Group #17949431 — subgroups

Subgroups — #17949431

Order 4352.

162 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 62 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×#11763738	C₂	1	1	yes	maximal, characteristic
2	D₁₇⋊#368	C₂	1	1	yes	maximal, characteristic
3	D₁₇×M₆₄	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587536	—	2	1	no	—
5	D₁₇⋊#108	—	2	1	no	—
6	#11763738	—	2	1	no	—
7	#11763738	—	2	1	no	—
8	#11763738	C₄	1	2	yes	—
9	#11763737	C₄	1	2	yes	—
10	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
11	#11762643	C₄	1	1	yes	characteristic
12	#11762642	—	2	1	no	—
13	C₃₂×D₁₇	—	2	1	no	—
14	C₁₇×M₆₄	C₄	1	1	yes	characteristic
15	D₁₇⋊M₁₆	—	4	1	no	—
16	C₂×#63243	—	4	1	no	—
17	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
18	#10587536	—	2	2	no	—
19	#10587536	—	2	2	no	—
20	#10587536	—	4	1	no	—
21	#10587535	—	4	1	no	—
22	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×#63214	Q₈	1	1	yes	characteristic
24	C₁₆×D₁₇	C₂×C₄	1	1	yes	characteristic
25	C₁₆×D₁₇	C₂×C₄	1	1	yes	characteristic
26	C₁₆×D₁₇	D₄	1	1	yes	characteristic
27	C₁₆×D₁₇	D₄	1	1	yes	characteristic
28	C₁₇×C₃₂	—	2	1	no	—
29	#10587465	—	2	1	no	—
30	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
31	#63247	—	8	1	no	—
32	C₂×#3624	—	8	1	no	—
33	#63243	—	4	2	no	—
34	#63243	—	4	2	no	—
35	#63243	—	8	1	no	—
36	#63242	—	8	1	no	—
37	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂×#3622	Q₁₆	1	1	yes	characteristic
39	C₈×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₈×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₈×D₁₇	SD₁₆	1	1	yes	characteristic
42	C₈×D₁₇	SD₁₆	1	1	yes	characteristic
43	C₁₆×C₁₇	C₄²	1	1	yes	characteristic
44	C₁₆×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
45	#63214	C₄⋊C₄	1	1	yes	characteristic
46	#63214	C₄⋊C₄	1	1	yes	characteristic
47	#7479	—	17	1	no	maximal, Sylow
48	C₂²×D₁₇	Q₃₂	1	1	yes	characteristic
49	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
50	C₂×Dic₁₇	D₁₆	1	1	yes	characteristic
51	C₄×D₁₇	#102	1	1	yes	characteristic
52	C₄×D₁₇	#102	1	1	yes	characteristic
53	C₄×D₁₇	SD₃₂	1	1	yes	characteristic
54	C₄×D₁₇	SD₃₂	1	1	yes	characteristic
55	#3624	—	8	2	no	—
56	#3624	—	8	2	no	—
57	C₈×C₁₇	#95	1	1	yes	characteristic
58	C₈×C₁₇	#106	1	1	yes	characteristic
59	#3622	#103	1	1	yes	characteristic
60	#3622	#103	1	1	yes	characteristic
61	C₂×M₆₄	—	17	1	no	—
62	C₂×#368	—	17	1	no	—
63	C₂×#368	—	17	1	no	—
64	C₂²×C₁₇	#366	1	1	yes	characteristic
65	C₂×D₁₇	—	4	1	no	—
66	C₂×D₁₇	#358	1	1	yes	characteristic
67	C₂×D₁₇	#358	1	1	yes	characteristic
68	C₄×C₁₇	#340	1	1	yes	characteristic
69	C₄×C₁₇	#367	1	1	yes	characteristic
70	Dic₁₇	#357	1	1	yes	characteristic
71	Dic₁₇	#357	1	1	yes	characteristic
72	C₂²×C₁₆	—	17	1	no	—
73	C₂×#108	—	34	1	no	—
74	C₂×#108	—	34	1	no	—
75	M₆₄	—	17	1	no	—
76	M₆₄	—	34	1	no	—
77	M₆₄	#595	1	1	yes	characteristic
78	C₂×C₃₂	—	34	1	no	—
79	#368	—	17	2	no	—
80	#368	—	17	2	no	—
81	#368	—	34	1	no	—
82	#368	—	34	1	no	—
83	C₂×C₁₇	—	2	1	no	—
84	C₂×C₁₇	#1380	1	1	yes	characteristic
85	D₁₇	—	2	1	no	—
86	D₁₇	—	2	1	no	—
87	C₂×M₁₆	—	68	1	no	—
88	C₂×M₁₆	—	68	1	no	—
89	C₂²×C₈	—	17	1	no	—
90	C₂×C₁₆	—	17	1	no	—
91	C₂×C₁₆	—	17	1	no	—
92	C₂×C₁₆	—	17	1	no	—
93	C₂×C₁₆	—	17	1	no	—
94	C₂×C₁₆	—	17	1	no	—
95	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
96	#108	—	34	2	no	—
97	#108	—	34	2	no	—
98	#108	—	68	1	no	—
99	#108	—	68	1	no	—
100	C₃₂	—	2	1	no	—
101	C₃₂	—	34	1	no	—
102	C₁₇	#7479	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₄	—	17	1	no	—
104	M₁₆	—	68	2	no	—
105	M₁₆	—	68	2	no	—
106	M₁₆	—	136	1	no	—
107	M₁₆	—	136	1	no	—
108	C₂×C₈	—	17	1	no	—
109	C₂×C₈	—	17	1	no	—
110	C₂×C₈	—	17	1	no	—
111	C₂×C₈	—	17	1	no	—
112	C₂×C₈	—	17	1	no	—
113	C₂×C₈	—	136	1	no	—
114	C₂×C₈	—	136	1	no	—
115	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
116	C₁₆	—	17	1	no	—
117	C₁₆	—	17	1	no	—
118	C₁₆	C₄×#595	1	1	yes	characteristic
119	C₁₆	C₂²⋊#595	1	1	yes	characteristic
120	C₂³	—	17	1	no	—
121	C₂×C₄	—	17	1	no	—
122	C₂×C₄	—	17	1	no	—
123	C₂×C₄	—	17	1	no	—
124	C₂×C₄	—	17	1	no	—
125	C₂×C₄	—	17	1	no	—
126	C₂×C₄	#10587534	1	1	yes	characteristic
127	C₈	—	17	1	no	—
128	C₈	—	17	1	no	—
129	C₈	—	136	2	no	—
130	C₈	—	136	2	no	—
131	C₈	#10587533	1	1	yes	characteristic
132	C₈	(C₂×#595)⋊C₄	1	1	yes	characteristic
133	C₂²	—	17	1	no	—
134	C₂²	—	17	1	no	—
135	C₂²	—	68	1	no	—
136	C₂²	#11763735	1	1	yes	characteristic
137	C₄	—	17	1	no	—
138	C₄	—	17	1	no	—
139	C₄	#11763683	1	1	yes	characteristic
140	C₄	#11763736	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	34	1	no	—
143	C₂	—	34	1	no	—
144	C₂	#14543842	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11763738

C₂

yes

maximal, characteristic

D₁₇⋊#368

C₂

yes

maximal, characteristic

D₁₇×M₆₄

C₂

yes

maximal, characteristic

C₂×#10587536

—

D₁₇⋊#108

—

#11763738

—

#11763738

—

#11763738

C₄

yes

—

#11763737

C₄

yes

—

C₂×C₁₆×D₁₇

C₂²

yes

characteristic

#11762643

C₄

yes

characteristic

#11762642

—

C₃₂×D₁₇

—

C₁₇×M₆₄

C₄

yes

characteristic

D₁₇⋊M₁₆

—

C₂×#63243

—

C₂×C₈×D₁₇

D₄

yes

—

—

—

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₂×#63214

Q₈

yes

characteristic

C₁₆×D₁₇

C₂×C₄

yes

characteristic

C₁₆×D₁₇

C₂×C₄

yes

characteristic

C₁₆×D₁₇

D₄

yes

characteristic

C₁₆×D₁₇

D₄

yes

characteristic

C₁₇×C₃₂

—

#10587465

—

C₂×C₄×D₁₇

D₈

yes

—

—

—

—

—

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₂×#3622

Q₁₆

yes

characteristic

C₈×D₁₇

C₂²⋊C₄

yes

characteristic

C₈×D₁₇

C₂²⋊C₄

yes

characteristic

C₈×D₁₇

SD₁₆

yes

characteristic

C₈×D₁₇

SD₁₆

yes

characteristic

C₁₆×C₁₇

C₄²

yes

characteristic

C₁₆×C₁₇

C₂²⋊C₄

yes

characteristic

#63214

C₄⋊C₄

yes

characteristic

#63214

C₄⋊C₄

yes

characteristic

#7479

—

maximal, Sylow

C₂²×D₁₇

Q₃₂

yes

characteristic

C₂×C₄×C₁₇

#107

yes

characteristic

C₂×Dic₁₇

D₁₆

yes

C₄×D₁₇

yes

C₄×D₁₇

yes

C₄×D₁₇

SD₃₂

yes

characteristic

C₄×D₁₇

SD₃₂

yes

characteristic

#3624

—

#3624

—

C₈×C₁₇

yes

C₈×C₁₇

yes

yes

yes

C₂×M₆₄

—

C₂×#368

—

C₂×#368

—

C₂²×C₁₇

#366

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

—

M₆₄

—

M₆₄

—

M₆₄

#595

yes

characteristic

C₂×C₃₂

—

—

—

—

—

C₂×C₁₇

—

C₂×C₁₇

#1380

yes

characteristic

D₁₇

—

D₁₇

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

—

—

—

—

100

C₃₂

—

101

C₃₂

—

102

C₁₇

#7479

yes

characteristic, complemented, Sylow

103

C₂²×C₄

—

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

136

—

107

M₁₆

—

136

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

136

—

114

C₂×C₈

—

136

—

115

C₂×C₈

C₄⋊#595

yes

characteristic

116

C₁₆

—

117

C₁₆

—

118

C₁₆

C₄×#595

yes

characteristic

119

C₁₆

C₂²⋊#595

yes

characteristic

120

C₂³

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

#10587534

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

136

—

130

C₈

—

136

—

131

C₈

#10587533

yes

characteristic

132

C₈

(C₂×#595)⋊C₄

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

#11763735

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

#11763683

yes

characteristic

140

C₄

#11763736

yes

characteristic

141

C₂

—

142

C₂

—

143

C₂

—

144

C₂

#14543842

yes

characteristic

	subgroup	G/H
center	C₂	#14543842
commutator	C₁₆×C₁₇	C₄²
Frattini	C₂×C₁₆	C₂×#595
Fitting	C₁₇×M₆₄	C₄
socle	C₂×C₁₇	#1380
prime radical	C₂²×D₁₇	Q₃₂
cosocle	C₂×C₁₆×D₁₇	C₂²
hypercenter	M₆₄	#595
hypocenter	C₁₇	#7479
LCS	G ▷ C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

subgroup

G/H

center

C₂

#14543842

commutator

C₁₆×C₁₇

C₄²

Frattini

C₂×C₁₆

C₂×#595

Fitting

C₁₇×M₆₄

C₄

socle

C₂×C₁₇

#1380

prime radical

C₂²×D₁₇

Q₃₂

cosocle

C₂×C₁₆×D₁₇

C₂²

M₆₄

C₁₇

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

N	H ≅ G/N	type
102. C₁₇	#7479	semidirect

H ≅ G/N

type

102. C₁₇

#7479

semidirect

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

subgroup	index
C₂×#11763738	2
D₁₇⋊#368	2
D₁₇×M₆₄	2
#7479	17

subgroup

index

C₂×#11763738

D₁₇⋊#368

D₁₇×M₆₄

#7479

Subgroups — #17949431

Conjugacy classes of subgroups

Complements

Maximal subgroups