Group #14528167 — subgroups

Subgroups — #14528167

Order 2176.

150 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 57 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	#11763105	C₂	1	1	yes	maximal, characteristic
3	#11763020	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₃₄	C₂	1	1	yes	maximal, characteristic
5	#11762557	C₂	1	1	yes	maximal, characteristic
6	#11762556	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#334	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₆₈	—	2	1	no	—
9	#10587555	C₂²	1	1	yes	characteristic
10	#10587555	C₂²	1	1	yes	characteristic
11	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
12	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	#10587489	—	2	1	no	—
16	#10587488	—	2	1	no	—
17	#10587488	—	2	1	no	—
18	#10587487	—	2	1	no	—
19	#10587487	—	2	1	no	—
20	#10587485	—	2	1	no	—
21	C₈×Dic₁₇	—	2	1	no	—
22	C₄⋊#3622	C₂²	1	1	yes	characteristic
23	C₂×Dic₃₄	—	2	1	no	—
24	C₂×Dic₃₄	—	2	1	no	—
25	C₂×Dic₃₄	D₄	1	1	yes	characteristic
26	C₂×C₈×C₁₇	—	2	1	no	—
27	C₂×C₈×C₁₇	—	2	1	no	—
28	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
29	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
30	C₄²×C₁₇	C₂³	1	1	yes	characteristic
31	C₄⋊Dic₁₇	—	2	1	no	—
32	C₄⋊Dic₁₇	—	2	1	no	—
33	C₄⋊Dic₁₇	—	2	1	no	—
34	C₄⋊Dic₁₇	—	2	1	no	—
35	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
36	#63225	—	2	1	no	—
37	C₄×Dic₁₇	—	2	1	no	—
38	C₂×#3622	—	2	1	no	—
39	Dic₆₈	—	4	2	no	—
40	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
43	C₂×Dic₁₇	—	2	1	no	—
44	C₂×Dic₁₇	—	2	1	no	—
45	C₂×Dic₁₇	—	4	1	no	—
46	C₂×Dic₁₇	—	4	1	no	—
47	Dic₃₄	—	2	1	no	—
48	Dic₃₄	—	4	1	no	—
49	Dic₃₄	—	4	1	no	—
50	Dic₃₄	—	8	1	no	—
51	Dic₃₄	SD₁₆	1	2	yes	—
52	C₈×C₁₇	—	2	1	no	—
53	C₈×C₁₇	—	4	1	no	—
54	C₈×C₁₇	—	4	1	no	—
55	C₈×C₁₇	Q₁₆	1	2	yes	complemented
56	#3622	—	4	1	no	—
57	#1672	—	17	1	no	maximal, Sylow
58	C₂²×C₁₇	#121	1	1	yes	characteristic
59	C₄×C₁₇	—	2	1	no	—
60	C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
61	C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
62	C₄×C₁₇	#135	1	1	yes	characteristic
63	C₄×C₁₇	#137	1	1	yes	characteristic
64	Dic₁₇	—	2	2	no	—
65	Dic₁₇	—	4	1	no	—
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	—	17	1	no	—
72	#334	D₁₇	1	1	yes	characteristic
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	—	17	1	no	—
82	#107	—	34	1	no	—
83	#106	—	34	1	no	—
84	#106	—	34	1	no	—
85	C₄⋊C₈	—	17	1	no	—
86	C₄⋊C₈	—	34	1	no	—
87	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
88	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
89	#103	—	34	1	no	—
90	#103	—	34	1	no	—
91	C₄×C₈	—	34	1	no	—
92	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
93	C₁₇	#1672	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₈	—	17	1	no	—
95	C₂×Q₈	—	34	1	no	—
96	C₂×Q₈	—	34	1	no	—
97	Q₁₆	—	68	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	34	1	no	—
101	C₂×C₈	D₆₈	1	1	yes	characteristic
102	C₂×C₈	D₆₈	1	1	yes	characteristic
103	C₄⋊C₄	—	17	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	—	34	1	no	—
109	C₄²	—	34	1	no	—
110	C₄²	C₂²×D₁₇	1	1	yes	characteristic
111	Q₈	—	17	2	no	—
112	Q₈	—	34	1	no	—
113	Q₈	—	68	1	no	—
114	Q₈	—	68	1	no	—
115	Q₈	—	136	1	no	—
116	C₂×C₄	—	34	1	no	—
117	C₂×C₄	—	34	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	D₄×D₁₇	1	1	yes	characteristic
122	C₂×C₄	#63256	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	4	1	no	—
126	C₈	—	68	1	no	—
127	C₈	Dic₆₈	1	2	yes	complemented
128	C₂²	#10587581	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	34	2	no	—
131	C₄	—	68	1	no	—
132	C₄	—	136	1	no	—
133	C₄	—	136	1	no	—
134	C₄	C₂×Dic₆₈	1	1	yes	characteristic
135	C₄	#10587595	1	1	yes	characteristic
136	C₄	SD₁₆×D₁₇	1	1	yes	characteristic
137	C₄	#10587602	1	1	yes	characteristic
138	C₂	#11763020	1	1	yes	characteristic
139	C₂	#11763039	1	1	yes	characteristic
140	C₂	#11763073	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763207

C₂

yes

maximal, characteristic

#11763105

C₂

yes

maximal, characteristic

#11763020

C₂

yes

maximal, characteristic

C₈×Dic₃₄

C₂

yes

maximal, characteristic

#11762557

C₂

yes

maximal, characteristic

#11762556

C₂

yes

maximal, characteristic

C₁₇×#334

C₂

yes

maximal, characteristic

C₂×Dic₆₈

—

#10587555

C₂²

yes

characteristic

#10587555

C₂²

yes

characteristic

C₄×Dic₃₄

C₂²

yes

characteristic

C₁₇×C₄⋊C₈

C₂²

yes

characteristic

C₁₇×C₄⋊C₈

C₂²

yes

characteristic

C₄×C₈×C₁₇

C₂²

yes

—

—

—

—

—

—

C₈×Dic₁₇

—

C₄⋊#3622

C₂²

yes

characteristic

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

D₄

yes

characteristic

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₄

yes

characteristic

C₂×C₈×C₁₇

D₄

yes

characteristic

C₄²×C₁₇

C₂³

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

D₄

yes

characteristic

#63225

—

C₄×Dic₁₇

—

C₂×#3622

—

Dic₆₈

—

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

—

Dic₃₄

SD₁₆

yes

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

Q₁₆

yes

—

—

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

C₂×SD₁₆

yes

characteristic

C₄×C₁₇

C₂×Q₁₆

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

#498

—

#477

—

#462

—

C₈×Q₈

—

#334

D₁₇

yes

characteristic

#333

—

#332

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

#107

—

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₁₇

yes

characteristic

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

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

D₆₈

yes

characteristic

102

C₂×C₈

D₆₈

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

—

110

C₄²

C₂²×D₁₇

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

136

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂×D₆₈

yes

characteristic

121

C₂×C₄

D₄×D₁₇

yes

characteristic

122

C₂×C₄

#63256

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

Dic₆₈

yes

complemented

128

C₂²

#10587581

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

136

—

133

C₄

—

136

—

134

C₄

C₂×Dic₆₈

yes

characteristic

135

C₄

#10587595

yes

characteristic

136

C₄

SD₁₆×D₁₇

yes

characteristic

137

C₄

#10587602

yes

characteristic

138

C₂

#11763020

yes

characteristic

139

C₂

#11763039

yes

characteristic

140

C₂

#11763073

yes

characteristic

	subgroup	G/H
center	C₂²	#10587581
commutator, cosocle	C₄²×C₁₇	C₂³
Frattini	C₄²	C₂²×D₁₇
Fitting	C₁₇×#334	C₂
socle, prime radical	C₂²×C₁₇	#121
hypercenter	#334	D₁₇
hypocenter	C₁₇	#1672
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#10587581

commutator, cosocle

C₄²×C₁₇

C₂³

Frattini

C₄²

C₂²×D₁₇

Fitting

C₁₇×#334

C₂

C₂²×C₁₇

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
55. C₈×C₁₇	Q₁₆	semidirect
93. C₁₇	#1672	semidirect
127. C₈	Dic₆₈	semidirect

H ≅ G/N

type

55. C₈×C₁₇

Q₁₆

semidirect

93. C₁₇

#1672

semidirect

127. C₈

Dic₆₈

semidirect

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

subgroup	index
#11763207	2
#11763105	2
#11763020	2
C₈×Dic₃₄	2
#11762557	2
#11762556	2
C₁₇×#334	2
#1672	17

subgroup

index

C₈×Dic₃₄

C₁₇×#334

Subgroups — #14528167

Conjugacy classes of subgroups

Complements

Maximal subgroups