Group #17953847 — subgroups

Subgroups — #17953847

Order 4352.

338 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 128 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	#14544725	C₂	1	1	yes	maximal, characteristic
2	#14544723	C₂	1	1	yes	maximal, characteristic
3	#14544340	C₂	1	2	yes	maximal
4	#14544339	C₂	1	2	yes	maximal
5	D₁₇×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587536	—	2	2	no	—
7	C₂×#10587535	—	2	2	no	—
8	#11763813	C₂²	1	2	yes	—
9	#11763813	C₂²	1	1	yes	characteristic
10	#11763813	C₂²	1	1	yes	characteristic
11	#11763671	—	2	4	no	—
12	#11763670	—	2	4	no	—
13	C₄×C₈×D₁₇	C₂²	1	2	yes	—
14	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
15	#11762531	C₄	1	2	yes	—
16	#11762530	C₄	1	1	yes	characteristic
17	C₁₇×#322	C₄	1	1	yes	characteristic
18	C₂×#63243	—	2	2	no	—
19	C₂×#63243	—	2	1	no	—
20	C₂×#63243	—	2	1	no	—
21	C₂×C₈×D₁₇	D₄	1	2	yes	—
22	C₂×C₈×D₁₇	Q₈	1	4	yes	—
23	C₄²×D₁₇	C₂³	1	1	yes	characteristic
24	C₄⋊#3624	—	2	2	no	—
25	C₄⋊#3624	—	2	2	no	—
26	C₄⋊#3624	—	2	2	no	—
27	C₄⋊#3624	—	2	2	no	—
28	#10587536	—	4	4	no	—
29	#10587535	—	4	4	no	—
30	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
32	C₈×Dic₁₇	D₄	1	4	yes	—
33	C₈×Dic₁₇	Q₈	1	2	yes	—
34	C₄×#3622	C₂×C₄	1	2	yes	—
35	C₄×#3622	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×D₁₇	C₂×Q₈	1	2	yes	—
38	C₂×#3624	—	4	2	no	—
39	C₂×#3624	—	4	1	no	—
40	C₂×#3624	—	4	1	no	—
41	#63243	—	4	4	no	—
42	#63243	—	4	2	no	—
43	#63243	—	4	2	no	—
44	C₂×C₈×C₁₇	C₄⋊C₄	1	4	yes	—
45	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
46	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
47	C₄×Dic₁₇	C₂×D₄	1	2	yes	—
48	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×#3622	C₄⋊C₄	1	4	yes	—
50	C₂×#3622	C₄⋊C₄	1	2	yes	—
51	C₈×D₁₇	—	4	4	no	—
52	C₈×D₁₇	SD₁₆	1	4	yes	—
53	C₈×D₁₇	SD₁₆	1	4	yes	—
54	#8825	—	17	1	no	maximal, Sylow
55	C₂²×D₁₇	C₄⋊Q₈	1	1	yes	characteristic
56	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
58	C₂×Dic₁₇	#127	1	1	yes	characteristic
59	C₂×Dic₁₇	C₄⋊Q₈	1	2	yes	—
60	C₄×D₁₇	—	2	4	no	—
61	C₄×D₁₇	C₂×SD₁₆	1	2	yes	—
62	C₄×D₁₇	C₂×SD₁₆	1	2	yes	—
63	#3624	—	8	2	no	—
64	#3624	—	8	1	no	—
65	#3624	—	8	1	no	—
66	C₈×C₁₇	—	2	4	no	—
67	C₈×C₁₇	#106	1	4	yes	—
68	#3622	—	2	4	no	—
69	#3622	#106	1	4	yes	—
70	#1848	—	17	1	no	—
71	#1848	—	17	1	no	—
72	#1566	—	17	2	no	—
73	#1566	—	17	2	no	—
74	C₂×#322	—	17	1	no	—
75	C₂²×C₁₇	#384	1	1	yes	characteristic
76	C₂×D₁₇	—	2	2	no	—
77	C₂×D₁₇	#498	1	2	yes	—
78	C₄×C₁₇	C₂×#106	1	2	yes	—
79	C₄×C₁₇	#430	1	4	yes	—
80	Dic₁₇	—	2	2	no	—
81	Dic₁₇	#492	1	1	yes	characteristic
82	Dic₁₇	#492	1	1	yes	characteristic
83	C₂×#108	—	34	2	no	—
84	C₂×#108	—	34	2	no	—
85	#423	—	17	1	no	—
86	#423	—	17	2	no	—
87	#423	—	17	1	no	—
88	C₂×C₄×C₈	—	17	2	no	—
89	C₂×C₄×C₈	—	17	1	no	—
90	#334	—	34	4	no	—
91	#334	—	34	4	no	—
92	#322	—	17	1	no	—
93	#322	—	17	2	no	—
94	#322	#595	1	1	yes	characteristic
95	C₂×C₁₇	#1844	1	1	yes	characteristic
96	C₂×C₁₇	#1849	1	2	yes	—
97	D₁₇	—	2	2	no	—
98	C₂×M₁₆	—	34	2	no	—
99	C₂×M₁₆	—	34	1	no	—
100	C₂×M₁₆	—	34	1	no	—
101	C₂²×C₈	—	17	2	no	—
102	C₂²×C₈	—	17	4	no	—
103	C₂×C₄²	—	17	1	no	—
104	#108	—	68	4	no	—
105	#108	—	68	4	no	—
106	C₄⋊C₈	—	34	2	no	—
107	C₄⋊C₈	—	34	2	no	—
108	C₄⋊C₈	—	34	2	no	—
109	C₄⋊C₈	—	34	2	no	—
110	C₄×C₈	—	17	4	no	—
111	C₄×C₈	—	17	2	no	—
112	C₄×C₈	—	17	1	no	—
113	C₄×C₈	—	17	2	no	—
114	C₄×C₈	C₂×#595	1	2	yes	—
115	C₄×C₈	C₂×#595	1	1	yes	characteristic
116	C₁₇	#8825	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₄	—	17	1	no	—
118	C₂²×C₄	—	17	2	no	—
119	M₁₆	—	68	4	no	—
120	M₁₆	—	68	2	no	—
121	M₁₆	—	68	2	no	—
122	C₂×C₈	—	17	2	no	—
123	C₂×C₈	—	17	4	no	—
124	C₂×C₈	—	17	4	no	—
125	C₂×C₈	—	17	4	no	—
126	C₂×C₈	—	68	4	no	—
127	C₂×C₈	—	68	2	no	—
128	C₂×C₈	—	68	1	no	—
129	C₂×C₈	—	68	1	no	—
130	C₂×C₈	C₄⋊#595	1	4	yes	—
131	C₂×C₈	C₄⋊#595	1	2	yes	—
132	C₄²	—	17	2	no	—
133	C₄²	—	17	1	no	—
134	C₄²	C₂²×#595	1	1	yes	characteristic
135	C₂³	—	17	1	no	—
136	C₂×C₄	—	17	1	no	—
137	C₂×C₄	—	17	2	no	—
138	C₂×C₄	—	17	2	no	—
139	C₂×C₄	—	17	2	no	—
140	C₂×C₄	—	34	4	no	—
141	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄⋊#595	1	2	yes	—
143	C₈	—	2	4	no	—
144	C₈	—	17	4	no	—
145	C₈	—	34	4	no	—
146	C₈	—	136	2	no	—
147	C₈	—	136	1	no	—
148	C₈	—	136	1	no	—
149	C₈	#10587533	1	4	yes	—
150	C₂²	—	17	2	no	—
151	C₂²	—	34	2	no	—
152	C₂²	#11763755	1	1	yes	characteristic
153	C₄	—	17	1	no	—
154	C₄	—	17	1	no	—
155	C₄	—	34	2	no	—
156	C₄	C₂×#10587533	1	2	yes	—
157	C₄	#11763824	1	4	yes	—
158	C₂	—	34	2	no	—
159	C₂	#14544719	1	1	yes	characteristic
160	C₂	#14544726	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14544725

C₂

yes

maximal, characteristic

#14544723

C₂

yes

maximal, characteristic

#14544340

C₂

yes

maximal

#14544339

C₂

yes

maximal

D₁₇×#322

C₂

yes

maximal, characteristic

C₂×#10587536

—

C₂×#10587535

—

#11763813

C₂²

yes

—

#11763813

C₂²

yes

characteristic

#11763813

C₂²

yes

characteristic

#11763671

—

#11763670

—

C₄×C₈×D₁₇

C₂²

yes

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762531

C₄

yes

—

#11762530

C₄

yes

characteristic

C₁₇×#322

C₄

yes

—

—

—

C₂×C₈×D₁₇

D₄

yes

—

C₂×C₈×D₁₇

Q₈

yes

—

C₄²×D₁₇

C₂³

yes

—

—

—

—

—

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₈×Dic₁₇

D₄

yes

—

C₈×Dic₁₇

Q₈

yes

—

C₄×#3622

C₂×C₄

yes

—

C₄×#3622

C₂×C₄

yes

characteristic

C₂×C₄×D₁₇

C₂×D₄

yes

characteristic

C₂×C₄×D₁₇

C₂×Q₈

yes

—

—

—

—

—

—

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄×Dic₁₇

C₂×D₄

yes

—

C₄×Dic₁₇

C₂×Q₈

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

—

C₂×#3622

C₄⋊C₄

yes

—

C₈×D₁₇

—

C₈×D₁₇

SD₁₆

yes

—

C₈×D₁₇

SD₁₆

yes

—

#8825

—

maximal, Sylow

C₂²×D₁₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

—

C₂×Dic₁₇

#127

yes

characteristic

C₂×Dic₁₇

C₄⋊Q₈

yes

—

C₄×D₁₇

—

C₄×D₁₇

C₂×SD₁₆

yes

—

C₄×D₁₇

C₂×SD₁₆

yes

—

#3624

—

#3624

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

#106

yes

—

#3622

—

#3622

#106

yes

—

—

—

—

—

—

C₂²×C₁₇

#384

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

#498

yes

—

C₄×C₁₇

C₂×#106

yes

—

C₄×C₁₇

#430

yes

—

Dic₁₇

—

Dic₁₇

yes

Dic₁₇

yes

—

—

—

—

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

—

—

—

—

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

—

D₁₇

—

C₂×M₁₆

—

C₂×M₁₆

—

100

C₂×M₁₆

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₂×C₄²

—

104

#108

—

105

#108

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

C₂×#595

yes

—

115

C₄×C₈

C₂×#595

yes

characteristic

116

C₁₇

#8825

yes

characteristic, complemented, Sylow

117

C₂²×C₄

—

118

C₂²×C₄

—

119

M₁₆

—

120

M₁₆

—

121

M₁₆

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

C₄⋊#595

yes

—

131

C₂×C₈

C₄⋊#595

yes

—

132

C₄²

—

133

C₄²

—

134

C₄²

C₂²×#595

yes

characteristic

135

C₂³

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

142

C₂×C₄

C₂×C₄⋊#595

yes

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

136

—

147

C₈

—

136

—

148

C₈

—

136

—

149

C₈

#10587533

yes

—

150

C₂²

—

151

C₂²

—

152

C₂²

#11763755

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

C₂×#10587533

yes

—

157

C₄

#11763824

yes

—

158

C₂

—

159

C₂

#14544719

yes

characteristic

160

C₂

#14544726

yes

—

	subgroup	G/H
center	C₂²	#11763755
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₄²	C₂²×#595
Fitting	C₁₇×#322	C₄
socle	C₂²×C₁₇	#384
prime radical	C₂²×D₁₇	C₄⋊Q₈
cosocle	C₄²×D₁₇	C₂³
hypercenter	#322	#595
hypocenter	C₁₇	#8825
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#11763755

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₄²

C₂²×#595

Fitting

C₁₇×#322

C₄

socle

C₂²×C₁₇

#384

prime radical

C₂²×D₁₇

C₄⋊Q₈

cosocle

C₄²×D₁₇

C₂³

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
116. C₁₇	#8825	semidirect

H ≅ G/N

type

116. C₁₇

#8825

semidirect

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

subgroup	index
#14544725	2
#14544723	2
#14544340	2
#14544339	2
D₁₇×#322	2
#8825	17

subgroup

index

D₁₇×#322

Subgroups — #17953847

Conjugacy classes of subgroups

Complements

Maximal subgroups