Group #14526261 — subgroups

Subgroups — C₁₇×#3255 #14526261

Order 2176.

342 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 176 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×#533	C₂	1	2	yes	maximal
2	C₁₇×#475	C₂	1	4	yes	maximal, complemented
3	C₁₇×#470	C₂	1	4	yes	maximal
4	C₁₇×#462	C₂	1	4	yes	maximal, complemented
5	C₁₇×#423	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₁₆×C₁₇	C₂²	1	4	yes	—
7	C₂×M₁₆×C₁₇	C₂²	1	2	yes	—
8	C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₁₇×#125	—	2	2	no	—
11	C₁₇×C₂²⋊Q₈	—	2	2	no	—
12	C₁₇×C₂²⋊Q₈	C₂²	1	4	yes	—
13	C₄×Q₈×C₁₇	—	2	2	no	—
14	C₄×Q₈×C₁₇	C₂²	1	4	yes	—
15	C₁₇×#117	—	2	2	no	—
16	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×#106	C₂²	1	4	yes	—
18	C₁₇×C₄⋊C₈	C₂²	1	4	yes	—
19	C₁₇×#103	C₂²	1	8	yes	—
20	C₂×Q₈×C₁₇	—	2	2	no	—
21	C₂×Q₈×C₁₇	C₂³	1	4	yes	—
22	C₂²×C₄×C₁₇	D₄	1	2	yes	—
23	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
24	Q₁₆×C₁₇	—	4	4	no	—
25	M₁₆×C₁₇	—	4	2	no	—
26	C₂×C₈×C₁₇	C₂³	1	4	yes	—
27	C₄⋊C₄×C₁₇	—	2	4	no	—
28	C₄⋊C₄×C₁₇	—	2	2	no	—
29	C₄⋊C₄×C₁₇	—	2	4	no	—
30	C₄⋊C₄×C₁₇	—	2	2	no	—
31	C₄⋊C₄×C₁₇	—	2	2	no	—
32	C₄⋊C₄×C₁₇	C₂³	1	4	yes	—
33	C₂²⋊C₄×C₁₇	—	2	4	no	—
34	C₄²×C₁₇	—	2	4	no	—
35	C₄²×C₁₇	D₄	1	1	yes	characteristic
36	C₄²×C₁₇	D₄	1	1	yes	characteristic
37	C₄²×C₁₇	C₂³	1	2	yes	—
38	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₁₇	—	2	4	no	—
40	Q₈×C₁₇	—	2	4	no	—
41	Q₈×C₁₇	—	4	4	no	—
42	C₂×C₄×C₁₇	—	2	4	no	—
43	C₂×C₄×C₁₇	—	2	1	no	—
44	C₂×C₄×C₁₇	—	2	1	no	—
45	C₂×C₄×C₁₇	—	2	4	no	—
46	C₂×C₄×C₁₇	—	2	4	no	—
47	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
51	C₈×C₁₇	—	2	4	no	—
52	#3255	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂²×C₁₇	—	2	1	no	—
54	C₂²×C₁₇	—	2	2	no	—
55	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₁₇	—	2	1	no	—
57	C₄×C₁₇	—	2	2	no	—
58	C₄×C₁₇	—	4	4	no	—
59	C₄×C₁₇	—	4	4	no	—
60	C₄×C₁₇	#137	1	4	yes	—
61	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
62	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
63	#533	C₂×C₁₇	1	2	yes	—
64	#475	C₂×C₁₇	1	4	yes	complemented
65	#470	C₂×C₁₇	1	4	yes	—
66	#462	C₂×C₁₇	1	4	yes	complemented
67	#423	C₂×C₁₇	1	1	yes	characteristic
68	C₂×C₁₇	—	4	1	no	—
69	C₂×C₁₇	#537	1	1	yes	characteristic
70	C₂×C₁₇	C₂×#137	1	2	yes	—
71	C₂×Q₁₆	C₂²×C₁₇	1	4	yes	—
72	C₂×M₁₆	C₂²×C₁₇	1	2	yes	—
73	C₄⋊Q₈	C₂²×C₁₇	1	2	yes	—
74	C₄⋊Q₈	C₂²×C₁₇	1	2	yes	—
75	#125	—	2	2	no	—
76	C₂²⋊Q₈	—	2	2	no	—
77	C₂²⋊Q₈	C₂²×C₁₇	1	4	yes	—
78	C₄×Q₈	—	2	2	no	—
79	C₄×Q₈	C₂²×C₁₇	1	4	yes	—
80	#117	—	2	2	no	—
81	C₂×C₄²	C₂²×C₁₇	1	1	yes	characteristic
82	#106	C₂²×C₁₇	1	4	yes	—
83	C₄⋊C₈	C₂²×C₁₇	1	4	yes	—
84	#103	C₂²×C₁₇	1	8	yes	—
85	C₁₇	#3255	1	1	yes	characteristic, direct product factor, Sylow
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	C₂³×C₁₇	1	4	yes	—
88	C₂²×C₄	D₄×C₁₇	1	2	yes	—
89	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
90	Q₁₆	—	4	4	no	—
91	M₁₆	—	4	2	no	—
92	C₂×C₈	C₂³×C₁₇	1	4	yes	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	4	no	—
98	C₄⋊C₄	C₂³×C₁₇	1	4	yes	—
99	C₂²⋊C₄	—	2	4	no	—
100	C₄²	—	2	4	no	—
101	C₄²	D₄×C₁₇	1	1	yes	characteristic
102	C₄²	D₄×C₁₇	1	1	yes	characteristic
103	C₄²	C₂³×C₁₇	1	2	yes	—
104	C₂³	C₂×D₄×C₁₇	1	1	yes	characteristic
105	Q₈	—	2	4	no	—
106	Q₈	—	2	4	no	—
107	Q₈	—	4	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
114	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
115	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
116	C₂×C₄	C₂⁴×C₁₇	1	1	yes	characteristic
117	C₈	—	2	4	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	2	no	—
120	C₂²	C₂²×D₄×C₁₇	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	2	no	—
123	C₄	—	4	4	no	—
124	C₄	—	4	4	no	—
125	C₄	C₁₇×#137	1	4	yes	—
126	C₄	C₁₇×H(5,𝔽₂)	1	1	yes	characteristic
127	C₄	C₁₇×2₋¹⁺⁴	1	1	yes	characteristic
128	C₂	—	4	1	no	—
129	C₂	C₁₇×#537	1	1	yes	characteristic
130	C₂	C₂×C₁₇×#137	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₇×#533

C₂

yes

maximal

C₁₇×#475

C₂

yes

maximal, complemented

C₁₇×#470

C₂

yes

maximal

C₁₇×#462

C₂

yes

maximal, complemented

C₁₇×#423

C₂

yes

maximal, characteristic

C₂×Q₁₆×C₁₇

C₂²

yes

—

C₂×M₁₆×C₁₇

C₂²

yes

—

C₁₇×C₄⋊Q₈

C₂²

yes

—

C₁₇×C₄⋊Q₈

C₂²

yes

—

C₁₇×#125

—

C₁₇×C₂²⋊Q₈

—

C₁₇×C₂²⋊Q₈

C₂²

yes

—

C₄×Q₈×C₁₇

—

C₄×Q₈×C₁₇

C₂²

yes

—

C₁₇×#117

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₁₇×#106

C₂²

yes

—

C₁₇×C₄⋊C₈

C₂²

yes

—

C₁₇×#103

C₂²

yes

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

C₂³

yes

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

Q₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

C₂³

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

C₂³

yes

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

D₄

yes

characteristic

C₄²×C₁₇

D₄

yes

characteristic

C₄²×C₁₇

C₂³

yes

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

Q₈×C₁₇

—

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂⁴

yes

characteristic

C₈×C₁₇

—

#3255

C₁₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂²×D₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#137

yes

—

C₄×C₁₇

H(5,𝔽₂)

yes

characteristic

C₄×C₁₇

2₋¹⁺⁴

yes

characteristic

#533

C₂×C₁₇

yes

—

#475

C₂×C₁₇

yes

complemented

#470

C₂×C₁₇

yes

—

#462

C₂×C₁₇

yes

complemented

#423

C₂×C₁₇

yes

characteristic

C₂×C₁₇

—

C₂×C₁₇

#537

yes

characteristic

C₂×C₁₇

C₂×#137

yes

—

C₂×Q₁₆

C₂²×C₁₇

yes

—

C₂×M₁₆

C₂²×C₁₇

yes

—

C₄⋊Q₈

C₂²×C₁₇

yes

—

C₄⋊Q₈

C₂²×C₁₇

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂²×C₁₇

yes

—

C₄×Q₈

—

C₄×Q₈

C₂²×C₁₇

yes

—

#117

—

C₂×C₄²

C₂²×C₁₇

yes

characteristic

#106

C₂²×C₁₇

yes

—

C₄⋊C₈

C₂²×C₁₇

yes

—

#103

C₂²×C₁₇

yes

—

C₁₇

#3255

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

C₂×Q₈

C₂³×C₁₇

yes

—

C₂²×C₄

D₄×C₁₇

yes

—

C₂²×C₄

C₂³×C₁₇

yes

characteristic

Q₁₆

—

M₁₆

—

C₂×C₈

C₂³×C₁₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂³×C₁₇

yes

—

C₂²⋊C₄

—

100

C₄²

—

101

C₄²

D₄×C₁₇

yes

characteristic

102

C₄²

D₄×C₁₇

yes

characteristic

103

C₄²

C₂³×C₁₇

yes

—

104

C₂³

C₂×D₄×C₁₇

yes

characteristic

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×D₄×C₁₇

yes

characteristic

114

C₂×C₄

C₂×D₄×C₁₇

yes

—

115

C₂×C₄

C₂×D₄×C₁₇

yes

—

116

C₂×C₄

C₂⁴×C₁₇

yes

characteristic

117

C₈

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂²×D₄×C₁₇

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

C₁₇×#137

yes

—

126

C₄

C₁₇×H(5,𝔽₂)

yes

characteristic

127

C₄

C₁₇×2₋¹⁺⁴

yes

characteristic

128

C₂

—

129

C₂

C₁₇×#537

yes

characteristic

130

C₂

C₂×C₁₇×#137

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₇
prime radical	C₂³×C₁₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₇

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₁₇

prime radical

C₂³×C₁₇

C₂×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

N	H ≅ G/N	type
2. C₁₇×#475	C₂	semidirect
4. C₁₇×#462	C₂	semidirect
52. #3255	C₁₇	direct (N × H)
64. #475	C₂×C₁₇	semidirect
66. #462	C₂×C₁₇	semidirect
85. C₁₇	#3255	direct (N × H)

H ≅ G/N

type

2. C₁₇×#475

C₂

semidirect

4. C₁₇×#462

C₂

semidirect

52. #3255

C₁₇

direct (N × H)

64. #475

C₂×C₁₇

semidirect

66. #462

C₂×C₁₇

semidirect

85. C₁₇

#3255

direct (N × H)

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

subgroup	index
C₁₇×#533	2
C₁₇×#475	2
C₁₇×#470	2
C₁₇×#462	2
C₁₇×#423	2
#3255	17

subgroup

index

C₁₇×#533

C₁₇×#475

C₁₇×#470

C₁₇×#462

C₁₇×#423

#3255

Subgroups — C₁₇×#3255 #14526261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C17×#3255 #14526261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#3255 #14526261