Group #14530194 — subgroups

Subgroups — #14530194

Order 2176.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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×#10587487	C₂	1	2	yes	maximal
2	#11762715	C₂	1	1	yes	maximal, characteristic
3	#11762679	C₂	1	1	yes	maximal, characteristic
4	#11762572	C₂	1	2	yes	maximal
5	C₁₇×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₄	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₇	C₂²	1	2	yes	—
8	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₇	—	2	1	no	—
10	C₂×C₄⋊Dic₁₇	—	2	1	no	—
11	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
13	#10587502	—	2	1	no	—
14	#10587502	—	2	1	no	—
15	#10587487	—	2	4	no	—
16	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
19	C₂²×Dic₁₇	—	2	1	no	—
20	C₂²×Dic₁₇	—	2	1	no	—
21	C₂²×Dic₁₇	—	2	2	no	—
22	C₂×Dic₃₄	—	2	2	no	—
23	C₂×Dic₃₄	—	2	2	no	—
24	C₂×Dic₃₄	—	2	1	no	—
25	C₂×Dic₃₄	D₄	1	2	yes	—
26	C₂×C₈×C₁₇	—	2	2	no	—
27	C₂×C₈×C₁₇	—	2	2	no	—
28	C₂×C₈×C₁₇	—	2	2	no	—
29	C₄²×C₁₇	—	2	2	no	—
30	C₄⋊Dic₁₇	—	2	2	no	—
31	C₄⋊Dic₁₇	—	2	2	no	—
32	C₄⋊Dic₁₇	—	2	2	no	—
33	C₄⋊Dic₁₇	—	4	2	no	—
34	C₄⋊Dic₁₇	—	4	2	no	—
35	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₇	—	2	1	no	—
37	C₂×C₄×C₁₇	—	2	1	no	—
38	C₂×C₄×C₁₇	—	2	1	no	—
39	C₂×C₄×C₁₇	—	2	1	no	—
40	C₂×C₄×C₁₇	SD₁₆	1	2	yes	—
41	C₂×C₄×C₁₇	Q₁₆	1	2	yes	—
42	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₇	#41	1	2	yes	—
44	C₂×C₄×C₁₇	#41	1	2	yes	—
45	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
46	C₂×Dic₁₇	—	2	2	no	—
47	C₂×Dic₁₇	—	4	1	no	—
48	C₂×Dic₁₇	—	4	1	no	—
49	C₂×Dic₁₇	—	4	2	no	—
50	C₂×Dic₁₇	—	4	1	no	—
51	C₂×Dic₁₇	—	4	2	no	—
52	C₂×Dic₁₇	—	4	1	no	—
53	C₂×Dic₁₇	—	4	2	no	—
54	C₂×Dic₁₇	—	4	1	no	—
55	Dic₃₄	—	2	4	no	—
56	Dic₃₄	—	4	2	no	—
57	C₈×C₁₇	—	4	2	no	—
58	#2072	—	17	1	no	maximal, Sylow
59	C₂²×C₁₇	#121	1	1	yes	characteristic
60	C₂²×C₁₇	#123	1	1	yes	characteristic
61	C₂²×C₁₇	#124	1	1	yes	characteristic
62	C₂²×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₂²×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
64	C₂²×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂²×C₁₇	#137	1	1	yes	characteristic
66	C₄×C₁₇	—	2	2	no	—
67	C₄×C₁₇	—	2	2	no	—
68	C₄×C₁₇	#123	1	2	yes	—
69	C₄×C₁₇	#126	1	2	yes	—
70	Dic₁₇	—	4	2	no	—
71	Dic₁₇	—	8	2	no	—
72	Dic₁₇	—	8	1	no	—
73	C₂×#103	—	17	2	no	—
74	#391	—	17	1	no	—
75	#384	—	17	1	no	—
76	#340	—	17	2	no	—
77	#336	D₁₇	1	1	yes	characteristic
78	C₂×C₁₇	#397	1	1	yes	characteristic
79	C₂×C₁₇	#460	1	1	yes	characteristic
80	C₂×C₁₇	#462	1	1	yes	characteristic
81	C₂×C₁₇	#483	1	1	yes	characteristic
82	C₂×C₁₇	#484	1	1	yes	characteristic
83	C₂×C₁₇	#487	1	1	yes	characteristic
84	C₂×C₁₇	#489	1	1	yes	characteristic
85	C₂²×Q₈	—	17	1	no	—
86	C₂²×C₈	C₂×D₁₇	1	2	yes	—
87	C₂×C₄⋊C₄	—	17	1	no	—
88	C₂×C₄⋊C₄	—	17	2	no	—
89	C₂×C₄⋊C₄	—	34	1	no	—
90	C₂×C₄⋊C₄	—	34	1	no	—
91	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
92	#103	—	34	4	no	—
93	#95	—	34	1	no	—
94	#95	—	34	1	no	—
95	C₁₇	#2072	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₈	—	17	2	no	—
97	C₂×Q₈	—	34	1	no	—
98	C₂×Q₈	—	34	2	no	—
99	C₂×Q₈	—	34	2	no	—
100	C₂²×C₄	—	34	1	no	—
101	C₂²×C₄	—	34	1	no	—
102	C₂²×C₄	—	34	2	no	—
103	C₂²×C₄	D₆₈	1	1	yes	characteristic
104	C₂²×C₄	D₆₈	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₄⋊C₄	—	34	2	no	—
110	C₄⋊C₄	—	34	2	no	—
111	C₄⋊C₄	—	34	2	no	—
112	C₄⋊C₄	—	68	2	no	—
113	C₄⋊C₄	—	68	2	no	—
114	C₄²	—	2	2	no	—
115	C₂³	C₂×D₆₈	1	1	yes	characteristic
116	Q₈	—	34	4	no	—
117	Q₈	—	68	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	34	2	no	—
123	C₂×C₄	—	68	1	no	—
124	C₂×C₄	—	68	1	no	—
125	C₂×C₄	—	68	1	no	—
126	C₂×C₄	—	68	2	no	—
127	C₂×C₄	—	68	2	no	—
128	C₂×C₄	—	68	2	no	—
129	C₂×C₄	—	68	1	no	—
130	C₂×C₄	—	68	1	no	—
131	C₂×C₄	#63219	1	2	yes	—
132	C₂×C₄	Dic₆₈	1	2	yes	—
133	C₂×C₄	#63252	1	2	yes	—
134	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
135	C₂×C₄	#63254	1	2	yes	—
136	C₂×C₄	#63256	1	1	yes	characteristic
137	C₈	—	4	2	no	—
138	C₂²	#10587561	1	1	yes	characteristic
139	C₂²	#10587572	1	1	yes	characteristic
140	C₂²	#10587581	1	1	yes	characteristic
141	C₂²	C₂×#63219	1	1	yes	characteristic
142	C₂²	C₂×Dic₆₈	1	1	yes	characteristic
143	C₂²	#10587594	1	1	yes	characteristic
144	C₂²	#10587595	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	68	2	no	—
148	C₄	—	136	2	no	—
149	C₄	—	136	1	no	—
150	C₄	#10587569	1	2	yes	—
151	C₄	#10587584	1	2	yes	—
152	C₂	#11762736	1	1	yes	characteristic
153	C₂	#11763010	1	1	yes	characteristic
154	C₂	#11763020	1	1	yes	characteristic
155	C₂	#11763140	1	1	yes	characteristic
156	C₂	#11763145	1	1	yes	characteristic
157	C₂	#11763160	1	1	yes	characteristic
158	C₂	#11763169	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587487

C₂

yes

maximal

#11762715

C₂

yes

maximal, characteristic

#11762679

C₂

yes

maximal, characteristic

#11762572

C₂

yes

maximal

C₁₇×#336

C₂

yes

maximal, characteristic

C₂²×Dic₃₄

C₂²

yes

characteristic

C₂²×C₈×C₁₇

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₂²

yes

—

#10587502

—

#10587502

—

#10587487

—

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

D₄

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

SD₁₆

yes

—

C₂×C₄×C₁₇

Q₁₆

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

—

C₂×C₄×C₁₇

#41

yes

—

C₂×C₄×C₁₇

#41

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₇

C₂×Q₁₆

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#123

yes

—

C₄×C₁₇

#126

yes

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

—

—

—

—

D₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×Q₈

—

C₂²×C₈

C₂×D₁₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₇

yes

—

—

—

C₁₇

#2072

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

D₆₈

yes

characteristic

104

C₂²×C₄

D₆₈

yes

characteristic

105

C₂²×C₄

C₂²×D₁₇

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄²

—

115

C₂³

C₂×D₆₈

yes

characteristic

116

Q₈

—

117

Q₈

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#63219

yes

—

132

C₂×C₄

Dic₆₈

yes

—

133

C₂×C₄

#63252

yes

—

134

C₂×C₄

D₄×D₁₇

yes

characteristic

135

C₂×C₄

#63254

yes

—

136

C₂×C₄

#63256

yes

characteristic

137

C₈

—

138

C₂²

#10587561

yes

characteristic

139

C₂²

#10587572

yes

characteristic

140

C₂²

#10587581

yes

characteristic

141

C₂²

C₂×#63219

yes

characteristic

142

C₂²

C₂×Dic₆₈

yes

characteristic

143

C₂²

#10587594

yes

characteristic

144

C₂²

#10587595

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

136

—

149

C₄

—

136

—

150

C₄

#10587569

yes

—

151

C₄

#10587584

yes

—

152

C₂

#11762736

yes

characteristic

153

C₂

#11763010

yes

characteristic

154

C₂

#11763020

yes

characteristic

155

C₂

#11763140

yes

characteristic

156

C₂

#11763145

yes

characteristic

157

C₂

#11763160

yes

characteristic

158

C₂

#11763169

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₈
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#336	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₄
hypercenter	#336	D₁₇
hypocenter	C₁₇	#2072
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₆₈

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#336

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₄

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
95. C₁₇	#2072	semidirect

H ≅ G/N

type

95. C₁₇

#2072

semidirect

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

subgroup	index
C₂×#10587487	2
#11762715	2
#11762679	2
#11762572	2
C₁₇×#336	2
#2072	17

subgroup

index

C₁₇×#336

Subgroups — #14530194

Conjugacy classes of subgroups

Complements

Maximal subgroups