Group #14541667 — subgroups

Subgroups — #14541667

Order 2176.

336 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 133 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	#11763539	C₂	1	2	yes	maximal
2	#11763213	C₂	1	2	yes	maximal
3	#11763103	C₂	1	4	yes	maximal
4	#11763101	C₂	1	2	yes	maximal
5	C₈×Dic₃₄	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₆₈	C₂	1	1	yes	maximal, characteristic
7	#11762903	C₂	1	2	yes	maximal
8	C₁₇×#500	C₂	1	1	yes	maximal, characteristic
9	C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
10	#10587630	—	2	4	no	—
11	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
12	#10587574	—	2	2	no	—
13	#10587574	C₂²	1	4	yes	—
14	#10587573	—	2	2	no	—
15	#10587573	C₂²	1	4	yes	—
16	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
17	C₄×Dic₃₄	C₂²	1	2	yes	—
18	C₁₇×#107	C₂²	1	2	yes	—
19	C₁₇×#107	C₂²	1	2	yes	—
20	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
21	#10587489	C₂²	1	1	yes	characteristic
22	#10587487	C₂²	1	2	yes	—
23	#10587485	C₂²	1	2	yes	—
24	C₈×Dic₁₇	C₂²	1	2	yes	—
25	#10587481	C₂²	1	4	yes	—
26	#10587478	C₂²	1	4	yes	—
27	C₄⋊#3622	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₁₇	—	2	2	no	—
29	C₂×Dic₃₄	—	2	4	no	—
30	C₂×Dic₃₄	D₄	1	1	yes	characteristic
31	C₂×Dic₃₄	C₂³	1	2	yes	—
32	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₇	—	2	2	no	—
35	C₄⋊C₄×C₁₇	C₂³	1	4	yes	—
36	C₄²×C₁₇	C₂³	1	1	yes	characteristic
37	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
38	C₄⋊Dic₁₇	C₂³	1	2	yes	—
39	#63225	—	2	2	no	—
40	#63225	—	2	4	no	—
41	#63225	—	2	4	no	—
42	#63225	D₄	1	2	yes	—
43	C₄×Dic₁₇	—	2	2	no	—
44	C₄×Dic₁₇	—	2	4	no	—
45	C₄×Dic₁₇	C₂³	1	2	yes	—
46	C₂×#3622	C₂³	1	2	yes	—
47	Dic₆₈	Q₈	1	4	yes	complemented
48	Q₈×C₁₇	—	8	2	no	—
49	C₂×C₄×C₁₇	—	2	4	no	—
50	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₁₇	—	2	2	no	—
54	C₂×Dic₁₇	—	2	2	no	—
55	C₂×Dic₁₇	C₂×D₄	1	2	yes	—
56	C₂×Dic₁₇	C₂×D₄	1	2	yes	—
57	Dic₃₄	—	2	2	no	—
58	Dic₃₄	—	8	4	no	—
59	Dic₃₄	Q₁₆	1	4	yes	complemented
60	Dic₃₄	C₂×Q₈	1	4	yes	—
61	C₈×C₁₇	—	2	1	no	—
62	C₈×C₁₇	C₂×Q₈	1	2	yes	—
63	#3622	—	2	2	no	—
64	Q₈×Q₁₆	—	17	1	no	maximal, Sylow
65	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
66	C₄×C₁₇	—	2	1	no	—
67	C₄×C₁₇	—	4	4	no	—
68	C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
69	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
70	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
71	Dic₁₇	—	2	4	no	—
72	Dic₁₇	—	2	2	no	—
73	Dic₁₇	—	4	2	no	—
74	Dic₁₇	C₂×Q₁₆	1	4	yes	—
75	Q₈²	—	17	2	no	—
76	#500	—	17	2	no	—
77	#500	D₁₇	1	1	yes	characteristic
78	#477	—	17	2	no	—
79	#477	—	17	4	no	—
80	C₈×Q₈	—	17	1	no	—
81	C₄×Q₁₆	—	17	2	no	—
82	C₄×Q₁₆	—	17	1	no	—
83	C₂×C₁₇	D₄×Q₈	1	1	yes	characteristic
84	C₂×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
85	C₂×C₁₇	#576	1	1	yes	characteristic
86	C₂×Q₁₆	—	17	1	no	—
87	C₄⋊Q₈	—	17	4	no	—
88	C₄⋊Q₈	—	34	2	no	—
89	C₄⋊Q₈	—	34	4	no	—
90	C₄⋊Q₈	C₂×D₁₇	1	2	yes	—
91	C₄×Q₈	—	17	1	no	—
92	C₄×Q₈	—	17	4	no	—
93	C₄×Q₈	—	17	2	no	—
94	C₄×Q₈	—	34	2	no	—
95	#107	—	17	1	no	—
96	#107	—	17	4	no	—
97	#107	C₂×D₁₇	1	2	yes	—
98	#107	C₂×D₁₇	1	2	yes	—
99	C₄⋊C₈	—	17	2	no	—
100	C₄⋊C₈	—	17	1	no	—
101	#103	—	17	2	no	—
102	#103	—	17	4	no	—
103	C₄×C₈	—	17	2	no	—
104	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
105	C₁₇	Q₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
106	C₂×Q₈	—	2	2	no	—
107	C₂×Q₈	—	17	1	no	—
108	C₂×Q₈	—	17	2	no	—
109	C₂×Q₈	—	34	4	no	—
110	Q₁₆	—	17	4	no	—
111	C₂×C₈	—	17	2	no	—
112	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
113	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	17	2	no	—
116	C₄⋊C₄	—	17	1	no	—
117	C₄⋊C₄	—	17	2	no	—
118	C₄⋊C₄	—	34	4	no	—
119	C₄⋊C₄	—	34	4	no	—
120	C₄⋊C₄	—	34	2	no	—
121	C₄⋊C₄	C₂²×D₁₇	1	4	yes	—
122	C₄²	—	17	2	no	—
123	C₄²	—	34	4	no	—
124	C₄²	—	34	2	no	—
125	C₄²	C₂²×D₁₇	1	1	yes	characteristic
126	Q₈	—	8	2	no	—
127	Q₈	—	17	4	no	—
128	Q₈	—	17	4	no	—
129	Q₈	—	34	2	no	—
130	Q₈	—	136	4	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	17	2	no	—
133	C₂×C₄	—	17	2	no	—
134	C₂×C₄	—	34	2	no	—
135	C₂×C₄	—	34	2	no	—
136	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
137	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	34	2	no	—
141	C₈	Q₈×D₁₇	1	2	yes	—
142	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	4	4	no	—
145	C₄	—	17	4	no	—
146	C₄	—	34	4	no	—
147	C₄	—	34	2	no	—
148	C₄	—	68	2	no	—
149	C₄	Q₁₆×D₁₇	1	2	yes	—
150	C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
151	C₄	#10587694	1	1	yes	characteristic
152	C₂	#11763474	1	1	yes	characteristic
153	C₂	C₂×Q₁₆×D₁₇	1	1	yes	characteristic
154	C₂	#11763610	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763539

C₂

yes

maximal

#11763213

C₂

yes

maximal

#11763103

C₂

yes

maximal

#11763101

C₂

yes

maximal

C₈×Dic₃₄

C₂

yes

maximal, characteristic

C₄×Dic₆₈

C₂

yes

maximal, characteristic

#11762903

C₂

yes

maximal

C₁₇×#500

C₂

yes

maximal, characteristic

C₁₇×C₄⋊Q₈

C₂²

yes

—

#10587630

—

C₂×Dic₆₈

C₂²

yes

characteristic

#10587574

—

#10587574

C₂²

yes

—

#10587573

—

#10587573

C₂²

yes

—

C₄×Dic₃₄

C₂²

yes

characteristic

C₄×Dic₃₄

C₂²

yes

—

C₁₇×#107

C₂²

yes

—

C₁₇×#107

C₂²

yes

—

C₄×C₈×C₁₇

C₂²

yes

characteristic

#10587489

C₂²

yes

characteristic

#10587487

C₂²

yes

—

#10587485

C₂²

yes

—

C₈×Dic₁₇

C₂²

yes

—

#10587481

C₂²

yes

—

#10587478

C₂²

yes

—

C₄⋊#3622

C₂²

yes

characteristic

C₂×Q₈×C₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

D₄

yes

characteristic

C₂×Dic₃₄

C₂³

yes

—

C₂×C₈×C₁₇

C₂³

yes

characteristic

C₂×C₈×C₁₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

C₂³

yes

—

C₄²×C₁₇

C₂³

yes

characteristic

C₄⋊Dic₁₇

D₄

yes

characteristic

C₄⋊Dic₁₇

C₂³

yes

—

—

—

—

D₄

yes

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

C₂³

yes

—

C₂×#3622

C₂³

yes

—

Dic₆₈

Q₈

yes

complemented

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂⁴

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂×D₄

yes

—

C₂×Dic₁₇

C₂×D₄

yes

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

Q₁₆

yes

complemented

Dic₃₄

C₂×Q₈

yes

—

C₈×C₁₇

—

C₈×C₁₇

C₂×Q₈

yes

—

#3622

—

Q₈×Q₁₆

—

maximal, Sylow

C₂²×C₁₇

C₂²×D₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×Q₁₆

yes

—

C₄×C₁₇

C₂²×Q₈

yes

characteristic

C₄×C₁₇

2₋¹⁺⁴

yes

characteristic

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

C₂×Q₁₆

yes

—

Q₈²

—

#500

—

#500

D₁₇

yes

characteristic

#477

—

#477

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₂×C₁₇

D₄×Q₈

yes

characteristic

C₂×C₁₇

C₂²×Q₁₆

yes

characteristic

C₂×C₁₇

#576

yes

characteristic

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₇

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#107

—

#107

—

#107

C₂×D₁₇

yes

—

#107

C₂×D₁₇

yes

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

#103

—

102

#103

—

103

C₄×C₈

—

104

C₄×C₈

C₂×D₁₇

yes

characteristic

105

C₁₇

Q₈×Q₁₆

yes

characteristic, complemented, Sylow

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

Q₁₆

—

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₁₇

yes

characteristic

113

C₂×C₈

C₂²×D₁₇

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

C₂²×D₁₇

yes

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

C₂²×D₁₇

yes

characteristic

126

Q₈

—

127

Q₈

—

128

Q₈

—

129

Q₈

—

130

Q₈

—

136

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

D₄×D₁₇

yes

characteristic

137

C₂×C₄

D₄×D₁₇

yes

characteristic

138

C₂×C₄

C₂³×D₁₇

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

Q₈×D₁₇

yes

—

142

C₂²

C₂×D₄×D₁₇

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

Q₁₆×D₁₇

yes

—

150

C₄

C₂×Q₈×D₁₇

yes

characteristic

151

C₄

#10587694

yes

characteristic

152

C₂

#11763474

yes

characteristic

153

C₂

C₂×Q₁₆×D₁₇

yes

characteristic

154

C₂

#11763610

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#500	C₂
socle, prime radical	C₂²×C₁₇	C₂²×D₄
hypercenter	#500	D₁₇
hypocenter	C₁₇	Q₈×Q₁₆
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₁₇

commutator, cosocle

C₂×C₄×C₁₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₇

Fitting

C₁₇×#500

C₂

socle, prime radical

C₂²×C₁₇

C₂²×D₄

hypercenter

#500

D₁₇

hypocenter

C₁₇

Q₈×Q₁₆

LCS

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
47. Dic₆₈	Q₈	semidirect
59. Dic₃₄	Q₁₆	semidirect
105. C₁₇	Q₈×Q₁₆	semidirect

H ≅ G/N

type

47. Dic₆₈

Q₈

semidirect

59. Dic₃₄

Q₁₆

semidirect

105. C₁₇

Q₈×Q₁₆

semidirect

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

subgroup	index
#11763539	2
#11763213	2
#11763103	2
#11763101	2
C₈×Dic₃₄	2
C₄×Dic₆₈	2
#11762903	2
C₁₇×#500	2
Q₈×Q₁₆	17

subgroup

index

C₈×Dic₃₄

C₄×Dic₆₈

#11762903

C₁₇×#500

Q₈×Q₁₆

Subgroups — #14541667

Conjugacy classes of subgroups

Complements

Maximal subgroups