Finite Group Databasebeta

← #16943903

Subgroups — #16943903

Order 4352.

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

normal subgroups only

This group has 408 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#14533242C211yesmaximal, characteristic
2#14527118C211yesmaximal, characteristic, complemented
3#14526729C212yesmaximal, complemented
4#14526715C212yesmaximal, complemented
5C17×#1458C211yesmaximal, characteristic, complemented
6C2×#10587616C411yescharacteristic
7#11762866—21no—
8#11762724—21no—
9#11762724—21no—
10#11762710C411yescharacteristic
11#11762710C2211yescharacteristic
12#11762700—21no—
13#11762700—21no—
14#11762678C2211yescharacteristic
15#11762674C411yescharacteristic
16C4×C22⋊Dic17C411yescharacteristic
17#11762535C2212yes—
18#11762533—21no—
19C2×C17×#117C2211yescharacteristic
20C17×#431—21no—
21C17×#324C2212yes—
22C17×#323—21no—
23C23×C4×C17C2×C411yescharacteristic
24C22×C22⋊D17—21no—
25C22×D68C2×C411yescharacteristic
26C17×#117—41no—
27C17×#117—41no—
28C2×C4⋊C4×C17C2×C411yescharacteristic
29C2×C4⋊C4×C17C2311yescharacteristic
30C2×C22⋊C4×C17D411yescharacteristic
31C2×C22⋊C4×C17D411yescharacteristic
32C2×C42×C17D411yescharacteristic
33C2×C42×C17D411yescharacteristic
34C2×C22⋊Dic17—21no—
35#10587616—41no—
36#10587616—41no—
37C2×#63227—21no—
38C2×#63227—21no—
39C2×#63227—21no—
40C2×#63227—21no—
41C2×#63227—21no—
42C2×C4⋊Dic17C2×C411yescharacteristic
43C2×#63225—21no—
44C2×#63225—21no—
45C2×C4×Dic17C2×C411yescharacteristic
46C2×C4×Dic17C2×C411yescharacteristic
47C17×C4⋊C8—41no—
48C17×C22⋊C8—22no—
49C4×C8×C17—41no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53C22⋊#3622—22no—
54C4⋊#3622—41no—
55#10587474—41no—
56C24×C17C22⋊C411yescharacteristic
57C23×D17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—22no—
61C22×C4×C17—41no—
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22×C411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C2×C22⋊D17—42no—
69C2×C22⋊D17—81no—
70C2×C22⋊D17—81no—
71C22×Dic17—21no—
72C22×Dic17—21no—
73C22×Dic17—21no—
74C2×D68—22no—
75C2×D68—41no—
76C2×D68—41no—
77C2×C8×C17—42no—
78C4⋊C4×C17—41no—
79C4⋊C4×C17—41no—
80C22⋊C4×C17—41no—
81C22⋊C4×C17—41no—
82C42×C17—21no—
83C42×C17—21no—
84C42×C17—21no—
85C42×C17—21no—
86C22⋊Dic17—42no—
87#63227—42no—
88#63227—42no—
89#63227—41no—
90#63227—41no—
91#63227—81no—
92#63227—81no—
93C4⋊Dic17—41no—
94#63225—81no—
95#63225—81no—
96C4×Dic17—22no—
97C4×Dic17—41no—
98C2×#3622—42no—
99#11554—171nomaximal, Sylow
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—41no—
104C23×C17—41no—
105C23×C17C2×C22⋊C411yescharacteristic
106C22×D17—41no—
107C22×D17—41no—
108C22×D17—41no—
109C22×D17—41no—
110C22×D17—41no—
111C22×D17—41no—
112C22×D17—41no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—42no—
122C2×C4×C17—42no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17#9911yescharacteristic
128C2×C4×C17#9911yescharacteristic
129C2×C4×C17#13112yes—
130C22⋊D17—84no—
131C2×Dic17—22no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140D68—82no—
141C8×C17—82no—
142#3622—82no—
143#2500—171no—
144#1464—171no—
145#1458D1711yescharacteristic, complemented
146#1325—172no—
147#1318—172no—
148C22×C17—21no—
149C22×C17—21no—
150C22×C17—41no—
151C22×C17—41no—
152C22×C17—41no—
153C22×C17—81no—
154C22×C17—81no—
155C22×C17#40811yescharacteristic
156C22×C17C2×#9911yescharacteristic
157C22×C17#41311yescharacteristic
158C2×D17—81no—
159C2×D17—81no—
160C2×D17—81no—
161C2×D17—81no—
162C2×D17—81no—
163C2×D17—81no—
164C2×D17—81no—
165C4×C17—22no—
166C4×C17—21no—
167C4×C17—21no—
168C4×C17—41no—
169C4×C17—41no—
170C4×C17—41no—
171C4×C17—41no—
172Dic17—42no—
173Dic17—81no—
174Dic17—81no—
175C2×#121—171no—
176C2×#117C2×D1711yescharacteristic
177#432—341no—
178#431—21no—
179#394—341no—
180#394—341no—
181#390—171no—
182#390—171no—
183#388—341no—
184#388—341no—
185#384—171no—
186#382—171no—
187C4×C22⋊C4—171no—
188#324—172no—
189#324C2×D1712yes—
190#323—21no—
191#323—341no—
192C2×C17—21no—
193C2×C17—21no—
194C2×C17—81no—
195C2×C17#146811yescharacteristic
196C2×C17#212811yescharacteristic
197C2×C17#213211yescharacteristic
198D17—161no—
199C22×D4—171no—
200C22×D4—341no—
201C23×C4C4×D1711yescharacteristic
202#121—681no—
203#121—681no—
204#117—41no—
205#117—41no—
206C2×C4⋊C4—171no—
207C2×C4⋊C4—341no—
208C2×C4⋊C4—341no—
209C2×C4⋊C4C4×D1711yescharacteristic
210C2×C4⋊C4C22×D1711yescharacteristic
211C2×C22⋊C4—341no—
212C2×C22⋊C4—341no—
213C2×C22⋊C4—341no—
214C2×C22⋊C4—341no—
215C2×C22⋊C4—341no—
216C2×C22⋊C4—341no—
217C2×C22⋊C4D6811yescharacteristic
218C2×C22⋊C4C22⋊D1711yescharacteristic
219C2×C42—171no—
220C2×C42—171no—
221C2×C42D6811yescharacteristic
222C2×C42C22⋊D1711yescharacteristic
223C4⋊C8—41no—
224C4⋊C8—681no—
225C22⋊C8—22no—
226C22⋊C8—342no—
227#97—681no—
228C4×C8—41no—
229#95—341no—
230#95—341no—
231#95—341no—
232C17#1155411yescharacteristic, complemented, Sylow
233C24—341no—
234C24#6322711yescharacteristic
235C2×D4—342no—
236C2×D4—681no—
237C2×D4—681no—
238C2×D4—682no—
239C2×D4—1361no—
240C2×D4—1361no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—41no—
245C22×C4—341no—
246C22×C4—341no—
247C22×C4—341no—
248C22×C4#6322711yescharacteristic
249C22×C4#6322711yescharacteristic
250C22×C4#6322711yescharacteristic
251C22×C4C2×C4×D1711yescharacteristic
252C22×C4C2×D6811yescharacteristic
253C22×C4C2×C22⋊D1711yescharacteristic
254C2×C8—42no—
255C2×C8—682no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—681no—
259C4⋊C4—1361no—
260C4⋊C4—1361no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—682no—
264C22⋊C4—682no—
265C22⋊C4—682no—
266C22⋊C4—681no—
267C22⋊C4—681no—
268C22⋊C4—1361no—
269C22⋊C4—1361no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—342no—
275C42—681no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—681no—
282C23—681no—
283C23—681no—
284C23—681no—
285C23—681no—
286C23—681no—
287C23—681no—
288C23C2×#6322711yescharacteristic
289D4—1362no—
290D4—1364no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—342no—
306C2×C4—681no—
307C2×C4—681no—
308C2×C4—681no—
309C2×C4—681no—
310C2×C4—681no—
311C2×C4—681no—
312C2×C4—681no—
313C2×C4—681no—
314C2×C4#1058747711yescharacteristic
315C2×C4#1058747711yescharacteristic
316C2×C4#1058758712yes—
317C8—82no—
318C8—1362no—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22—1361no—
327C22—1361no—
328C22—1361no—
329C22—1361no—
330C22—1361no—
331C22—1361no—
332C22—1361no—
333C22#1176277511yescharacteristic
334C22C2×#1058747711yescharacteristic
335C22#1176279111yescharacteristic
336C4—21no—
337C4—21no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—682no—
344C4—1361no—
345C4—1361no—
346C2—21no—
347C2—21no—
348C2—81no—
349C2—2721no—
350C2#1452713011yescharacteristic
351C2#1453045511yescharacteristic
352C2#1453046711yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1458C2
socleC22×C17C2×#99
prime radicalC2×#10587616C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1458D17
hypocenterC17#11554
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
2. #14527118C2semidirect
3. #14526729C2semidirect
4. #14526715C2semidirect
5. C17×#1458C2semidirect
145. #1458D17semidirect
232. C17#11554semidirect

Maximal subgroups

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

subgroupindex
#145332422
#145271182
#145267292
#145267152
C17×#14582
#1155417