Group #14529697 — subgroups

Subgroups — #14529697

Order 2176.

276 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 89 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	#11762819	C₂	1	2	yes	maximal
3	C₂×#10587495	C₂	1	2	yes	maximal
4	#11762764	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₇×#108	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₄	C₂²	1	2	yes	—
7	C₂×M₁₆×C₁₇	C₂²	1	2	yes	—
8	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
9	#10587612	C₂²	1	1	yes	characteristic
10	C₂×#63223	C₂²	1	1	yes	characteristic
11	C₂×Dic₆₈	—	2	1	no	—
12	C₂×Dic₆₈	—	2	1	no	—
13	C₂×Dic₆₈	—	2	2	no	—
14	C₂×Dic₆₈	C₄	1	4	yes	complemented
15	C₁₇×#108	—	2	2	no	—
16	#10587495	—	2	4	no	—
17	#10587487	—	2	4	no	—
18	#10587486	—	2	1	no	—
19	C₈×Dic₁₇	—	2	1	no	—
20	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
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	M₁₆×C₁₇	—	2	2	no	—
29	M₁₆×C₁₇	—	2	2	no	—
30	C₂×C₈×C₁₇	—	2	2	no	—
31	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₇	D₄	1	2	yes	—
35	C₂²⋊Dic₁₇	—	2	1	no	—
36	C₄⋊Dic₁₇	D₄	1	2	yes	—
37	C₄×Dic₁₇	—	2	1	no	—
38	#63223	—	2	1	no	—
39	#63223	D₄	1	2	yes	—
40	C₂×#3622	—	2	1	no	—
41	Dic₆₈	—	2	4	no	—
42	Dic₆₈	—	2	4	no	—
43	C₂³×C₁₇	#41	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₂²×C₄	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	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
50	C₂×Dic₁₇	—	2	4	no	—
51	C₂×Dic₁₇	—	2	2	no	—
52	C₂×Dic₁₇	—	4	2	no	—
53	C₂×Dic₁₇	—	4	2	no	—
54	Dic₃₄	—	2	4	no	—
55	Dic₃₄	—	2	4	no	—
56	Dic₃₄	—	4	2	no	—
57	Dic₃₄	—	4	2	no	—
58	C₈×C₁₇	—	4	2	no	—
59	C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
60	C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
61	#3622	—	2	2	no	—
62	#1979	—	17	1	no	maximal, Sylow
63	C₂²×C₁₇	—	2	2	no	—
64	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
65	C₂²×C₁₇	#121	1	1	yes	characteristic
66	C₂²×C₁₇	#124	1	1	yes	characteristic
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	#127	1	1	yes	characteristic
71	Dic₁₇	—	4	4	no	—
72	Dic₁₇	—	4	2	no	—
73	C₂²×Q₁₆	—	17	1	no	—
74	C₂×#108	D₁₇	1	1	yes	characteristic
75	#419	—	17	2	no	—
76	C₂×#101	—	17	2	no	—
77	#405	—	17	1	no	—
78	C₂×C₁₇	—	2	1	no	—
79	C₂×C₁₇	—	2	1	no	—
80	C₂×C₁₇	#390	1	1	yes	characteristic
81	C₂×C₁₇	#473	1	2	yes	—
82	C₂²×Q₈	—	17	2	no	—
83	C₂×Q₁₆	—	17	4	no	—
84	C₂×Q₁₆	—	34	1	no	—
85	C₂×Q₁₆	—	34	1	no	—
86	C₂×Q₁₆	—	34	2	no	—
87	C₂×M₁₆	—	17	1	no	—
88	C₂×M₁₆	C₂×D₁₇	1	2	yes	—
89	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
90	#117	—	17	1	no	—
91	#108	—	2	2	no	—
92	#103	—	34	4	no	—
93	#101	—	34	4	no	—
94	#97	—	34	1	no	—
95	C₄×C₈	—	34	1	no	—
96	C₁₇	#1979	1	1	yes	characteristic, complemented, Sylow
97	C₂×Q₈	—	17	4	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	C₂×Q₈	—	34	2	no	—
103	C₂²×C₄	—	34	2	no	—
104	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
105	Q₁₆	—	34	4	no	—
106	Q₁₆	—	34	4	no	—
107	M₁₆	—	2	2	no	—
108	M₁₆	—	2	2	no	—
109	M₁₆	—	17	2	no	—
110	M₁₆	—	34	1	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	34	1	no	—
113	C₂×C₈	C₄×D₁₇	1	2	yes	—
114	C₂×C₈	D₆₈	1	1	yes	characteristic
115	C₂×C₈	D₆₈	1	1	yes	characteristic
116	C₂×C₈	C₂²⋊D₁₇	1	2	yes	—
117	C₄⋊C₄	—	17	2	no	—
118	C₂²⋊C₄	—	34	1	no	—
119	C₄²	—	34	1	no	—
120	C₂³	#63256	1	1	yes	characteristic
121	Q₈	—	34	4	no	—
122	Q₈	—	34	4	no	—
123	Q₈	—	68	2	no	—
124	Q₈	—	68	2	no	—
125	C₂×C₄	—	34	4	no	—
126	C₂×C₄	—	34	2	no	—
127	C₂×C₄	—	68	2	no	—
128	C₂×C₄	—	68	2	no	—
129	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
130	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
131	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
132	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
133	C₂×C₄	#63256	1	1	yes	characteristic
134	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
135	C₈	—	4	2	no	—
136	C₈	—	34	2	no	—
137	C₈	#63227	1	2	yes	—
138	C₈	#63227	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	#10587579	1	1	yes	characteristic
141	C₂²	#10587581	1	1	yes	characteristic
142	C₂²	#10587633	1	1	yes	characteristic
143	C₄	—	68	4	no	—
144	C₄	—	68	2	no	—
145	C₄	#10587579	1	1	yes	characteristic
146	C₄	#10587581	1	1	yes	characteristic
147	C₄	C₂×#63227	1	1	yes	characteristic
148	C₄	#10587626	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	#11762710	1	1	yes	characteristic
152	C₂	#11763085	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₆₈

C₂

yes

maximal, characteristic

#11762819

C₂

yes

maximal

C₂×#10587495

C₂

yes

maximal

#11762764

C₂

yes

maximal, characteristic

C₂×C₁₇×#108

C₂

yes

maximal, characteristic

C₂²×Dic₃₄

C₂²

yes

—

C₂×M₁₆×C₁₇

C₂²

yes

—

C₂²×C₈×C₁₇

C₂²

yes

characteristic

#10587612

C₂²

yes

characteristic

C₂×#63223

C₂²

yes

characteristic

C₂×Dic₆₈

—

C₂×Dic₆₈

—

C₂×Dic₆₈

—

C₂×Dic₆₈

C₄

yes

complemented

C₁₇×#108

—

#10587495

—

#10587487

—

#10587486

—

C₈×Dic₁₇

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

C₂×C₄

yes

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×C₄

yes

—

C₂×C₈×C₁₇

D₄

yes

characteristic

C₂×C₈×C₁₇

D₄

yes

characteristic

C₂×C₈×C₁₇

D₄

yes

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

D₄

yes

—

C₄×Dic₁₇

—

#63223

—

#63223

D₄

yes

—

C₂×#3622

—

Dic₆₈

—

Dic₆₈

—

C₂³×C₁₇

#41

yes

characteristic

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

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

C₂²⋊C₄

yes

—

C₈×C₁₇

C₂²⋊C₄

yes

—

#3622

—

#1979

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

C₄×D₄

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₇

C₄×D₄

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

C₂²×Q₁₆

—

C₂×#108

D₁₇

yes

—

—

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#390

yes

characteristic

C₂×C₁₇

#473

yes

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₇

yes

—

C₂²×C₈

C₂×D₁₇

yes

—

—

—

—

—

C₄×C₈

—

C₁₇

#1979

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂²×D₁₇

yes

characteristic

105

Q₁₆

—

106

Q₁₆

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₄×D₁₇

yes

—

114

C₂×C₈

D₆₈

yes

characteristic

115

C₂×C₈

D₆₈

yes

characteristic

116

C₂×C₈

C₂²⋊D₁₇

yes

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₂³

#63256

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

130

C₂×C₄

C₂×D₆₈

yes

characteristic

131

C₂×C₄

D₄×D₁₇

yes

characteristic

132

C₂×C₄

D₄×D₁₇

yes

characteristic

133

C₂×C₄

#63256

yes

characteristic

134

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

#63227

yes

—

138

C₈

#63227

yes

—

139

C₂²

—

140

C₂²

#10587579

yes

characteristic

141

C₂²

#10587581

yes

characteristic

142

C₂²

#10587633

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

#10587579

yes

characteristic

146

C₄

#10587581

yes

characteristic

147

C₄

C₂×#63227

yes

characteristic

148

C₄

#10587626

yes

characteristic

149

C₂

—

150

C₂

—

151

C₂

#11762710

yes

characteristic

152

C₂

#11763085

yes

—

	subgroup	G/H
center	C₂²	#10587581
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂×C₁₇×#108	C₂
socle	C₂²×C₁₇	#121
prime radical	C₂³×C₁₇	#41
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	C₂×#108	D₁₇
hypocenter	C₁₇	#1979
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂²

#10587581

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₂×C₁₇×#108

C₂

C₂²×C₁₇

C₂³×C₁₇

C₂²×C₄×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
14. C₂×Dic₆₈	C₄	semidirect
96. C₁₇	#1979	semidirect

H ≅ G/N

type

14. C₂×Dic₆₈

C₄

semidirect

96. C₁₇

#1979

semidirect

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

subgroup	index
C₂²×Dic₆₈	2
#11762819	2
C₂×#10587495	2
#11762764	2
C₂×C₁₇×#108	2
#1979	17

subgroup

index

C₂²×Dic₆₈

C₂×C₁₇×#108

Subgroups — #14529697

Conjugacy classes of subgroups

Complements

Maximal subgroups