Group #16946117 — subgroups

Subgroups — #16946117

Order 4352.

218 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 95 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	#14530686	C₂	1	2	yes	maximal
2	C₈×Dic₆₈	C₂	1	1	yes	maximal, characteristic
3	#14526762	C₂	1	2	yes	maximal
4	C₈×#63214	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1370	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	C₄×#63214	C₂²	1	1	yes	characteristic
10	C₄×#63214	C₂²	1	1	yes	characteristic
11	C₁₆×Dic₁₇	—	2	2	no	—
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	1	yes	characteristic
28	C₂×#63214	D₄	1	1	yes	characteristic
29	C₄⋊#3622	C₂×C₄	1	2	yes	—
30	C₂×Dic₃₄	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₁₇	—	2	1	no	—
32	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
36	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
37	C₄⋊Dic₁₇	C₂×C₈	1	2	yes	—
38	#63225	—	2	2	no	—
39	C₄×Dic₁₇	—	2	2	no	—
40	C₂×#3622	—	2	2	no	—
41	Dic₆₈	C₁₆	1	4	yes	complemented
42	C₁₆×C₁₇	—	4	2	no	—
43	#63214	—	2	2	no	—
44	#63214	—	2	1	no	—
45	#63214	Q₁₆	1	2	yes	complemented
46	C₁₆×Q₁₆	—	17	1	no	maximal, Sylow
47	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
49	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
50	C₂×Dic₁₇	—	2	2	no	—
51	C₂×Dic₁₇	—	2	2	no	—
52	Dic₃₄	—	2	2	no	—
53	Dic₃₄	C₂×C₁₆	1	4	yes	—
54	C₈×C₁₇	—	2	2	no	—
55	C₈×C₁₇	—	2	1	no	—
56	C₈×C₁₇	—	2	1	no	—
57	C₈×C₁₇	C₂×C₁₆	1	2	yes	—
58	C₈×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
59	C₈×C₁₇	#135	1	1	yes	characteristic
60	#3622	—	4	2	no	—
61	Q₈×C₁₆	—	17	2	no	—
62	C₈×Q₁₆	—	17	1	no	—
63	#1370	D₁₇	1	1	yes	characteristic
64	Q₈⋊C₁₆	—	17	2	no	—
65	C₈×C₁₆	—	17	1	no	—
66	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
67	C₄×C₁₇	—	2	1	no	—
68	C₄×C₁₇	C₄×Q₁₆	1	1	yes	characteristic
69	C₄×C₁₇	#443	1	1	yes	characteristic
70	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
71	C₄×C₁₇	#504	1	1	yes	characteristic
72	Dic₁₇	—	2	4	no	—
73	Dic₁₇	—	4	2	no	—
74	C₈×Q₈	—	17	2	no	—
75	C₄×Q₁₆	—	17	1	no	—
76	C₄⋊C₁₆	—	34	2	no	—
77	C₄⋊C₁₆	C₂×D₁₇	1	2	yes	—
78	C₄×C₁₆	—	17	1	no	—
79	C₄×C₁₆	—	17	1	no	—
80	C₄×C₁₆	—	34	2	no	—
81	#335	—	17	1	no	—
82	Q₈⋊C₈	—	17	2	no	—
83	C₈²	C₂×D₁₇	1	1	yes	characteristic
84	C₂×C₁₇	C₈×Q₁₆	1	1	yes	characteristic
85	C₂×C₁₇	D₄×C₁₆	1	1	yes	characteristic
86	C₂×C₁₇	#2170	1	1	yes	characteristic
87	C₂×Q₁₆	—	17	1	no	—
88	C₄×Q₈	—	17	2	no	—
89	C₂×C₁₆	—	2	2	no	—
90	C₂×C₁₆	—	17	1	no	—
91	C₂×C₁₆	—	17	1	no	—
92	C₂×C₁₆	—	34	1	no	—
93	#107	—	17	1	no	—
94	C₄⋊C₈	—	17	2	no	—
95	C₄⋊C₈	—	34	2	no	—
96	#103	—	17	2	no	—
97	C₄×C₈	—	34	2	no	—
98	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
99	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
100	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
101	C₁₇	C₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
102	C₂×Q₈	—	17	2	no	—
103	Q₁₆	—	17	4	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	34	2	no	—
106	C₂×C₈	C₈×D₁₇	1	1	yes	characteristic
107	C₂×C₈	C₈×D₁₇	1	1	yes	characteristic
108	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
109	C₂×C₈	#63256	1	1	yes	characteristic
110	C₄⋊C₄	—	17	2	no	—
111	C₄⋊C₄	—	34	2	no	—
112	C₄²	—	34	2	no	—
113	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
114	C₁₆	—	4	2	no	—
115	C₁₆	—	17	2	no	—
116	C₁₆	—	34	2	no	—
117	C₁₆	—	34	1	no	—
118	Q₈	—	17	4	no	—
119	Q₈	—	34	2	no	—
120	C₂×C₄	—	34	2	no	—
121	C₂×C₄	—	34	2	no	—
122	C₂×C₄	#10587579	1	1	yes	characteristic
123	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
124	C₂×C₄	#10587593	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	2	1	no	—
127	C₈	—	2	1	no	—
128	C₈	—	68	2	no	—
129	C₈	C₁₆×D₁₇	1	2	yes	—
130	C₈	#10587598	1	1	yes	characteristic
131	C₈	Q₁₆×D₁₇	1	1	yes	characteristic
132	C₂²	#11762874	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	34	4	no	—
135	C₄	—	68	2	no	—
136	C₄	#11762903	1	1	yes	characteristic
137	C₄	#11762925	1	1	yes	characteristic
138	C₄	C₂×C₁₆×D₁₇	1	1	yes	characteristic
139	C₄	#11763227	1	1	yes	characteristic
140	C₂	#14527590	1	1	yes	characteristic
141	C₂	#14530603	1	1	yes	characteristic
142	C₂	#14530620	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14530686

C₂

yes

maximal

C₈×Dic₆₈

C₂

yes

maximal, characteristic

#14526762

C₂

yes

maximal

C₈×#63214

C₂

yes

maximal, characteristic

C₁₇×#1370

C₂

yes

maximal, characteristic

C₈×Dic₃₄

C₂²

yes

—

C₄×Dic₆₈

C₄

yes

characteristic

#11762627

—

C₄×#63214

C₂²

yes

characteristic

C₄×#63214

C₂²

yes

characteristic

C₁₆×Dic₁₇

—

#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

characteristic

C₂×#63214

D₄

yes

characteristic

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

yes

complemented

C₁₆×C₁₇

—

#63214

—

#63214

—

#63214

Q₁₆

yes

complemented

C₁₆×Q₁₆

—

maximal, Sylow

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

C₂×C₁₆

yes

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₂×C₁₆

yes

—

C₈×C₁₇

C₂×Q₁₆

yes

C₈×C₁₇

yes

—

Q₈×C₁₆

—

C₈×Q₁₆

—

#1370

D₁₇

yes

characteristic

Q₈⋊C₁₆

—

C₈×C₁₆

—

C₂²×C₁₇

C₈×D₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

C₄×Q₁₆

yes

characteristic

C₄×C₁₇

#443

yes

characteristic

C₄×C₁₇

C₂²×C₁₆

yes

characteristic

C₄×C₁₇

#504

yes

characteristic

Dic₁₇

—

Dic₁₇

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₁₇

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₁₇

yes

characteristic

C₂×C₁₇

C₈×Q₁₆

yes

characteristic

C₂×C₁₇

D₄×C₁₆

yes

characteristic

C₂×C₁₇

#2170

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

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

100

C₄×C₈

C₂²×D₁₇

yes

characteristic

101

C₁₇

C₁₆×Q₁₆

yes

characteristic, complemented, Sylow

102

C₂×Q₈

—

103

Q₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₈×D₁₇

yes

characteristic

107

C₂×C₈

C₈×D₁₇

yes

characteristic

108

C₂×C₈

D₄×D₁₇

yes

characteristic

109

C₂×C₈

#63256

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

113

C₄²

C₂×C₄×D₁₇

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

117

C₁₆

—

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

#10587579

yes

characteristic

123

C₂×C₄

C₂×C₈×D₁₇

yes

characteristic

124

C₂×C₄

#10587593

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

C₁₆×D₁₇

yes

—

130

C₈

#10587598

yes

characteristic

131

C₈

Q₁₆×D₁₇

yes

characteristic

132

C₂²

#11762874

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#11762903

yes

characteristic

137

C₄

#11762925

yes

characteristic

138

C₄

C₂×C₁₆×D₁₇

yes

characteristic

139

C₄

#11763227

yes

characteristic

140

C₂

#14527590

yes

characteristic

141

C₂

#14530603

yes

characteristic

142

C₂

#14530620

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₇
commutator	C₄×C₁₇	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1370	C₂
socle, prime radical	C₂²×C₁₇	C₈×D₄
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	#1370	D₁₇
hypocenter	C₁₇	C₁₆×Q₁₆
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

subgroup

G/H

center

C₂×C₈

D₄×D₁₇

commutator

C₄×C₁₇

C₂²×C₁₆

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#1370

C₂

socle, prime radical

C₂²×C₁₇

C₈×D₄

cosocle

C₄×C₈×C₁₇

C₂³

hypercenter

#1370

D₁₇

hypocenter

C₁₇

C₁₆×Q₁₆

LCS

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

N	H ≅ G/N	type
41. Dic₆₈	C₁₆	semidirect
45. #63214	Q₁₆	semidirect
101. C₁₇	C₁₆×Q₁₆	semidirect

H ≅ G/N

type

41. Dic₆₈

C₁₆

semidirect

45. #63214

Q₁₆

semidirect

101. C₁₇

C₁₆×Q₁₆

semidirect

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

subgroup	index
#14530686	2
C₈×Dic₆₈	2
#14526762	2
C₈×#63214	2
C₁₇×#1370	2
C₁₆×Q₁₆	17

subgroup

index

#14530686

C₈×Dic₆₈

#14526762

C₈×#63214

C₁₇×#1370

C₁₆×Q₁₆

Subgroups — #16946117

Conjugacy classes of subgroups

Complements

Maximal subgroups