#11763682	1	1	yes	characteristic
144	C₄	#11763721	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	17	2	no	—
147	C₂	—	34	1	no	—
148	C₂	#14543841	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11763740

C₂

yes

maximal, characteristic

D₁₇⋊#368

C₂

yes

maximal, characteristic

D₁₇×#368

C₂

yes

maximal, characteristic

D₁₇⋊#108

—

#11763740

—

#11763740

C₄

yes

—

C₂×#10587467

—

#11763738

—

#11763737

C₄

yes

—

C₂×C₁₆×D₁₇

C₂²

yes

characteristic

#108×D₁₇

—

#11762639

C₄

yes

characteristic

#11762638

—

C₁₇×#368

C₄

yes

characteristic

D₁₇⋊M₁₆

—

M₁₆×D₁₇

—

C₂×C₈×D₁₇

D₄

yes

characteristic

#10587536

—

#10587535

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₁₇×#108

—

#10587493

—

#10587490

—

C₂×#63214

D₄

yes

characteristic

C₁₆×D₁₇

C₂×C₄

yes

characteristic

C₁₆×D₁₇

C₂×C₄

yes

characteristic

C₁₆×D₁₇

D₄

yes

characteristic

C₁₆×D₁₇

Q₈

yes

characteristic

#10587467

—

C₂×C₄×D₁₇

D₈

yes

—

—

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

characteristic

#63223

—

C₂×#3622

SD₁₆

yes

characteristic

#63218

—

C₈×D₁₇

—

C₈×D₁₇

C₂²⋊C₄

yes

characteristic

C₈×D₁₇

C₄⋊C₄

yes

characteristic

C₈×D₁₇

SD₁₆

yes

characteristic

C₈×D₁₇

Q₁₆

yes

characteristic

C₁₆×C₁₇

C₄²

yes

characteristic

C₁₆×C₁₇

C₄⋊C₄

yes

characteristic

#63214

C₂²⋊C₄

yes

characteristic

#63214

C₄⋊C₄

yes

characteristic

#7472

—

maximal, Sylow

C₂²×D₁₇

Q₃₂

yes

characteristic

C₂×C₄×C₁₇

#102

yes

characteristic

C₂×Dic₁₇

SD₃₂

yes

C₄×D₁₇

yes

C₄×D₁₇

yes

C₄×D₁₇

D₁₆

yes

characteristic

C₄×D₁₇

SD₃₂

yes

characteristic

#3624

—

C₈×C₁₇

—

C₈×C₁₇

yes

C₈×C₁₇

yes

—

yes

yes

C₂²×C₃₂

—

C₂×#368

—

C₂×#368

—

C₂²×C₁₇

#358

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

yes

C₂×D₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

yes

Dic₁₇

yes

C₂²×C₁₆

—

C₂×#108

—

C₂×#108

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

—

—

—

—

yes

C₂×C₁₇

—

C₂×C₁₇

#1380

yes

characteristic

D₁₇

—

D₁₇

#1425

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

—

—

100

#108

—

101

#108

—

102

#108

—

103

C₃₂

—

104

C₁₇

#7472

yes

characteristic, complemented, Sylow

105

C₂²×C₄

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

136

—

110

M₁₆

—

136

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

136

—

117

C₂×C₈

—

136

—

118

C₂×C₈

C₂²⋊#595

yes

characteristic

119

C₁₆

—

120

C₁₆

—

121

C₁₆

C₄×#595

yes

characteristic

122

C₁₆

C₄⋊#595

yes

characteristic

123

C₂³

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#10587547

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

136

—

134

C₈

—

136

—

135

C₈

(C₂×#595)⋊C₄

yes

characteristic

136

C₈

#10587549

yes

characteristic

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

#11763722

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

#11763682

yes

characteristic

144

C₄

#11763721

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#14543841

yes

characteristic

	subgroup	G/H
center	C₄	#11763721
commutator	C₁₆×C₁₇	C₄²
Frattini	C₂×C₁₆	C₂×#595
Fitting	C₁₇×#368	C₄
socle	C₂×C₁₇	#1380
prime radical	C₂²×D₁₇	Q₃₂
cosocle	C₂×C₁₆×D₁₇	C₂²
hypercenter	#368	#595
hypocenter	C₁₇	#7472
LCS	G ▷ C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

subgroup

G/H

center

C₄

#11763721

commutator

C₁₆×C₁₇

C₄²

Frattini

C₂×C₁₆

C₂×#595

Fitting

C₁₇×#368

C₄

socle

C₂×C₁₇

#1380

prime radical

C₂²×D₁₇

Q₃₂

cosocle

C₂×C₁₆×D₁₇

C₂²

C₁₇

G ▷ C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

N	H ≅ G/N	type
104. C₁₇	#7472	semidirect

H ≅ G/N

type

104. C₁₇

#7472

semidirect

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

subgroup	index
C₂×#11763740	2
D₁₇⋊#368	2
D₁₇×#368	2
#7472	17

subgroup

index

C₂×#11763740

D₁₇⋊#368

D₁₇×#368

#7472

Subgroups — #17949412

Conjugacy classes of subgroups

Complements

Maximal subgroups