Group #14530091 — subgroups

Subgroups — #14530091

Order 2176.

264 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 63 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	#11763609	C₂	1	1	yes	maximal, characteristic
2	#11762837	C₂	1	1	yes	maximal, characteristic, complemented
3	#11762803	C₂	1	2	yes	maximal, complemented
4	#11762782	C₂	1	2	yes	maximal, complemented
5	C₁₇×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#63252	C₂²	1	2	yes	—
7	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
8	C₁₇×#117	C₂²	1	2	yes	—
9	#10587612	C₂²	1	1	yes	characteristic
10	C₂²×#3622	C₂²	1	1	yes	characteristic
11	#10587595	—	2	2	no	—
12	#10587594	—	2	2	no	—
13	#10587590	—	2	2	no	—
14	#10587485	—	2	2	no	—
15	#10587481	—	2	2	no	—
16	#10587480	—	2	2	no	—
17	#10587477	—	2	4	no	—
18	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
19	C₂×C₂²⋊D₁₇	—	2	2	no	—
20	#63252	—	2	2	no	—
21	#63252	—	2	2	no	—
22	#63252	—	2	2	no	—
23	#63252	—	2	2	no	—
24	C₂×D₆₈	D₄	1	2	yes	—
25	C₂×C₄×D₁₇	—	2	2	no	—
26	C₂×Dic₃₄	D₄	1	2	yes	—
27	M₁₆×C₁₇	—	2	2	no	—
28	C₂×C₈×C₁₇	D₄	1	2	yes	—
29	C₄⋊C₄×C₁₇	—	2	2	no	—
30	C₂²⋊C₄×C₁₇	—	2	2	no	—
31	C₄²×C₁₇	—	2	2	no	—
32	C₂²⋊Dic₁₇	D₄	1	2	yes	—
33	C₄⋊Dic₁₇	—	2	1	no	—
34	C₄×Dic₁₇	—	2	1	no	—
35	C₂×#3622	—	2	2	no	—
36	C₂×#3622	—	2	1	no	—
37	Dic₆₈	—	2	2	no	—
38	D₁₃₆	—	2	2	no	—
39	#63219	—	2	4	no	—
40	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₁₇	—	2	2	no	—
42	C₂×C₄×C₁₇	—	4	2	no	—
43	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₇	#41	1	2	yes	—
46	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
47	C₂²⋊D₁₇	—	4	2	no	—
48	C₂²⋊D₁₇	—	4	2	no	—
49	C₂×Dic₁₇	—	2	2	no	—
50	C₂×Dic₁₇	—	2	2	no	—
51	D₆₈	—	2	2	no	—
52	D₆₈	—	2	2	no	—
53	C₄×D₁₇	—	4	2	no	—
54	C₄×D₁₇	—	4	2	no	—
55	Dic₃₄	—	2	2	no	—
56	Dic₃₄	—	2	2	no	—
57	C₈×C₁₇	—	2	2	no	—
58	#3622	—	4	1	no	—
59	#2053	—	17	1	no	maximal, Sylow
60	C₂²×C₁₇	—	4	1	no	—
61	C₂²×C₁₇	#121	1	2	yes	—
62	C₂²×C₁₇	#121	1	1	yes	characteristic
63	C₂×D₁₇	—	4	2	no	—
64	C₂×D₁₇	—	8	2	no	—
65	C₄×C₁₇	—	8	2	no	—
66	C₄×C₁₇	#120	1	1	yes	characteristic
67	C₄×C₁₇	#123	1	2	yes	—
68	C₄×C₁₇	#124	1	1	yes	characteristic
69	Dic₁₇	—	4	2	no	—
70	Dic₁₇	—	4	2	no	—
71	#575	—	17	1	no	—
72	#424	—	17	1	no	—
73	#416	—	17	2	no	—
74	#410	—	17	2	no	—
75	H(3,ℤ/4)	D₁₇	1	1	yes	characteristic, complemented
76	C₂×C₁₇	—	2	1	no	—
77	C₂×C₁₇	—	2	2	no	—
78	C₂×C₁₇	#394	1	1	yes	characteristic
79	D₁₇	—	8	2	no	—
80	C₂×#41	—	17	2	no	—
81	#137	—	34	2	no	—
82	AGL(1,ℤ/8)	—	34	2	no	—
83	#135	—	34	2	no	—
84	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
85	C₂²×C₈	—	17	1	no	—
86	#117	—	17	1	no	—
87	#117	C₂×D₁₇	1	2	yes	—
88	C₄⋊C₈	—	34	2	no	—
89	#103	—	34	2	no	—
90	#102	—	34	2	no	—
91	#99	—	34	4	no	—
92	C₁₇	#2053	1	1	yes	characteristic, complemented, Sylow
93	#41	—	34	2	no	—
94	#41	—	34	2	no	—
95	#41	—	34	2	no	—
96	#41	—	34	2	no	—
97	C₂×Q₈	—	17	2	no	—
98	C₂×D₄	—	17	2	no	—
99	C₂×D₄	—	34	2	no	—
100	C₂²×C₄	—	34	2	no	—
101	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
102	Q₁₆	—	34	2	no	—
103	SD₁₆	—	34	4	no	—
104	D₈	—	34	2	no	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	—	34	1	no	—
107	C₂×C₈	—	34	2	no	—
108	C₂×C₈	D₆₈	1	2	yes	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	34	1	no	—
111	C₂²⋊C₄	—	2	2	no	—
112	C₂²⋊C₄	—	17	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	—	34	1	no	—
115	C₂³	—	34	2	no	—
116	C₂³	D₄×D₁₇	1	1	yes	characteristic
117	Q₈	—	34	2	no	—
118	Q₈	—	34	2	no	—
119	D₄	—	34	2	no	—
120	D₄	—	34	2	no	—
121	D₄	—	68	2	no	—
122	D₄	—	68	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	34	2	no	—
125	C₂×C₄	—	34	2	no	—
126	C₂×C₄	—	68	2	no	—
127	C₂×C₄	—	68	2	no	—
128	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
129	C₂×C₄	#63252	1	2	yes	—
130	C₂×C₄	D₄×D₁₇	1	2	yes	—
131	C₂×C₄	#63256	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	68	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	—	68	2	no	—
136	C₂²	—	136	2	no	—
137	C₂²	#10587570	1	2	yes	—
138	C₂²	#10587581	1	1	yes	characteristic
139	C₄	—	8	2	no	—
140	C₄	—	68	2	no	—
141	C₄	—	68	2	no	—
142	C₄	#10587561	1	1	yes	characteristic
143	C₄	#10587568	1	1	yes	characteristic
144	C₄	#10587580	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	136	2	no	—
148	C₂	#11762727	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763609

C₂

yes

maximal, characteristic

#11762837

C₂

yes

maximal, characteristic, complemented

#11762803

C₂

yes

maximal, complemented

#11762782

C₂

yes

maximal, complemented

C₁₇×H(3,ℤ/4)

C₂

yes

maximal, characteristic, complemented

C₂×#63252

C₂²

yes

—

C₂×M₁₆×C₁₇

C₂²

yes

characteristic

C₁₇×#117

C₂²

yes

—

#10587612

C₂²

yes

characteristic

C₂²×#3622

C₂²

yes

—

—

—

—

—

—

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂×C₂²⋊D₁₇

—

—

—

—

—

C₂×D₆₈

D₄

yes

—

C₂×C₄×D₁₇

—

C₂×Dic₃₄

D₄

yes

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

D₄

yes

—

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₂²⋊Dic₁₇

D₄

yes

—

C₄⋊Dic₁₇

—

C₄×Dic₁₇

—

C₂×#3622

—

C₂×#3622

—

Dic₆₈

—

D₁₃₆

—

#63219

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂²×D₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

#41

yes

—

C₂×C₄×C₁₇

#41

yes

characteristic

C₂²⋊D₁₇

—

C₂²⋊D₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

D₆₈

—

D₆₈

—

C₄×D₁₇

—

C₄×D₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

#3622

—

#2053

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

#121

yes

—

C₂²×C₁₇

#121

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#120

yes

characteristic

C₄×C₁₇

#123

yes

—

C₄×C₁₇

#124

yes

characteristic

Dic₁₇

—

Dic₁₇

—

—

—

—

—

D₁₇

yes

characteristic, complemented

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#394

yes

characteristic

D₁₇

—

C₂×#41

—

#137

—

AGL(1,ℤ/8)

—

#135

—

C₂×M₁₆

C₂×D₁₇

yes

characteristic

C₂²×C₈

—

#117

—

#117

C₂×D₁₇

yes

—

C₄⋊C₈

—

#103

—

#102

—

#99

—

C₁₇

#2053

yes

characteristic, complemented, Sylow

—

—

—

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂²×D₁₇

yes

characteristic

102

Q₁₆

—

103

SD₁₆

—

104

D₈

—

105

M₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

D₆₈

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

—

115

C₂³

—

116

C₂³

D₄×D₁₇

yes

characteristic

117

Q₈

—

118

Q₈

—

119

D₄

—

120

D₄

—

121

D₄

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×D₆₈

yes

characteristic

129

C₂×C₄

#63252

yes

—

130

C₂×C₄

D₄×D₁₇

yes

—

131

C₂×C₄

#63256

yes

characteristic

132

C₈

—

133

C₈

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

136

—

137

C₂²

#10587570

yes

—

138

C₂²

#10587581

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

#10587561

yes

characteristic

143

C₄

#10587568

yes

characteristic

144

C₄

#10587580

yes

—

145

C₂

—

146

C₂

—

147

C₂

—

136

—

148

C₂

#11762727

yes

characteristic

	subgroup	G/H
center	C₄	#10587561
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×H(3,ℤ/4)	C₂
socle	C₂×C₁₇	#394
prime radical	#11763609	C₂
hypercenter	H(3,ℤ/4)	D₁₇
hypocenter	C₁₇	#2053
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#10587561

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×H(3,ℤ/4)

C₂

C₂×C₁₇

C₂

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
2. #11762837	C₂	semidirect
3. #11762803	C₂	semidirect
4. #11762782	C₂	semidirect
5. C₁₇×H(3,ℤ/4)	C₂	semidirect
75. H(3,ℤ/4)	D₁₇	semidirect
92. C₁₇	#2053	semidirect

H ≅ G/N

type

2. #11762837

C₂

semidirect

3. #11762803

C₂

semidirect

4. #11762782

C₂

semidirect

5. C₁₇×H(3,ℤ/4)

C₂

semidirect

75. H(3,ℤ/4)

D₁₇

semidirect

92. C₁₇

#2053

semidirect

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

subgroup	index
#11763609	2
#11762837	2
#11762803	2
#11762782	2
C₁₇×H(3,ℤ/4)	2
#2053	17

subgroup

index

C₁₇×H(3,ℤ/4)

Subgroups — #14530091

Conjugacy classes of subgroups

Complements

Maximal subgroups