Group #14530833 — subgroups

Subgroups — #14530833

Order 2176.

154 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 49 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	C₂×Dic₁₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587500	C₂	1	1	yes	maximal, characteristic
3	#11763187	C₂	1	1	yes	maximal, characteristic
4	#11762903	C₂	1	1	yes	maximal, characteristic
5	#11762627	C₂	1	1	yes	maximal, characteristic
6	#11762612	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#358	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
9	#10587630	—	2	1	no	—
10	C₂×#63231	—	2	1	no	—
11	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
12	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
13	#10587573	—	2	1	no	—
14	C₂×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
15	C₁₇×#107	C₂²	1	1	yes	characteristic
16	#10587500	—	2	2	no	—
17	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
18	C₂×#63214	C₂²	1	1	yes	characteristic
19	#10587481	—	2	1	no	—
20	Dic₁₃₆	—	2	2	no	—
21	C₂×Q₈×C₁₇	—	2	1	no	—
22	C₂×Dic₃₄	—	2	1	no	—
23	C₂×Dic₃₄	—	2	1	no	—
24	Q₁₆×C₁₇	—	2	1	no	—
25	Q₁₆×C₁₇	—	2	1	no	—
26	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₁₇	—	2	1	no	—
28	#63231	—	4	2	no	—
29	#63225	—	4	1	no	—
30	#63225	—	4	1	no	—
31	#63225	—	4	1	no	—
32	C₄×Dic₁₇	—	4	1	no	—
33	C₄×Dic₁₇	D₄	1	1	yes	characteristic
34	C₂×#3622	D₄	1	1	yes	characteristic
35	Dic₆₈	—	2	1	no	—
36	Dic₆₈	—	2	1	no	—
37	Dic₆₈	—	2	1	no	—
38	Dic₆₈	D₄	1	2	yes	—
39	C₁₆×C₁₇	—	2	1	no	—
40	#63214	—	2	1	no	—
41	Q₈×C₁₇	—	4	1	no	—
42	Q₈×C₁₇	—	4	1	no	—
43	C₂×C₄×C₁₇	—	4	1	no	—
44	C₂×C₄×C₁₇	—	4	1	no	—
45	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₁₇	—	4	1	no	—
47	C₂×Dic₁₇	—	4	1	no	—
48	C₂×Dic₁₇	D₈	1	1	yes	characteristic
49	C₂×Dic₁₇	D₈	1	1	yes	characteristic
50	Dic₃₄	—	2	2	no	—
51	Dic₃₄	—	4	1	no	—
52	Dic₃₄	—	4	1	no	—
53	Dic₃₄	—	4	1	no	—
54	C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
55	C₈×C₁₇	#41	1	1	yes	characteristic
56	#3622	—	2	1	no	—
57	#2209	—	17	1	no	maximal, Sylow
58	C₂²×C₁₇	C₂×D₈	1	1	yes	characteristic
59	C₄×C₁₇	—	8	1	no	—
60	C₄×C₁₇	—	8	1	no	—
61	C₄×C₁₇	#121	1	1	yes	characteristic
62	C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
63	Dic₁₇	—	2	1	no	—
64	Dic₁₇	—	4	2	no	—
65	Dic₁₇	—	8	1	no	—
66	Dic₁₇	Q₃₂	1	2	yes	complemented
67	C₂×Q₃₂	—	17	1	no	—
68	C₂×Q₃₂	—	17	1	no	—
69	#494	—	17	1	no	—
70	C₄×Q₁₆	—	17	1	no	—
71	C₄⋊C₁₆	—	17	1	no	—
72	#358	—	17	1	no	—
73	#358	D₁₇	1	1	yes	characteristic
74	C₂×C₁₇	#459	1	1	yes	characteristic
75	C₂×C₁₇	C₂×Q₃₂	1	1	yes	characteristic
76	C₂×C₁₇	#510	1	1	yes	characteristic
77	C₂×Q₁₆	—	17	1	no	—
78	C₂×Q₁₆	—	17	1	no	—
79	C₂×Q₁₆	—	34	1	no	—
80	C₂×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
81	C₄⋊Q₈	—	34	1	no	—
82	C₄×Q₈	—	34	1	no	—
83	Q₃₂	—	34	2	no	—
84	Q₃₂	—	34	2	no	—
85	C₂×C₁₆	—	17	1	no	—
86	C₂×C₁₆	C₂×D₁₇	1	1	yes	characteristic
87	#107	C₂×D₁₇	1	1	yes	characteristic
88	#103	—	34	1	no	—
89	C₄×C₈	—	17	1	no	—
90	C₁₇	#2209	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	34	1	no	—
93	C₂×Q₈	—	34	1	no	—
94	Q₁₆	—	2	1	no	—
95	Q₁₆	—	2	1	no	—
96	Q₁₆	—	17	2	no	—
97	Q₁₆	—	34	1	no	—
98	Q₁₆	—	34	1	no	—
99	Q₁₆	—	34	1	no	—
100	Q₁₆	—	68	2	no	—
101	C₂×C₈	—	17	1	no	—
102	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	68	1	no	—
105	C₄⋊C₄	—	68	1	no	—
106	C₄⋊C₄	—	68	1	no	—
107	C₄²	—	17	1	no	—
108	C₄²	—	68	1	no	—
109	C₁₆	—	2	1	no	—
110	C₁₆	—	34	1	no	—
111	Q₈	—	4	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	34	2	no	—
114	Q₈	—	68	1	no	—
115	Q₈	—	68	1	no	—
116	Q₈	—	68	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	17	1	no	—
120	C₂×C₄	—	17	1	no	—
121	C₂×C₄	—	68	1	no	—
122	C₂×C₄	—	68	1	no	—
123	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
124	C₈	—	34	1	no	—
125	C₈	#63252	1	1	yes	characteristic
126	C₈	D₄×D₁₇	1	1	yes	characteristic
127	C₂²	D₈×D₁₇	1	1	yes	characteristic
128	C₄	—	8	1	no	—
129	C₄	—	8	1	no	—
130	C₄	—	17	2	no	—
131	C₄	—	34	1	no	—
132	C₄	—	68	2	no	—
133	C₄	—	136	1	no	—
134	C₄	#10587570	1	1	yes	characteristic
135	C₄	#10587600	1	1	yes	characteristic
136	C₂	#11763003	1	1	yes	characteristic
137	C₂	Q₃₂×D₁₇	1	1	yes	characteristic
138	C₂	#11763248	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₃₆

C₂

yes

maximal, characteristic

C₂×#10587500

C₂

yes

maximal, characteristic

#11763187

C₂

yes

maximal, characteristic

#11762903

C₂

yes

maximal, characteristic

#11762627

C₂

yes

maximal, characteristic

#11762612

C₂

yes

maximal, characteristic

C₁₇×#358

C₂

yes

maximal, characteristic

C₂×Q₁₆×C₁₇

C₂²

yes

characteristic

#10587630

—

C₂×#63231

—

C₂×Dic₆₈

C₂²

yes

characteristic

C₂×Dic₆₈

C₂²

yes

characteristic

#10587573

—

C₂×C₁₆×C₁₇

C₂²

yes

characteristic

C₁₇×#107

C₂²

yes

characteristic

#10587500

—

C₈×Dic₁₇

C₂²

yes

characteristic

C₂×#63214

C₂²

yes

characteristic

#10587481

—

Dic₁₃₆

—

C₂×Q₈×C₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

Q₁₆×C₁₇

—

Q₁₆×C₁₇

—

C₂×C₈×C₁₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₇

—

—

—

—

—

C₄×Dic₁₇

—

C₄×Dic₁₇

D₄

yes

characteristic

C₂×#3622

D₄

yes

characteristic

Dic₆₈

—

Dic₆₈

—

Dic₆₈

—

Dic₆₈

D₄

yes

—

C₁₆×C₁₇

—

#63214

—

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

D₈

yes

characteristic

C₂×Dic₁₇

D₈

yes

characteristic

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

C₂×D₄

yes

C₈×C₁₇

yes

—

—

C₂²×C₁₇

C₂×D₈

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

Q₃₂

yes

complemented

C₂×Q₃₂

—

C₂×Q₃₂

—

#494

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

#358

—

#358

D₁₇

yes

characteristic

C₂×C₁₇

#459

yes

characteristic

C₂×C₁₇

C₂×Q₃₂

yes

characteristic

C₂×C₁₇

#510

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₁₇

yes

characteristic

C₄⋊Q₈

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₁₇

yes

characteristic

#107

C₂×D₁₇

yes

characteristic

#103

—

C₄×C₈

—

C₁₇

#2209

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

100

Q₁₆

—

101

C₂×C₈

—

102

C₂×C₈

C₂²×D₁₇

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄²

—

108

C₄²

—

109

C₁₆

—

110

C₁₆

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

Q₈

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

D₄×D₁₇

yes

characteristic

124

C₈

—

125

C₈

#63252

yes

characteristic

126

C₈

D₄×D₁₇

yes

characteristic

127

C₂²

D₈×D₁₇

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

136

—

134

C₄

#10587570

yes

characteristic

135

C₄

#10587600

yes

characteristic

136

C₂

#11763003

yes

characteristic

137

C₂

Q₃₂×D₁₇

yes

characteristic

138

C₂

#11763248

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₇
commutator, cosocle	C₂×C₈×C₁₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₇
Fitting	C₁₇×#358	C₂
socle, prime radical	C₂²×C₁₇	C₂×D₈
hypercenter	#358	D₁₇
hypocenter	C₁₇	#2209
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₁₇

commutator, cosocle

C₂×C₈×C₁₇

C₂³

Frattini

C₂×C₈

C₂²×D₁₇

Fitting

C₁₇×#358

C₂

socle, prime radical

C₂²×C₁₇

C₂×D₈

D₁₇

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
66. Dic₁₇	Q₃₂	semidirect
90. C₁₇	#2209	semidirect

H ≅ G/N

type

66. Dic₁₇

Q₃₂

semidirect

90. C₁₇

#2209

semidirect

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

subgroup	index
C₂×Dic₁₃₆	2
C₂×#10587500	2
#11763187	2
#11762903	2
#11762627	2
#11762612	2
C₁₇×#358	2
#2209	17

subgroup

index

C₂×Dic₁₃₆

C₁₇×#358

Subgroups — #14530833

Conjugacy classes of subgroups

Complements

Maximal subgroups