Group #17021997 — subgroups

Subgroups — C₂×#14528394 #17021997

Order 4352.

750 conjugacy classes of subgroups; only the 267 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 267 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×#11763214	C₂	1	2	yes	maximal
2	C₂×#11763160	C₂	1	4	yes	maximal
3	#14528394	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₈²×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587555	C₂²	1	4	yes	—
6	#11763214	—	2	4	no	—
7	#11763214	C₂²	1	8	yes	—
8	#11763160	—	2	8	no	—
9	#11763160	C₂²	1	16	yes	—
10	C₂×#10587489	—	2	4	no	—
11	C₂×#10587487	—	2	8	no	—
12	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₇	C₂²	1	2	yes	—
14	C₈²×C₁₇	C₂²	1	4	yes	—
15	C₂²×Dic₃₄	—	2	4	no	—
16	C₂²×C₈×C₁₇	D₄	1	4	yes	—
17	C₂²×C₈×C₁₇	D₄	1	2	yes	—
18	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₇	—	2	4	no	—
20	C₂×C₄⋊Dic₁₇	—	2	4	no	—
21	#10587555	—	2	4	no	—
22	#10587555	—	2	4	no	—
23	#10587555	—	2	4	no	—
24	#10587555	C₂³	1	8	yes	—
25	C₄×C₈×C₁₇	D₄	1	2	yes	—
26	C₄×C₈×C₁₇	D₄	1	4	yes	—
27	C₄×C₈×C₁₇	C₂³	1	2	yes	—
28	C₄×C₈×C₁₇	C₂³	1	4	yes	—
29	#10587489	—	2	16	no	—
30	#10587487	—	2	32	no	—
31	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
33	C₂²×Dic₁₇	—	4	4	no	—
34	C₂×Dic₃₄	—	2	8	no	—
35	C₂×Dic₃₄	—	4	16	no	—
36	C₂×Dic₃₄	—	4	4	no	—
37	C₂×C₈×C₁₇	—	2	8	no	—
38	C₂×C₈×C₁₇	Q₁₆	1	8	yes	—
39	C₂×C₈×C₁₇	C₂×D₄	1	8	yes	—
40	C₂×C₈×C₁₇	C₂×D₄	1	4	yes	—
41	C₄²×C₁₇	C₂×D₄	1	2	yes	—
42	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
43	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
44	C₄⋊Dic₁₇	—	2	8	no	—
45	C₄⋊Dic₁₇	—	2	8	no	—
46	C₄⋊Dic₁₇	—	4	4	no	—
47	C₄⋊Dic₁₇	—	4	4	no	—
48	C₂×#1710	—	17	1	no	maximal, Sylow
49	C₂³×C₁₇	#127	1	1	yes	characteristic
50	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
51	C₂×C₄×C₁₇	#127	1	2	yes	—
52	C₂×C₄×C₁₇	C₂×Q₁₆	1	4	yes	—
53	C₂×C₄×C₁₇	#135	1	8	yes	—
54	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₇	C₂²×D₄	1	2	yes	—
56	C₂×Dic₁₇	—	4	8	no	—
57	C₂×Dic₁₇	—	8	8	no	—
58	Dic₃₄	—	4	16	no	—
59	Dic₃₄	—	8	8	no	—
60	C₈×C₁₇	—	2	8	no	—
61	C₈×C₁₇	C₂×Q₁₆	1	8	yes	—
62	C₂×#500	—	17	2	no	—
63	C₂×#487	—	17	4	no	—
64	#1710	—	17	8	no	—
65	C₂×C₈²	D₁₇	1	1	yes	characteristic
66	C₂²×C₁₇	#494	1	2	yes	—
67	C₂²×C₁₇	#495	1	4	yes	—
68	C₂²×C₁₇	C₂×#127	1	1	yes	characteristic
69	C₄×C₁₇	#494	1	2	yes	—
70	C₄×C₁₇	#495	1	4	yes	—
71	C₄×C₁₇	C₂²×Q₁₆	1	2	yes	—
72	C₄×C₁₇	C₂×#135	1	4	yes	—
73	Dic₁₇	—	8	8	no	—
74	C₂×C₄⋊Q₈	—	17	4	no	—
75	#500	—	17	8	no	—
76	#500	—	34	4	no	—
77	#487	—	17	16	no	—
78	#487	—	34	8	no	—
79	C₂×#107	—	34	4	no	—
80	C₂×#103	—	34	8	no	—
81	C₂×C₄×C₈	C₂×D₁₇	1	2	yes	—
82	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
83	C₈²	C₂×D₁₇	1	4	yes	—
84	C₂×C₁₇	#1710	1	4	yes	complemented
85	C₂×C₁₇	C₂×#494	1	1	yes	characteristic
86	C₂×C₁₇	C₂×#495	1	2	yes	—
87	C₂²×Q₈	—	34	4	no	—
88	C₂²×C₈	D₆₈	1	4	yes	—
89	C₂²×C₈	D₆₈	1	2	yes	—
90	C₄⋊Q₈	—	17	8	no	—
91	C₄⋊Q₈	—	34	4	no	—
92	C₄⋊Q₈	—	34	4	no	—
93	C₄⋊Q₈	—	34	4	no	—
94	C₂×C₄⋊C₄	—	34	4	no	—
95	C₂×C₄⋊C₄	—	34	4	no	—
96	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
97	#107	—	34	16	no	—
98	#103	—	34	32	no	—
99	C₄×C₈	D₆₈	1	4	yes	—
100	C₄×C₈	D₆₈	1	2	yes	—
101	C₄×C₈	C₂²×D₁₇	1	4	yes	—
102	C₄×C₈	C₂²×D₁₇	1	2	yes	—
103	C₁₇	C₂×#1710	1	1	yes	characteristic, complemented, Sylow
104	C₂×Q₈	—	34	8	no	—
105	C₂×Q₈	—	68	4	no	—
106	C₂×Q₈	—	68	16	no	—
107	C₂²×C₄	—	68	4	no	—
108	C₂²×C₄	C₂×D₆₈	1	2	yes	—
109	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
110	C₂×C₈	—	2	8	no	—
111	C₂×C₈	Dic₆₈	1	8	yes	—
112	C₂×C₈	C₂×D₆₈	1	8	yes	—
113	C₂×C₈	C₂×D₆₈	1	4	yes	—
114	C₄⋊C₄	—	34	8	no	—
115	C₄⋊C₄	—	34	8	no	—
116	C₄⋊C₄	—	68	4	no	—
117	C₄⋊C₄	—	68	4	no	—
118	C₄²	C₂×D₆₈	1	1	yes	characteristic
119	C₄²	C₂×D₆₈	1	2	yes	—
120	C₄²	C₂³×D₁₇	1	1	yes	characteristic
121	C₂³	#10587560	1	1	yes	characteristic
122	Q₈	—	68	16	no	—
123	Q₈	—	136	8	no	—
124	C₂×C₄	—	68	8	no	—
125	C₂×C₄	—	136	8	no	—
126	C₂×C₄	#10587560	1	1	yes	characteristic
127	C₂×C₄	#10587560	1	2	yes	—
128	C₂×C₄	#10587590	1	8	yes	—
129	C₂×C₄	C₂×Dic₆₈	1	4	yes	—
130	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
131	C₂×C₄	C₂²×D₆₈	1	2	yes	—
132	C₈	—	2	8	no	—
133	C₈	C₂×Dic₆₈	1	8	yes	—
134	C₂²	#11763189	1	2	yes	—
135	C₂²	#11763193	1	4	yes	—
136	C₂²	C₂×#10587560	1	1	yes	characteristic
137	C₄	—	136	8	no	—
138	C₄	#11763189	1	2	yes	—
139	C₄	#11763193	1	4	yes	—
140	C₄	C₂²×Dic₆₈	1	2	yes	—
141	C₄	C₂×#10587590	1	4	yes	—
142	C₂	#14528394	1	4	yes	direct product factor
143	C₂	C₂×#11763189	1	1	yes	characteristic
144	C₂	C₂×#11763193	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11763214

C₂

yes

maximal

C₂×#11763160

C₂

yes

maximal

#14528394

C₂

yes

maximal, direct product factor

C₂×C₈²×C₁₇

C₂

yes

maximal, characteristic

C₂×#10587555

C₂²

yes

—

#11763214

—

#11763214

C₂²

yes

—

#11763160

—

#11763160

C₂²

yes

—

C₂×#10587489

—

C₂×#10587487

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₇

C₂²

yes

—

C₈²×C₁₇

C₂²

yes

—

C₂²×Dic₃₄

—

C₂²×C₈×C₁₇

D₄

yes

—

C₂²×C₈×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

—

—

—

C₂³

yes

—

C₄×C₈×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

C₂³

yes

—

C₄×C₈×C₁₇

C₂³

yes

—

#10587489

—

#10587487

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×D₄

yes

—

C₂²×Dic₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

Q₁₆

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₄²×C₁₇

C₂×D₄

yes

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

C₂⁴

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

—

C₂×C₄×C₁₇

C₂×Q₁₆

yes

—

C₂×C₄×C₁₇

#135

yes

—

C₂×C₄×C₁₇

C₂²×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂²×D₄

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

C₂×Q₁₆

yes

—

C₂×#500

—

C₂×#487

—

#1710

—

C₂×C₈²

D₁₇

yes

characteristic

C₂²×C₁₇

#494

yes

—

C₂²×C₁₇

#495

yes

—

C₂²×C₁₇

C₂×#127

yes

characteristic

C₄×C₁₇

#494

yes

—

C₄×C₁₇

#495

yes

—

C₄×C₁₇

C₂²×Q₁₆

yes

—

C₄×C₁₇

C₂×#135

yes

—

Dic₁₇

—

C₂×C₄⋊Q₈

—

—

—

—

—

—

—

C₂×C₄×C₈

C₂×D₁₇

yes

—

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

C₈²

C₂×D₁₇

yes

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

—

C₂²×Q₈

—

C₂²×C₈

D₆₈

yes

—

C₂²×C₈

D₆₈

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₁₇

yes

characteristic

#107

—

#103

—

C₄×C₈

D₆₈

yes

—

100

C₄×C₈

D₆₈

yes

—

101

C₄×C₈

C₂²×D₁₇

yes

—

102

C₄×C₈

C₂²×D₁₇

yes

—

103

C₁₇

C₂×#1710

yes

characteristic, complemented, Sylow

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂²×C₄

—

108

C₂²×C₄

C₂×D₆₈

yes

—

109

C₂²×C₄

C₂×D₆₈

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

Dic₆₈

yes

—

112

C₂×C₈

C₂×D₆₈

yes

—

113

C₂×C₈

C₂×D₆₈

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄²

C₂×D₆₈

yes

characteristic

119

C₄²

C₂×D₆₈

yes

—

120

C₄²

C₂³×D₁₇

yes

characteristic

121

C₂³

#10587560

yes

characteristic

122

Q₈

—

123

Q₈

—

136

—

124

C₂×C₄

—

125

C₂×C₄

—

136

—

126

C₂×C₄

#10587560

yes

characteristic

127

C₂×C₄

#10587560

yes

—

128

C₂×C₄

#10587590

yes

—

129

C₂×C₄

C₂×Dic₆₈

yes

—

130

C₂×C₄

C₂²×D₆₈

yes

characteristic

131

C₂×C₄

C₂²×D₆₈

yes

—

132

C₈

—

133

C₈

C₂×Dic₆₈

yes

—

134

C₂²

#11763189

yes

—

135

C₂²

#11763193

yes

—

136

C₂²

C₂×#10587560

yes

characteristic

137

C₄

—

136

—

138

C₄

#11763189

yes

—

139

C₄

#11763193

yes

—

140

C₄

C₂²×Dic₆₈

yes

—

141

C₄

C₂×#10587590

yes

—

142

C₂

#14528394

yes

direct product factor

143

C₂

C₂×#11763189

yes

characteristic

144

C₂

C₂×#11763193

yes

—

	subgroup	G/H
center	C₂³	#10587560
commutator, cosocle	C₄²×C₁₇	C₂⁴
Frattini	C₄²	C₂³×D₁₇
Fitting	C₂×C₈²×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	#127
hypercenter	C₂×C₈²	D₁₇
hypocenter	C₁₇	C₂×#1710
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂³

#10587560

commutator, cosocle

C₄²×C₁₇

C₂⁴

Frattini

C₄²

C₂³×D₁₇

Fitting

C₂×C₈²×C₁₇

C₂

socle, prime radical

C₂³×C₁₇

#127

hypercenter

C₂×C₈²

D₁₇

hypocenter

C₁₇

C₂×#1710

LCS

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
3. #14528394	C₂	direct (N × H)
84. C₂×C₁₇	#1710	semidirect
103. C₁₇	C₂×#1710	semidirect
142. C₂	#14528394	direct (N × H)

H ≅ G/N

type

3. #14528394

C₂

direct (N × H)

84. C₂×C₁₇

#1710

semidirect

103. C₁₇

C₂×#1710

semidirect

142. C₂

#14528394

direct (N × H)

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

subgroup	index
C₂×#11763214	2
C₂×#11763160	2
#14528394	2
C₂×C₈²×C₁₇	2
C₂×#1710	17

subgroup

index

C₂×C₈²×C₁₇

Subgroups — C₂×#14528394 #17021997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14528394 #17021997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14528394 #17021997