Group #16946174 — subgroups

Subgroups — #16946174

Order 4352.

200 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 85 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	#14530689	C₂	1	2	yes	maximal
2	C₈×Dic₆₈	C₂	1	1	yes	maximal, characteristic
3	#14526762	C₂	1	2	yes	maximal
4	#14526702	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1367	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₃₄	C₂²	1	2	yes	—
7	C₄×Dic₆₈	C₄	1	1	yes	characteristic
8	#11762627	—	2	2	no	—
9	#11762584	—	2	2	no	—
10	C₄×#63214	C₂²	1	1	yes	characteristic
11	C₄×#63214	C₂²	1	1	yes	characteristic
12	#11762562	C₄	1	1	yes	characteristic
13	#11762541	C₄	1	2	yes	—
14	C₁₇×C₄⋊C₁₆	C₂²	1	2	yes	—
15	C₈²×C₁₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₆₈	C₈	1	1	yes	characteristic
17	C₄×Dic₃₄	C₂×C₄	1	2	yes	—
18	C₂×C₁₆×C₁₇	—	2	2	no	—
19	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
22	#10587489	C₈	1	1	yes	characteristic
23	#10587487	C₈	1	2	yes	—
24	#10587485	—	2	2	no	—
25	C₈×Dic₁₇	—	2	2	no	—
26	C₂×#63214	—	2	1	no	—
27	C₂×#63214	D₄	1	2	yes	—
28	C₄⋊#3622	C₂×C₄	1	2	yes	—
29	C₂×Dic₃₄	C₂×C₈	1	2	yes	—
30	C₂×C₈×C₁₇	—	2	1	no	—
31	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
35	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
36	C₄⋊Dic₁₇	C₂×C₈	1	2	yes	—
37	#63225	—	2	2	no	—
38	C₄×Dic₁₇	—	2	2	no	—
39	C₂×#3622	—	2	2	no	—
40	Dic₆₈	—	2	2	no	—
41	C₁₆×C₁₇	—	4	2	no	—
42	#63214	—	2	2	no	—
43	#63214	—	4	1	no	—
44	#12061	—	17	1	no	maximal, Sylow
45	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
47	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
48	C₂×Dic₁₇	—	2	2	no	—
49	C₂×Dic₁₇	—	2	2	no	—
50	Dic₃₄	—	2	2	no	—
51	Dic₃₄	—	2	2	no	—
52	C₈×C₁₇	—	2	1	no	—
53	C₈×C₁₇	—	2	1	no	—
54	C₈×C₁₇	—	4	1	no	—
55	C₈×C₁₇	M₃₂	1	2	yes	—
56	C₈×C₁₇	#137	1	2	yes	—
57	#3622	—	4	2	no	—
58	#2183	—	17	2	no	—
59	C₈×Q₁₆	—	17	1	no	—
60	#1367	D₁₇	1	1	yes	characteristic
61	Q₈⋊C₁₆	—	17	2	no	—
62	#1312	—	17	1	no	—
63	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
64	C₄×C₁₇	—	2	1	no	—
65	C₄×C₁₇	#441	1	1	yes	characteristic
66	C₄×C₁₇	#444	1	1	yes	characteristic
67	C₄×C₁₇	C₂×M₃₂	1	1	yes	characteristic
68	C₄×C₁₇	#504	1	1	yes	characteristic
69	Dic₁₇	—	4	2	no	—
70	Dic₁₇	—	4	2	no	—
71	C₈×Q₈	—	17	2	no	—
72	C₄×Q₁₆	—	17	1	no	—
73	C₄⋊C₁₆	—	34	2	no	—
74	C₄⋊C₁₆	C₂×D₁₇	1	2	yes	—
75	#346	—	34	2	no	—
76	C₄×C₁₆	—	17	1	no	—
77	C₄×C₁₆	—	17	1	no	—
78	#335	—	17	1	no	—
79	Q₈⋊C₈	—	17	2	no	—
80	C₈²	C₂×D₁₇	1	1	yes	characteristic
81	C₂×C₁₇	#1579	1	1	yes	characteristic
82	C₂×C₁₇	#2169	1	1	yes	characteristic
83	C₂×C₁₇	#2171	1	1	yes	characteristic
84	C₂×Q₁₆	—	17	1	no	—
85	C₄×Q₈	—	17	2	no	—
86	C₂×C₁₆	—	2	2	no	—
87	C₂×C₁₆	—	17	2	no	—
88	C₂×C₁₆	—	34	1	no	—
89	#107	—	17	1	no	—
90	C₄⋊C₈	—	17	2	no	—
91	C₄⋊C₈	—	34	2	no	—
92	#103	—	17	2	no	—
93	C₄×C₈	—	34	2	no	—
94	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
95	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
96	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
97	C₁₇	#12061	1	1	yes	characteristic, complemented, Sylow
98	C₂×Q₈	—	17	2	no	—
99	Q₁₆	—	34	2	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	34	2	no	—
102	C₂×C₈	C₈×D₁₇	1	1	yes	characteristic
103	C₂×C₈	C₈×D₁₇	1	1	yes	characteristic
104	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
105	C₂×C₈	#63256	1	1	yes	characteristic
106	C₄⋊C₄	—	17	2	no	—
107	C₄⋊C₄	—	34	2	no	—
108	C₄²	—	34	2	no	—
109	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
110	C₁₆	—	4	2	no	—
111	C₁₆	—	34	2	no	—
112	C₁₆	—	68	1	no	—
113	Q₈	—	34	2	no	—
114	Q₈	—	34	2	no	—
115	C₂×C₄	—	34	2	no	—
116	C₂×C₄	—	34	2	no	—
117	C₂×C₄	#10587579	1	1	yes	characteristic
118	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
119	C₂×C₄	#10587593	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	2	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	68	2	no	—
124	C₈	#10587469	1	2	yes	—
125	C₈	#10587601	1	2	yes	—
126	C₂²	#11762874	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	68	2	no	—
129	C₄	—	68	2	no	—
130	C₄	#11762915	1	1	yes	characteristic
131	C₄	#11762930	1	1	yes	characteristic
132	C₄	C₂×#10587469	1	1	yes	characteristic
133	C₄	#11763227	1	1	yes	characteristic
134	C₂	#14527602	1	1	yes	characteristic
135	C₂	#14530615	1	1	yes	characteristic
136	C₂	#14530625	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14530689

C₂

yes

maximal

C₈×Dic₆₈

C₂

yes

maximal, characteristic

#14526762

C₂

yes

maximal

#14526702

C₂

yes

maximal, characteristic

C₁₇×#1367

C₂

yes

maximal, characteristic

C₈×Dic₃₄

C₂²

yes

—

C₄×Dic₆₈

C₄

yes

—

—

C₂²

yes

characteristic

C₄×#63214

C₂²

yes

characteristic

#11762562

C₄

yes

characteristic

#11762541

C₄

yes

—

C₁₇×C₄⋊C₁₆

C₂²

yes

—

C₈²×C₁₇

C₂²

yes

characteristic

C₂×Dic₆₈

C₈

yes

characteristic

C₄×Dic₃₄

C₂×C₄

yes

—

C₂×C₁₆×C₁₇

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂³

yes

characteristic

#10587489

C₈

yes

characteristic

#10587487

C₈

yes

—

#10587485

—

C₈×Dic₁₇

—

C₂×#63214

—

C₂×#63214

D₄

yes

—

C₄⋊#3622

C₂×C₄

yes

—

C₂×Dic₃₄

C₂×C₈

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

#41

yes

characteristic

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄⋊Dic₁₇

C₂×C₈

yes

—

#63225

—

C₄×Dic₁₇

—

C₂×#3622

—

Dic₆₈

—

C₁₆×C₁₇

—

—

—

—

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₇

#131

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

M₃₂

yes

—

C₈×C₁₇

#137

yes

—

#3622

—

#2183

—

C₈×Q₁₆

—

#1367

D₁₇

yes

characteristic

Q₈⋊C₁₆

—

#1312

—

C₂²×C₁₇

C₈×D₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

C₂×M₃₂

yes

characteristic

C₄×C₁₇

#504

yes

characteristic

Dic₁₇

—

Dic₁₇

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₁₇

yes

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

C₄×D₁₇

yes

characteristic

C₄×C₈

C₄×D₁₇

yes

characteristic

C₄×C₈

C₂²×D₁₇

yes

characteristic

C₁₇

#12061

yes

characteristic, complemented, Sylow

C₂×Q₈

—

Q₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₈×D₁₇

yes

characteristic

103

C₂×C₈

C₈×D₁₇

yes

characteristic

104

C₂×C₈

D₄×D₁₇

yes

characteristic

105

C₂×C₈

#63256

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

—

109

C₄²

C₂×C₄×D₁₇

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#10587579

yes

characteristic

118

C₂×C₄

C₂×C₈×D₁₇

yes

characteristic

119

C₂×C₄

#10587593

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

#10587469

yes

—

125

C₈

#10587601

yes

—

126

C₂²

#11762874

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#11762915

yes

characteristic

131

C₄

#11762930

yes

characteristic

132

C₄

C₂×#10587469

yes

characteristic

133

C₄

#11763227

yes

characteristic

134

C₂

#14527602

yes

characteristic

135

C₂

#14530615

yes

characteristic

136

C₂

#14530625

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1367	C₂
socle, prime radical	C₂²×C₁₇	C₈×D₄
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	#1367	D₁₇
hypocenter	C₁₇	#12061
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

subgroup

G/H

center

C₂×C₈

D₄×D₁₇

commutator

C₂×C₄×C₁₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#1367

C₂

socle, prime radical

C₂²×C₁₇

C₈×D₄

cosocle

C₄×C₈×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

N	H ≅ G/N	type
97. C₁₇	#12061	semidirect

H ≅ G/N

type

97. C₁₇

#12061

semidirect

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

subgroup	index
#14530689	2
C₈×Dic₆₈	2
#14526762	2
#14526702	2
C₁₇×#1367	2
#12061	17

subgroup

index

C₈×Dic₆₈

C₁₇×#1367

Subgroups — #16946174

Conjugacy classes of subgroups

Complements

Maximal subgroups