Group #16928137 — subgroups

Subgroups — #16928137

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	#14527543	C₂	1	1	yes	maximal, characteristic
3	#14527542	C₂	1	2	yes	maximal
4	C₂×#11762531	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1567	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	#11762562	—	2	2	no	—
11	#11762561	—	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×#335	—	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	D₈	1	4	yes	—
50	C₂×#3622	Q₁₆	1	4	yes	—
51	#8826	—	17	1	no	maximal, Sylow
52	C₂³×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
53	C₂×C₄×C₁₇	—	2	2	no	—
54	C₂×C₄×C₁₇	—	2	2	no	—
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	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
59	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
62	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
63	C₈×C₁₇	—	2	4	no	—
64	C₈×C₁₇	—	8	2	no	—
65	#3622	—	2	4	no	—
66	#3622	—	8	2	no	—
67	#3622	#107	1	8	yes	—
68	#1848	—	17	2	no	—
69	#1567	—	17	2	no	—
70	#1567	—	17	1	no	—
71	#1567	D₁₇	1	1	yes	characteristic
72	C₂×#322	—	17	1	no	—
73	C₂²×C₁₇	—	2	1	no	—
74	C₂²×C₁₇	—	2	1	no	—
75	C₂²×C₁₇	#384	1	1	yes	characteristic
76	C₂²×C₁₇	#500	1	2	yes	—
77	C₄×C₁₇	—	2	1	no	—
78	C₄×C₁₇	—	2	1	no	—
79	C₄×C₁₇	C₂×#107	1	2	yes	—
80	C₄×C₁₇	#430	1	2	yes	—
81	C₄×C₁₇	#430	1	2	yes	—
82	C₄×C₁₇	#493	1	1	yes	characteristic
83	C₄×C₁₇	#494	1	1	yes	characteristic
84	C₂×#108	—	34	4	no	—
85	#423	—	17	2	no	—
86	#423	C₂×D₁₇	1	2	yes	—
87	C₂×C₄×C₈	—	17	1	no	—
88	C₂×C₄×C₈	—	17	1	no	—
89	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
90	#335	—	2	2	no	—
91	#335	—	34	4	no	—
92	#335	—	34	2	no	—
93	#322	—	17	4	no	—
94	C₂×C₁₇	—	2	2	no	—
95	C₂×C₁₇	#1845	1	1	yes	characteristic
96	C₂×C₁₇	#1849	1	1	yes	characteristic
97	C₂×C₁₇	#1849	1	1	yes	characteristic
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	34	2	no	—
100	C₂²×C₈	—	17	2	no	—
101	C₂²×C₈	—	17	2	no	—
102	C₂²×C₈	Dic₃₄	1	2	yes	—
103	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
104	#108	—	68	8	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	34	2	no	—
108	C₄⋊C₈	—	34	2	no	—
109	C₄×C₈	—	17	2	no	—
110	C₄×C₈	—	17	2	no	—
111	C₄×C₈	—	17	2	no	—
112	C₄×C₈	—	17	2	no	—
113	C₄×C₈	C₄×D₁₇	1	2	yes	—
114	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
115	C₁₇	#8826	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
117	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
118	C₂²×C₄	Q₈×D₁₇	1	1	yes	characteristic
119	M₁₆	—	4	4	no	—
120	M₁₆	—	68	4	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	4	2	no	—
123	C₂×C₈	—	17	4	no	—
124	C₂×C₈	—	17	4	no	—
125	C₂×C₈	—	17	4	no	—
126	C₂×C₈	—	17	4	no	—
127	C₂×C₈	—	68	2	no	—
128	C₂×C₈	—	68	2	no	—
129	C₂×C₈	#63225	1	4	yes	—
130	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
131	C₄²	D₄×D₁₇	1	1	yes	characteristic
132	C₄²	Q₈×D₁₇	1	1	yes	characteristic
133	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
134	C₂³	#10587574	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	#10587574	1	1	yes	characteristic
138	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
139	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
140	C₂×C₄	D₈×D₁₇	1	2	yes	—
141	C₂×C₄	Q₁₆×D₁₇	1	2	yes	—
142	C₂×C₄	C₂×#63225	1	1	yes	characteristic
143	C₂×C₄	#10587626	1	1	yes	characteristic
144	C₂×C₄	#10587630	1	1	yes	characteristic
145	C₈	—	2	4	no	—
146	C₈	—	8	2	no	—
147	C₈	—	17	8	no	—
148	C₈	—	34	4	no	—
149	C₈	—	136	2	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#11762678	1	1	yes	characteristic
153	C₂²	#11763213	1	2	yes	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	#107×D₁₇	1	2	yes	—
157	C₄	#11762856	1	2	yes	—
158	C₄	#11762857	1	2	yes	—
159	C₄	#11763184	1	1	yes	characteristic
160	C₄	#11763187	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	#14528942	1	1	yes	characteristic
163	C₂	#14528957	1	1	yes	characteristic
164	C₂	#14528958	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14528955

C₂

yes

maximal

#14527543

C₂

yes

maximal, characteristic

#14527542

C₂

yes

maximal

C₂×#11762531

C₂

yes

maximal, characteristic

C₁₇×#1567

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×#335

—

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

D₈

yes

—

C₂×#3622

Q₁₆

yes

—

#8826

—

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×D₈

yes

—

C₂×C₄×C₁₇

C₂×Q₁₆

yes

—

C₈×C₁₇

—

C₈×C₁₇

—

—

—

yes

—

—

—

—

D₁₇

yes

characteristic

C₂×#322

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#384

yes

characteristic

C₂²×C₁₇

#500

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×#107

yes

—

C₄×C₁₇

#430

yes

—

C₄×C₁₇

#430

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

—

—

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

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

Dic₃₄

yes

—

103

C₂×C₄²

C₂²×D₁₇

yes

characteristic

104

#108

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

C₄×D₁₇

yes

—

114

C₄×C₈

C₂²⋊D₁₇

yes

—

115

C₁₇

#8826

yes

characteristic, complemented, Sylow

116

C₂²×C₄

C₂×Dic₃₄

yes

characteristic

117

C₂²×C₄

D₄×D₁₇

yes

characteristic

118

C₂²×C₄

Q₈×D₁₇

yes

characteristic

119

M₁₆

—

120

M₁₆

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

#63225

yes

—

130

C₄²

C₂×C₄×D₁₇

yes

characteristic

131

C₄²

D₄×D₁₇

yes

characteristic

132

C₄²

Q₈×D₁₇

yes

characteristic

133

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

134

C₂³

#10587574

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#10587574

yes

characteristic

138

C₂×C₄

C₄⋊C₄×D₁₇

yes

characteristic

139

C₂×C₄

C₄⋊C₄×D₁₇

yes

characteristic

140

C₂×C₄

D₈×D₁₇

yes

—

141

C₂×C₄

Q₁₆×D₁₇

yes

—

142

C₂×C₄

C₂×#63225

yes

characteristic

143

C₂×C₄

#10587626

yes

characteristic

144

C₂×C₄

#10587630

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

136

—

150

C₂²

—

151

C₂²

—

152

C₂²

#11762678

yes

characteristic

153

C₂²

#11763213

yes

—

154

C₄

—

155

C₄

—

156

C₄

#107×D₁₇

yes

—

157

C₄

#11762856

yes

—

158

C₄

#11762857

yes

—

159

C₄

#11763184

yes

characteristic

160

C₄

#11763187

yes

characteristic

161

C₂

—

162

C₂

#14528942

yes

characteristic

163

C₂

#14528957

yes

characteristic

164

C₂

#14528958

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10587626
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1567	C₂
socle	C₂²×C₁₇	#384
prime radical	C₂³×C₁₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1567	D₁₇
hypocenter	C₁₇	#8826
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

subgroup

G/H

center

C₂×C₄

#10587626

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1567

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 ◁ #1567

N	H ≅ G/N	type
115. C₁₇	#8826	semidirect

H ≅ G/N

type

115. C₁₇

#8826

semidirect

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

subgroup	index
#14528955	2
#14527543	2
#14527542	2
C₂×#11762531	2
C₁₇×#1567	2
#8826	17

subgroup

index

C₁₇×#1567

Subgroups — #16928137

Conjugacy classes of subgroups

Complements

Maximal subgroups