Finite Group Databasebeta

← #16951160

Subgroups — #16951160

Order 4352.

492 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 103 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#14531896C211yesmaximal, characteristic
2#14529188C211yesmaximal, characteristic
3#14529001C212yesmaximal, complemented
4#14526910C212yesmaximal, complemented
5C2×C17×#355C211yesmaximal, characteristic
6#11763287C2211yescharacteristic
7#11762851—22no—
8#11762814—21no—
9C2×#10587507C2211yescharacteristic
10#11762716C2212yes—
11#11762702—22no—
12#11762689—21no—
13C22⋊C4×Dic17—22no—
14#11762655—21no—
15#11762570—22no—
16#11762569—21no—
17C2×C17×#124C2211yescharacteristic
18C2×C17×#101C2212yes—
19C17×#355C414yescomplemented
20C22×Q8×C17C2311yescharacteristic
21C22×D4×C17D411yescharacteristic
22C23×Dic17—21no—
23C2×M16×C17—22no—
24C17×#124—22no—
25C17×#124C2×C412yes—
26C17×#124Q812yes—
27C2×C22⋊C4×C17—22no—
28C2×C42×C17D411yescharacteristic
29C2×C22⋊Dic17—22no—
30C2×C22⋊Dic17—21no—
31Q8×Dic17—42no—
32#10587612—21no—
33#10587612—22no—
34C2×C4⋊Dic17D412yes—
35C2×C4×Dic17—22no—
36C2×C4×Dic17D412yes—
37C2×#63223—21no—
38C17×#101—22no—
39C17×#101C2×C414yescomplemented
40#10587507—22no—
41#10587503—42no—
42#10587502—21no—
43#10587502—21no—
44#10587502—22no—
45#10587502—22no—
46#10587489—42no—
47#10587488—42no—
48C24×C17—21no—
49C2×Q8×C17—41no—
50C2×Q8×C17—41no—
51C2×Q8×C17C22×C411yescharacteristic
52C2×Q8×C17C2×Q811yescharacteristic
53C2×Q8×C17#4112yes—
54C2×D4×C17—21no—
55C2×D4×C17—41no—
56C2×D4×C17—41no—
57C2×D4×C17C4⋊C412yes—
58C22×C4×C17—21no—
59C22×C4×C17C2×D411yescharacteristic
60C22×C4×C17C2×D412yes—
61C22×Dic17—22no—
62C22×Dic17—22no—
63C22×Dic17—44no—
64C22×Dic17—81no—
65M16×C17—24no—
66M16×C17—42no—
67C2×C8×C17—42no—
68C22⋊C4×C17—24no—
69C22⋊C4×C17—24no—
70C42×C17—21no—
71C42×C17C4⋊C412yes—
72C22⋊Dic17—44no—
73C22⋊Dic17—41no—
74C22⋊Dic17—81no—
75C4⋊Dic17—22no—
76C4⋊Dic17—22no—
77C4⋊Dic17—22no—
78C4⋊Dic17—42no—
79C4×Dic17—44no—
80C4×Dic17—42no—
81C4×Dic17—41no—
82C4×Dic17—42no—
83#63223—41no—
84#63223—41no—
85C2×#3622—41no—
86#12972—171nomaximal, Sylow
87C23×C17—22no—
88C23×C17—42no—
89C23×C17—42no—
90C23×C17—41no—
91C23×C17—41no—
92C23×C17#12011yescharacteristic
93Q8×C17—41no—
94Q8×C17—41no—
95Q8×C17—42no—
96D4×C17—42no—
97D4×C17—81no—
98C2×C4×C17—22no—
99C2×C4×C17—21no—
100C2×C4×C17—22no—
101C2×C4×C17—21no—
102C2×C4×C17—22no—
103C2×C4×C17—42no—
104C2×C4×C17C2×C4⋊C411yescharacteristic
105C2×C4×C17C4×D412yes—
106C2×C4×C17C22⋊Q812yes—
107C2×C4×C17#12311yescharacteristic
108C2×Dic17—24no—
109C2×Dic17—42no—
110C2×Dic17—42no—
111C2×Dic17—42no—
112C2×Dic17—42no—
113C2×Dic17—42no—
114C2×Dic17—42no—
115C2×Dic17—84no—
116C8×C17—44no—
117#3622—81no—
118#2372—171no—
119C2×#355D1711yescharacteristic
120#1888—171no—
121#1857—172no—
122#1394—172no—
123C22×C17—22no—
124C22×C17—44no—
125C22×C17—42no—
126C22×C17—81no—
127C22×C17—81no—
128C22×C17—82no—
129C22×C17#38511yescharacteristic
130C22×C17B(4,𝔽2)11yescharacteristic
131C22×C17#45811yescharacteristic
132C4×C17—21no—
133C4×C17—22no—
134C4×C17—21no—
135C4×C17—22no—
136C4×C17—42no—
137Dic17—44no—
138Dic17—82no—
139Dic17—82no—
140C2×#124C2×D1711yescharacteristic
141#519—171no—
142#428—342no—
143#418—341no—
144C2×#101—171no—
145C2×#101C2×D1712yes—
146#391—172no—
147#388—342no—
148#386—341no—
149C4×C22⋊C4—342no—
150#376—341no—
151#355Dic1714yescomplemented
152#339—342no—
153#339—341no—
154C2×C17—21no—
155C2×C17—21no—
156C2×C17—82no—
157C2×C17#186711yescharacteristic
158C2×C17#220412yes—
159C22×Q8C22×D1711yescharacteristic
160C22×D4D6811yescharacteristic
161C23×C4—341no—
162C2×M16—22no—
163C2×M16—341no—
164#124—22no—
165#124Dic3412yes—
166#124C2×Dic1712yes—
167C4×Q8—682no—
168#117—341no—
169#117—342no—
170C2×C4⋊C4—172no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—342no—
173C2×C22⋊C4—341no—
174C2×C42—172no—
175C2×C42—342no—
176C2×C42D6811yescharacteristic
177#107—682no—
178#106—682no—
179#102—682no—
180#101—22no—
181#101—342no—
182#101C2×Dic1714yescomplemented
183#95—341no—
184#95—341no—
185#95—342no—
186#95—342no—
187C17#1297211yescharacteristic, complemented, Sylow
188C24—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8C2×Dic3411yescharacteristic
192C2×Q8#6325412yes—
193C2×Q8C22×Dic1711yescharacteristic
194C2×D4—21no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4C4⋊Dic1712yes—
198C22×C4—21no—
199C22×C4—342no—
200C22×C4—342no—
201C22×C4—684no—
202C22×C4—1361no—
203C22×C4C2×D6811yescharacteristic
204C22×C4D4×D1712yes—
205M16—24no—
206M16—42no—
207M16—681no—
208M16—681no—
209C2×C8—42no—
210C2×C8—681no—
211C4⋊C4—342no—
212C4⋊C4—342no—
213C4⋊C4—342no—
214C4⋊C4—682no—
215C22⋊C4—24no—
216C22⋊C4—24no—
217C22⋊C4—684no—
218C22⋊C4—681no—
219C22⋊C4—1361no—
220C42—21no—
221C42—682no—
222C42—682no—
223C42—681no—
224C42—684no—
225C42C4⋊Dic1712yes—
226C23—22no—
227C23—42no—
228C23—42no—
229C23—41no—
230C23—41no—
231C23#1058756811yescharacteristic
232Q8—41no—
233Q8—42no—
234Q8—41no—
235D4—42no—
236D4—81no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—344no—
244C2×C4—682no—
245C2×C4—682no—
246C2×C4—682no—
247C2×C4—682no—
248C2×C4—682no—
249C2×C4—682no—
250C2×C4—1364no—
251C2×C4#1058756412yes—
252C2×C4#1058757211yescharacteristic
253C2×C4C2×C4⋊Dic1711yescharacteristic
254C2×C4D4×Dic1712yes—
255C8—44no—
256C8—1361no—
257C22—22no—
258C22—44no—
259C22—42no—
260C22—81no—
261C22—81no—
262C22—82no—
263C22#1176268411yescharacteristic
264C22#1176299711yescharacteristic
265C22#1176300011yescharacteristic
266C4—22no—
267C4—21no—
268C4—22no—
269C4—21no—
270C4—42no—
271C4—684no—
272C4—1362no—
273C4—1362no—
274C2—21no—
275C2—21no—
276C2—82no—
277C2#1452906711yescharacteristic
278C2#1453081112yes—
subgroupG/H
centerC22#11762997
commutatorC2×Q8×C17C22×C4
FrattiniC22×Q8C22×D17
FittingC2×C17×#355C2
socleC22×C17B(4,𝔽2)
prime radicalC22×D4×C17D4
cosocleC22×Q8×C17C23
hypercenterC2×#355D17
hypocenterC17#12972
LCSG ▷ C2×Q8×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#355

Complements

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

NH ≅ G/Ntype
3. #14529001C2semidirect
4. #14526910C2semidirect
19. C17×#355C4semidirect
39. C17×#101C2×C4semidirect
151. #355Dic17semidirect
182. #101C2×Dic17semidirect
187. C17#12972semidirect

Maximal subgroups

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

subgroupindex
#145318962
#145291882
#145290012
#145269102
C2×C17×#3552
#1297217