Finite Group Databasebeta

← #16938665

Subgroups — #16938665

Order 4352.

500 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 79 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#14539535C211yesmaximal, characteristic
2#14529247C212yesmaximal, complemented
3#14527299C212yesmaximal, complemented
4#14526661C211yesmaximal, characteristic, complemented
5C17×#1851C211yesmaximal, characteristic, complemented
6C2×#10587616C2212yes—
7#11763123—22no—
8#11763045—21no—
9#11763045—21no—
10#11762945—22no—
11C2×#10587494—22no—
12#11762773C2212yes—
13#11762577—22no—
14#11762542—22no—
15#11762542—22no—
16C22×M16×C17C2211yescharacteristic
17C2×C17×C4⋊C8—21no—
18C2×C17×C22⋊C8C2211yescharacteristic
19C4×C22⋊C4×C17C2211yescharacteristic
20C17×#336—21no—
21C23×C4×C17C2311yescharacteristic
22C22×C22⋊D17—22no—
23C22×D68D412yes—
24C2×M16×C17—22no—
25C2×M16×C17—42no—
26C22×C8×C17—21no—
27C22×C8×C17D411yescharacteristic
28C22×C8×C17D411yescharacteristic
29C2×C22⋊C4×C17—21no—
30C2×C42×C17—21no—
31#10587616—22no—
32#10587616—22no—
33#10587616—22no—
34#10587616—22no—
35C2×#63227—22no—
36C2×C4⋊Dic17D412yes—
37C2×#63223—22no—
38C2×D136—21no—
39C2×D136—21no—
40C17×C4⋊C8—42no—
41C17×C22⋊C8—22no—
42C17×#95D411yescharacteristic
43C17×#95D411yescharacteristic
44C22⋊#3622—22no—
45C22⋊#3622—22no—
46#10587494—44no—
47#10587492—22no—
48#10587492—22no—
49#10587489—21no—
50#10587489—21no—
51C24×C17#4111yescharacteristic
52C23×D17—22no—
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17—41no—
56C22×C4×C17—41no—
57C22×C4×C17C2×D411yescharacteristic
58C22×C4×C17C2×D412yes—
59C22×C4×C17C2×D411yescharacteristic
60C22×C4×C17#4112yes—
61C2×C22⋊D17—42no—
62C2×C22⋊D17—42no—
63C2×C22⋊D17—82no—
64C2×C22⋊D17—82no—
65C22×Dic17—22no—
66C2×D68—22no—
67C2×D68—22no—
68C2×D68—42no—
69C2×D68—42no—
70M16×C17—24no—
71M16×C17—42no—
72C2×C8×C17—22no—
73C2×C8×C17—21no—
74C2×C8×C17—21no—
75C2×C8×C17—41no—
76C2×C8×C17—41no—
77C2×C8×C17—41no—
78C2×C8×C17—41no—
79C22⋊C4×C17—42no—
80C42×C17—41no—
81C42×C17—41no—
82#63227—42no—
83#63227—42no—
84C4⋊Dic17—22no—
85C4⋊Dic17—22no—
86#63223—42no—
87#63223—42no—
88C2×#3622—42no—
89D136—81no—
90D136—81no—
91#10557—171nomaximal, Sylow
92C23×C17—21no—
93C23×C17—21no—
94C23×C17—41no—
95C23×C17—41no—
96C23×C17#12111yescharacteristic
97C23×C17#12312yes—
98C22×D17—42no—
99C22×D17—44no—
100C22×D17—82no—
101C22×D17—82no—
102C2×C4×C17—21no—
103C2×C4×C17—22no—
104C2×C4×C17—21no—
105C2×C4×C17—22no—
106C2×C4×C17—22no—
107C2×C4×C17—41no—
108C2×C4×C17—42no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17#12011yescharacteristic
115C2×C4×C17#12112yes—
116C2×C4×C17#12411yescharacteristic
117C22⋊D17—84no—
118C22⋊D17—82no—
119C22⋊D17—82no—
120C2×Dic17—42no—
121C2×Dic17—82no—
122D68—42no—
123D68—42no—
124D68—82no—
125C8×C17—42no—
126C8×C17—41no—
127C8×C17—41no—
128#3622—82no—
129#3186—171no—
130#1900—172no—
131#1851D1711yescharacteristic, complemented
132#1507—172no—
133#1293—171no—
134C22×C17—21no—
135C22×C17—21no—
136C22×C17—22no—
137C22×C17—22no—
138C22×C17—22no—
139C22×C17—81no—
140C22×C17—81no—
141C22×C17#39411yescharacteristic
142C22×C17#47211yescharacteristic
143C22×C17#47211yescharacteristic
144C2×D17—82no—
145C2×D17—82no—
146C2×D17—82no—
147C2×D17—164no—
148C4×C17—22no—
149C4×C17—21no—
150C4×C17—42no—
151C4×C17—81no—
152C4×C17B(4,𝔽2)11yescharacteristic
153C4×C17#45811yescharacteristic
154Dic17—82no—
155C22×M16C2×D1711yescharacteristic
156C2×#121—172no—
157#480—342no—
158#466—341no—
159#466—341no—
160#447—342no—
161C2×C4⋊C8—21no—
162C2×#100—342no—
163#407—172no—
164C2×C22⋊C8C2×D1711yescharacteristic
165C4×C22⋊C4C2×D1711yescharacteristic
166#343—342no—
167#336—21no—
168#327—342no—
169#327—342no—
170C2×C17—21no—
171C2×C17—21no—
172C2×C17—42no—
173C2×C17#204411yescharacteristic
174C2×C17#204511yescharacteristic
175C2×C17#205011yescharacteristic
176D17—162no—
177C22×D4—172no—
178C22×D4—342no—
179C23×C4C22×D1711yescharacteristic
180C2×D8—341no—
181C2×D8—341no—
182C2×M16—22no—
183C2×M16—42no—
184C2×M16—342no—
185C22×C8—21no—
186C22×C8C22⋊D1711yescharacteristic
187C22×C8C22⋊D1711yescharacteristic
188#121—342no—
189#121—342no—
190#121—342no—
191#121—342no—
192C2×C4⋊C4—172no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—342no—
195C2×C42—21no—
196#107—341no—
197#107—341no—
198C4⋊C8—42no—
199#102—342no—
200#102—342no—
201#100—684no—
202C22⋊C8—22no—
203C22⋊C8—342no—
204C22⋊C8—342no—
205#95D6811yescharacteristic
206#95D6811yescharacteristic
207C17#1055711yescharacteristic, complemented, Sylow
208C24—342no—
209C24#6325211yescharacteristic
210C2×D4—342no—
211C2×D4—342no—
212C2×D4—682no—
213C2×D4—682no—
214C2×D4—682no—
215C2×D4—682no—
216C2×D4—1362no—
217C2×D4—1362no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—342no—
223C22×C4C2×D6811yescharacteristic
224C22×C4D4×D1712yes—
225C22×C4#6325612yes—
226C22×C4C2×C22⋊D1711yescharacteristic
227D8—1361no—
228D8—1361no—
229M16—24no—
230M16—42no—
231M16—682no—
232M16—682no—
233C2×C8—22no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—682no—
241C4⋊C4—342no—
242C4⋊C4—342no—
243C22⋊C4—42no—
244C22⋊C4—682no—
245C22⋊C4—682no—
246C42—41no—
247C42—41no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—684no—
253C23—682no—
254C23—1362no—
255C23—1362no—
256C23#1058758012yes—
257C23#1058761611yescharacteristic
258D4—682no—
259D4—682no—
260D4—1362no—
261D4—1364no—
262D4—1362no—
263D4—1362no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—682no—
277C2×C4—1362no—
278C2×C4#1058758112yes—
279C2×C4#1058762311yescharacteristic
280C2×C4#1058763311yescharacteristic
281C8—42no—
282C8—41no—
283C8—41no—
284C8—1362no—
285C22—21no—
286C22—21no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—81no—
291C22—81no—
292C22—2724no—
293C22—1362no—
294C22—1362no—
295C22—1362no—
296C22#1176272611yescharacteristic
297C22#1176307911yescharacteristic
298C22#1176307911yescharacteristic
299C4—22no—
300C4—21no—
301C4—42no—
302C4—81no—
303C4—1362no—
304C4#1176299511yescharacteristic
305C4#1176299811yescharacteristic
306C2—21no—
307C2—21no—
308C2—42no—
309C2—2722no—
310C2#1453003511yescharacteristic
311C2#1453004011yescharacteristic
312C2#1453007111yescharacteristic
subgroupG/H
centerC22#11762726
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC17×#1851C2
socleC22×C17#394
prime radical#14539535C2
hypercenter#1851D17
hypocenterC17#10557
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
2. #14529247C2semidirect
3. #14527299C2semidirect
4. #14526661C2semidirect
5. C17×#1851C2semidirect
131. #1851D17semidirect
207. C17#10557semidirect

Maximal subgroups

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

subgroupindex
#145395352
#145292472
#145272992
#145266612
C17×#18512
#1055717