Finite Group Databasebeta

← #16945459

Subgroups — #16945459

Order 4352.

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

normal subgroups only

This group has 508 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#14533737C211yesmaximal, characteristic
2#14527102C211yesmaximal, characteristic
3#14526807C214yesmaximal, complemented
4C17×#1799C211yesmaximal, characteristic
5C2×#10587555C411yescharacteristic, complemented
6C2×#10587610—21no—
7C2×#10587610—21no—
8C2×C22⋊#3622—21no—
9#11762745C412yescomplemented
10#11762745C2212yes—
11#11762712—21no—
12#11762678—21no—
13C4×C4⋊Dic17C411yescharacteristic, complemented
14#11762535—21no—
15#11762533C2212yes—
16C2×C4×D4×C17C2211yescharacteristic
17C2×C17×#99—21no—
18C17×#380—21no—
19C17×#342C2212yes—
20C22×D4×C17D411yescharacteristic
21C23×C4×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C22×Dic34—21no—
24C4×D4×C17—41no—
25C4×D4×C17—41no—
26C2×C4⋊C4×C17—41no—
27C2×C4⋊C4×C17D411yescharacteristic
28C2×C22⋊C4×C17—22no—
29C2×C22⋊C4×C17C2×C411yescharacteristic
30C2×C22⋊C4×C17C2311yescharacteristic
31C2×C42×C17C2×C411yescharacteristic
32C2×C22⋊Dic17—22no—
33C2×C22⋊Dic17—22no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17C2×C412yes—
37#10587610—44no—
38#10587610—44no—
39C2×#63225—22no—
40C2×#63225—22no—
41C2×C4×Dic17—21no—
42C22×#3622—41no—
43#10587555—41no—
44#10587555—41no—
45C17×#99—42no—
46C17×#95—41no—
47#10587502C2×C412yes—
48C22⋊#3622—22no—
49C22⋊#3622—41no—
50C24×C17C22⋊C411yescharacteristic
51C24×C17C2×D411yescharacteristic
52C2×D4×C17—41no—
53C2×D4×C17—41no—
54C2×D4×C17SD1612yes—
55C2×D4×C17Q1612yes—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—42no—
61C22×C4×C17—42no—
62C22×C4×C17—42no—
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22×C411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×Dic17—22no—
69C22×Dic17—22no—
70C2×Dic34—42no—
71C2×Dic34—42no—
72C2×Dic34—41no—
73C2×Dic34—41no—
74C4⋊C4×C17—41no—
75C4⋊C4×C17—82no—
76C22⋊C4×C17—41no—
77C22⋊C4×C17—41no—
78C22⋊C4×C17—42no—
79C22⋊C4×C17—42no—
80C42×C17—21no—
81C42×C17—21no—
82C22⋊Dic17—44no—
83C22⋊Dic17—44no—
84C4⋊Dic17—24no—
85C4⋊Dic17—42no—
86C4⋊Dic17—42no—
87#63225—44no—
88#63225—44no—
89C4×Dic17—42no—
90C2×#3622—42no—
91C2×#3622—82no—
92#11901—171nomaximal, Sylow
93C23×C17—21no—
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—21no—
97C23×C17—41no—
98C23×C17—41no—
99C23×C17—41no—
100C23×C17—41no—
101C23×C17#10312yes—
102C23×C17C2×C22⋊C411yescharacteristic
103C23×C17C2×SD1611yescharacteristic
104C23×C17C2×Q1611yescharacteristic
105D4×C17—42no—
106D4×C17—42no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—22no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—81no—
119C2×C4×C17—81no—
120C2×C4×C17—82no—
121C2×C4×C17—82no—
122C2×C4×C17—82no—
123C2×C4×C17#9911yescharacteristic
124C2×C4×C17#9911yescharacteristic
125C2×C4×C17#10312yes—
126C2×Dic17—24no—
127C2×Dic17—42no—
128C2×Dic17—42no—
129C2×Dic17—42no—
130C2×Dic17—42no—
131C2×Dic17—42no—
132Dic34—82no—
133Dic34—82no—
134#3622—82no—
135#2542—171no—
136#1799D1711yescharacteristic
137#1459—171no—
138#1352—174no—
139C22×C17—21no—
140C22×C17—21no—
141C22×C17—21no—
142C22×C17—22no—
143C22×C17—21no—
144C22×C17—42no—
145C22×C17—41no—
146C22×C17—41no—
147C22×C17—41no—
148C22×C17—41no—
149C22×C17—41no—
150C22×C17—41no—
151C22×C17—81no—
152C22×C17—81no—
153C22×C17C2×#9911yescharacteristic
154C22×C17C2×#10311yescharacteristic
155C22×C17#42111yescharacteristic
156C4×C17—21no—
157C4×C17—22no—
158C4×C17—41no—
159C4×C17—41no—
160C4×C17—41no—
161C4×C17—162no—
162C4×C17#35312yes—
163Dic17—44no—
164Dic17—82no—
165C2×C4⋊Q8—171no—
166C2×C22⋊Q8—341no—
167C2×C22⋊Q8—341no—
168C2×C4×D4C2×D1711yescharacteristic
169C2×#99—21no—
170C2×C22⋊C8—341no—
171#399—172no—
172#399—172no—
173#390—341no—
174#384—341no—
175#380—21no—
176C4×C4⋊C4—171no—
177#342C2×D1712yes—
178#324—341no—
179#323—172no—
180C2×C17—21no—
181C2×C17—21no—
182C2×C17—41no—
183C2×C17—42no—
184C2×C17—41no—
185C2×C17#152011yescharacteristic
186C2×C17#212311yescharacteristic
187C2×C17C2×#35311yescharacteristic
188C22×Q8—341no—
189C22×D4C22⋊D1711yescharacteristic
190C23×C4D6811yescharacteristic
191C23×C4C22⋊D1711yescharacteristic
192C22×C8—681no—
193C4⋊Q8—681no—
194C4⋊Q8—681no—
195C22⋊Q8—684no—
196C22⋊Q8—684no—
197C4×D4—41no—
198C4×D4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—172no—
201C2×C4⋊C4—341no—
202C2×C4⋊C4—342no—
203C2×C4⋊C4—342no—
204C2×C4⋊C4—341no—
205C2×C4⋊C4D6811yescharacteristic
206C2×C22⋊C4—22no—
207C2×C22⋊C4—342no—
208C2×C22⋊C4—342no—
209C2×C22⋊C4C4×D1711yescharacteristic
210C2×C22⋊C4C22×D1711yescharacteristic
211C2×C42—341no—
212C2×C42C4×D1711yescharacteristic
213#99—42no—
214C22⋊C8—342no—
215C22⋊C8—681no—
216#95—41no—
217#95—172no—
218C17#1190111yescharacteristic, complemented, Sylow
219C24#6322711yescharacteristic
220C24C2×C22⋊D1711yescharacteristic
221C2×Q8—682no—
222C2×Q8—682no—
223C2×Q8—681no—
224C2×Q8—681no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4#6322912yes—
228C2×D4#6323112yes—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—342no—
237C22×C4—342no—
238C22×C4#6322711yescharacteristic
239C22×C4#6322711yescharacteristic
240C22×C4#6322711yescharacteristic
241C22×C4C2×C4×D1711yescharacteristic
242C22×C4C2×D6811yescharacteristic
243C2×C8—682no—
244C2×C8—1362no—
245C4⋊C4—41no—
246C4⋊C4—82no—
247C4⋊C4—344no—
248C4⋊C4—682no—
249C4⋊C4—682no—
250C4⋊C4—684no—
251C4⋊C4—684no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—684no—
257C22⋊C4—684no—
258C42—21no—
259C42—21no—
260C42—682no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#1058748112yes—
270C23C2×#6322711yescharacteristic
271C23C2×#6322911yescharacteristic
272C23C2×#6323111yescharacteristic
273Q8—1362no—
274Q8—1362no—
275D4—42no—
276D4—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—344no—
294C2×C4—682no—
295C2×C4—682no—
296C2×C4—682no—
297C2×C4—682no—
298C2×C4—682no—
299C2×C4#1058747711yescharacteristic
300C2×C4#1058747711yescharacteristic
301C2×C4#1058748112yes—
302C8—1362no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—21no—
308C22—42no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22—81no—
317C22C2×#1058747711yescharacteristic
318C22C2×#1058748111yescharacteristic
319C22#1176282711yescharacteristic
320C4—22no—
321C4—21no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—162no—
326C4—684no—
327C4—1362no—
328C4#1176260112yes—
329C2—21no—
330C2—21no—
331C2—41no—
332C2—42no—
333C2—41no—
334C2#1452735511yescharacteristic
335C2#1453043711yescharacteristic
336C2C2×#1176260111yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#1799C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C22⋊C4×C17C23
hypercenter#1799D17
hypocenterC17#11901
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #14526807C2semidirect
5. C2×#10587555C4semidirect
9. #11762745C4semidirect
13. C4×C4⋊Dic17C4semidirect
218. C17#11901semidirect

Maximal subgroups

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

subgroupindex
#145337372
#145271022
#145268072
C17×#17992
#1190117