Finite Group Databasebeta

← #16947378

Subgroups — #16947378

Order 4352.

442 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533813C211yesmaximal, characteristic
2#14527758C212yesmaximal
3#14526813C212yesmaximal
4#14526730C211yesmaximal, characteristic
5C17×#2060C211yesmaximal, characteristic
6C2×#10587574—22no—
7#11762964—22no—
8#11762729—21no—
9#11762729C2212yes—
10#11762705—22no—
11C4×C4⋊Dic17—21no—
12#11762546—22no—
13#11762535C2212yes—
14C17×#544C2211yescharacteristic
15C17×#428—21no—
16C17×#384C2211yescharacteristic
17C17×#343C2211yescharacteristic
18C17×#339—21no—
19C22×Dic34—21no—
20C22×Dic34D412yes—
21C2×M16×C17—21no—
22C17×C4⋊Q8—41no—
23C17×#125—41no—
24C17×C22⋊Q8—22no—
25C17×#117—21no—
26C2×C4⋊C4×C17—22no—
27C2×C4⋊C4×C17D411yescharacteristic
28C2×C4⋊C4×C17C2311yescharacteristic
29C2×C42×C17D411yescharacteristic
30C2×#63231—42no—
31C2×C4⋊Dic17—21no—
32C2×C4⋊Dic17—21no—
33C2×#63225—22no—
34C2×#63225—22no—
35C2×C4×Dic17—22no—
36#10587574—48no—
37C17×#107—41no—
38C17×#106—41no—
39#10587502—21no—
40#10587502—22no—
41#10587502D412yes—
42C22⋊#3622—22no—
43#10587481—42no—
44C2×Q8×C17—22no—
45C22×C4×C17—21no—
46C22×C4×C17—21no—
47C22×C4×C17C2×D412yes—
48C22×C4×C17C2×D411yescharacteristic
49C22×Dic17—22no—
50C22×Dic17—22no—
51C2×Dic34—42no—
52C2×Dic34—44no—
53C2×Dic34—44no—
54C2×Dic34—81no—
55M16×C17—22no—
56M16×C17—41no—
57C2×C8×C17—41no—
58C4⋊C4×C17—22no—
59C4⋊C4×C17—21no—
60C4⋊C4×C17—21no—
61C4⋊C4×C17—41no—
62C4⋊C4×C17—41no—
63C4⋊C4×C17—42no—
64C4⋊C4×C17—44no—
65C22⋊C4×C17—41no—
66C42×C17—21no—
67C42×C17—41no—
68C42×C17Q1612yescomplemented
69#63231—84no—
70C4⋊Dic17—24no—
71C4⋊Dic17—24no—
72#63225—44no—
73#63225—44no—
74C4×Dic17—44no—
75C2×#3622—42no—
76#12256—171nomaximal, Sylow
77C23×C17#12011yescharacteristic
78Q8×C17—82no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—21no—
83C2×C4×C17—21no—
84C2×C4×C17—22no—
85C2×C4×C17—42no—
86C2×C4×C17—42no—
87C2×C4×C17—41no—
88C2×C4×C17—41no—
89C2×C4×C17—41no—
90C2×C4×C17C2×Q1611yescharacteristic
91C2×C4×C17#13711yescharacteristic
92C2×Dic17—24no—
93C2×Dic17—24no—
94C2×Dic17—42no—
95C2×Dic17—42no—
96C2×Dic17—42no—
97C2×Dic17—42no—
98Dic34—44no—
99Dic34—82no—
100Dic34—84no—
101C8×C17—42no—
102#3622—82no—
103#2551—171no—
104#2060D1711yescharacteristic
105#1605—172no—
106#1354—172no—
107#1325—171no—
108C22×C17—21no—
109C22×C17—21no—
110C22×C17#45111yescharacteristic
111C22×C17#45611yescharacteristic
112C22×C17B(4,𝔽2)11yescharacteristic
113C4×C17—24no—
114C4×C17—21no—
115C4×C17—21no—
116C4×C17—42no—
117C4×C17—81no—
118C4×C17—81no—
119C4×C17—81no—
120Dic17—44no—
121Dic17—44no—
122#544C2×D1711yescharacteristic
123C2×C4⋊Q8—342no—
124#451—342no—
125#428—21no—
126#395—172no—
127#395—341no—
128#389—342no—
129#384C2×D1711yescharacteristic
130C4×C4⋊C4—341no—
131#343C2×D1711yescharacteristic
132#339—21no—
133#328—342no—
134#324—172no—
135C2×C17—21no—
136C2×C17—21no—
137C2×C17#160511yescharacteristic
138C2×C17#219611yescharacteristic
139C2×C17#220511yescharacteristic
140C22×Q8—172no—
141C22×Q8—341no—
142C2×Q16—682no—
143C2×M16—21no—
144C4⋊Q8—41no—
145C4⋊Q8—688no—
146#125—41no—
147C22⋊Q8—22no—
148#117—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—341no—
151C2×C4⋊C4—341no—
152C2×C4⋊C4—342no—
153C2×C4⋊C4—342no—
154C2×C4⋊C4C22⋊D1711yescharacteristic
155C2×C4⋊C4C22×D1711yescharacteristic
156C2×C42—342no—
157C2×C42C22⋊D1711yescharacteristic
158#107—41no—
159#106—41no—
160#103—682no—
161C22⋊C8—342no—
162#95—172no—
163#95—341no—
164#95—342no—
165C17#1225611yescharacteristic, complemented, Sylow
166C2×Q8—22no—
167C2×Q8—682no—
168C2×Q8—684no—
169C2×Q8—684no—
170C2×Q8—1361no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—342no—
174C22×C4—342no—
175C22×C4D4×D1712yes—
176C22×C4C2×C22⋊D1711yescharacteristic
177Q16—1364no—
178M16—22no—
179M16—41no—
180C2×C8—41no—
181C2×C8—682no—
182C4⋊C4—22no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—44no—
188C4⋊C4—42no—
189C4⋊C4—344no—
190C4⋊C4—344no—
191C4⋊C4—684no—
192C4⋊C4—684no—
193C22⋊C4—41no—
194C42—21no—
195C42—41no—
196C42—684no—
197C42#6323112yescomplemented
198C23#1058762311yescharacteristic
199Q8—82no—
200Q8—684no—
201Q8—1362no—
202Q8—1364no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—344no—
215C2×C4—344no—
216C2×C4—682no—
217C2×C4—682no—
218C2×C4—682no—
219C2×C4—682no—
220C2×C4C2×#6323111yescharacteristic
221C2×C4#1058763711yescharacteristic
222C8—42no—
223C8—1362no—
224C22—21no—
225C22—21no—
226C22#1176296411yescharacteristic
227C22#1176299111yescharacteristic
228C22#1176299511yescharacteristic
229C4—24no—
230C4—21no—
231C4—21no—
232C4—42no—
233C4—81no—
234C4—81no—
235C4—81no—
236C4—684no—
237C4—684no—
238C2—21no—
239C2—21no—
240C2#1452775711yescharacteristic
241C2#1453076711yescharacteristic
242C2#1453081311yescharacteristic
subgroupG/H
centerC22#11762995
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2060C2
socleC22×C17B(4,𝔽2)
prime radicalC23×C17#120
hypercenter#2060D17
hypocenterC17#12256
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2060

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
68. C42×C17Q16semidirect
165. C17#12256semidirect
197. C42#63231semidirect

Maximal subgroups

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

subgroupindex
#145338132
#145277582
#145268132
#145267302
C17×#20602
#1225617