Group #14524807 — subgroups

Subgroups — C₁₇×#1801 #14524807

Order 2176.

272 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 124 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×C₄×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic
2	C₁₇×#423	C₂	1	2	yes	maximal
3	C₂×C₁₇×#101	C₂	1	2	yes	maximal
4	C₁₇×#343	C₂	1	2	yes	maximal
5	C₂²×Q₈×C₁₇	C₂²	1	1	yes	characteristic
6	C₂×M₁₆×C₁₇	C₂²	1	4	yes	—
7	C₄×Q₈×C₁₇	—	2	2	no	—
8	C₄×Q₈×C₁₇	—	2	2	no	—
9	C₂×C₄⋊C₄×C₁₇	C₄	1	2	yes	—
10	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₁₇	C₄	1	2	yes	—
12	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
13	C₁₇×C₄⋊C₈	—	2	2	no	—
14	C₁₇×C₄⋊C₈	—	2	2	no	—
15	C₁₇×#101	—	2	4	no	—
16	C₂×Q₈×C₁₇	—	2	2	no	—
17	C₂×Q₈×C₁₇	—	2	2	no	—
18	C₂×Q₈×C₁₇	D₄	1	2	yes	—
19	C₂×Q₈×C₁₇	Q₈	1	2	yes	—
20	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
25	M₁₆×C₁₇	—	2	4	no	—
26	M₁₆×C₁₇	—	2	4	no	—
27	C₂×C₈×C₁₇	—	2	4	no	—
28	C₄⋊C₄×C₁₇	—	2	4	no	—
29	C₄⋊C₄×C₁₇	—	2	2	no	—
30	C₄²×C₁₇	—	2	2	no	—
31	C₄²×C₁₇	—	2	2	no	—
32	C₄²×C₁₇	D₄	1	2	yes	—
33	C₄²×C₁₇	D₄	1	2	yes	—
34	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₁₇	—	2	4	no	—
36	Q₈×C₁₇	—	2	4	no	—
37	C₂×C₄×C₁₇	—	2	2	no	—
38	C₂×C₄×C₁₇	—	2	2	no	—
39	C₂×C₄×C₁₇	—	2	2	no	—
40	C₂×C₄×C₁₇	—	2	2	no	—
41	C₂×C₄×C₁₇	—	4	1	no	—
42	C₂×C₄×C₁₇	—	4	2	no	—
43	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₄⋊C₄	1	4	yes	—
47	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₁₇	#41	1	2	yes	—
51	C₈×C₁₇	—	4	4	no	—
52	#1801	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂²×C₁₇	—	2	1	no	—
54	C₂²×C₁₇	—	2	1	no	—
55	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₇	#117	1	1	yes	characteristic
58	C₄×C₁₇	—	2	4	no	—
59	C₄×C₁₇	—	2	1	no	—
60	C₄×C₁₇	—	2	2	no	—
61	C₄×C₁₇	—	4	2	no	—
62	C₄×C₁₇	#101	1	2	yes	complemented
63	C₄×C₁₇	#121	1	2	yes	—
64	C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
65	C₂×C₄×Q₈	C₂×C₁₇	1	1	yes	characteristic
66	#423	C₂×C₁₇	1	2	yes	—
67	C₂×#101	C₂×C₁₇	1	2	yes	—
68	#343	C₂×C₁₇	1	2	yes	—
69	C₂×C₁₇	—	2	1	no	—
70	C₂×C₁₇	—	2	1	no	—
71	C₂×C₁₇	#380	1	1	yes	characteristic
72	C₂×C₁₇	C₂×#101	1	1	yes	characteristic
73	C₂×C₁₇	#413	1	1	yes	characteristic
74	C₂²×Q₈	C₂²×C₁₇	1	1	yes	characteristic
75	C₂×M₁₆	C₂²×C₁₇	1	4	yes	—
76	C₄×Q₈	—	2	2	no	—
77	C₄×Q₈	—	2	2	no	—
78	C₂×C₄⋊C₄	C₄×C₁₇	1	2	yes	—
79	C₂×C₄⋊C₄	C₂²×C₁₇	1	1	yes	characteristic
80	C₂×C₄²	C₄×C₁₇	1	2	yes	—
81	C₂×C₄²	C₂²×C₁₇	1	1	yes	characteristic
82	C₄⋊C₈	—	2	2	no	—
83	C₄⋊C₈	—	2	2	no	—
84	#101	—	2	4	no	—
85	C₁₇	#1801	1	1	yes	characteristic, direct product factor, Sylow
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	D₄×C₁₇	1	2	yes	—
89	C₂×Q₈	Q₈×C₁₇	1	2	yes	—
90	C₂²×C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
91	C₂²×C₄	C₂×C₄×C₁₇	1	2	yes	—
92	C₂²×C₄	C₂×C₄×C₁₇	1	2	yes	—
93	C₂²×C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
94	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
95	M₁₆	—	2	4	no	—
96	M₁₆	—	2	4	no	—
97	C₂×C₈	—	2	4	no	—
98	C₄⋊C₄	—	2	4	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄²	—	2	2	no	—
101	C₄²	—	2	2	no	—
102	C₄²	D₄×C₁₇	1	2	yes	—
103	C₄²	D₄×C₁₇	1	2	yes	—
104	C₂³	C₂²×C₄×C₁₇	1	1	yes	characteristic
105	Q₈	—	2	4	no	—
106	Q₈	—	2	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
114	C₂×C₄	C₂²⋊C₄×C₁₇	1	2	yes	—
115	C₂×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
116	C₂×C₄	C₄⋊C₄×C₁₇	1	4	yes	—
117	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
118	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
119	C₂×C₄	C₂×Q₈×C₁₇	1	1	yes	characteristic
120	C₂×C₄	#41×C₁₇	1	2	yes	—
121	C₈	—	4	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
125	C₂²	C₂×C₄⋊C₄×C₁₇	1	1	yes	characteristic
126	C₂²	C₁₇×#117	1	1	yes	characteristic
127	C₄	—	2	4	no	—
128	C₄	—	2	1	no	—
129	C₄	—	2	2	no	—
130	C₄	—	4	2	no	—
131	C₄	C₁₇×#101	1	2	yes	complemented
132	C₄	C₁₇×#121	1	2	yes	—
133	C₄	C₁₇×C₂²⋊Q₈	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	C₁₇×#380	1	1	yes	characteristic
137	C₂	C₂×C₁₇×#101	1	1	yes	characteristic
138	C₂	C₁₇×#413	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Q₈×C₁₇

C₂

yes

maximal, characteristic

C₁₇×#423

C₂

yes

maximal

C₂×C₁₇×#101

C₂

yes

maximal

C₁₇×#343

C₂

yes

maximal

C₂²×Q₈×C₁₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₇

C₂²

yes

—

C₄×Q₈×C₁₇

—

C₄×Q₈×C₁₇

—

C₂×C₄⋊C₄×C₁₇

C₄

yes

—

C₂×C₄⋊C₄×C₁₇

C₂²

yes

characteristic

C₂×C₄²×C₁₇

C₄

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×#101

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

D₄

yes

—

C₂×Q₈×C₁₇

Q₈

yes

—

C₂²×C₄×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×C₄

yes

—

C₂²×C₄×C₁₇

C₂×C₄

yes

—

C₂²×C₄×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂³

yes

characteristic

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

D₄

yes

—

C₄²×C₁₇

D₄

yes

—

C₂³×C₁₇

C₂²×C₄

yes

characteristic

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

#41

yes

—

C₈×C₁₇

—

#1801

C₁₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₇

#117

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#101

yes

complemented

C₄×C₁₇

#121

yes

—

C₄×C₁₇

C₂²⋊Q₈

yes

—

C₂×C₄×Q₈

C₂×C₁₇

yes

characteristic

#423

C₂×C₁₇

yes

—

C₂×#101

C₂×C₁₇

yes

—

#343

C₂×C₁₇

yes

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×Q₈

C₂²×C₁₇

yes

characteristic

C₂×M₁₆

C₂²×C₁₇

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₄×C₁₇

yes

—

C₂×C₄⋊C₄

C₂²×C₁₇

yes

characteristic

C₂×C₄²

C₄×C₁₇

yes

—

C₂×C₄²

C₂²×C₁₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#101

—

C₁₇

#1801

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₄×C₁₇

yes

—

C₂×Q₈

Q₈×C₁₇

yes

—

C₂²×C₄

C₂×C₄×C₁₇

yes

characteristic

C₂²×C₄

C₂×C₄×C₁₇

yes

—

C₂²×C₄

C₂×C₄×C₁₇

yes

—

C₂²×C₄

C₂×C₄×C₁₇

yes

characteristic

C₂²×C₄

C₂³×C₁₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

—

101

C₄²

—

102

C₄²

D₄×C₁₇

yes

—

103

C₄²

D₄×C₁₇

yes

—

104

C₂³

C₂²×C₄×C₁₇

yes

characteristic

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²⋊C₄×C₁₇

yes

characteristic

114

C₂×C₄

C₂²⋊C₄×C₁₇

yes

—

115

C₂×C₄

C₂²⋊C₄×C₁₇

yes

characteristic

116

C₂×C₄

C₄⋊C₄×C₁₇

yes

—

117

C₂×C₄

C₂×D₄×C₁₇

yes

—

118

C₂×C₄

C₂×D₄×C₁₇

yes

characteristic

119

C₂×C₄

C₂×Q₈×C₁₇

yes

characteristic

120

C₂×C₄

#41×C₁₇

yes

—

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×C₂²⋊C₄×C₁₇

yes

characteristic

125

C₂²

C₂×C₄⋊C₄×C₁₇

yes

characteristic

126

C₂²

C₁₇×#117

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

C₁₇×#101

yes

complemented

132

C₄

C₁₇×#121

yes

—

133

C₄

C₁₇×C₂²⋊Q₈

yes

—

134

C₂

—

135

C₂

—

136

C₂

C₁₇×#380

yes

characteristic

137

C₂

C₂×C₁₇×#101

yes

characteristic

138

C₂

C₁₇×#413

yes

characteristic

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

subgroup

G/H

center, socle

C₂²×C₁₇

C₂×C₂²⋊C₄

commutator

C₂³

C₂²×C₄×C₁₇

Frattini, cosocle

C₂²×C₄

C₂³×C₁₇

prime radical

C₂³×C₁₇

C₂²×C₄

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

N	H ≅ G/N	type
52. #1801	C₁₇	direct (N × H)
62. C₄×C₁₇	#101	semidirect
85. C₁₇	#1801	direct (N × H)
131. C₄	C₁₇×#101	semidirect

H ≅ G/N

type

52. #1801

C₁₇

direct (N × H)

62. C₄×C₁₇

#101

semidirect

85. C₁₇

#1801

direct (N × H)

131. C₄

C₁₇×#101

semidirect

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

subgroup	index
C₂×C₄×Q₈×C₁₇	2
C₁₇×#423	2
C₂×C₁₇×#101	2
C₁₇×#343	2
#1801	17

subgroup

index

C₂×C₄×Q₈×C₁₇

C₁₇×#423

C₂×C₁₇×#101

C₁₇×#343

#1801

Subgroups — C₁₇×#1801 #14524807

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C17×#1801 #14524807

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#1801 #14524807