Finite Group Databasebeta

← #16932023

Subgroups — #16932023

Order 4352.

554 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 107 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#14533737C211yesmaximal, characteristic
2#14527502C212yesmaximal, complemented
3#14527125C211yesmaximal, characteristic
4#14526641C212yesmaximal, complemented
5C17×#2973C211yesmaximal, characteristic
6C2×#10587555C2211yescharacteristic
7C2×#10587610—21no—
8C2×#10587610—21no—
9#11762866C2212yes—
10#11762745—21no—
11#11762745—21no—
12#11762712C412yes—
13#11762678C412yes—
14C4×C4⋊Dic17C2211yescharacteristic
15#11762556—24no—
16#11762533—21no—
17#11762533—21no—
18C2×C4×D4×C17C2211yescharacteristic
19C17×#435—24no—
20C17×#432C2212yes—
21C17×#407—21no—
22C2×C17×C22⋊C8—21no—
23C22×D4×C17C2×C411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C22×Dic34—21no—
27C2×M16×C17—22no—
28C22×C8×C17—22no—
29C4×D4×C17—22no—
30C4×D4×C17—22no—
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C22⋊C4×C17D411yescharacteristic
33C2×C22⋊C4×C17D411yescharacteristic
34C2×C42×C17C2311yescharacteristic
35C2×C22⋊Dic17—22no—
36C2×C22⋊Dic17—22no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17C2×C412yes—
40#10587610—44no—
41#10587610—44no—
42C2×#63225—22no—
43C2×#63225—22no—
44C2×C4×Dic17C2×C412yes—
45#10587555—22no—
46#10587555—22no—
47C17×C4⋊C8D414yes—
48C17×C22⋊C8—22no—
49C17×C22⋊C8—22no—
50C17×C22⋊C8—22no—
51C17×#97—22no—
52#10587502—21no—
53C22⋊#3622—22no—
54C4⋊#3622—22no—
55#10587474—22no—
56C24×C17C22⋊C411yescharacteristic
57C24×C17C22⋊C411yescharacteristic
58C2×D4×C17—22no—
59C2×D4×C17—22no—
60C2×D4×C17—41no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—42no—
66C22×C4×C17—42no—
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22×C411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C22×Dic17—22no—
73C22×Dic17—22no—
74C2×Dic34—42no—
75C2×Dic34—42no—
76C2×Dic34—41no—
77C2×Dic34—41no—
78M16×C17—82no—
79C2×C8×C17—24no—
80C2×C8×C17—82no—
81C4⋊C4×C17—22no—
82C22⋊C4×C17—22no—
83C22⋊C4×C17—22no—
84C42×C17C2×D412yes—
85C42×C17#4112yes—
86C22⋊Dic17—44no—
87C22⋊Dic17—44no—
88C4⋊Dic17—24no—
89C4⋊Dic17—24no—
90C4⋊Dic17—42no—
91#63225—44no—
92#63225—44no—
93C4×Dic17—42no—
94C2×#3622—42no—
95#9455—171nomaximal, Sylow
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—41no—
102C23×C17—41no—
103C23×C17—41no—
104C23×C17—41no—
105C23×C17#10112yes—
106C23×C17C2×C22⋊C411yescharacteristic
107D4×C17—44no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—22no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—41no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—81no—
119C2×C4×C17—81no—
120C2×C4×C17C4×D412yes—
121C2×C4×C17#12011yescharacteristic
122C2×C4×C17#12112yes—
123C2×C4×C17#12311yescharacteristic
124C2×Dic17—24no—
125C2×Dic17—42no—
126C2×Dic17—42no—
127C2×Dic17—42no—
128C2×Dic17—42no—
129C2×Dic17—42no—
130Dic34—82no—
131Dic34—82no—
132C8×C17—44no—
133#3622—82no—
134#2973D1711yescharacteristic
135#2542—171no—
136#1556—172no—
137#1466—171no—
138#1285—172no—
139C22×C17—21no—
140C22×C17—21no—
141C22×C17—22no—
142C22×C17—42no—
143C22×C17—42no—
144C22×C17—41no—
145C22×C17—41no—
146C22×C17—41no—
147C22×C17—41no—
148C22×C17—41no—
149C22×C17—81no—
150C22×C17—81no—
151C22×C17#38611yescharacteristic
152C22×C17C2×#10111yescharacteristic
153C22×C17#41311yescharacteristic
154C4×C17—21no—
155C4×C17—22no—
156C4×C17—22no—
157C4×C17—41no—
158C4×C17—41no—
159C4×C17#45312yes—
160Dic17—44no—
161Dic17—82no—
162C2×C4⋊Q8—171no—
163C2×C22⋊Q8—341no—
164C2×C22⋊Q8—341no—
165C2×C4×D4C2×D1711yescharacteristic
166#435—24no—
167#432—172no—
168#432C2×D1712yes—
169#407—21no—
170C2×C22⋊C8—21no—
171#399—341no—
172#399—341no—
173#390—172no—
174#384—172no—
175C4×C4⋊C4—171no—
176#332—344no—
177#323—341no—
178#323—341no—
179C2×C17—21no—
180C2×C17—21no—
181C2×C17—42no—
182C2×C17—81no—
183C2×C17#188711yescharacteristic
184C2×C17#190111yescharacteristic
185C2×C17#190911yescharacteristic
186C22×Q8—341no—
187C22×D4C4×D1711yescharacteristic
188C23×C4C22⋊D1711yescharacteristic
189C23×C4C22⋊D1711yescharacteristic
190C2×M16—22no—
191C22×C8—22no—
192C4⋊Q8—342no—
193C4⋊Q8—342no—
194C22⋊Q8—684no—
195C22⋊Q8—684no—
196C4×D4—22no—
197C4×D4—22no—
198C2×C4⋊C4—172no—
199C2×C4⋊C4—341no—
200C2×C4⋊C4—341no—
201C2×C4⋊C4—342no—
202C2×C4⋊C4—342no—
203C2×C4⋊C4C4×D1711yescharacteristic
204C2×C22⋊C4—342no—
205C2×C22⋊C4—342no—
206C2×C22⋊C4D6811yescharacteristic
207C2×C22⋊C4D6811yescharacteristic
208C2×C42—172no—
209C2×C42C22×D1711yescharacteristic
210C4⋊C8—342no—
211C4⋊C8C22⋊D1714yes—
212C22⋊C8—22no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—342no—
216#97—22no—
217#97—342no—
218#95—341no—
219C17#945511yescharacteristic, complemented, Sylow
220C24#6322711yescharacteristic
221C24#6322711yescharacteristic
222C2×Q8—682no—
223C2×Q8—682no—
224C2×Q8—681no—
225C2×Q8—681no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—342no—
236C22×C4—342no—
237C22×C4#6322711yescharacteristic
238C22×C4#6322711yescharacteristic
239C22×C4C2×C4×D1711yescharacteristic
240C22×C4C2×D6811yescharacteristic
241C22×C4C2×C22⋊D1711yescharacteristic
242M16—82no—
243C2×C8—24no—
244C2×C8—82no—
245C2×C8—682no—
246C4⋊C4—22no—
247C4⋊C4—344no—
248C4⋊C4—344no—
249C4⋊C4—684no—
250C4⋊C4—684no—
251C4⋊C4—682no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—684no—
255C22⋊C4—684no—
256C42—682no—
257C42#6325212yes—
258C42C2×C22⋊D1712yes—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23#1058749512yes—
269C23C2×#6322711yescharacteristic
270Q8—1362no—
271Q8—1362no—
272D4—44no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—344no—
286C2×C4—682no—
287C2×C4—682no—
288C2×C4—682no—
289C2×C4—682no—
290C2×C4—682no—
291C2×C4C4×C22⋊D1712yes—
292C2×C4#1058761511yescharacteristic
293C2×C4#1058761612yes—
294C2×C4#1058763911yescharacteristic
295C8—44no—
296C8—1362no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22—81no—
309C22#1176268811yescharacteristic
310C22C2×#1058749511yescharacteristic
311C22#1176279111yescharacteristic
312C4—22no—
313C4—21no—
314C4—22no—
315C4—41no—
316C4—41no—
317C4—684no—
318C4—1362no—
319C4#1176297712yes—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—81no—
324C2#1452918211yescharacteristic
325C2#1452925211yescharacteristic
326C2#1452930611yescharacteristic
subgroupG/H
centerC22#11762688
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2973C2
socleC22×C17#386
prime radicalC22×D4×C17C2×C4
cosocleC2×C42×C17C23
hypercenter#2973D17
hypocenterC17#9455
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #14527502C2semidirect
4. #14526641C2semidirect
219. C17#9455semidirect

Maximal subgroups

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

subgroupindex
#145337372
#145275022
#145271252
#145266412
C17×#29732
#945517