Group #14540129 — subgroups

Subgroups — #14540129

Order 2176.

348 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 137 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	#11763366	C₂	1	2	yes	maximal
2	#11763105	C₂	1	4	yes	maximal, complemented
3	#11763050	C₂	1	4	yes	maximal
4	#11763020	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₁₇	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
8	#10587612	—	2	2	no	—
9	#10587610	—	2	2	no	—
10	#10587610	C₂²	1	4	yes	—
11	C₂×Dic₆₈	C₂²	1	4	yes	—
12	#10587556	—	2	2	no	—
13	#10587555	C₂²	1	2	yes	—
14	#10587555	C₂²	1	2	yes	—
15	C₄×Dic₃₄	—	2	2	no	—
16	C₄×Dic₃₄	C₂²	1	4	yes	—
17	C₁₇×C₄⋊C₈	C₂²	1	4	yes	—
18	#10587489	C₂²	1	4	yes	—
19	#10587487	C₂²	1	8	yes	—
20	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
23	C₂×Dic₃₄	—	2	2	no	—
24	C₂×Dic₃₄	C₂³	1	4	yes	—
25	C₂×C₈×C₁₇	—	4	2	no	—
26	C₂×C₈×C₁₇	C₂³	1	4	yes	—
27	C₄²×C₁₇	D₄	1	2	yes	—
28	C₄²×C₁₇	C₂³	1	2	yes	—
29	C₂²⋊Dic₁₇	—	2	4	no	—
30	C₄⋊Dic₁₇	—	2	2	no	—
31	C₄⋊Dic₁₇	—	2	2	no	—
32	C₄⋊Dic₁₇	—	2	2	no	—
33	C₄⋊Dic₁₇	C₂³	1	4	yes	—
34	#63225	—	2	4	no	—
35	#63225	—	2	4	no	—
36	C₄×Dic₁₇	—	2	4	no	—
37	Dic₆₈	—	4	4	no	—
38	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₇	—	2	1	no	—
40	C₂×C₄×C₁₇	—	2	1	no	—
41	C₂×C₄×C₁₇	—	4	1	no	—
42	C₂×C₄×C₁₇	Q₁₆	1	4	yes	complemented
43	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₁₇	—	2	4	no	—
48	C₂×Dic₁₇	—	2	4	no	—
49	Dic₃₄	—	2	4	no	—
50	Dic₃₄	—	2	4	no	—
51	Dic₃₄	—	4	4	no	—
52	C₈×C₁₇	—	2	4	no	—
53	#3249	—	17	1	no	maximal, Sylow
54	C₂²×C₁₇	—	2	1	no	—
55	C₂²×C₁₇	—	2	1	no	—
56	C₂²×C₁₇	C₂×Q₁₆	1	2	yes	—
57	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₁₇	—	2	1	no	—
59	C₄×C₁₇	—	2	2	no	—
60	C₄×C₁₇	C₂×Q₁₆	1	4	yes	—
61	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
62	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
63	Dic₁₇	—	4	4	no	—
64	Dic₁₇	—	4	4	no	—
65	#533	—	17	2	no	—
66	#477	—	17	4	no	—
67	#467	—	17	4	no	—
68	#462	—	17	4	no	—
69	C₂×C₄⋊C₈	D₁₇	1	1	yes	characteristic
70	C₂×C₁₇	—	2	2	no	—
71	C₂×C₁₇	#537	1	1	yes	characteristic
72	C₂×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
73	C₂×C₁₇	#575	1	1	yes	characteristic
74	C₂×Q₁₆	—	17	4	no	—
75	C₂²×C₈	C₂×D₁₇	1	2	yes	—
76	C₄⋊Q₈	—	17	2	no	—
77	C₄⋊Q₈	—	17	2	no	—
78	#125	—	34	2	no	—
79	C₂²⋊Q₈	—	17	4	no	—
80	C₂²⋊Q₈	—	34	2	no	—
81	C₄×Q₈	—	17	4	no	—
82	C₄×Q₈	—	34	2	no	—
83	#117	—	34	2	no	—
84	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
85	#107	—	17	4	no	—
86	C₄⋊C₈	C₂×D₁₇	1	4	yes	—
87	#103	—	17	8	no	—
88	C₁₇	#3249	1	1	yes	characteristic, complemented, Sylow
89	C₂×Q₈	—	17	4	no	—
90	C₂×Q₈	—	34	2	no	—
91	C₂²×C₄	D₆₈	1	1	yes	characteristic
92	C₂²×C₄	D₆₈	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
94	Q₁₆	—	68	4	no	—
95	C₂×C₈	—	4	2	no	—
96	C₂×C₈	C₂²×D₁₇	1	4	yes	—
97	C₄⋊C₄	—	17	4	no	—
98	C₄⋊C₄	—	34	2	no	—
99	C₄⋊C₄	—	34	2	no	—
100	C₄⋊C₄	—	34	4	no	—
101	C₄⋊C₄	—	34	4	no	—
102	C₄⋊C₄	—	34	2	no	—
103	C₂²⋊C₄	—	34	4	no	—
104	C₄²	—	34	4	no	—
105	C₄²	D₆₈	1	2	yes	—
106	C₄²	C₂²×D₁₇	1	2	yes	—
107	C₂³	C₂×D₆₈	1	1	yes	characteristic
108	Q₈	—	34	4	no	—
109	Q₈	—	34	4	no	—
110	Q₈	—	68	4	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	34	4	no	—
115	C₂×C₄	—	34	4	no	—
116	C₂×C₄	Dic₆₈	1	4	yes	complemented
117	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
118	C₂×C₄	C₂×D₆₈	1	2	yes	—
119	C₂×C₄	C₂×D₆₈	1	2	yes	—
120	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
121	C₈	—	2	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	C₂×Dic₆₈	1	2	yes	—
125	C₂²	C₂²×D₆₈	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	1	no	—
128	C₄	—	68	4	no	—
129	C₄	—	68	4	no	—
130	C₄	C₂×Dic₆₈	1	4	yes	—
131	C₄	#10587691	1	1	yes	characteristic
132	C₄	#10587694	1	1	yes	characteristic
133	C₂	—	2	2	no	—
134	C₂	#11763390	1	1	yes	characteristic
135	C₂	C₂²×Dic₆₈	1	1	yes	characteristic
136	C₂	#11763609	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763366

C₂

yes

maximal

#11763105

C₂

yes

maximal, complemented

#11763050

C₂

yes

maximal

#11763020

C₂

yes

maximal, complemented

C₂×C₁₇×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂²×C₈×C₁₇

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

—

—

C₂²

yes

—

C₂×Dic₆₈

C₂²

yes

—

#10587556

—

#10587555

C₂²

yes

—

#10587555

C₂²

yes

—

C₄×Dic₃₄

—

C₄×Dic₃₄

C₂²

yes

—

C₁₇×C₄⋊C₈

C₂²

yes

—

#10587489

C₂²

yes

—

#10587487

C₂²

yes

—

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂×Dic₃₄

—

C₂×Dic₃₄

C₂³

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂³

yes

—

C₄²×C₁₇

D₄

yes

—

C₄²×C₁₇

C₂³

yes

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂³

yes

—

#63225

—

#63225

—

C₄×Dic₁₇

—

Dic₆₈

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

Q₁₆

yes

complemented

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂⁴

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

#3249

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×Q₁₆

yes

—

C₂²×C₁₇

C₂²×D₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×Q₁₆

yes

—

C₄×C₁₇

H(5,𝔽₂)

yes

characteristic

C₄×C₁₇

2₋¹⁺⁴

yes

characteristic

Dic₁₇

—

Dic₁₇

—

—

—

—

—

C₂×C₄⋊C₈

D₁₇

yes

characteristic

C₂×C₁₇

—

C₂×C₁₇

#537

yes

characteristic

C₂×C₁₇

C₂²×Q₁₆

yes

characteristic

C₂×C₁₇

#575

yes

characteristic

C₂×Q₁₆

—

C₂²×C₈

C₂×D₁₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

#107

—

C₄⋊C₈

C₂×D₁₇

yes

—

#103

—

C₁₇

#3249

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

D₆₈

yes

characteristic

C₂²×C₄

D₆₈

yes

characteristic

C₂²×C₄

C₂²×D₁₇

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

C₂²×D₁₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₄²

D₆₈

yes

—

106

C₄²

C₂²×D₁₇

yes

—

107

C₂³

C₂×D₆₈

yes

characteristic

108

Q₈

—

109

Q₈

—

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

Dic₆₈

yes

complemented

117

C₂×C₄

C₂×D₆₈

yes

characteristic

118

C₂×C₄

C₂×D₆₈

yes

—

119

C₂×C₄

C₂×D₆₈

yes

—

120

C₂×C₄

C₂³×D₁₇

yes

characteristic

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×Dic₆₈

yes

—

125

C₂²

C₂²×D₆₈

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

C₂×Dic₆₈

yes

—

131

C₄

#10587691

yes

characteristic

132

C₄

#10587694

yes

characteristic

133

C₂

—

134

C₂

#11763390

yes

characteristic

135

C₂

C₂²×Dic₆₈

yes

characteristic

136

C₂

#11763609

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₆₈
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₂×C₁₇×C₄⋊C₈	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂³×C₁₇	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₁₇
hypocenter	C₁₇	#3249
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂²

C₂²×D₆₈

commutator, cosocle

C₂×C₄×C₁₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₇

Fitting

C₂×C₁₇×C₄⋊C₈

C₂

socle

C₂²×C₁₇

C₂²×D₄

prime radical

C₂³×C₁₇

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₁₇

hypocenter

C₁₇

#3249

LCS

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
2. #11763105	C₂	semidirect
4. #11763020	C₂	semidirect
42. C₂×C₄×C₁₇	Q₁₆	semidirect
88. C₁₇	#3249	semidirect
116. C₂×C₄	Dic₆₈	semidirect

H ≅ G/N

type

2. #11763105

C₂

semidirect

4. #11763020

C₂

semidirect

42. C₂×C₄×C₁₇

Q₁₆

semidirect

88. C₁₇

#3249

semidirect

116. C₂×C₄

Dic₆₈

semidirect

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

subgroup	index
#11763366	2
#11763105	2
#11763050	2
#11763020	2
C₂×C₁₇×C₄⋊C₈	2
#3249	17

subgroup

index

C₂×C₁₇×C₄⋊C₈

Subgroups — #14540129

Conjugacy classes of subgroups

Complements

Maximal subgroups