Finite Group Databasebeta

← #16938723

Subgroups — #16938723

Order 4352.

490 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 95 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#14529768C212yesmaximal
3#14529205C212yesmaximal
4#14528961C211yesmaximal, characteristic
5C17×#1289C211yesmaximal, characteristic
6C23×#3622C2211yescharacteristic
7C2×#10587610C2212yes—
8#11763145—22no—
9#11763050—21no—
10#11763050—21no—
11#11762967—22no—
12C2×#10587485—21no—
13C2×#10587481—22no—
14C4×C22⋊Dic17C2211yescharacteristic
15#11762575—22no—
16#11762575—22no—
17#11762571—22no—
18#11762563—21no—
19C2×C17×C22⋊C8C2211yescharacteristic
20C17×#380C2212yes—
21C23×C4×C17C2311yescharacteristic
22C22×Dic34D412yes—
23C22×C8×C17D411yescharacteristic
24C22×C8×C17D411yescharacteristic
25C2×C4⋊C4×C17—22no—
26C2×C22⋊C4×C17—22no—
27C2×C22⋊Dic17—22no—
28C2×C22⋊Dic17D411yescharacteristic
29C2×C22⋊Dic17D411yescharacteristic
30C2×C4⋊Dic17D412yes—
31#10587610—22no—
32#10587610—22no—
33#10587610—22no—
34#10587610—22no—
35C2×#63225—22no—
36C2×C4×Dic17—21no—
37C22×#3622—22no—
38C22×#3622—21no—
39C22×#3622—42no—
40C2×Dic68—21no—
41C2×Dic68—21no—
42C17×C22⋊C8—22no—
43C17×#95—22no—
44#10587502—21no—
45#10587489—21no—
46#10587489—21no—
47#10587487—22no—
48#10587487—22no—
49#10587485—42no—
50#10587481—44no—
51C24×C17C2×D411yescharacteristic
52C22×C4×C17—41no—
53C22×C4×C17—41no—
54C22×C4×C17—42no—
55C22×C4×C17C2×D411yescharacteristic
56C22×C4×C17C2×D412yes—
57C22×C4×C17#4112yes—
58C22×C4×C17#4111yescharacteristic
59C22×Dic17—21no—
60C22×Dic17—21no—
61C22×Dic17—22no—
62C22×Dic17—22no—
63C2×Dic34—22no—
64C2×Dic34—22no—
65C2×Dic34—42no—
66C2×Dic34—42no—
67C2×C8×C17—21no—
68C2×C8×C17—21no—
69C2×C8×C17—41no—
70C2×C8×C17—41no—
71C4⋊C4×C17—42no—
72C4⋊C4×C17—42no—
73C22⋊C4×C17—42no—
74C22⋊C4×C17—42no—
75C22⋊Dic17—21no—
76C22⋊Dic17—22no—
77C22⋊Dic17—42no—
78C22⋊Dic17—42no—
79C22⋊Dic17Q1612yescomplemented
80C4⋊Dic17—22no—
81C4⋊Dic17—22no—
82#63225—42no—
83#63225—42no—
84C4×Dic17—41no—
85C4×Dic17—41no—
86C2×#3622—24no—
87C2×#3622—22no—
88C2×#3622—42no—
89C2×#3622—41no—
90C2×#3622—41no—
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—22no—
102C2×C4×C17—21no—
103C2×C4×C17—22no—
104C2×C4×C17—21no—
105C2×C4×C17—42no—
106C2×C4×C17—41no—
107C2×C4×C17—82no—
108C2×C4×C17—82no—
109C2×C4×C17—82no—
110C2×C4×C17#12011yescharacteristic
111C2×C4×C17#12312yes—
112C2×C4×C17#12411yescharacteristic
113C2×Dic17—22no—
114C2×Dic17—41no—
115C2×Dic17—41no—
116C2×Dic17—41no—
117C2×Dic17—41no—
118C2×Dic17—42no—
119C2×Dic17—42no—
120C2×Dic17—41no—
121C2×Dic17—82no—
122C2×Dic17—82no—
123Dic34—44no—
124Dic34—42no—
125Dic34—42no—
126C8×C17—41no—
127C8×C17—41no—
128#3622—42no—
129#3189—171no—
130#1996—172no—
131#1892—172no—
132#1850—171no—
133#1289D1711yescharacteristic
134C22×C17—21no—
135C22×C17—22no—
136C22×C17—21no—
137C22×C17—22no—
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—162no—
149C4×C17B(4,𝔽2)11yescharacteristic
150C4×C17#45811yescharacteristic
151Dic17—42no—
152Dic17—81no—
153Dic17—82no—
154Dic17—82no—
155C23×C8—171no—
156C2×C22⋊Q8—172no—
157#484—342no—
158#467—341no—
159#467—341no—
160#451—342no—
161C2×C4⋊C8—341no—
162C2×#103—342no—
163C2×C22⋊C8C2×D1711yescharacteristic
164#380C2×D1712yes—
165C4×C22⋊C4—171no—
166#342—342no—
167#342—342no—
168#340—342no—
169#336—341no—
170C2×C17—21no—
171C2×C17—21no—
172C2×C17—22no—
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—341no—
182C22×C8—342no—
183C22×C8—682no—
184C22×C8D6811yescharacteristic
185C22×C8D6811yescharacteristic
186C22⋊Q8—342no—
187C22⋊Q8—342no—
188C22⋊Q8—342no—
189C22⋊Q8—342no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—172no—
192C2×C4⋊C4—342no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—171no—
195C2×C22⋊C4—171no—
196C2×C22⋊C4—342no—
197C2×C42—341no—
198#107—341no—
199#107—341no—
200C4⋊C8—682no—
201#103—342no—
202#103—342no—
203#103—684no—
204C22⋊C8—22no—
205#95—22no—
206#95—341no—
207C17#1056611yescharacteristic, complemented, Sylow
208C24D4×D1711yescharacteristic
209C2×Q8—342no—
210C2×Q8—342no—
211C2×Q8—682no—
212C2×Q8—682no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—341no—
217C22×C4—341no—
218C22×C4—342no—
219C22×C4—342no—
220C22×C4C2×D6811yescharacteristic
221C22×C4#6325212yes—
222C22×C4D4×D1712yes—
223C22×C4#6325611yescharacteristic
224Q16—1361no—
225Q16—1361no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—344no—
231C2×C8—342no—
232C2×C8—681no—
233C2×C8—682no—
234C2×C8—681no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—342no—
238C4⋊C4—342no—
239C4⋊C4—682no—
240C4⋊C4—682no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—172no—
244C22⋊C4—341no—
245C22⋊C4—342no—
246C22⋊C4—682no—
247C22⋊C4—682no—
248C42—681no—
249C42—681no—
250C23—41no—
251C23—41no—
252C23#1058757012yes—
253C23#1058758111yescharacteristic
254C23#1058759811yescharacteristic
255C23#1058760112yes—
256C23Q16×D1711yescharacteristic
257Q8—684no—
258Q8—682no—
259Q8—682no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—342no—
270C2×C4—681no—
271C2×C4—681no—
272C2×C4—681no—
273C2×C4—681no—
274C2×C4—682no—
275C2×C4—682no—
276C2×C4—681no—
277C2×C4—1362no—
278C2×C4—1362no—
279C2×C4#1058756111yescharacteristic
280C2×C4#1058756811yescharacteristic
281C2×C4#1058758012yes—
282C8—41no—
283C8—41no—
284C8—682no—
285C22—22no—
286C22—21no—
287C22—22no—
288C22—21no—
289C22—41no—
290C22—42no—
291C22—41no—
292C22#1176272711yescharacteristic
293C22#1176301812yes—
294C22#1176303012yes—
295C22#1176305111yescharacteristic
296C22#1176306811yescharacteristic
297C4—21no—
298C4—22no—
299C4—162no—
300C4—682no—
301C4—1361no—
302C4—1362no—
303C4—1362no—
304C4#1176299711yescharacteristic
305C4#1176300011yescharacteristic
306C2—22no—
307C2—21no—
308C2—21no—
309C2—22no—
310C2#1453003611yescharacteristic
311C2#1453009111yescharacteristic
312C2#1453010811yescharacteristic
subgroupG/H
centerC22#11762727
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC17×#1289C2
socleC22×C17#394
prime radicalC24×C17C2×D4
hypercenter#1289D17
hypocenterC17#10566
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
79. C22⋊Dic17Q16semidirect
207. C17#10566semidirect

Maximal subgroups

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

subgroupindex
#145395592
#145297682
#145292052
#145289612
C17×#12892
#1056617