Group #16950721 — subgroups

Subgroups — #16950721

Order 4352.

234 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 111 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	#14528908	C₂	1	2	yes	maximal
2	#14526888	C₂	1	1	yes	maximal, characteristic
3	#14526886	C₂	1	1	yes	maximal, characteristic
4	#14526880	C₂	1	2	yes	maximal
5	C₁₇×#2108	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587490	C₂²	1	2	yes	—
7	#11762847	C₂²	1	2	yes	—
8	#11762580	—	2	2	no	—
9	#11762565	—	2	2	no	—
10	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
12	C₁₇×#405	C₂²	1	1	yes	characteristic
13	C₁₇×#346	C₄	1	2	yes	complemented
14	C₄×C₁₆×C₁₇	C₄	1	2	yes	complemented
15	C₂×M₁₆×C₁₇	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
17	C₁₇×#117	D₄	1	1	yes	characteristic
18	#10587612	—	2	2	no	—
19	C₂×#63223	—	2	2	no	—
20	C₁₇×M₃₂	—	2	2	no	—
21	C₂×C₁₆×C₁₇	—	2	2	no	—
22	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
25	C₁₇×#97	Q₈	1	2	yes	—
26	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
27	#10587490	—	2	2	no	—
28	#10587490	—	2	2	no	—
29	#10587489	—	2	2	no	—
30	#10587488	—	2	2	no	—
31	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
32	M₁₆×C₁₇	—	2	2	no	—
33	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
34	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈×C₁₇	#41	1	2	yes	—
37	C₂×C₈×C₁₇	#41	1	2	yes	—
38	C₄⋊C₄×C₁₇	—	2	1	no	—
39	C₂²⋊C₄×C₁₇	—	2	1	no	—
40	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
41	C₂²⋊Dic₁₇	—	4	2	no	—
42	C₄⋊Dic₁₇	—	4	2	no	—
43	C₄×Dic₁₇	—	4	2	no	—
44	#63223	—	4	2	no	—
45	#63223	—	4	2	no	—
46	C₂×#3622	—	4	2	no	—
47	C₁₆×C₁₇	—	2	1	no	—
48	C₁₆×C₁₇	—	2	1	no	—
49	C₁₆×C₁₇	—	2	2	no	—
50	#12893	—	17	1	no	maximal, Sylow
51	C₂³×C₁₇	#135	1	1	yes	characteristic
52	C₂×C₄×C₁₇	—	2	2	no	—
53	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₇	#124	1	2	yes	—
55	C₂×C₄×C₁₇	#135	1	2	yes	—
56	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
57	C₂×Dic₁₇	—	8	2	no	—
58	C₈×C₁₇	—	2	2	no	—
59	C₈×C₁₇	#117	1	1	yes	characteristic
60	C₈×C₁₇	#117	1	1	yes	characteristic
61	C₈×C₁₇	#125	1	2	yes	—
62	#3622	—	8	2	no	—
63	#2108	D₁₇	1	1	yes	characteristic
64	#1833	—	17	2	no	—
65	#1385	—	17	1	no	—
66	#1384	—	17	1	no	—
67	#1382	—	17	2	no	—
68	C₂²×C₁₇	—	4	1	no	—
69	C₂²×C₁₇	#427	1	1	yes	characteristic
70	C₂²×C₁₇	#488	1	2	yes	—
71	C₄×C₁₇	—	4	2	no	—
72	C₄×C₁₇	#382	1	1	yes	characteristic
73	C₄×C₁₇	#427	1	1	yes	characteristic
74	C₄×C₁₇	#488	1	2	yes	—
75	Dic₁₇	—	16	2	no	—
76	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
77	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
78	C₂×#108	—	17	2	no	—
79	#427	—	17	2	no	—
80	#405	C₂×D₁₇	1	1	yes	characteristic
81	#346	Dic₁₇	1	2	yes	complemented
82	C₄×C₁₆	Dic₁₇	1	2	yes	complemented
83	#344	—	34	2	no	—
84	H(3,ℤ/4)	—	34	2	no	—
85	C₂×C₁₇	—	2	2	no	—
86	C₂×C₁₇	—	2	1	no	—
87	C₂×C₁₇	#1835	1	1	yes	characteristic
88	C₂×M₁₆	—	34	2	no	—
89	C₂×M₁₆	D₆₈	1	1	yes	characteristic
90	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
91	#117	—	34	2	no	—
92	#117	D₆₈	1	1	yes	characteristic
93	M₃₂	—	2	2	no	—
94	C₂×C₁₆	—	2	2	no	—
95	C₂×C₁₆	C₂×Dic₁₇	1	1	yes	characteristic
96	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
97	C₂×C₁₆	C₂×Dic₁₇	1	1	yes	characteristic
98	#108	—	34	2	no	—
99	#108	—	34	2	no	—
100	#107	—	34	2	no	—
101	#106	—	34	2	no	—
102	#97	Dic₃₄	1	2	yes	—
103	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
104	C₁₇	#12893	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
106	M₁₆	—	2	2	no	—
107	M₁₆	—	68	2	no	—
108	M₁₆	—	68	2	no	—
109	C₂×C₈	—	68	2	no	—
110	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
111	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
112	C₂×C₈	#63252	1	2	yes	—
113	C₂×C₈	#63252	1	2	yes	—
114	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	68	2	no	—
117	C₂²⋊C₄	—	2	1	no	—
118	C₂²⋊C₄	—	68	2	no	—
119	C₄²	—	68	2	no	—
120	C₄²	C₄⋊Dic₁₇	1	2	yes	—
121	C₁₆	—	2	1	no	—
122	C₁₆	—	2	1	no	—
123	C₁₆	—	2	2	no	—
124	C₂³	#10587590	1	1	yes	characteristic
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	136	2	no	—
127	C₂×C₄	#10587561	1	2	yes	—
128	C₂×C₄	#10587590	1	2	yes	—
129	C₂×C₄	#10587590	1	1	yes	characteristic
130	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₈	—	136	2	no	—
133	C₈	#10587556	1	2	yes	—
134	C₈	#10587612	1	1	yes	characteristic
135	C₈	#10587612	1	1	yes	characteristic
136	C₂²	—	4	1	no	—
137	C₂²	#11762847	1	1	yes	characteristic
138	C₂²	#11763164	1	2	yes	—
139	C₄	—	4	2	no	—
140	C₄	—	272	2	no	—
141	C₄	#11762675	1	1	yes	characteristic
142	C₄	#11762847	1	1	yes	characteristic
143	C₄	#11763164	1	2	yes	—
144	C₂	—	2	1	no	—
145	C₂	—	2	2	no	—
146	C₂	#14528915	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14528908

C₂

yes

maximal

#14526888

C₂

yes

maximal, characteristic

#14526886

C₂

yes

maximal, characteristic

#14526880

C₂

yes

maximal

C₁₇×#2108

C₂

yes

maximal, characteristic

C₂×#10587490

C₂²

yes

—

#11762847

C₂²

yes

—

#11762580

—

#11762565

—

C₂×C₁₇×M₃₂

C₂²

yes

characteristic

C₂²×C₁₆×C₁₇

C₂²

yes

characteristic

C₁₇×#405

C₂²

yes

characteristic

C₁₇×#346

C₄

yes

complemented

C₄×C₁₆×C₁₇

C₄

yes

complemented

C₂×M₁₆×C₁₇

D₄

yes

characteristic

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₁₇×#117

D₄

yes

characteristic

#10587612

—

C₂×#63223

—

C₁₇×M₃₂

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₇

C₂×C₄

yes

—

C₁₇×#97

Q₈

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

—

—

—

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

M₁₆×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₇

#41

yes

—

C₂×C₈×C₁₇

#41

yes

—

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

C₄⋊C₄

yes

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄×Dic₁₇

—

#63223

—

#63223

—

C₂×#3622

—

C₁₆×C₁₇

—

C₁₆×C₁₇

—

C₁₆×C₁₇

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

#124

yes

—

C₂×C₄×C₁₇

#135

yes

—

C₂×C₄×C₁₇

#135

yes

characteristic

C₂×Dic₁₇

—

C₈×C₁₇

—

C₈×C₁₇

yes

C₈×C₁₇

yes

C₈×C₁₇

yes

—

#3622

—

#2108

D₁₇

yes

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

#427

yes

characteristic

C₂²×C₁₇

#488

yes

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

Dic₁₇

—

C₂×M₃₂

C₂×D₁₇

yes

characteristic

C₂²×C₁₆

C₂×D₁₇

yes

—

—

C₂×D₁₇

yes

characteristic

#346

Dic₁₇

yes

complemented

C₄×C₁₆

Dic₁₇

yes

complemented

#344

—

H(3,ℤ/4)

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#1835

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

D₆₈

yes

characteristic

C₂²×C₈

C₂²×D₁₇

yes

characteristic

#117

—

#117

D₆₈

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₁₇

yes

characteristic

C₂×C₁₆

C₂×Dic₁₇

yes

—

C₂×C₁₆

C₂×Dic₁₇

yes

characteristic

#108

—

#108

—

100

#107

—

101

#106

—

102

#97

Dic₃₄

yes

—

103

C₄×C₈

C₂×Dic₁₇

yes

—

104

C₁₇

#12893

yes

characteristic, complemented, Sylow

105

C₂²×C₄

C₂×D₆₈

yes

characteristic

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

C₂×C₈

—

110

C₂×C₈

C₄⋊Dic₁₇

yes

—

111

C₂×C₈

C₂×Dic₃₄

yes

characteristic

112

C₂×C₈

#63252

yes

—

113

C₂×C₈

#63252

yes

—

114

C₂×C₈

C₂²×Dic₁₇

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₂²⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₄²

C₄⋊Dic₁₇

yes

—

121

C₁₆

—

122

C₁₆

—

123

C₁₆

—

124

C₂³

#10587590

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

136

—

127

C₂×C₄

#10587561

yes

—

128

C₂×C₄

#10587590

yes

—

129

C₂×C₄

#10587590

yes

characteristic

130

C₂×C₄

C₂×C₄⋊Dic₁₇

yes

characteristic

131

C₈

—

132

C₈

—

136

—

133

C₈

#10587556

yes

—

134

C₈

#10587612

yes

characteristic

135

C₈

#10587612

yes

characteristic

136

C₂²

—

137

C₂²

#11762847

yes

characteristic

138

C₂²

#11763164

yes

—

139

C₄

—

140

C₄

—

272

—

141

C₄

#11762675

yes

characteristic

142

C₄

#11762847

yes

characteristic

143

C₄

#11763164

yes

—

144

C₂

—

145

C₂

—

146

C₂

#14528915

yes

characteristic

	subgroup	G/H
center	C₄	#11762847
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#2108	C₂
socle	C₂×C₁₇	#1835
prime radical	C₂³×C₁₇	#135
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#2108	D₁₇
hypocenter	C₁₇	#12893
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ #117 ◁ #405 ◁ #2108

subgroup

G/H

center

C₄

#11762847

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₁₇×#2108

C₂

C₂×C₁₇

C₂³×C₁₇

C₂²×C₈×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ #117 ◁ #405 ◁ #2108

N	H ≅ G/N	type
13. C₁₇×#346	C₄	semidirect
14. C₄×C₁₆×C₁₇	C₄	semidirect
81. #346	Dic₁₇	semidirect
82. C₄×C₁₆	Dic₁₇	semidirect
104. C₁₇	#12893	semidirect

H ≅ G/N

type

13. C₁₇×#346

C₄

semidirect

14. C₄×C₁₆×C₁₇

C₄

semidirect

81. #346

Dic₁₇

semidirect

82. C₄×C₁₆

Dic₁₇

semidirect

104. C₁₇

#12893

semidirect

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

subgroup	index
#14528908	2
#14526888	2
#14526886	2
#14526880	2
C₁₇×#2108	2
#12893	17

subgroup

index

C₁₇×#2108

Subgroups — #16950721

Conjugacy classes of subgroups

Complements

Maximal subgroups