Group #14528140 — subgroups

Subgroups — #14528140

Order 2176.

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

normal subgroups only

This group has 204 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	#11763186	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763005	C₂	1	2	yes	maximal, complemented
3	C₈×D₆₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#11762552	C₂	1	2	yes	maximal, complemented
5	C₁₇×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₃₆	—	2	2	no	—
7	C₂×D₁₃₆	—	2	1	no	—
8	C₂×D₁₃₆	—	2	1	no	—
9	C₂×C₈×D₁₇	—	2	1	no	—
10	#10587560	C₂²	1	2	yes	—
11	C₄×D₆₈	C₂²	1	1	yes	characteristic
12	C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
13	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	#10587492	—	2	2	no	—
15	#10587491	—	2	1	no	—
16	C₄⋊#3622	C₂²	1	1	yes	characteristic
17	C₂×D₆₈	—	2	2	no	—
18	C₂×D₆₈	—	2	2	no	—
19	C₂×D₆₈	—	2	2	no	—
20	C₂×D₆₈	D₄	1	1	yes	characteristic
21	C₂×C₄×D₁₇	—	2	1	no	—
22	C₂×C₈×C₁₇	—	2	2	no	—
23	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
24	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
25	C₄²×C₁₇	C₂³	1	1	yes	characteristic
26	#63227	—	2	1	no	—
27	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
28	C₂×#3622	—	2	1	no	—
29	D₁₃₆	—	4	2	no	—
30	D₁₃₆	—	4	4	no	—
31	D₁₃₆	—	4	2	no	—
32	C₈×D₁₇	—	4	2	no	—
33	C₂²×D₁₇	—	2	1	no	—
34	C₂²×D₁₇	—	4	2	no	—
35	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
38	C₂×Dic₁₇	—	2	1	no	—
39	D₆₈	—	2	1	no	—
40	D₆₈	—	4	2	no	—
41	D₆₈	—	4	2	no	—
42	D₆₈	—	4	2	no	—
43	D₆₈	—	4	2	no	—
44	D₆₈	—	8	2	no	—
45	D₆₈	D₈	1	2	yes	complemented
46	C₄×D₁₇	—	4	1	no	—
47	C₄×D₁₇	—	4	1	no	—
48	C₈×C₁₇	—	2	1	no	—
49	C₈×C₁₇	—	4	2	no	—
50	C₈×C₁₇	D₈	1	2	yes	complemented
51	#3622	—	4	1	no	—
52	#1667	—	17	1	no	maximal, Sylow
53	C₂²×C₁₇	#121	1	1	yes	characteristic
54	C₂×D₁₇	—	2	2	no	—
55	C₂×D₁₇	—	4	1	no	—
56	C₂×D₁₇	—	4	1	no	—
57	C₂×D₁₇	—	8	2	no	—
58	C₂×D₁₇	—	8	2	no	—
59	C₂×D₁₇	—	8	2	no	—
60	C₄×C₁₇	—	2	1	no	—
61	C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
62	C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
63	C₄×C₁₇	#135	1	1	yes	characteristic
64	C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
65	Dic₁₇	—	4	1	no	—
66	#493	—	17	1	no	—
67	#459	—	17	2	no	—
68	C₈×D₄	—	17	1	no	—
69	#335	D₁₇	1	1	yes	characteristic, complemented
70	#331	—	17	2	no	—
71	C₂×C₁₇	#459	1	1	yes	characteristic
72	C₂×C₁₇	#466	1	1	yes	characteristic
73	C₂×C₁₇	#472	1	1	yes	characteristic
74	D₁₇	—	4	2	no	—
75	D₁₇	—	16	2	no	—
76	C₂×D₈	—	34	2	no	—
77	C₂×D₈	—	34	1	no	—
78	C₂×D₈	—	34	1	no	—
79	C₂²×C₈	—	34	1	no	—
80	#127	—	17	2	no	—
81	C₄×D₄	—	17	1	no	—
82	C₄⋊C₈	—	17	1	no	—
83	C₄⋊C₈	C₂×D₁₇	1	2	yes	—
84	#102	—	34	2	no	—
85	C₂²⋊C₈	—	34	1	no	—
86	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
87	C₁₇	#1667	1	1	yes	characteristic, complemented, Sylow
88	C₂×D₄	—	17	1	no	—
89	C₂×D₄	—	34	2	no	—
90	C₂×D₄	—	34	2	no	—
91	C₂×D₄	—	34	2	no	—
92	C₂²×C₄	—	34	1	no	—
93	D₈	—	68	2	no	—
94	D₈	—	68	4	no	—
95	D₈	—	68	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	34	1	no	—
98	C₂×C₈	—	68	2	no	—
99	C₂×C₈	D₆₈	1	1	yes	characteristic
100	C₂×C₈	D₆₈	1	1	yes	characteristic
101	C₄⋊C₄	—	17	1	no	—
102	C₂²⋊C₄	—	34	1	no	—
103	C₄²	C₂²×D₁₇	1	1	yes	characteristic
104	C₂³	—	34	1	no	—
105	C₂³	—	68	2	no	—
106	D₄	—	17	2	no	—
107	D₄	—	34	1	no	—
108	D₄	—	68	2	no	—
109	D₄	—	68	2	no	—
110	D₄	—	68	2	no	—
111	D₄	—	68	2	no	—
112	D₄	—	136	2	no	—
113	C₂×C₄	—	34	1	no	—
114	C₂×C₄	—	68	1	no	—
115	C₂×C₄	—	68	1	no	—
116	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
117	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
118	C₂×C₄	#63256	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	—	4	2	no	—
121	C₈	—	68	1	no	—
122	C₈	D₁₃₆	1	2	yes	complemented
123	C₂²	—	34	2	no	—
124	C₂²	—	68	1	no	—
125	C₂²	—	68	1	no	—
126	C₂²	—	136	2	no	—
127	C₂²	—	136	2	no	—
128	C₂²	—	136	2	no	—
129	C₂²	#10587581	1	1	yes	characteristic
130	C₄	—	2	1	no	—
131	C₄	—	68	1	no	—
132	C₄	C₂×D₁₃₆	1	1	yes	characteristic
133	C₄	#10587594	1	1	yes	characteristic
134	C₄	D₈×D₁₇	1	1	yes	characteristic
135	C₄	#10587605	1	1	yes	characteristic
136	C₂	—	272	2	no	—
137	C₂	—	68	2	no	—
138	C₂	#11763005	1	1	yes	characteristic
139	C₂	#11763046	1	1	yes	characteristic
140	C₂	#11763080	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763186

C₂

yes

maximal, characteristic, complemented

#11763005

C₂

yes

maximal, complemented

C₈×D₆₈

C₂

yes

maximal, characteristic, complemented

#11762552

C₂

yes

maximal, complemented

C₁₇×#335

C₂

yes

maximal, characteristic, complemented

C₂×D₁₃₆

—

C₂×D₁₃₆

—

C₂×D₁₃₆

—

C₂×C₈×D₁₇

—

#10587560

C₂²

yes

—

C₄×D₆₈

C₂²

yes

characteristic

C₁₇×C₄⋊C₈

C₂²

yes

—

C₄×C₈×C₁₇

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

D₄

yes

characteristic

C₂×C₄×D₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₄

yes

characteristic

C₂×C₈×C₁₇

D₄

yes

characteristic

C₄²×C₁₇

C₂³

yes

characteristic

#63227

—

C₄⋊Dic₁₇

D₄

yes

characteristic

C₂×#3622

—

D₁₃₆

—

D₁₃₆

—

D₁₃₆

—

C₈×D₁₇

—

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

characteristic

C₂×Dic₁₇

—

D₆₈

—

D₆₈

—

D₆₈

—

D₆₈

—

D₆₈

—

D₆₈

—

D₆₈

D₈

yes

complemented

C₄×D₁₇

—

C₄×D₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

D₈

yes

—

—

C₂²×C₁₇

yes

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×D₈

yes

characteristic

C₄×C₁₇

C₂×D₈

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

#493

—

#459

—

C₈×D₄

—

#335

D₁₇

yes

characteristic, complemented

#331

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

D₁₇

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

—

#127

—

C₄×D₄

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₁₇

yes

—

#102

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₁₇

yes

characteristic

C₁₇

#1667

yes

characteristic, complemented, Sylow

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

D₈

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₆₈

yes

characteristic

100

C₂×C₈

D₆₈

yes

characteristic

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

C₂²×D₁₇

yes

characteristic

104

C₂³

—

105

C₂³

—

106

D₄

—

107

D₄

—

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

—

112

D₄

—

136

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂×D₆₈

yes

characteristic

117

C₂×C₄

D₄×D₁₇

yes

characteristic

118

C₂×C₄

#63256

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

D₁₃₆

yes

complemented

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

136

—

127

C₂²

—

136

—

128

C₂²

—

136

—

129

C₂²

#10587581

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

C₂×D₁₃₆

yes

characteristic

133

C₄

#10587594

yes

characteristic

134

C₄

D₈×D₁₇

yes

characteristic

135

C₄

#10587605

yes

characteristic

136

C₂

—

272

—

137

C₂

—

138

C₂

#11763005

yes

characteristic

139

C₂

#11763046

yes

characteristic

140

C₂

#11763080

yes

characteristic

	subgroup	G/H
center	C₂²	#10587581
commutator, cosocle	C₄²×C₁₇	C₂³
Frattini	C₄²	C₂²×D₁₇
Fitting	C₁₇×#335	C₂
socle	C₂²×C₁₇	#121
hypercenter	#335	D₁₇
hypocenter	C₁₇	#1667
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#10587581

commutator, cosocle

C₄²×C₁₇

C₂³

Frattini

C₄²

C₂²×D₁₇

Fitting

C₁₇×#335

C₂

C₂²×C₁₇

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
1. #11763186	C₂	semidirect
2. #11763005	C₂	semidirect
3. C₈×D₆₈	C₂	semidirect
4. #11762552	C₂	semidirect
5. C₁₇×#335	C₂	semidirect
45. D₆₈	D₈	semidirect
50. C₈×C₁₇	D₈	semidirect
69. #335	D₁₇	semidirect
87. C₁₇	#1667	semidirect
122. C₈	D₁₃₆	semidirect

H ≅ G/N

type

1. #11763186

C₂

semidirect

2. #11763005

C₂

semidirect

3. C₈×D₆₈

C₂

semidirect

4. #11762552

C₂

semidirect

5. C₁₇×#335

C₂

semidirect

45. D₆₈

D₈

semidirect

50. C₈×C₁₇

D₈

semidirect

69. #335

D₁₇

semidirect

87. C₁₇

#1667

semidirect

122. C₈

D₁₃₆

semidirect

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

subgroup	index
#11763186	2
#11763005	2
C₈×D₆₈	2
#11762552	2
C₁₇×#335	2
#1667	17

subgroup

index

C₈×D₆₈

C₁₇×#335

Subgroups — #14528140

Conjugacy classes of subgroups

Complements

Maximal subgroups