Group #14530107 — subgroups

Subgroups — #14530107

Order 2176.

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

normal subgroups only

This group has 288 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	C₂×#10587487	C₂	1	2	yes	maximal
3	#11762715	C₂	1	2	yes	maximal
4	#11762572	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₇×C₄⋊C₈	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	4	no	—
12	C₂×Dic₆₈	—	2	2	no	—
13	C₁₇×C₄⋊C₈	—	2	2	no	—
14	#10587502	—	2	2	no	—
15	#10587502	—	2	2	no	—
16	#10587487	—	2	4	no	—
17	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
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	D₄	1	4	yes	—
28	C₂×C₈×C₁₇	—	2	2	no	—
29	C₂×C₈×C₁₇	—	2	2	no	—
30	C₂×C₈×C₁₇	—	2	1	no	—
31	C₂×C₈×C₁₇	D₄	1	2	yes	—
32	C₄²×C₁₇	—	2	1	no	—
33	C₄²×C₁₇	—	2	1	no	—
34	C₄⋊Dic₁₇	—	2	2	no	—
35	C₄⋊Dic₁₇	—	2	2	no	—
36	Dic₆₈	—	2	8	no	—
37	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₇	—	2	1	no	—
39	C₂×C₄×C₁₇	—	2	1	no	—
40	C₂×C₄×C₁₇	—	2	1	no	—
41	C₂×C₄×C₁₇	—	2	2	no	—
42	C₂×C₄×C₁₇	Q₁₆	1	2	yes	complemented
43	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
46	C₂×C₄×C₁₇	#41	1	2	yes	—
47	C₂×Dic₁₇	—	2	4	no	—
48	C₂×Dic₁₇	—	4	2	no	—
49	C₂×Dic₁₇	—	4	2	no	—
50	C₂×Dic₁₇	—	4	2	no	—
51	C₂×Dic₁₇	—	4	2	no	—
52	C₂×Dic₁₇	—	4	2	no	—
53	Dic₃₄	—	2	4	no	—
54	Dic₃₄	—	2	4	no	—
55	Dic₃₄	—	4	2	no	—
56	Dic₃₄	—	4	2	no	—
57	C₈×C₁₇	—	2	2	no	—
58	C₈×C₁₇	—	4	1	no	—
59	#2056	—	17	1	no	maximal, Sylow
60	C₂²×C₁₇	#121	1	2	yes	—
61	C₂²×C₁₇	#121	1	1	yes	characteristic
62	C₂²×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
63	C₂²×C₁₇	#135	1	1	yes	characteristic
64	C₂²×C₁₇	#137	1	2	yes	—
65	C₄×C₁₇	—	2	2	no	—
66	C₄×C₁₇	—	4	1	no	—
67	C₄×C₁₇	#120	1	1	yes	characteristic
68	C₄×C₁₇	#123	1	2	yes	—
69	C₄×C₁₇	#124	1	1	yes	characteristic
70	Dic₁₇	—	4	4	no	—
71	Dic₁₇	—	8	2	no	—
72	C₂²×Q₁₆	—	17	1	no	—
73	C₂×C₄⋊C₈	D₁₇	1	1	yes	characteristic
74	C₂×#103	—	17	2	no	—
75	#391	—	17	2	no	—
76	#340	—	17	1	no	—
77	C₂×C₁₇	#394	1	1	yes	characteristic
78	C₂×C₁₇	#462	1	2	yes	—
79	C₂×C₁₇	#464	1	2	yes	—
80	C₂×C₁₇	#467	1	1	yes	characteristic
81	C₂×C₁₇	#470	1	1	yes	characteristic
82	C₂²×Q₈	—	17	2	no	—
83	C₂×Q₁₆	—	34	4	no	—
84	C₂×Q₁₆	—	34	2	no	—
85	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
87	C₂×C₄⋊C₄	—	17	2	no	—
88	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
89	C₄⋊C₈	—	2	2	no	—
90	#103	—	34	4	no	—
91	#95	—	34	2	no	—
92	#95	—	34	2	no	—
93	C₁₇	#2056	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₈	—	17	4	no	—
95	C₂×Q₈	—	34	2	no	—
96	C₂×Q₈	—	34	2	no	—
97	C₂×Q₈	—	34	2	no	—
98	C₂×Q₈	—	34	2	no	—
99	C₂×Q₈	—	34	2	no	—
100	C₂²×C₄	—	34	2	no	—
101	C₂²×C₄	—	34	2	no	—
102	C₂²×C₄	D₆₈	1	1	yes	characteristic
103	C₂²×C₄	D₆₈	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
105	Q₁₆	—	34	8	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	C₂²⋊D₁₇	1	2	yes	—
110	C₄⋊C₄	—	34	2	no	—
111	C₄⋊C₄	—	34	2	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	2	1	no	—
114	C₂³	C₂×D₆₈	1	1	yes	characteristic
115	Q₈	—	34	4	no	—
116	Q₈	—	34	4	no	—
117	Q₈	—	68	2	no	—
118	Q₈	—	68	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	34	4	no	—
124	C₂×C₄	—	68	2	no	—
125	C₂×C₄	—	68	2	no	—
126	C₂×C₄	—	68	2	no	—
127	C₂×C₄	—	68	2	no	—
128	C₂×C₄	—	68	2	no	—
129	C₂×C₄	Dic₆₈	1	2	yes	complemented
130	C₂×C₄	#63252	1	1	yes	characteristic
131	C₂×C₄	D₄×D₁₇	1	2	yes	—
132	C₂×C₄	#63256	1	2	yes	—
133	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	—	4	1	no	—
136	C₂²	#10587581	1	2	yes	—
137	C₂²	#10587590	1	1	yes	characteristic
138	C₂²	C₂×Dic₆₈	1	1	yes	characteristic
139	C₂²	#10587595	1	2	yes	—
140	C₂²	#10587616	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	4	1	no	—
143	C₄	—	68	4	no	—
144	C₄	—	136	2	no	—
145	C₄	#10587580	1	2	yes	—
146	C₄	#10587623	1	1	yes	characteristic
147	C₄	#10587633	1	1	yes	characteristic
148	C₂	#11762726	1	1	yes	characteristic
149	C₂	#11763020	1	2	yes	—
150	C₂	#11763032	1	2	yes	—
151	C₂	#11763050	1	1	yes	characteristic
152	C₂	#11763067	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₆₈

C₂

yes

maximal, characteristic

C₂×#10587487

C₂

yes

maximal

#11762715

C₂

yes

maximal

#11762572

C₂

yes

maximal, characteristic

C₂×C₁₇×C₄⋊C₈

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

—

C₁₇×C₄⋊C₈

—

#10587502

—

#10587502

—

#10587487

—

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

D₄

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

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

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

Q₁₆

yes

complemented

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

#2056

—

maximal, Sylow

C₂²×C₁₇

#121

yes

—

C₂²×C₁₇

#121

yes

characteristic

C₂²×C₁₇

C₂×Q₁₆

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#120

yes

characteristic

C₄×C₁₇

#123

yes

—

C₄×C₁₇

#124

yes

characteristic

Dic₁₇

—

Dic₁₇

—

C₂²×Q₁₆

—

C₂×C₄⋊C₈

D₁₇

yes

—

—

—

C₂×C₁₇

#394

yes

characteristic

C₂×C₁₇

#462

yes

—

C₂×C₁₇

#464

yes

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×Q₈

—

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

C₄⋊C₈

—

#103

—

#95

—

#95

—

C₁₇

#2056

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

D₆₈

yes

characteristic

103

C₂²×C₄

D₆₈

yes

characteristic

104

C₂²×C₄

C₂²×D₁₇

yes

characteristic

105

Q₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂²⋊D₁₇

yes

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

113

C₄²

—

114

C₂³

C₂×D₆₈

yes

characteristic

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

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

#63252

yes

characteristic

131

C₂×C₄

D₄×D₁₇

yes

—

132

C₂×C₄

#63256

yes

—

133

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

134

C₈

—

135

C₈

—

136

C₂²

#10587581

yes

—

137

C₂²

#10587590

yes

characteristic

138

C₂²

C₂×Dic₆₈

yes

characteristic

139

C₂²

#10587595

yes

—

140

C₂²

#10587616

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

136

—

145

C₄

#10587580

yes

—

146

C₄

#10587623

yes

characteristic

147

C₄

#10587633

yes

characteristic

148

C₂

#11762726

yes

characteristic

149

C₂

#11763020

yes

—

150

C₂

#11763032

yes

—

151

C₂

#11763050

yes

characteristic

152

C₂

#11763067

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₈
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂×C₁₇×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₁₇
hypocenter	C₁₇	#2056
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₆₈

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₂×C₁₇×C₄⋊C₈

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₁₇

hypocenter

C₁₇

#2056

LCS

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
42. C₂×C₄×C₁₇	Q₁₆	semidirect
93. C₁₇	#2056	semidirect
129. C₂×C₄	Dic₆₈	semidirect

H ≅ G/N

type

42. C₂×C₄×C₁₇

Q₁₆

semidirect

93. C₁₇

#2056

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×#10587487	2
#11762715	2
#11762572	2
C₂×C₁₇×C₄⋊C₈	2
#2056	17

subgroup

index

C₂²×Dic₆₈

C₂×C₁₇×C₄⋊C₈

Subgroups — #14530107

Conjugacy classes of subgroups

Complements

Maximal subgroups