Group #14529645 — subgroups

Subgroups — #14529645

Order 2176.

312 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂²×Dic₆₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587489	C₂	1	1	yes	maximal, characteristic
3	C₂×#10587487	C₂	1	2	yes	maximal
4	#11762715	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₄	C₂²	1	2	yes	—
7	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
11	C₂×Dic₆₈	—	2	2	no	—
12	C₂×Dic₆₈	—	2	2	no	—
13	C₂×Dic₆₈	C₄	1	4	yes	complemented
14	C₄×C₈×C₁₇	—	2	2	no	—
15	#10587502	—	2	4	no	—
16	#10587489	—	2	2	no	—
17	#10587487	—	2	4	no	—
18	C₂²×C₄×C₁₇	D₄	1	2	yes	—
19	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
20	C₂²×Dic₁₇	—	2	2	no	—
21	C₂²×Dic₁₇	—	2	2	no	—
22	C₂×Dic₃₄	—	2	2	no	—
23	C₂×Dic₃₄	—	2	2	no	—
24	C₂×Dic₃₄	—	2	2	no	—
25	C₂×Dic₃₄	—	2	2	no	—
26	C₂×Dic₃₄	—	2	2	no	—
27	C₂×Dic₃₄	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₁₇	—	2	1	no	—
29	C₂×C₈×C₁₇	—	2	1	no	—
30	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₁₇	D₄	1	2	yes	—
32	C₂×C₈×C₁₇	D₄	1	2	yes	—
33	C₂×C₈×C₁₇	D₄	1	2	yes	—
34	C₄²×C₁₇	—	2	1	no	—
35	C₄²×C₁₇	—	2	1	no	—
36	C₄⋊Dic₁₇	—	2	2	no	—
37	C₄⋊Dic₁₇	—	2	2	no	—
38	Dic₆₈	—	2	8	no	—
39	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₇	—	2	2	no	—
41	C₂×C₄×C₁₇	—	2	2	no	—
42	C₂×C₄×C₁₇	Q₁₆	1	4	yes	complemented
43	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
48	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
49	C₂×Dic₁₇	—	2	4	no	—
50	C₂×Dic₁₇	—	4	2	no	—
51	C₂×Dic₁₇	—	4	2	no	—
52	C₂×Dic₁₇	—	4	2	no	—
53	C₂×Dic₁₇	—	4	4	no	—
54	Dic₃₄	—	2	4	no	—
55	Dic₃₄	—	2	4	no	—
56	Dic₃₄	—	4	2	no	—
57	Dic₃₄	—	4	2	no	—
58	C₈×C₁₇	—	2	2	no	—
59	C₈×C₁₇	C₂²⋊C₄	1	4	yes	—
60	#1969	—	17	1	no	maximal, Sylow
61	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
62	C₂²×C₁₇	#121	1	1	yes	characteristic
63	C₂²×C₁₇	#127	1	1	yes	characteristic
64	C₂²×C₁₇	C₂×Q₁₆	1	2	yes	—
65	C₂²×C₁₇	#135	1	2	yes	—
66	C₄×C₁₇	—	2	4	no	—
67	C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
69	C₄×C₁₇	#121	1	1	yes	characteristic
70	C₄×C₁₇	#124	1	1	yes	characteristic
71	Dic₁₇	—	4	4	no	—
72	Dic₁₇	—	8	2	no	—
73	C₂²×Q₁₆	—	17	1	no	—
74	C₂×#107	—	17	1	no	—
75	C₂×#103	—	17	2	no	—
76	C₂×C₄×C₈	D₁₇	1	1	yes	characteristic
77	#391	—	17	2	no	—
78	C₂×C₁₇	#390	1	1	yes	characteristic
79	C₂×C₁₇	C₄×Q₁₆	1	2	yes	—
80	C₂×C₁₇	#467	1	2	yes	—
81	C₂×C₁₇	#494	1	1	yes	characteristic
82	C₂×C₁₇	#495	1	1	yes	characteristic
83	C₂²×Q₈	—	17	2	no	—
84	C₂×Q₁₆	—	17	4	no	—
85	C₂×Q₁₆	—	34	2	no	—
86	C₂×Q₁₆	—	34	2	no	—
87	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
88	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
89	C₂×C₄⋊C₄	—	17	2	no	—
90	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
91	#107	—	34	2	no	—
92	#103	—	34	4	no	—
93	C₄×C₈	—	2	2	no	—
94	#95	—	34	4	no	—
95	C₁₇	#1969	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₈	—	17	4	no	—
97	C₂×Q₈	—	34	2	no	—
98	C₂×Q₈	—	34	2	no	—
99	C₂×Q₈	—	34	2	no	—
100	C₂×Q₈	—	34	2	no	—
101	C₂×Q₈	—	34	2	no	—
102	C₂²×C₄	—	34	2	no	—
103	C₂²×C₄	—	34	2	no	—
104	C₂²×C₄	D₆₈	1	2	yes	—
105	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
106	Q₁₆	—	34	8	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	C₄×D₁₇	1	2	yes	—
110	C₂×C₈	D₆₈	1	2	yes	—
111	C₂×C₈	D₆₈	1	2	yes	—
112	C₂×C₈	C₂²⋊D₁₇	1	2	yes	—
113	C₄⋊C₄	—	34	2	no	—
114	C₄⋊C₄	—	34	2	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	2	1	no	—
117	C₂³	C₂×D₆₈	1	1	yes	characteristic
118	Q₈	—	34	4	no	—
119	Q₈	—	34	4	no	—
120	Q₈	—	68	2	no	—
121	Q₈	—	68	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	34	4	no	—
125	C₂×C₄	—	68	2	no	—
126	C₂×C₄	—	68	2	no	—
127	C₂×C₄	—	68	2	no	—
128	C₂×C₄	—	68	4	no	—
129	C₂×C₄	Dic₆₈	1	4	yes	complemented
130	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
131	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
132	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
133	C₂×C₄	#63252	1	1	yes	characteristic
134	C₂×C₄	#63252	1	1	yes	characteristic
135	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	#63227	1	4	yes	—
138	C₂²	C₄×D₆₈	1	1	yes	characteristic
139	C₂²	#10587560	1	1	yes	characteristic
140	C₂²	#10587590	1	2	yes	—
141	C₂²	C₂×Dic₆₈	1	2	yes	—
142	C₂²	#10587616	1	1	yes	characteristic
143	C₄	—	2	4	no	—
144	C₄	—	68	4	no	—
145	C₄	—	136	2	no	—
146	C₄	C₄×D₆₈	1	1	yes	characteristic
147	C₄	#10587561	1	1	yes	characteristic
148	C₄	C₂×#63227	1	1	yes	characteristic
149	C₄	#10587616	1	1	yes	characteristic
150	C₂	#11762711	1	1	yes	characteristic
151	C₂	C₄×Dic₆₈	1	2	yes	—
152	C₂	#11763050	1	2	yes	—
153	C₂	#11763189	1	1	yes	characteristic
154	C₂	#11763193	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₆₈

C₂

yes

maximal, characteristic

C₂×#10587489

C₂

yes

maximal, characteristic

C₂×#10587487

C₂

yes

maximal

#11762715

C₂

yes

maximal

C₂×C₄×C₈×C₁₇

C₂

yes

maximal, characteristic

C₂²×Dic₃₄

C₂²

yes

—

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×Dic₆₈

—

C₂×Dic₆₈

—

C₂×Dic₆₈

C₄

yes

complemented

C₄×C₈×C₁₇

—

#10587502

—

#10587489

—

#10587487

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

C₂×C₄

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×C₄

yes

—

C₂×C₈×C₁₇

D₄

yes

—

C₂×C₈×C₁₇

D₄

yes

—

C₂×C₈×C₁₇

D₄

yes

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

Dic₆₈

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

Q₁₆

yes

complemented

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

C₂²⋊C₄

yes

—

#1969

—

maximal, Sylow

C₂²×C₁₇

C₄×D₄

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

C₂×Q₁₆

yes

—

C₂²×C₁₇

#135

yes

—

C₄×C₁₇

—

C₄×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₇

C₄×D₄

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

C₂²×Q₁₆

—

C₂×#107

—

C₂×#103

—

C₂×C₄×C₈

D₁₇

yes

characteristic

#391

—

C₂×C₁₇

#390

yes

characteristic

C₂×C₁₇

C₄×Q₁₆

yes

—

C₂×C₁₇

#467

yes

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

C₂×D₁₇

yes

characteristic

C₂²×C₈

C₂×D₁₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

#107

—

#103

—

C₄×C₈

—

#95

—

C₁₇

#1969

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

D₆₈

yes

—

105

C₂²×C₄

C₂²×D₁₇

yes

characteristic

106

Q₁₆

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₄×D₁₇

yes

—

110

C₂×C₈

D₆₈

yes

—

111

C₂×C₈

D₆₈

yes

—

112

C₂×C₈

C₂²⋊D₁₇

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₂³

C₂×D₆₈

yes

characteristic

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

Dic₆₈

yes

complemented

130

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

131

C₂×C₄

C₂×D₆₈

yes

characteristic

132

C₂×C₄

C₂×D₆₈

yes

characteristic

133

C₂×C₄

#63252

yes

characteristic

134

C₂×C₄

#63252

yes

characteristic

135

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

136

C₈

—

137

C₈

#63227

yes

—

138

C₂²

C₄×D₆₈

yes

characteristic

139

C₂²

#10587560

yes

characteristic

140

C₂²

#10587590

yes

—

141

C₂²

C₂×Dic₆₈

yes

—

142

C₂²

#10587616

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

136

—

146

C₄

C₄×D₆₈

yes

characteristic

147

C₄

#10587561

yes

characteristic

148

C₄

C₂×#63227

yes

characteristic

149

C₄

#10587616

yes

characteristic

150

C₂

#11762711

yes

characteristic

151

C₂

C₄×Dic₆₈

yes

—

152

C₂

#11763050

yes

—

153

C₂

#11763189

yes

characteristic

154

C₂

#11763193

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₈
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂×C₄×C₈×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₄
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₇
hypocenter	C₁₇	#1969
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₆₈

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₂×C₄×C₈×C₁₇

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₄

cosocle

C₂²×C₄×C₁₇

C₂³

hypercenter

C₂×C₄×C₈

D₁₇

hypocenter

C₁₇

#1969

LCS

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
13. C₂×Dic₆₈	C₄	semidirect
42. C₂×C₄×C₁₇	Q₁₆	semidirect
95. C₁₇	#1969	semidirect
129. C₂×C₄	Dic₆₈	semidirect

H ≅ G/N

type

13. C₂×Dic₆₈

C₄

semidirect

42. C₂×C₄×C₁₇

Q₁₆

semidirect

95. C₁₇

#1969

semidirect

129. C₂×C₄

Dic₆₈

semidirect

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

subgroup	index
C₂²×Dic₆₈	2
C₂×#10587489	2
C₂×#10587487	2
#11762715	2
C₂×C₄×C₈×C₁₇	2
#1969	17

subgroup

index

C₂²×Dic₆₈

C₂×C₄×C₈×C₁₇

Subgroups — #14529645

Conjugacy classes of subgroups

Complements

Maximal subgroups