Group #14525031 — subgroups

Subgroups — C₁₇×#2025 #14525031

Order 2176.

294 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 100 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×C₁₇×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₇×#102	C₂	1	2	yes	maximal
3	C₁₇×#390	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#340	C₂	1	2	yes	maximal, complemented
5	C₁₇×#336	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×C₁₇	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₇	C₂²	1	2	yes	—
8	C₁₇×#125	—	2	4	no	—
9	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
10	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
12	C₂×C₂²⋊C₄×C₁₇	—	2	2	no	—
13	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×#102	—	2	4	no	—
15	C₂⁴×C₁₇	—	2	1	no	—
16	C₂×D₄×C₁₇	—	2	2	no	—
17	C₂×D₄×C₁₇	—	2	2	no	—
18	C₂×D₄×C₁₇	—	2	1	no	—
19	C₂×D₄×C₁₇	Q₈	1	2	yes	—
20	C₂²×C₄×C₁₇	—	2	2	no	—
21	C₂²×C₄×C₁₇	—	2	1	no	—
22	C₂²×C₄×C₁₇	D₄	1	2	yes	—
23	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₁₇	—	2	2	no	—
25	C₂×C₈×C₁₇	—	2	2	no	—
26	C₂×C₈×C₁₇	—	2	2	no	—
27	C₄⋊C₄×C₁₇	—	2	8	no	—
28	C₄⋊C₄×C₁₇	—	2	2	no	—
29	C₄⋊C₄×C₁₇	—	2	2	no	—
30	C₄⋊C₄×C₁₇	D₄	1	4	yes	—
31	C₂²⋊C₄×C₁₇	—	4	4	no	—
32	C₄²×C₁₇	—	2	2	no	—
33	C₂³×C₁₇	—	2	2	no	—
34	C₂³×C₁₇	—	4	1	no	—
35	C₂³×C₁₇	—	4	1	no	—
36	C₂³×C₁₇	—	4	2	no	—
37	C₂³×C₁₇	—	4	2	no	—
38	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
39	D₄×C₁₇	—	2	4	no	—
40	D₄×C₁₇	—	4	2	no	—
41	C₂×C₄×C₁₇	—	2	2	no	—
42	C₂×C₄×C₁₇	—	2	4	no	—
43	C₂×C₄×C₁₇	—	2	2	no	—
44	C₂×C₄×C₁₇	—	2	2	no	—
45	C₂×C₄×C₁₇	—	4	2	no	—
46	C₂×C₄×C₁₇	—	4	2	no	—
47	C₂×C₄×C₁₇	—	4	1	no	—
48	C₂×C₄×C₁₇	—	4	2	no	—
49	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₇	#41	1	2	yes	—
52	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
53	C₈×C₁₇	—	4	2	no	—
54	#2025	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂²×C₁₇	—	4	2	no	—
56	C₂²×C₁₇	—	4	4	no	—
57	C₂²×C₁₇	—	8	2	no	—
58	C₂²×C₁₇	—	8	1	no	—
59	C₂²×C₁₇	—	8	1	no	—
60	C₂²×C₁₇	#120	1	1	yes	characteristic
61	C₂²×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂²×C₁₇	#124	1	1	yes	characteristic
63	C₂²×C₁₇	#135	1	2	yes	—
64	C₂²×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
65	C₄×C₁₇	—	4	2	no	—
66	C₄×C₁₇	—	4	4	no	—
67	C₄×C₁₇	—	8	1	no	—
68	C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
69	C₄×C₁₇	#124	1	2	yes	—
70	C₂×#125	C₂×C₁₇	1	1	yes	characteristic, complemented
71	C₂×#102	C₂×C₁₇	1	2	yes	—
72	#390	C₂×C₁₇	1	1	yes	characteristic
73	#340	C₂×C₁₇	1	2	yes	complemented
74	#336	C₂×C₁₇	1	1	yes	characteristic, complemented
75	C₂×C₁₇	—	8	2	no	—
76	C₂×C₁₇	#393	1	1	yes	characteristic
77	C₂×C₁₇	#449	1	2	yes	—
78	C₂×C₁₇	#478	1	2	yes	—
79	C₂×C₁₇	#488	1	1	yes	characteristic
80	C₂×C₁₇	#490	1	1	yes	characteristic
81	C₂²×D₄	C₂²×C₁₇	1	1	yes	characteristic
82	C₂²×C₈	C₂²×C₁₇	1	2	yes	—
83	#125	—	2	4	no	—
84	C₂×C₄⋊C₄	—	2	2	no	—
85	C₂×C₄⋊C₄	C₂²×C₁₇	1	1	yes	characteristic
86	C₂×C₄⋊C₄	C₂²×C₁₇	1	2	yes	—
87	C₂×C₂²⋊C₄	—	2	2	no	—
88	C₂×C₄²	C₂²×C₁₇	1	1	yes	characteristic
89	#102	—	2	4	no	—
90	C₁₇	#2025	1	1	yes	characteristic, direct product factor, Sylow
91	C₂⁴	—	2	1	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	Q₈×C₁₇	1	2	yes	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	D₄×C₁₇	1	2	yes	—
99	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₄⋊C₄	—	2	8	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	D₄×C₁₇	1	4	yes	—
107	C₂²⋊C₄	—	4	4	no	—
108	C₄²	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	4	2	no	—
112	C₂³	—	4	2	no	—
113	C₂³	—	4	1	no	—
114	C₂³	C₂×D₄×C₁₇	1	1	yes	characteristic
115	D₄	—	2	4	no	—
116	D₄	—	4	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
126	C₂×C₄	C₂×Q₈×C₁₇	1	1	yes	characteristic
127	C₂×C₄	#41×C₁₇	1	2	yes	—
128	C₂×C₄	#41×C₁₇	1	1	yes	characteristic
129	C₈	—	4	2	no	—
130	C₂²	—	4	2	no	—
131	C₂²	—	4	4	no	—
132	C₂²	—	8	2	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	C₁₇×#120	1	1	yes	characteristic
136	C₂²	C₁₇×C₂²⋊Q₈	1	1	yes	characteristic
137	C₂²	C₁₇×#124	1	1	yes	characteristic
138	C₂²	C₁₇×#135	1	2	yes	—
139	C₂²	C₁₇×AGL(1,ℤ/8)	1	2	yes	—
140	C₄	—	4	2	no	—
141	C₄	—	4	4	no	—
142	C₄	—	8	1	no	—
143	C₄	C₁₇×C₂²⋊Q₈	1	2	yes	—
144	C₄	C₁₇×#124	1	2	yes	—
145	C₂	—	8	2	no	—
146	C₂	C₁₇×#393	1	1	yes	characteristic
147	C₂	C₁₇×#449	1	2	yes	—
148	C₂	C₁₇×#478	1	2	yes	—
149	C₂	C₁₇×#488	1	1	yes	characteristic
150	C₂	C₁₇×#490	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₇×#125

C₂

yes

maximal, characteristic, complemented

C₂×C₁₇×#102

C₂

yes

maximal

C₁₇×#390

C₂

yes

maximal, characteristic

C₁₇×#340

C₂

yes

maximal, complemented

C₁₇×#336

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₇

C₂²

yes

characteristic

C₂²×C₈×C₁₇

C₂²

yes

—

C₁₇×#125

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₇

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₁₇×#102

—

C₂⁴×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

Q₈

yes

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

D₄

yes

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

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

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

#41

yes

—

C₂×C₄×C₁₇

#41

yes

characteristic

C₈×C₁₇

—

#2025

C₁₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#120

yes

characteristic

C₂²×C₁₇

C₂²⋊Q₈

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₂²×C₁₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂²⋊Q₈

yes

—

C₄×C₁₇

#124

yes

—

C₂×#125

C₂×C₁₇

yes

characteristic, complemented

C₂×#102

C₂×C₁₇

yes

—

#390

C₂×C₁₇

yes

characteristic

#340

C₂×C₁₇

yes

complemented

#336

C₂×C₁₇

yes

characteristic, complemented

C₂×C₁₇

—

C₂×C₁₇

#393

yes

characteristic

C₂×C₁₇

#449

yes

—

C₂×C₁₇

#478

yes

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×D₄

C₂²×C₁₇

yes

characteristic

C₂²×C₈

C₂²×C₁₇

yes

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₁₇

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₁₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂²×C₁₇

yes

characteristic

#102

—

C₁₇

#2025

yes

characteristic, direct product factor, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

Q₈×C₁₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₁₇

yes

—

C₂²×C₄

C₂³×C₁₇

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

D₄×C₁₇

yes

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

—

114

C₂³

C₂×D₄×C₁₇

yes

characteristic

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂×D₄×C₁₇

yes

—

126

C₂×C₄

C₂×Q₈×C₁₇

yes

characteristic

127

C₂×C₄

#41×C₁₇

yes

—

128

C₂×C₄

#41×C₁₇

yes

characteristic

129

C₈

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

C₁₇×#120

yes

characteristic

136

C₂²

C₁₇×C₂²⋊Q₈

yes

characteristic

137

C₂²

C₁₇×#124

yes

characteristic

138

C₂²

C₁₇×#135

yes

—

139

C₂²

C₁₇×AGL(1,ℤ/8)

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

C₁₇×C₂²⋊Q₈

yes

—

144

C₄

C₁₇×#124

yes

—

145

C₂

—

146

C₂

C₁₇×#393

yes

characteristic

147

C₂

C₁₇×#449

yes

—

148

C₂

C₁₇×#478

yes

—

149

C₂

C₁₇×#488

yes

characteristic

150

C₂

C₁₇×#490

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₁₇	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₇
prime radical	C₂²×D₄×C₁₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₇

C₂×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₁₇

prime radical

C₂²×D₄×C₁₇

C₂²

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

N	H ≅ G/N	type
1. C₂×C₁₇×#125	C₂	semidirect
4. C₁₇×#340	C₂	semidirect
5. C₁₇×#336	C₂	semidirect
54. #2025	C₁₇	direct (N × H)
70. C₂×#125	C₂×C₁₇	semidirect
73. #340	C₂×C₁₇	semidirect
74. #336	C₂×C₁₇	semidirect
90. C₁₇	#2025	direct (N × H)

H ≅ G/N

type

1. C₂×C₁₇×#125

C₂

semidirect

4. C₁₇×#340

C₂

semidirect

5. C₁₇×#336

C₂

semidirect

54. #2025

C₁₇

direct (N × H)

70. C₂×#125

C₂×C₁₇

semidirect

73. #340

C₂×C₁₇

semidirect

74. #336

C₂×C₁₇

semidirect

90. C₁₇

#2025

direct (N × H)

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

subgroup	index
C₂×C₁₇×#125	2
C₂×C₁₇×#102	2
C₁₇×#390	2
C₁₇×#340	2
C₁₇×#336	2
#2025	17

subgroup

index

C₂×C₁₇×#125

C₂×C₁₇×#102

C₁₇×#390

C₁₇×#340

C₁₇×#336

#2025

Subgroups — C₁₇×#2025 #14525031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C17×#2025 #14525031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#2025 #14525031