Group #14524946 — subgroups

Subgroups — C₁₇×#1940 #14524946

Order 2176.

248 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 136 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×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₇×C₄⋊C₈	C₂	1	2	yes	maximal
3	C₂×C₁₇×#97	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊C₄×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#336	C₂	1	2	yes	maximal
6	C₂²×C₈×C₁₇	C₂²	1	2	yes	—
7	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₁₇	C₄	1	2	yes	—
10	C₂×C₄²×C₁₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
12	C₁₇×C₄⋊C₈	—	2	4	no	—
13	C₁₇×C₄⋊C₈	—	2	2	no	—
14	C₁₇×#97	—	2	2	no	—
15	C₁₇×#95	C₄	1	1	yes	characteristic
16	C₁₇×#95	C₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₁₇	—	2	1	no	—
23	C₂×C₈×C₁₇	—	2	1	no	—
24	C₂×C₈×C₁₇	—	2	1	no	—
25	C₂×C₈×C₁₇	—	2	1	no	—
26	C₂×C₈×C₁₇	—	2	2	no	—
27	C₂×C₈×C₁₇	—	2	2	no	—
28	C₂×C₈×C₁₇	—	2	2	no	—
29	C₂×C₈×C₁₇	D₄	1	2	yes	—
30	C₂×C₈×C₁₇	Q₈	1	2	yes	—
31	C₄⋊C₄×C₁₇	—	2	4	no	—
32	C₄²×C₁₇	—	2	2	no	—
33	C₄²×C₁₇	—	2	2	no	—
34	C₄²×C₁₇	D₄	1	2	yes	—
35	C₄²×C₁₇	Q₈	1	2	yes	—
36	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
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	—	2	2	no	—
42	C₂×C₄×C₁₇	—	2	2	no	—
43	C₂×C₄×C₁₇	—	2	2	no	—
44	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₇	M₁₆	1	4	yes	—
47	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
52	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
53	C₈×C₁₇	—	2	2	no	—
54	C₈×C₁₇	—	2	2	no	—
55	C₈×C₁₇	—	4	2	no	—
56	#1940	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
59	C₂²×C₁₇	C₄×Q₈	1	1	yes	characteristic
60	C₂²×C₁₇	C₂×M₁₆	1	2	yes	—
61	C₂²×C₁₇	#131	1	2	yes	—
62	C₄×C₁₇	—	2	2	no	—
63	C₄×C₁₇	—	2	2	no	—
64	C₄×C₁₇	—	2	4	no	—
65	C₄×C₁₇	—	4	2	no	—
66	C₄×C₁₇	#121	1	1	yes	characteristic
67	C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
68	C₄×C₁₇	#125	1	1	yes	characteristic
69	C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄⋊C₈	C₂×C₁₇	1	2	yes	—
71	C₂×C₄⋊C₈	C₂×C₁₇	1	1	yes	characteristic
72	C₂×#97	C₂×C₁₇	1	1	yes	characteristic
73	C₄×C₄⋊C₄	C₂×C₁₇	1	1	yes	characteristic
74	#336	C₂×C₁₇	1	2	yes	—
75	C₂×C₁₇	#389	1	1	yes	characteristic
76	C₂×C₁₇	#423	1	1	yes	characteristic
77	C₂×C₁₇	#424	1	1	yes	characteristic
78	C₂×C₁₇	#435	1	2	yes	—
79	C₂×C₁₇	#446	1	2	yes	—
80	C₂²×C₈	C₂²×C₁₇	1	2	yes	—
81	C₂²×C₈	C₂²×C₁₇	1	1	yes	characteristic
82	C₂²×C₈	C₂²×C₁₇	1	1	yes	characteristic
83	C₂×C₄⋊C₄	C₄×C₁₇	1	2	yes	—
84	C₂×C₄²	C₂²×C₁₇	1	2	yes	—
85	C₂×C₄²	C₂²×C₁₇	1	1	yes	characteristic
86	C₄⋊C₈	—	2	4	no	—
87	C₄⋊C₈	—	2	2	no	—
88	#97	—	2	2	no	—
89	#95	C₄×C₁₇	1	1	yes	characteristic
90	#95	C₄×C₁₇	1	1	yes	characteristic
91	C₁₇	#1940	1	1	yes	characteristic, direct product factor, Sylow
92	C₂²×C₄	C₂×C₄×C₁₇	1	2	yes	—
93	C₂²×C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
94	C₂²×C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
95	C₂²×C₄	C₂×C₄×C₁₇	1	2	yes	—
96	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	D₄×C₁₇	1	2	yes	—
105	C₂×C₈	Q₈×C₁₇	1	2	yes	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄²	—	2	2	no	—
108	C₄²	—	2	2	no	—
109	C₄²	D₄×C₁₇	1	2	yes	—
110	C₄²	Q₈×C₁₇	1	2	yes	—
111	C₂³	C₂²×C₄×C₁₇	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	C₄⋊C₄×C₁₇	1	2	yes	—
120	C₂×C₄	C₄⋊C₄×C₁₇	1	2	yes	—
121	C₂×C₄	M₁₆×C₁₇	1	4	yes	—
122	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
123	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
124	C₂×C₄	C₂×Q₈×C₁₇	1	1	yes	characteristic
125	C₂×C₄	C₂×Q₈×C₁₇	1	1	yes	characteristic
126	C₂×C₄	#41×C₁₇	1	1	yes	characteristic
127	C₂×C₄	#41×C₁₇	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	—	2	2	no	—
130	C₈	—	4	2	no	—
131	C₂²	C₂×C₄⋊C₄×C₁₇	1	1	yes	characteristic
132	C₂²	C₄×D₄×C₁₇	1	1	yes	characteristic
133	C₂²	C₄×Q₈×C₁₇	1	1	yes	characteristic
134	C₂²	C₂×M₁₆×C₁₇	1	2	yes	—
135	C₂²	C₁₇×#131	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	2	4	no	—
139	C₄	—	4	2	no	—
140	C₄	C₁₇×#121	1	1	yes	characteristic
141	C₄	C₁₇×C₂²⋊Q₈	1	1	yes	characteristic
142	C₄	C₁₇×#125	1	1	yes	characteristic
143	C₄	C₁₇×C₄⋊Q₈	1	1	yes	characteristic
144	C₂	C₁₇×#389	1	1	yes	characteristic
145	C₂	C₁₇×#423	1	1	yes	characteristic
146	C₂	C₁₇×#424	1	1	yes	characteristic
147	C₂	C₁₇×#435	1	2	yes	—
148	C₂	C₁₇×#446	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₇×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂×C₁₇×C₄⋊C₈

C₂

yes

maximal

C₂×C₁₇×#97

C₂

yes

maximal, characteristic

C₄×C₄⋊C₄×C₁₇

C₂

yes

maximal, characteristic

C₁₇×#336

C₂

yes

maximal

C₂²×C₈×C₁₇

C₂²

yes

—

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇

C₄

yes

—

C₂×C₄²×C₁₇

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×#97

—

C₁₇×#95

C₄

yes

characteristic

C₁₇×#95

C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×C₄

yes

—

C₂²×C₄×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×C₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₄

yes

—

C₂×C₈×C₁₇

Q₈

yes

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

D₄

yes

—

C₄²×C₁₇

Q₈

yes

—

C₂³×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₇

M₁₆

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

#1940

C₁₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₇

C₄×D₄

yes

characteristic

C₂²×C₁₇

C₄×Q₈

yes

characteristic

C₂²×C₁₇

C₂×M₁₆

yes

—

C₂²×C₁₇

#131

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#121

yes

characteristic

C₄×C₁₇

C₂²⋊Q₈

yes

characteristic

C₄×C₁₇

#125

yes

characteristic

C₄×C₁₇

C₄⋊Q₈

yes

characteristic

C₂×C₄⋊C₈

C₂×C₁₇

yes

—

C₂×C₄⋊C₈

C₂×C₁₇

yes

characteristic

C₂×#97

C₂×C₁₇

yes

characteristic

C₄×C₄⋊C₄

C₂×C₁₇

yes

characteristic

#336

C₂×C₁₇

yes

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

—

C₂×C₁₇

#446

yes

—

C₂²×C₈

C₂²×C₁₇

yes

—

C₂²×C₈

C₂²×C₁₇

yes

characteristic

C₂²×C₈

C₂²×C₁₇

yes

characteristic

C₂×C₄⋊C₄

C₄×C₁₇

yes

—

C₂×C₄²

C₂²×C₁₇

yes

—

C₂×C₄²

C₂²×C₁₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#95

C₄×C₁₇

yes

characteristic

#95

C₄×C₁₇

yes

characteristic

C₁₇

#1940

yes

characteristic, direct product factor, Sylow

C₂²×C₄

C₂×C₄×C₁₇

yes

—

C₂²×C₄

C₂×C₄×C₁₇

yes

characteristic

C₂²×C₄

C₂×C₄×C₁₇

yes

characteristic

C₂²×C₄

C₂×C₄×C₁₇

yes

—

C₂²×C₄

C₂³×C₁₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

D₄×C₁₇

yes

—

105

C₂×C₈

Q₈×C₁₇

yes

—

106

C₄⋊C₄

—

107

C₄²

—

108

C₄²

—

109

C₄²

D₄×C₁₇

yes

—

110

C₄²

Q₈×C₁₇

yes

—

111

C₂³

C₂²×C₄×C₁₇

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₄⋊C₄×C₁₇

yes

—

120

C₂×C₄

C₄⋊C₄×C₁₇

yes

—

121

C₂×C₄

M₁₆×C₁₇

yes

—

122

C₂×C₄

C₂×D₄×C₁₇

yes

characteristic

123

C₂×C₄

C₂×D₄×C₁₇

yes

characteristic

124

C₂×C₄

C₂×Q₈×C₁₇

yes

characteristic

125

C₂×C₄

C₂×Q₈×C₁₇

yes

characteristic

126

C₂×C₄

#41×C₁₇

yes

characteristic

127

C₂×C₄

#41×C₁₇

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₂²

C₂×C₄⋊C₄×C₁₇

yes

characteristic

132

C₂²

C₄×D₄×C₁₇

yes

characteristic

133

C₂²

C₄×Q₈×C₁₇

yes

characteristic

134

C₂²

C₂×M₁₆×C₁₇

yes

—

135

C₂²

C₁₇×#131

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

C₁₇×#121

yes

characteristic

141

C₄

C₁₇×C₂²⋊Q₈

yes

characteristic

142

C₄

C₁₇×#125

yes

characteristic

143

C₄

C₁₇×C₄⋊Q₈

yes

characteristic

144

C₂

C₁₇×#389

yes

characteristic

145

C₂

C₁₇×#423

yes

characteristic

146

C₂

C₁₇×#424

yes

characteristic

147

C₂

C₁₇×#435

yes

—

148

C₂

C₁₇×#446

yes

—

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

subgroup

G/H

center

C₂²×C₄×C₁₇

C₂³

commutator

C₂³

C₂²×C₄×C₁₇

Frattini, cosocle

C₂²×C₄

C₂³×C₁₇

socle, prime radical

C₂³×C₁₇

C₂²×C₄

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₁₇ ◁ G

N	H ≅ G/N	type
56. #1940	C₁₇	direct (N × H)
91. C₁₇	#1940	direct (N × H)

H ≅ G/N

type

56. #1940

C₁₇

direct (N × H)

91. C₁₇

#1940

direct (N × H)

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

subgroup	index
C₂×C₁₇×C₄⋊C₈	2
C₂×C₁₇×#97	2
C₄×C₄⋊C₄×C₁₇	2
C₁₇×#336	2
#1940	17

subgroup

index

C₂×C₁₇×C₄⋊C₈

C₂×C₁₇×#97

C₄×C₄⋊C₄×C₁₇

C₁₇×#336

#1940

Subgroups — C₁₇×#1940 #14524946

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C17×#1940 #14524946

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#1940 #14524946