Group #16865488 — subgroups

Subgroups — C₁₇×#7744 #16865488

Order 4352.

448 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 94 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	C₁₇×#2810	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₇×#2041	C₂	1	3	yes	maximal
3	C₁₇×#1271	C₂	1	3	yes	maximal, complemented
4	C₂×C₁₇×#127	C₂²	1	1	yes	characteristic
5	C₂×C₁₇×#102	—	2	3	no	—
6	C₂×C₁₇×#102	—	2	3	no	—
7	C₁₇×#400	—	2	3	no	—
8	C₁₇×#389	C₂²	1	3	yes	—
9	C₁₇×#387	—	2	3	no	—
10	C₁₇×#336	C₂²	1	3	yes	—
11	C₂²×D₄×C₁₇	—	2	3	no	—
12	C₂²×C₈×C₁₇	—	2	3	no	—
13	C₁₇×#127	—	4	2	no	—
14	C₂×C₄⋊C₄×C₁₇	—	2	3	no	—
15	C₂×C₄⋊C₄×C₁₇	D₄	1	3	yes	—
16	C₂×C₄⋊C₄×C₁₇	D₄	1	3	yes	—
17	C₂×C₄²×C₁₇	—	2	1	no	—
18	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
19	C₁₇×#102	—	4	6	no	—
20	C₁₇×#102	—	4	6	no	—
21	C₁₇×#95	—	2	3	no	—
22	C₁₇×#95	—	2	3	no	—
23	C₁₇×#95	—	2	1	no	—
24	C₁₇×#95	—	2	3	no	—
25	C₂⁴×C₁₇	—	4	1	no	—
26	C₂×D₄×C₁₇	—	2	6	no	—
27	C₂×D₄×C₁₇	—	4	3	no	—
28	C₂×D₄×C₁₇	—	4	3	no	—
29	C₂×D₄×C₁₇	—	4	3	no	—
30	C₂×D₄×C₁₇	—	4	3	no	—
31	C₂×D₄×C₁₇	—	4	3	no	—
32	C₂²×C₄×C₁₇	—	2	3	no	—
33	C₂²×C₄×C₁₇	—	2	3	no	—
34	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
35	C₂×C₈×C₁₇	—	4	3	no	—
36	C₂×C₈×C₁₇	—	4	3	no	—
37	C₂×C₈×C₁₇	—	4	3	no	—
38	C₄⋊C₄×C₁₇	—	2	3	no	—
39	C₄⋊C₄×C₁₇	—	2	3	no	—
40	C₄⋊C₄×C₁₇	—	4	3	no	—
41	C₄⋊C₄×C₁₇	—	4	3	no	—
42	C₄⋊C₄×C₁₇	—	4	3	no	—
43	C₄²×C₁₇	—	4	1	no	—
44	C₄²×C₁₇	—	8	1	no	—
45	#7744	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂³×C₁₇	—	4	2	no	—
47	C₂³×C₁₇	—	8	3	no	—
48	C₂³×C₁₇	—	8	3	no	—
49	C₂³×C₁₇	#120	1	1	yes	characteristic
50	D₄×C₁₇	—	4	6	no	—
51	D₄×C₁₇	—	4	6	no	—
52	D₄×C₁₇	—	8	3	no	—
53	D₄×C₁₇	—	8	3	no	—
54	C₂×C₄×C₁₇	—	2	3	no	—
55	C₂×C₄×C₁₇	—	2	3	no	—
56	C₂×C₄×C₁₇	—	4	3	no	—
57	C₂×C₄×C₁₇	—	4	3	no	—
58	C₂×C₄×C₁₇	—	4	3	no	—
59	C₂×C₄×C₁₇	—	4	3	no	—
60	C₂×C₄×C₁₇	—	4	3	no	—
61	C₂×C₄×C₁₇	—	4	3	no	—
62	C₂×C₄×C₁₇	#135	1	3	yes	—
63	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	3	yes	—
64	C₈×C₁₇	—	8	3	no	—
65	#2810	C₂×C₁₇	1	1	yes	characteristic, complemented
66	#2041	C₂×C₁₇	1	3	yes	—
67	#1271	C₂×C₁₇	1	3	yes	complemented
68	C₂²×C₁₇	—	8	6	no	—
69	C₂²×C₁₇	—	16	3	no	—
70	C₂²×C₁₇	—	16	1	no	—
71	C₂²×C₁₇	#449	1	3	yes	—
72	C₂²×C₁₇	#454	1	3	yes	—
73	C₂²×C₁₇	B(4,𝔽₂)	1	1	yes	characteristic
74	C₄×C₁₇	—	2	3	no	—
75	C₄×C₁₇	—	2	3	no	—
76	C₄×C₁₇	—	8	3	no	—
77	C₄×C₁₇	—	8	3	no	—
78	C₂×#127	C₂²×C₁₇	1	1	yes	characteristic
79	C₂×#102	—	2	3	no	—
80	C₂×#102	—	2	3	no	—
81	#400	—	2	3	no	—
82	#389	C₂²×C₁₇	1	3	yes	—
83	#387	—	2	3	no	—
84	#336	C₂²×C₁₇	1	3	yes	—
85	C₂×C₁₇	—	16	2	no	—
86	C₂×C₁₇	#1597	1	3	yes	—
87	C₂×C₁₇	#1628	1	3	yes	—
88	C₂×C₁₇	#1658	1	1	yes	characteristic
89	C₂²×D₄	—	2	3	no	—
90	C₂²×C₈	—	2	3	no	—
91	#127	—	4	2	no	—
92	C₂×C₄⋊C₄	—	2	3	no	—
93	C₂×C₄⋊C₄	D₄×C₁₇	1	3	yes	—
94	C₂×C₄⋊C₄	D₄×C₁₇	1	3	yes	—
95	C₂×C₄²	—	2	1	no	—
96	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
97	#102	—	4	6	no	—
98	#102	—	4	6	no	—
99	#95	—	2	3	no	—
100	#95	—	2	3	no	—
101	#95	—	2	1	no	—
102	#95	—	2	3	no	—
103	C₁₇	#7744	1	1	yes	characteristic, direct product factor, Sylow
104	C₂⁴	—	4	1	no	—
105	C₂×D₄	—	2	6	no	—
106	C₂×D₄	—	4	3	no	—
107	C₂×D₄	—	4	3	no	—
108	C₂×D₄	—	4	3	no	—
109	C₂×D₄	—	4	3	no	—
110	C₂×D₄	—	4	3	no	—
111	C₂²×C₄	—	2	3	no	—
112	C₂²×C₄	—	2	3	no	—
113	C₂²×C₄	C₂×D₄×C₁₇	1	3	yes	—
114	C₂×C₈	—	4	3	no	—
115	C₂×C₈	—	4	3	no	—
116	C₂×C₈	—	4	3	no	—
117	C₄⋊C₄	—	2	3	no	—
118	C₄⋊C₄	—	2	3	no	—
119	C₄⋊C₄	—	4	3	no	—
120	C₄⋊C₄	—	4	3	no	—
121	C₄⋊C₄	—	4	3	no	—
122	C₄²	—	4	1	no	—
123	C₄²	—	8	1	no	—
124	C₂³	—	4	2	no	—
125	C₂³	—	8	3	no	—
126	C₂³	—	8	3	no	—
127	C₂³	C₁₇×#120	1	1	yes	characteristic
128	D₄	—	4	6	no	—
129	D₄	—	4	6	no	—
130	D₄	—	8	3	no	—
131	D₄	—	8	3	no	—
132	C₂×C₄	—	2	3	no	—
133	C₂×C₄	—	2	3	no	—
134	C₂×C₄	—	4	3	no	—
135	C₂×C₄	—	4	3	no	—
136	C₂×C₄	—	4	3	no	—
137	C₂×C₄	—	4	3	no	—
138	C₂×C₄	—	4	3	no	—
139	C₂×C₄	—	4	3	no	—
140	C₂×C₄	C₁₇×#135	1	3	yes	—
141	C₂×C₄	C₁₇×AGL(1,ℤ/8)	1	3	yes	—
142	C₈	—	8	3	no	—
143	C₂²	—	8	6	no	—
144	C₂²	—	16	3	no	—
145	C₂²	—	16	1	no	—
146	C₂²	C₁₇×#449	1	3	yes	—
147	C₂²	C₁₇×#454	1	3	yes	—
148	C₂²	C₁₇×B(4,𝔽₂)	1	1	yes	characteristic
149	C₄	—	2	3	no	—
150	C₄	—	2	3	no	—
151	C₄	—	8	3	no	—
152	C₄	—	8	3	no	—
153	C₂	—	16	2	no	—
154	C₂	C₁₇×#1597	1	3	yes	—
155	C₂	C₁₇×#1628	1	3	yes	—
156	C₂	C₁₇×#1658	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₇×#2810

C₂

yes

maximal, characteristic, complemented

C₁₇×#2041

C₂

yes

maximal

C₁₇×#1271

C₂

yes

maximal, complemented

C₂×C₁₇×#127

C₂²

yes

characteristic

C₂×C₁₇×#102

—

C₂×C₁₇×#102

—

C₁₇×#400

—

C₁₇×#389

C₂²

yes

—

C₁₇×#387

—

C₁₇×#336

C₂²

yes

—

C₂²×D₄×C₁₇

—

C₂²×C₈×C₁₇

—

C₁₇×#127

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

D₄

yes

—

C₂×C₄⋊C₄×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₁₇×#102

—

C₁₇×#102

—

C₁₇×#95

—

C₁₇×#95

—

C₁₇×#95

—

C₁₇×#95

—

C₂⁴×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

#7744

C₁₇

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

#120

yes

characteristic

D₄×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#135

yes

—

C₂×C₄×C₁₇

AGL(1,ℤ/8)

yes

—

C₈×C₁₇

—

#2810

C₂×C₁₇

yes

characteristic, complemented

#2041

C₂×C₁₇

yes

—

#1271

C₂×C₁₇

yes

complemented

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#449

yes

—

C₂²×C₁₇

#454

yes

—

C₂²×C₁₇

B(4,𝔽₂)

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₂×#127

C₂²×C₁₇

yes

—

—

—

C₂²×C₁₇

yes

—

#387

—

#336

C₂²×C₁₇

yes

—

C₂×C₁₇

—

C₂×C₁₇

#1597

yes

—

C₂×C₁₇

#1628

yes

—

C₂×C₁₇

#1658

yes

characteristic

C₂²×D₄

—

C₂²×C₈

—

#127

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄×C₁₇

yes

—

C₂×C₄⋊C₄

D₄×C₁₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³×C₁₇

yes

—

—

—

100

#95

—

101

#95

—

102

#95

—

103

C₁₇

#7744

yes

characteristic, direct product factor, Sylow

104

C₂⁴

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂×D₄×C₁₇

yes

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄²

—

123

C₄²

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

C₁₇×#120

yes

characteristic

128

D₄

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

C₁₇×#135

yes

—

141

C₂×C₄

C₁₇×AGL(1,ℤ/8)

yes

—

142

C₈

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₁₇×#449

yes

—

147

C₂²

C₁₇×#454

yes

—

148

C₂²

C₁₇×B(4,𝔽₂)

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₂

—

154

C₂

C₁₇×#1597

yes

—

155

C₂

C₁₇×#1628

yes

—

156

C₂

C₁₇×#1658

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₁₇	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₇
prime radical	C₂×C₁₇×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₇

#120

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₇

prime radical

C₂×C₁₇×#127

C₂²

LCS

G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

N	H ≅ G/N	type
1. C₁₇×#2810	C₂	semidirect
3. C₁₇×#1271	C₂	semidirect
45. #7744	C₁₇	direct (N × H)
65. #2810	C₂×C₁₇	semidirect
67. #1271	C₂×C₁₇	semidirect
103. C₁₇	#7744	direct (N × H)

H ≅ G/N

type

1. C₁₇×#2810

C₂

semidirect

3. C₁₇×#1271

C₂

semidirect

45. #7744

C₁₇

direct (N × H)

65. #2810

C₂×C₁₇

semidirect

67. #1271

C₂×C₁₇

semidirect

103. C₁₇

#7744

direct (N × H)

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

subgroup	index
C₁₇×#2810	2
C₁₇×#2041	2
C₁₇×#1271	2
#7744	17

subgroup

index

C₁₇×#2810

C₁₇×#2041

C₁₇×#1271

#7744

Subgroups — C₁₇×#7744 #16865488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C17×#7744 #16865488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#7744 #16865488