Group #14544409 — subgroups

Subgroups — #14544409

Order 2176.

152 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 56 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	#11763860	C₂	1	1	yes	maximal, characteristic
2	Q₈×#3624	C₂	1	1	yes	maximal, characteristic
3	#11763671	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊#3624	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊#3624	C₂	1	1	yes	maximal, characteristic
6	#11763651	C₂	1	1	yes	maximal, characteristic
7	Q₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₁₆×C₁₇	C₄	1	1	yes	characteristic
9	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
10	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
11	#10587544	—	2	1	no	—
12	#10587542	—	2	1	no	—
13	C₄⋊#3624	—	2	1	no	—
14	C₄⋊#3624	C₂²	1	1	yes	characteristic
15	C₄⋊#3624	C₂²	1	1	yes	characteristic
16	C₄×#3624	—	2	1	no	—
17	C₄×#3624	C₂²	1	1	yes	characteristic
18	C₄×#3624	C₂²	1	1	yes	characteristic
19	#10587506	C₄	1	1	yes	characteristic
20	#10587506	C₄	1	1	yes	characteristic
21	#10587489	C₄	1	1	yes	characteristic
22	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×#3624	—	2	1	no	—
26	C₂×#3624	—	2	1	no	—
27	C₂×#3624	—	2	1	no	—
28	C₂×#3624	D₄	1	1	yes	characteristic
29	C₂×#3624	D₄	1	1	yes	characteristic
30	Q₁₆×C₁₇	—	2	2	no	—
31	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₄⋊Dic₁₇	—	2	1	no	—
33	C₄⋊Dic₁₇	—	2	1	no	—
34	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
35	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₁₇	—	2	1	no	—
37	C₄×Dic₁₇	—	2	1	no	—
38	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
39	C₂×#3622	C₂×C₄	1	1	yes	characteristic
40	Q₈×C₁₇	—	2	1	no	—
41	Q₈×C₁₇	—	2	1	no	—
42	Q₈×C₁₇	—	2	1	no	—
43	Q₈×C₁₇	M₁₆	1	2	yes	—
44	C₂×C₄×C₁₇	—	2	1	no	—
45	C₂×C₄×C₁₇	—	2	1	no	—
46	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₁₇	—	2	1	no	—
48	C₂×Dic₁₇	—	2	1	no	—
49	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₁₇	#41	1	1	yes	characteristic
51	#3624	—	2	2	no	—
52	#3624	—	2	1	no	—
53	#3624	—	2	1	no	—
54	#3624	—	4	1	no	—
55	#3624	—	4	1	no	—
56	C₈×C₁₇	—	2	1	no	—
57	#3622	—	2	1	no	—
58	#1587	—	17	1	no	maximal, Sylow
59	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
60	C₄×C₁₇	—	2	2	no	—
61	C₄×C₁₇	—	4	1	no	—
62	C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
63	C₄×C₁₇	#131	1	1	yes	characteristic
64	Dic₁₇	—	2	1	no	—
65	Dic₁₇	—	4	1	no	—
66	Dic₁₇	—	4	1	no	—
67	Dic₁₇	#137	1	1	yes	characteristic
68	Dic₁₇	#137	1	1	yes	characteristic
69	#446	—	17	1	no	—
70	C₈×Q₈	—	17	1	no	—
71	C₄×Q₁₆	—	17	1	no	—
72	#334	—	17	1	no	—
73	Q₈⋊C₈	—	17	1	no	—
74	Q₈⋊C₈	—	17	1	no	—
75	#322	—	17	1	no	—
76	C₂×C₁₇	#435	1	1	yes	characteristic
77	C₂×C₁₇	#441	1	1	yes	characteristic
78	C₂×C₁₇	#443	1	1	yes	characteristic
79	C₂×Q₁₆	#595	1	1	yes	characteristic
80	C₄×Q₈	—	17	1	no	—
81	C₄×Q₈	—	17	1	no	—
82	#107	—	17	1	no	—
83	C₄⋊C₈	—	17	1	no	—
84	C₄⋊C₈	—	17	1	no	—
85	C₄⋊C₈	—	34	1	no	—
86	C₄⋊C₈	—	34	1	no	—
87	#103	—	17	1	no	—
88	#103	—	17	1	no	—
89	#97	—	34	1	no	—
90	C₄×C₈	—	17	1	no	—
91	C₄×C₈	—	17	1	no	—
92	C₄×C₈	—	17	1	no	—
93	C₄×C₈	—	34	1	no	—
94	C₁₇	#1587	1	1	yes	characteristic, complemented, Sylow
95	C₂×Q₈	C₂×#595	1	1	yes	characteristic
96	C₂×Q₈	C₂×#595	1	1	yes	characteristic
97	Q₁₆	—	2	2	no	—
98	C₂×C₈	—	17	1	no	—
99	C₂×C₈	—	17	1	no	—
100	C₂×C₈	—	17	1	no	—
101	C₂×C₈	—	34	1	no	—
102	C₂×C₈	—	34	1	no	—
103	C₂×C₈	—	34	1	no	—
104	C₂×C₈	C₂×#595	1	1	yes	characteristic
105	C₄⋊C₄	—	17	1	no	—
106	C₄⋊C₄	—	17	1	no	—
107	C₄⋊C₄	—	34	1	no	—
108	C₄⋊C₄	—	34	1	no	—
109	C₄²	—	17	1	no	—
110	C₄²	—	34	1	no	—
111	C₄²	—	34	1	no	—
112	Q₈	—	2	1	no	—
113	Q₈	—	2	1	no	—
114	Q₈	—	2	1	no	—
115	Q₈	#63247	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	17	1	no	—
119	C₂×C₄	—	17	1	no	—
120	C₂×C₄	—	34	1	no	—
121	C₂×C₄	—	34	1	no	—
122	C₂×C₄	C₂²×#595	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	34	1	no	—
125	C₈	—	34	1	no	—
126	C₈	—	34	2	no	—
127	C₈	—	34	1	no	—
128	C₈	—	68	1	no	—
129	C₈	—	68	1	no	—
130	C₂²	D₄×#595	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	4	1	no	—
133	C₄	—	17	1	no	—
134	C₄	—	17	1	no	—
135	C₄	—	34	1	no	—
136	C₄	—	68	1	no	—
137	C₄	—	68	1	no	—
138	C₄	#10587678	1	1	yes	characteristic
139	C₄	C₂×#63247	1	1	yes	characteristic
140	C₂	#11763840	1	1	yes	characteristic
141	C₂	#11763849	1	1	yes	characteristic
142	C₂	#11763853	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763860

C₂

yes

maximal, characteristic

Q₈×#3624

C₂

yes

maximal, characteristic

#11763671

C₂

yes

maximal, characteristic

Q₈⋊#3624

C₂

yes

maximal, characteristic

Q₈⋊#3624

C₂

yes

maximal, characteristic

#11763651

C₂

yes

maximal, characteristic

Q₁₆×Dic₁₇

C₂

yes

maximal, characteristic

C₂×Q₁₆×C₁₇

C₄

yes

characteristic

Q₈×Dic₁₇

C₂²

yes

characteristic

Q₈×Dic₁₇

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄⋊#3624

C₂²

yes

characteristic

C₄×#3624

—

C₄×#3624

C₂²

yes

characteristic

C₄×#3624

C₂²

yes

characteristic

#10587506

C₄

yes

characteristic

#10587506

C₄

yes

characteristic

#10587489

C₄

yes

characteristic

C₈×Dic₁₇

C₂²

yes

characteristic

C₂×Q₈×C₁₇

C₂×C₄

yes

characteristic

C₂×Q₈×C₁₇

C₂×C₄

yes

—

—

—

D₄

yes

characteristic

C₂×#3624

D₄

yes

characteristic

Q₁₆×C₁₇

—

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂×C₄

yes

characteristic

C₄⋊Dic₁₇

C₂×C₄

yes

characteristic

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

C₂³

yes

characteristic

C₂×#3622

C₂×C₄

yes

characteristic

Q₈×C₁₇

—

Q₈×C₁₇

—

Q₈×C₁₇

—

Q₈×C₁₇

M₁₆

yes

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂×D₄

yes

C₂×Dic₁₇

yes

—

—

—

—

—

C₈×C₁₇

—

#3622

—

#1587

—

maximal, Sylow

C₂²×C₁₇

C₄×D₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×M₁₆

yes

characteristic

C₄×C₁₇

#131

yes

characteristic

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

yes

Dic₁₇

yes

—

C₈×Q₈

—

C₄×Q₁₆

—

#334

—

Q₈⋊C₈

—

Q₈⋊C₈

—

#322

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

yes

C₄×Q₈

—

C₄×Q₈

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₁₇

#1587

yes

characteristic, complemented, Sylow

C₂×Q₈

yes

C₂×Q₈

yes

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₂×#595

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

#63247

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂²×#595

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₂²

D₄×#595

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#10587678

yes

characteristic

139

C₄

C₂×#63247

yes

characteristic

140

C₂

#11763840

yes

characteristic

141

C₂

#11763849

yes

characteristic

142

C₂

#11763853

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#595
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#595
Fitting	C₂×Q₁₆×C₁₇	C₄
socle, prime radical	C₂²×C₁₇	C₄×D₄
cosocle	C₄×Dic₁₇	C₂³
hypercenter	C₂×Q₁₆	#595
hypocenter	C₁₇	#1587
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂²

D₄×#595

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#595

Fitting

C₂×Q₁₆×C₁₇

C₄

socle, prime radical

C₂²×C₁₇

C₄×D₄

cosocle

C₄×Dic₁₇

C₂³

C₂×Q₁₆

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
94. C₁₇	#1587	semidirect

H ≅ G/N

type

94. C₁₇

#1587

semidirect

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

subgroup	index
#11763860	2
Q₈×#3624	2
#11763671	2
Q₈⋊#3624	2
#11763651	2
Q₁₆×Dic₁₇	2
#1587	17

subgroup

index

Q₁₆×Dic₁₇

Subgroups — #14544409

Conjugacy classes of subgroups

Complements

Maximal subgroups