Group #14528166 — subgroups

Subgroups — #14528166

Order 2176.

150 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 61 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	#11763207	C₂	1	1	yes	maximal, characteristic
2	#11763104	C₂	1	1	yes	maximal, characteristic
3	#11763019	C₂	1	1	yes	maximal, characteristic
4	#11762560	C₂	1	1	yes	maximal, characteristic
5	#11762557	C₂	1	1	yes	maximal, characteristic
6	#11762554	C₂	1	1	yes	maximal, characteristic
7	C₈×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×#63231	—	2	1	no	—
10	#10587555	C₂²	1	1	yes	characteristic
11	#10587555	C₂²	1	1	yes	characteristic
12	C₁₇×C₄⋊C₈	—	2	1	no	—
13	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₇	—	2	1	no	—
15	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	#10587506	—	2	1	no	—
17	#10587488	—	2	1	no	—
18	#10587488	—	2	1	no	—
19	#10587481	—	2	1	no	—
20	#10587478	—	2	1	no	—
21	C₄⋊#3622	C₂²	1	1	yes	characteristic
22	C₄⋊#3622	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₁₇	D₄	1	1	yes	characteristic
24	C₂×Dic₃₄	—	2	1	no	—
25	C₂×Dic₃₄	—	2	1	no	—
26	C₂×C₈×C₁₇	—	2	1	no	—
27	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
28	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
29	C₄⋊C₄×C₁₇	—	2	1	no	—
30	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
31	C₄²×C₁₇	—	2	1	no	—
32	C₄²×C₁₇	C₂³	1	1	yes	characteristic
33	#63231	—	4	2	no	—
34	C₄⋊Dic₁₇	—	2	1	no	—
35	C₄⋊Dic₁₇	—	2	1	no	—
36	C₄⋊Dic₁₇	—	2	1	no	—
37	C₄⋊Dic₁₇	—	2	1	no	—
38	C₂×#3622	—	2	1	no	—
39	C₂×#3622	—	2	1	no	—
40	Q₈×C₁₇	—	2	1	no	—
41	Q₈×C₁₇	SD₁₆	1	2	yes	—
42	C₂×C₄×C₁₇	—	2	1	no	—
43	C₂×C₄×C₁₇	—	2	1	no	—
44	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
47	C₂×Dic₁₇	—	4	1	no	—
48	C₂×Dic₁₇	—	4	1	no	—
49	Dic₃₄	—	4	1	no	—
50	Dic₃₄	—	4	1	no	—
51	Dic₃₄	—	8	1	no	—
52	C₈×C₁₇	—	2	1	no	—
53	C₈×C₁₇	—	4	1	no	—
54	C₈×C₁₇	Q₁₆	1	2	yes	complemented
55	#3622	—	4	1	no	—
56	#3622	—	4	1	no	—
57	#1672	—	17	1	no	maximal, Sylow
58	C₂²×C₁₇	#121	1	1	yes	characteristic
59	C₄×C₁₇	—	2	2	no	—
60	C₄×C₁₇	—	2	1	no	—
61	C₄×C₁₇	—	4	1	no	—
62	C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
64	C₄×C₁₇	#135	1	1	yes	characteristic
65	C₄×C₁₇	#137	1	1	yes	characteristic
66	Dic₁₇	—	8	1	no	—
67	Dic₁₇	—	8	1	no	—
68	#498	—	17	1	no	—
69	#477	—	17	1	no	—
70	#462	—	17	1	no	—
71	C₈×Q₈	D₁₇	1	1	yes	characteristic
72	#334	—	17	1	no	—
73	#333	—	17	1	no	—
74	#332	—	17	1	no	—
75	C₂×C₁₇	#462	1	1	yes	characteristic
76	C₂×C₁₇	#465	1	1	yes	characteristic
77	C₂×C₁₇	#471	1	1	yes	characteristic
78	C₂×Q₁₆	—	34	1	no	—
79	C₄⋊Q₈	—	17	1	no	—
80	C₄⋊Q₈	—	17	1	no	—
81	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
82	#107	—	34	1	no	—
83	#106	—	34	1	no	—
84	#106	—	34	1	no	—
85	C₄⋊C₈	—	2	1	no	—
86	C₄⋊C₈	—	17	1	no	—
87	C₄⋊C₈	—	17	1	no	—
88	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
89	#103	—	34	1	no	—
90	#103	—	34	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
93	C₁₇	#1672	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₈	—	34	1	no	—
95	C₂×Q₈	—	34	1	no	—
96	C₂×Q₈	D₆₈	1	1	yes	characteristic
97	Q₁₆	—	68	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	34	1	no	—
100	C₂×C₈	—	34	1	no	—
101	C₂×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
102	C₂×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	34	1	no	—
105	C₄⋊C₄	—	34	1	no	—
106	C₄⋊C₄	—	34	1	no	—
107	C₄⋊C₄	—	34	1	no	—
108	C₄⋊C₄	D₆₈	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	C₂²×D₁₇	1	1	yes	characteristic
111	Q₈	—	2	1	no	—
112	Q₈	—	68	1	no	—
113	Q₈	—	68	1	no	—
114	Q₈	—	136	1	no	—
115	Q₈	#63219	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	68	1	no	—
119	C₂×C₄	—	68	1	no	—
120	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
121	C₂×C₄	#63252	1	1	yes	characteristic
122	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	68	1	no	—
126	C₈	—	68	1	no	—
127	C₈	#63231	1	2	yes	complemented
128	C₂²	#10587616	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	—	2	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	136	1	no	—
133	C₄	—	136	1	no	—
134	C₄	C₂×#63219	1	1	yes	characteristic
135	C₄	#10587590	1	1	yes	characteristic
136	C₄	C₂×#63231	1	1	yes	characteristic
137	C₄	#10587637	1	1	yes	characteristic
138	C₂	#11763019	1	1	yes	characteristic
139	C₂	#11763038	1	1	yes	characteristic
140	C₂	#11763072	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763207

C₂

yes

maximal, characteristic

#11763104

C₂

yes

maximal, characteristic

#11763019

C₂

yes

maximal, characteristic

#11762560

C₂

yes

maximal, characteristic

#11762557

C₂

yes

maximal, characteristic

#11762554

C₂

yes

maximal, characteristic

C₈×Q₈×C₁₇

C₂

yes

maximal, characteristic

C₄×Q₈×C₁₇

C₂²

yes

characteristic

C₂×#63231

—

#10587555

C₂²

yes

characteristic

#10587555

C₂²

yes

characteristic

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

C₂²

yes

characteristic

C₄×C₈×C₁₇

—

C₄×C₈×C₁₇

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₄⋊#3622

C₂²

yes

characteristic

C₂×Q₈×C₁₇

D₄

yes

characteristic

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₄

yes

characteristic

C₂×C₈×C₁₇

D₄

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

D₄

yes

characteristic

C₄²×C₁₇

—

C₄²×C₁₇

C₂³

yes

characteristic

#63231

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₂×#3622

—

C₂×#3622

—

Q₈×C₁₇

—

Q₈×C₁₇

SD₁₆

yes

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

Q₁₆

yes

—

—

—

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×SD₁₆

yes

characteristic

C₄×C₁₇

C₂×Q₁₆

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

#498

—

#477

—

#462

—

C₈×Q₈

D₁₇

yes

—

—

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

C₂×D₁₇

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₁₇

yes

characteristic

#103

—

#103

—

C₄×C₈

—

C₄×C₈

C₂×D₁₇

yes

characteristic

C₁₇

#1672

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₆₈

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂²⋊D₁₇

yes

characteristic

102

C₂×C₈

C₂²⋊D₁₇

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

D₆₈

yes

characteristic

109

C₄²

—

110

C₄²

C₂²×D₁₇

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

136

—

115

Q₈

#63219

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂×D₆₈

yes

characteristic

121

C₂×C₄

#63252

yes

characteristic

122

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

#63231

yes

complemented

128

C₂²

#10587616

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

136

—

133

C₄

—

136

—

134

C₄

C₂×#63219

yes

characteristic

135

C₄

#10587590

yes

characteristic

136

C₄

C₂×#63231

yes

characteristic

137

C₄

#10587637

yes

characteristic

138

C₂

#11763019

yes

characteristic

139

C₂

#11763038

yes

characteristic

140

C₂

#11763072

yes

characteristic

	subgroup	G/H
center	C₂²	#10587616
commutator, cosocle	C₄²×C₁₇	C₂³
Frattini	C₄²	C₂²×D₁₇
Fitting	C₈×Q₈×C₁₇	C₂
socle, prime radical	C₂²×C₁₇	#121
hypercenter	C₈×Q₈	D₁₇
hypocenter	C₁₇	#1672
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#10587616

commutator, cosocle

C₄²×C₁₇

C₂³

Frattini

C₄²

C₂²×D₁₇

Fitting

C₈×Q₈×C₁₇

C₂

socle, prime radical

C₂²×C₁₇

#121

hypercenter

C₈×Q₈

D₁₇

hypocenter

C₁₇

#1672

LCS

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
54. C₈×C₁₇	Q₁₆	semidirect
93. C₁₇	#1672	semidirect
127. C₈	#63231	semidirect

H ≅ G/N

type

54. C₈×C₁₇

Q₁₆

semidirect

93. C₁₇

#1672

semidirect

127. C₈

#63231

semidirect

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

subgroup	index
#11763207	2
#11763104	2
#11763019	2
#11762560	2
#11762557	2
#11762554	2
C₈×Q₈×C₁₇	2
#1672	17

subgroup

index

C₈×Q₈×C₁₇

Subgroups — #14528166

Conjugacy classes of subgroups

Complements

Maximal subgroups