Finite Group Databasebeta

← #16938722

Subgroups — #16938722

Order 4352.

490 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 99 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#14539559C211yesmaximal, characteristic
2#14529769C212yesmaximal
3#14529207C212yesmaximal
4#14526652C211yesmaximal, characteristic
5C17×#1850C211yesmaximal, characteristic
6C2×#10587610C2212yes—
7#11763145—22no—
8#11763050—21no—
9#11763050—21no—
10#11762967—22no—
11C2×#10587487—22no—
12#11762669C2212yes—
13#11762575—22no—
14#11762575—22no—
15#11762572—22no—
16C23×C8×C17C2211yescharacteristic
17C2×C17×C4⋊C8—21no—
18C2×C17×C22⋊C8C2211yescharacteristic
19C4×C22⋊C4×C17C2211yescharacteristic
20C17×#336—21no—
21C23×C4×C17C2311yescharacteristic
22C22×Dic34D412yes—
23C22×C8×C17—21no—
24C22×C8×C17—22no—
25C22×C8×C17—42no—
26C22×C8×C17D411yescharacteristic
27C22×C8×C17D411yescharacteristic
28C2×C22⋊C4×C17D411yescharacteristic
29C2×C22⋊C4×C17D411yescharacteristic
30C2×C42×C17—21no—
31C2×C22⋊Dic17—22no—
32C2×C22⋊Dic17—22no—
33C2×C4⋊Dic17—22no—
34C2×C4⋊Dic17D412yes—
35#10587610—22no—
36#10587610—22no—
37#10587610—22no—
38#10587610—22no—
39C2×#63225—22no—
40C2×Dic68—21no—
41C2×Dic68—21no—
42C17×C4⋊C8—42no—
43C17×C22⋊C8—22no—
44C17×#95—21no—
45#10587502—22no—
46#10587489—21no—
47#10587489—21no—
48#10587487—22no—
49#10587487—22no—
50#10587487—44no—
51C24×C17C2×D411yescharacteristic
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17—41no—
55C22×C4×C17—41no—
56C22×C4×C17C2×D412yes—
57C22×C4×C17C2×D411yescharacteristic
58C22×C4×C17#4111yescharacteristic
59C22×C4×C17#4112yes—
60C22×Dic17—22no—
61C22×Dic17—22no—
62C22×Dic17—42no—
63C2×Dic34—22no—
64C2×Dic34—22no—
65C2×Dic34—42no—
66C2×Dic34—42no—
67C2×C8×C17—24no—
68C2×C8×C17—22no—
69C2×C8×C17—21no—
70C2×C8×C17—21no—
71C2×C8×C17—41no—
72C2×C8×C17—42no—
73C2×C8×C17—41no—
74C2×C8×C17—41no—
75C2×C8×C17—41no—
76C22⋊C4×C17—21no—
77C22⋊C4×C17—22no—
78C22⋊C4×C17Q1612yescomplemented
79C42×C17—41no—
80C42×C17—41no—
81C22⋊Dic17—42no—
82C22⋊Dic17—42no—
83C22⋊Dic17—42no—
84C22⋊Dic17—42no—
85C4⋊Dic17—22no—
86C4⋊Dic17—22no—
87C4⋊Dic17—42no—
88C4⋊Dic17—42no—
89#63225—42no—
90#63225—42no—
91Dic68—81no—
92Dic68—81no—
93#10566—171nomaximal, Sylow
94C23×C17—41no—
95C23×C17—41no—
96C23×C17#12112yes—
97C23×C17#12111yescharacteristic
98C23×C17C2×Q1611yescharacteristic
99C23×C17#13511yescharacteristic
100C23×C17#13712yes—
101C2×C4×C17—21no—
102C2×C4×C17—22no—
103C2×C4×C17—21no—
104C2×C4×C17—22no—
105C2×C4×C17—22no—
106C2×C4×C17—41no—
107C2×C4×C17—42no—
108C2×C4×C17—41no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17#12011yescharacteristic
114C2×C4×C17#12312yes—
115C2×C4×C17#12411yescharacteristic
116C2×Dic17—42no—
117C2×Dic17—42no—
118C2×Dic17—82no—
119C2×Dic17—82no—
120C2×Dic17—82no—
121C2×Dic17—82no—
122C2×Dic17—82no—
123Dic34—44no—
124Dic34—42no—
125Dic34—42no—
126C8×C17—42no—
127C8×C17—41no—
128C8×C17—41no—
129#3189—171no—
130#1996—172no—
131#1892—172no—
132#1850D1711yescharacteristic
133#1289—171no—
134C22×C17—22no—
135C22×C17—21no—
136C22×C17—22no—
137C22×C17—21no—
138C22×C17—41no—
139C22×C17—41no—
140C22×C17—42no—
141C22×C17#39411yescharacteristic
142C22×C17#46212yes—
143C22×C17#46412yes—
144C22×C17#46711yescharacteristic
145C22×C17#47011yescharacteristic
146C4×C17—22no—
147C4×C17—21no—
148C4×C17—42no—
149C4×C17—81no—
150C4×C17B(4,𝔽2)11yescharacteristic
151C4×C17#45811yescharacteristic
152Dic17—82no—
153Dic17—82no—
154Dic17—162no—
155C23×C8C2×D1711yescharacteristic
156C2×C22⋊Q8—172no—
157#484—342no—
158#467—341no—
159#467—341no—
160#451—342no—
161C2×C4⋊C8—21no—
162C2×#103—342no—
163C2×C22⋊C8C2×D1711yescharacteristic
164#380—172no—
165C4×C22⋊C4C2×D1711yescharacteristic
166#342—342no—
167#342—342no—
168#340—342no—
169#336—21no—
170C2×C17—22no—
171C2×C17—21no—
172C2×C17—21no—
173C2×C17—22no—
174C2×C17#204411yescharacteristic
175C2×C17#205311yescharacteristic
176C2×C17#205611yescharacteristic
177C22×Q8—172no—
178C23×C4C22×D1711yescharacteristic
179C2×Q16—341no—
180C2×Q16—341no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8—42no—
184C22×C8C22⋊D1711yescharacteristic
185C22×C8C22⋊D1711yescharacteristic
186C22⋊Q8—342no—
187C22⋊Q8—342no—
188C22⋊Q8—342no—
189C22⋊Q8—342no—
190C2×C4⋊C4—172no—
191C2×C4⋊C4—342no—
192C2×C4⋊C4—342no—
193C2×C22⋊C4—342no—
194C2×C22⋊C4—342no—
195C2×C22⋊C4D6811yescharacteristic
196C2×C22⋊C4D6811yescharacteristic
197C2×C42—21no—
198#107—341no—
199#107—341no—
200C4⋊C8—42no—
201#103—342no—
202#103—342no—
203#103—684no—
204C22⋊C8—22no—
205#95—21no—
206#95—342no—
207C17#1056611yescharacteristic, complemented, Sylow
208C24C2×D6811yescharacteristic
209C2×Q8—342no—
210C2×Q8—342no—
211C2×Q8—682no—
212C2×Q8—682no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—342no—
218C22×C4—342no—
219C22×C4—682no—
220C22×C4#6325211yescharacteristic
221C22×C4D4×D1712yes—
222C22×C4#6325612yes—
223C22×C4C2×C22⋊D1711yescharacteristic
224Q16—1361no—
225Q16—1361no—
226C2×C8—24no—
227C2×C8—22no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—41no—
231C2×C8—42no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C4⋊C4—342no—
236C4⋊C4—342no—
237C4⋊C4—682no—
238C4⋊C4—682no—
239C4⋊C4—682no—
240C4⋊C4—682no—
241C22⋊C4—21no—
242C22⋊C4—22no—
243C22⋊C4—682no—
244C22⋊C4—682no—
245C22⋊C4—682no—
246C22⋊C4—682no—
247C22⋊C4Dic6812yescomplemented
248C42—41no—
249C42—41no—
250C23—41no—
251C23—41no—
252C23#1058758112yes—
253C23#1058759011yescharacteristic
254C23C2×Dic6811yescharacteristic
255C23#1058759512yes—
256C23#1058761611yescharacteristic
257Q8—684no—
258Q8—682no—
259Q8—682no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—682no—
273C2×C4—682no—
274C2×C4—1362no—
275C2×C4—1362no—
276C2×C4—1362no—
277C2×C4—1362no—
278C2×C4—1362no—
279C2×C4#1058758012yes—
280C2×C4#1058762311yescharacteristic
281C2×C4#1058763311yescharacteristic
282C8—42no—
283C8—41no—
284C8—41no—
285C22—21no—
286C22—22no—
287C22—21no—
288C22—22no—
289C22—41no—
290C22—41no—
291C22—42no—
292C22#1176272611yescharacteristic
293C22#1176302012yes—
294C22#1176303212yes—
295C22#1176305011yescharacteristic
296C22#1176306711yescharacteristic
297C4—22no—
298C4—21no—
299C4—42no—
300C4—81no—
301C4—2722no—
302C4—1362no—
303C4—1362no—
304C4#1176299511yescharacteristic
305C4#1176299811yescharacteristic
306C2—21no—
307C2—21no—
308C2—22no—
309C2—22no—
310C2#1453003511yescharacteristic
311C2#1453009011yescharacteristic
312C2#1453010711yescharacteristic
subgroupG/H
centerC22#11762726
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC17×#1850C2
socleC22×C17#394
prime radicalC24×C17C2×D4
hypercenter#1850D17
hypocenterC17#10566
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
78. C22⋊C4×C17Q16semidirect
207. C17#10566semidirect
247. C22⋊C4Dic68semidirect

Maximal subgroups

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

subgroupindex
#145395592
#145297692
#145292072
#145266522
C17×#18502
#1056617