Group #16928134 — subgroups

Subgroups — #16928134

Order 4352.

338 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 147 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	#14528955	C₂	1	2	yes	maximal
2	#14527540	C₂	1	1	yes	maximal, characteristic
3	#14527539	C₂	1	2	yes	maximal
4	C₂×#11762531	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1566	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587493	—	2	4	no	—
7	#11762833	C₂²	1	2	yes	—
8	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
9	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
10	#11762560	—	2	2	no	—
11	#11762559	—	2	4	no	—
12	#11762531	C₄	1	4	yes	—
13	C₁₇×#423	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₁₇×#334	—	2	2	no	—
16	C₂×M₁₆×C₁₇	—	2	2	no	—
17	C₂²×C₈×C₁₇	Q₈	1	2	yes	—
18	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
19	C₂×#63223	—	2	2	no	—
20	C₂²×#3622	D₄	1	2	yes	—
21	C₂²×#3622	Q₈	1	2	yes	—
22	C₁₇×C₄⋊C₈	—	2	2	no	—
23	C₁₇×C₄⋊C₈	—	2	2	no	—
24	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₁₇	D₄	1	2	yes	—
26	#10587493	—	4	8	no	—
27	C₄⋊#3622	—	2	2	no	—
28	C₄⋊#3622	—	2	2	no	—
29	C₄×#3622	C₂×C₄	1	2	yes	—
30	C₄×#3622	C₂×C₄	1	2	yes	—
31	C₄×#3622	D₄	1	2	yes	—
32	C₄×#3622	Q₈	1	2	yes	—
33	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
36	M₁₆×C₁₇	—	4	4	no	—
37	C₂×C₈×C₁₇	—	4	2	no	—
38	C₂×C₈×C₁₇	—	4	2	no	—
39	C₂×C₈×C₁₇	C₄⋊C₄	1	4	yes	—
40	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
43	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
44	#63223	—	4	4	no	—
45	C₂×#3622	—	4	2	no	—
46	C₂×#3622	—	4	2	no	—
47	C₂×#3622	C₄⋊C₄	1	4	yes	—
48	C₂×#3622	C₄⋊C₄	1	4	yes	—
49	C₂×#3622	SD₁₆	1	8	yes	—
50	#8825	—	17	1	no	maximal, Sylow
51	C₂³×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₄×C₁₇	—	2	2	no	—
53	C₂×C₄×C₁₇	—	2	2	no	—
54	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
58	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₁₇	C₂×SD₁₆	1	4	yes	—
61	C₈×C₁₇	—	2	4	no	—
62	C₈×C₁₇	—	8	2	no	—
63	#3622	—	2	4	no	—
64	#3622	—	8	2	no	—
65	#3622	#106	1	8	yes	—
66	#1848	—	17	2	no	—
67	#1566	—	17	2	no	—
68	#1566	—	17	1	no	—
69	#1566	D₁₇	1	1	yes	characteristic
70	C₂×#322	—	17	1	no	—
71	C₂²×C₁₇	—	2	1	no	—
72	C₂²×C₁₇	—	2	1	no	—
73	C₂²×C₁₇	#384	1	1	yes	characteristic
74	C₂²×C₁₇	#498	1	2	yes	—
75	C₄×C₁₇	—	2	1	no	—
76	C₄×C₁₇	—	2	1	no	—
77	C₄×C₁₇	C₂×#106	1	2	yes	—
78	C₄×C₁₇	#430	1	2	yes	—
79	C₄×C₁₇	#430	1	2	yes	—
80	C₄×C₁₇	#492	1	2	yes	—
81	C₂×#108	—	34	4	no	—
82	#423	—	17	2	no	—
83	#423	C₂×D₁₇	1	2	yes	—
84	C₂×C₄×C₈	—	17	1	no	—
85	C₂×C₄×C₈	—	17	1	no	—
86	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
87	#334	—	2	2	no	—
88	#334	—	34	4	no	—
89	#334	—	34	2	no	—
90	#322	—	17	4	no	—
91	C₂×C₁₇	—	2	2	no	—
92	C₂×C₁₇	#1844	1	1	yes	characteristic
93	C₂×C₁₇	#1849	1	1	yes	characteristic
94	C₂×C₁₇	#1849	1	1	yes	characteristic
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	34	2	no	—
97	C₂²×C₈	—	17	2	no	—
98	C₂²×C₈	—	17	2	no	—
99	C₂²×C₈	Dic₃₄	1	2	yes	—
100	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
101	#108	—	68	8	no	—
102	C₄⋊C₈	—	2	2	no	—
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	34	2	no	—
105	C₄⋊C₈	—	34	2	no	—
106	C₄×C₈	—	17	2	no	—
107	C₄×C₈	—	17	2	no	—
108	C₄×C₈	—	17	2	no	—
109	C₄×C₈	—	17	2	no	—
110	C₄×C₈	C₄×D₁₇	1	2	yes	—
111	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
112	C₁₇	#8825	1	1	yes	characteristic, complemented, Sylow
113	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
114	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
115	C₂²×C₄	Q₈×D₁₇	1	1	yes	characteristic
116	M₁₆	—	4	4	no	—
117	M₁₆	—	68	4	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	17	4	no	—
121	C₂×C₈	—	17	8	no	—
122	C₂×C₈	—	17	4	no	—
123	C₂×C₈	—	68	2	no	—
124	C₂×C₈	—	68	2	no	—
125	C₂×C₈	#63225	1	4	yes	—
126	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
127	C₄²	D₄×D₁₇	1	1	yes	characteristic
128	C₄²	Q₈×D₁₇	1	1	yes	characteristic
129	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
130	C₂³	#10587574	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	#10587574	1	1	yes	characteristic
134	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
135	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
136	C₂×C₄	SD₁₆×D₁₇	1	4	yes	—
137	C₂×C₄	C₂×#63225	1	1	yes	characteristic
138	C₂×C₄	#10587626	1	1	yes	characteristic
139	C₂×C₄	#10587630	1	1	yes	characteristic
140	C₈	—	2	4	no	—
141	C₈	—	8	2	no	—
142	C₈	—	17	8	no	—
143	C₈	—	34	4	no	—
144	C₈	—	136	2	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#11762678	1	1	yes	characteristic
148	C₂²	#11763206	1	2	yes	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	#106×D₁₇	1	2	yes	—
152	C₄	#11762856	1	2	yes	—
153	C₄	#11762857	1	2	yes	—
154	C₄	#11763180	1	2	yes	—
155	C₂	—	2	2	no	—
156	C₂	#14528939	1	1	yes	characteristic
157	C₂	#14528957	1	1	yes	characteristic
158	C₂	#14528958	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14528955

C₂

yes

maximal

#14527540

C₂

yes

maximal, characteristic

#14527539

C₂

yes

maximal

C₂×#11762531

C₂

yes

maximal, characteristic

C₁₇×#1566

C₂

yes

maximal, characteristic

C₂×#10587493

—

#11762833

C₂²

yes

—

C₂×C₄×#3622

C₂²

yes

characteristic

C₂×C₄×#3622

C₂²

yes

—

—

C₄

yes

—

C₁₇×#423

C₂²

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#334

—

C₂×M₁₆×C₁₇

—

C₂²×C₈×C₁₇

Q₈

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×#63223

—

C₂²×#3622

D₄

yes

—

C₂²×#3622

Q₈

yes

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

D₄

yes

—

—

—

—

C₂×C₄

yes

—

C₄×#3622

C₂×C₄

yes

—

C₄×#3622

D₄

yes

—

C₄×#3622

Q₈

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₇

C₂×Q₈

yes

characteristic

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

C₂×Q₈

yes

—

—

—

C₄⋊C₄

yes

—

C₂×#3622

C₄⋊C₄

yes

—

C₂×#3622

SD₁₆

yes

—

#8825

—

maximal, Sylow

C₂³×C₁₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂×SD₁₆

yes

—

C₈×C₁₇

—

C₈×C₁₇

—

—

—

yes

—

—

—

—

D₁₇

yes

characteristic

C₂×#322

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#384

yes

characteristic

C₂²×C₁₇

#498

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×#106

yes

—

C₄×C₁₇

#430

yes

—

C₄×C₁₇

#430

yes

—

C₄×C₁₇

#492

yes

—

C₂×#108

—

#423

—

#423

C₂×D₁₇

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₇

yes

—

—

—

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

Dic₃₄

yes

—

100

C₂×C₄²

C₂²×D₁₇

yes

characteristic

101

#108

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

C₄×D₁₇

yes

—

111

C₄×C₈

C₂²⋊D₁₇

yes

—

112

C₁₇

#8825

yes

characteristic, complemented, Sylow

113

C₂²×C₄

C₂×Dic₃₄

yes

characteristic

114

C₂²×C₄

D₄×D₁₇

yes

characteristic

115

C₂²×C₄

Q₈×D₁₇

yes

characteristic

116

M₁₆

—

117

M₁₆

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

#63225

yes

—

126

C₄²

C₂×C₄×D₁₇

yes

characteristic

127

C₄²

D₄×D₁₇

yes

characteristic

128

C₄²

Q₈×D₁₇

yes

characteristic

129

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

130

C₂³

#10587574

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

#10587574

yes

characteristic

134

C₂×C₄

C₄⋊C₄×D₁₇

yes

characteristic

135

C₂×C₄

C₄⋊C₄×D₁₇

yes

characteristic

136

C₂×C₄

SD₁₆×D₁₇

yes

—

137

C₂×C₄

C₂×#63225

yes

characteristic

138

C₂×C₄

#10587626

yes

characteristic

139

C₂×C₄

#10587630

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

136

—

145

C₂²

—

146

C₂²

—

147

C₂²

#11762678

yes

characteristic

148

C₂²

#11763206

yes

—

149

C₄

—

150

C₄

—

151

C₄

#106×D₁₇

yes

—

152

C₄

#11762856

yes

—

153

C₄

#11762857

yes

—

154

C₄

#11763180

yes

—

155

C₂

—

156

C₂

#14528939

yes

characteristic

157

C₂

#14528957

yes

characteristic

158

C₂

#14528958

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10587626
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1566	C₂
socle	C₂²×C₁₇	#384
prime radical	C₂³×C₁₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1566	D₁₇
hypocenter	C₁₇	#8825
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#10587626

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1566

C₂

socle

C₂²×C₁₇

#384

prime radical

C₂³×C₁₇

C₄⋊Q₈

cosocle

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
112. C₁₇	#8825	semidirect

H ≅ G/N

type

112. C₁₇

#8825

semidirect

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

subgroup	index
#14528955	2
#14527540	2
#14527539	2
C₂×#11762531	2
C₁₇×#1566	2
#8825	17

subgroup

index

C₁₇×#1566

Subgroups — #16928134

Conjugacy classes of subgroups

Complements

Maximal subgroups