Finite Group Databasebeta

← #16945223

Subgroups — #16945223

Order 4352.

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

normal subgroups only

This group has 432 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#14534350C211yesmaximal, characteristic
2#14528867C211yesmaximal, characteristic, complemented
3#14526812C212yesmaximal, complemented
4#14526788C212yesmaximal, complemented
5C17×#1458C211yesmaximal, characteristic, complemented
6C2×C4×D68C411yescharacteristic
7#11762773—21no—
8#11762748C2211yescharacteristic
9#11762745C411yescharacteristic
10#11762726C2211yescharacteristic
11#11762718C411yescharacteristic
12#11762700—21no—
13#11762688—21no—
14#11762685—21no—
15#11762680—21no—
16#11762677C411yescharacteristic
17#11762578—21no—
18#11762578C2212yes—
19C2×C17×#117C2211yescharacteristic
20C17×#431—21no—
21C17×#324C2212yes—
22C17×#323—21no—
23C23×C4×C17D411yescharacteristic
24C22×C22⋊D17—21no—
25C22×D68C2×C411yescharacteristic
26C22×C4×D17—21no—
27C17×#117—41no—
28C17×#117—41no—
29C2×C4⋊C4×C17D411yescharacteristic
30C2×C4⋊C4×C17C2311yescharacteristic
31C2×C22⋊C4×C17C2×C411yescharacteristic
32C2×C22⋊C4×C17D411yescharacteristic
33C2×C42×C17C2×C411yescharacteristic
34C2×C42×C17D411yescharacteristic
35C2×C22⋊Dic17—21no—
36C2×#63227—21no—
37C2×#63227—21no—
38C2×#63227—21no—
39C2×#63227—21no—
40C2×C4⋊Dic17C2×C411yescharacteristic
41C2×#63225—21no—
42C2×#63223—22no—
43C4×D68—41no—
44C4×D68—41no—
45C17×C4⋊C8—41no—
46C17×C22⋊C8—22no—
47C4×C8×C17—41no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502C2×C411yescharacteristic
53#10587502C2×C411yescharacteristic
54C22⋊#3622—41no—
55C22⋊#3622—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—42no—
70C2×C22⋊D17—81no—
71C22×Dic17—21no—
72C22×Dic17—21no—
73C22×Dic17—21no—
74C2×D68—41no—
75C2×D68—42no—
76C2×C4×D17—42no—
77C2×C4×D17—81no—
78C2×C4×D17—81no—
79C2×C8×C17—42no—
80C4⋊C4×C17—41no—
81C4⋊C4×C17—41no—
82C22⋊C4×C17—41no—
83C22⋊C4×C17—41no—
84C42×C17—21no—
85C42×C17—21no—
86C42×C17—21no—
87C42×C17—21no—
88C22⋊Dic17—81no—
89#63227—42no—
90#63227—41no—
91#63227—41no—
92#63227—42no—
93#63227—81no—
94C4⋊Dic17—41no—
95#63225—81no—
96#63223—42no—
97#63223—42no—
98C2×#3622—42no—
99#11847—171nomaximal, Sylow
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—41no—
104C23×C17—41no—
105C23×C17C2×C22⋊C411yescharacteristic
106C22×D17—22no—
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#13512yes—
130C22⋊D17—84no—
131C2×Dic17—41no—
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—42no—
141D68—42no—
142C4×D17—84no—
143C8×C17—82no—
144#3622—82no—
145#2598—171no—
146#1822—171no—
147#1458D1711yescharacteristic, complemented
148#1353—172no—
149#1345—172no—
150C22×C17—21no—
151C22×C17—21no—
152C22×C17—42no—
153C22×C17—41no—
154C22×C17—41no—
155C22×C17—41no—
156C22×C17—81no—
157C22×C17C2×#9911yescharacteristic
158C22×C17#41611yescharacteristic
159C22×C17#42111yescharacteristic
160C2×D17—42no—
161C2×D17—42no—
162C2×D17—42no—
163C2×D17—81no—
164C2×D17—81no—
165C2×D17—81no—
166C2×D17—81no—
167C4×C17—22no—
168C4×C17—21no—
169C4×C17—21no—
170C4×C17—41no—
171C4×C17—41no—
172C4×C17—41no—
173C4×C17—41no—
174Dic17—81no—
175Dic17—81no—
176Dic17—81no—
177C2×C4×D4—171no—
178C2×#117C2×D1711yescharacteristic
179#431—21no—
180#407—341no—
181#400—171no—
182#399—171no—
183#394—171no—
184#392—171no—
185#388—341no—
186#386—341no—
187#386—341no—
188#385—341no—
189#383—171no—
190#343—172no—
191#343—341no—
192#324C2×D1712yes—
193#323—21no—
194C2×C17—21no—
195C2×C17—21no—
196C2×C17—81no—
197C2×C17#151611yescharacteristic
198C2×C17#212011yescharacteristic
199C2×C17#213411yescharacteristic
200D17—82no—
201C22×D4—171no—
202C22×D4—341no—
203C23×C4—341no—
204C23×C4D6811yescharacteristic
205C2×M16—342no—
206C4×D4—681no—
207C4×D4—681no—
208#117—41no—
209#117—41no—
210C2×C4⋊C4—171no—
211C2×C4⋊C4—341no—
212C2×C4⋊C4D6811yescharacteristic
213C2×C4⋊C4C22×D1711yescharacteristic
214C2×C22⋊C4—341no—
215C2×C22⋊C4—341no—
216C2×C22⋊C4—341no—
217C2×C22⋊C4—341no—
218C2×C22⋊C4—341no—
219C2×C22⋊C4C4×D1711yescharacteristic
220C2×C22⋊C4C22⋊D1711yescharacteristic
221C2×C42C4×D1711yescharacteristic
222C2×C42C22⋊D1711yescharacteristic
223C4⋊C8—41no—
224C22⋊C8—22no—
225C22⋊C8—681no—
226C22⋊C8—681no—
227C4×C8—41no—
228#95—171no—
229#95—171no—
230#95—341no—
231#95—341no—
232#95—341no—
233#95—341no—
234C17#1184711yescharacteristic, complemented, Sylow
235C24—341no—
236C24#6322711yescharacteristic
237C2×D4—681no—
238C2×D4—682no—
239C2×D4—682no—
240C2×D4—682no—
241C2×D4—1361no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4—341no—
247C22×C4—341no—
248C22×C4—341no—
249C22×C4—682no—
250C22×C4—1361no—
251C22×C4—1361no—
252C22×C4#6322711yescharacteristic
253C22×C4#6322711yescharacteristic
254C22×C4#6322711yescharacteristic
255C22×C4C2×C4×D1711yescharacteristic
256C22×C4C2×D6811yescharacteristic
257C22×C4C2×C22⋊D1711yescharacteristic
258M16—682no—
259M16—682no—
260C2×C8—42no—
261C2×C8—682no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—681no—
265C4⋊C4—1361no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—682no—
269C22⋊C4—681no—
270C22⋊C4—681no—
271C22⋊C4—682no—
272C22⋊C4—1361no—
273C22⋊C4—1361no—
274C42—21no—
275C42—21no—
276C42—21no—
277C42—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—342no—
284C23—681no—
285C23—681no—
286C23—681no—
287C23—681no—
288C23—681no—
289C23—681no—
290C23C2×#6322711yescharacteristic
291D4—682no—
292D4—682no—
293D4—1364no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—681no—
309C2×C4—681no—
310C2×C4—681no—
311C2×C4—681no—
312C2×C4—681no—
313C2×C4—681no—
314C2×C4—681no—
315C2×C4—681no—
316C2×C4—681no—
317C2×C4—1364no—
318C2×C4#1058747711yescharacteristic
319C2×C4#1058747711yescharacteristic
320C2×C4#1058759012yes—
321C8—82no—
322C8—1362no—
323C22—21no—
324C22—21no—
325C22—42no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—81no—
330C22—682no—
331C22—682no—
332C22—682no—
333C22—1361no—
334C22—1361no—
335C22—1361no—
336C22—1361no—
337C22C2×#1058747711yescharacteristic
338C22#1176280611yescharacteristic
339C22#1176282511yescharacteristic
340C4—22no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—1361no—
348C4—1361no—
349C4—1361no—
350C2—21no—
351C2—21no—
352C2—81no—
353C2—1362no—
354C2#1452734111yescharacteristic
355C2#1453042311yescharacteristic
356C2#1453047511yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1458C2
socleC22×C17C2×#99
prime radical#11762718C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1458D17
hypocenterC17#11847
LCSG ▷ C22×C4×C17 ▷ C23×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. #14528867C2semidirect
3. #14526812C2semidirect
4. #14526788C2semidirect
5. C17×#1458C2semidirect
147. #1458D17semidirect
234. C17#11847semidirect

Maximal subgroups

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

subgroupindex
#145343502
#145288672
#145268122
#145267882
C17×#14582
#1184717