Finite Group Databasebeta

← #16944536

Subgroups — #16944536

Order 4352.

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

normal subgroups only

This group has 552 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#14534534C211yesmaximal, characteristic
2#14528523C211yesmaximal, characteristic, complemented
3#14526788C214yesmaximal, complemented
4C17×#1458C211yesmaximal, characteristic, complemented
5C22×#63223—21no—
6C2×#10587560C411yescharacteristic
7C2×#10587616—21no—
8#11762726—21no—
9#11762726C2212yes—
10#11762718—21no—
11#11762718C412yes—
12#11762711—21no—
13#11762675C411yescharacteristic
14#11762578—21no—
15#11762578C2212yes—
16C2×C17×#117C2211yescharacteristic
17C17×#431—21no—
18C17×#324C2212yes—
19C17×#323—21no—
20C23×C4×C17D411yescharacteristic
21C22×C22⋊D17—22no—
22C22×D68—21no—
23C22×D68—21no—
24C22×D68C2×C412yes—
25C17×#117—41no—
26C17×#117—41no—
27C2×C4⋊C4×C17D411yescharacteristic
28C2×C4⋊C4×C17C2311yescharacteristic
29C2×C22⋊C4×C17C2×C411yescharacteristic
30C2×C22⋊C4×C17D411yescharacteristic
31C2×C42×C17C2×C411yescharacteristic
32C2×C42×C17D411yescharacteristic
33#10587616—42no—
34#10587616—42no—
35C2×#63227—22no—
36C2×#63227—22no—
37C2×#63227—22no—
38C2×C4⋊Dic17—21no—
39C2×#63223—24no—
40C2×#63223—22no—
41C2×#63223—41no—
42C2×#63223—42no—
43C22×#3622—41no—
44#10587560—41no—
45#10587560—41no—
46C17×C4⋊C8—41no—
47C17×C22⋊C8—22no—
48C4×C8×C17—41no—
49#10587502—21no—
50#10587502C2×C412yes—
51C24×C17C22⋊C411yescharacteristic
52C23×D17—22no—
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17—22no—
56C22×C4×C17C22⋊C411yescharacteristic
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17D812yes—
60C22×C4×C17SD1612yes—
61C22×C4×C17C22×C411yescharacteristic
62C22×C4×C17C2×D411yescharacteristic
63C22×C4×C17C2×D411yescharacteristic
64C2×C22⋊D17—44no—
65C2×C22⋊D17—82no—
66C2×C22⋊D17—82no—
67C22×Dic17—22no—
68C2×D68—42no—
69C2×D68—42no—
70C2×D68—41no—
71C2×D68—41no—
72C2×D68—42no—
73C2×D68—42no—
74C2×D68—82no—
75C2×D68—82no—
76C2×C8×C17—42no—
77C4⋊C4×C17—41no—
78C4⋊C4×C17—41no—
79C22⋊C4×C17—41no—
80C22⋊C4×C17—41no—
81C42×C17—21no—
82C42×C17—21no—
83C42×C17—21no—
84C42×C17—21no—
85#63227—42no—
86#63227—42no—
87#63227—82no—
88#63227—82no—
89C4⋊Dic17—42no—
90#63223—24no—
91#63223—44no—
92#63223—42no—
93C2×#3622—42no—
94C2×#3622—44no—
95#11699—171nomaximal, Sylow
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—41no—
99C23×C17—41no—
100C23×C17#10212yes—
101C23×C17C2×C22⋊C411yescharacteristic
102C22×D17—42no—
103C22×D17—42no—
104C22×D17—42no—
105C22×D17—42no—
106C22×D17—42no—
107C22×D17—42no—
108C22×D17—42no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—22no—
112C2×C4×C17—22no—
113C2×C4×C17—22no—
114C2×C4×C17—22no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17#9911yescharacteristic
124C2×C4×C17#9911yescharacteristic
125C2×C4×C17#10212yes—
126C2×C4×C17#10412yes—
127C2×C4×C17C2×D811yescharacteristic
128C2×C4×C17C2×SD1611yescharacteristic
129C22⋊D17—88no—
130C2×Dic17—42no—
131C2×Dic17—42no—
132C2×Dic17—42no—
133D68—84no—
134D68—84no—
135D68—82no—
136D68—82no—
137C8×C17—82no—
138#3622—44no—
139#2619—171no—
140#1742—171no—
141#1458D1711yescharacteristic, complemented
142#1345—174no—
143C22×C17—21no—
144C22×C17—21no—
145C22×C17—22no—
146C22×C17—42no—
147C22×C17—41no—
148C22×C17—41no—
149C22×C17—81no—
150C22×C17C2×#9911yescharacteristic
151C22×C17C2×#10211yescharacteristic
152C22×C17C2×#10411yescharacteristic
153C2×D17—82no—
154C2×D17—82no—
155C2×D17—82no—
156C2×D17—82no—
157C2×D17—82no—
158C2×D17—82no—
159C2×D17—82no—
160C4×C17—21no—
161C4×C17—21no—
162C4×C17—41no—
163C4×C17—41no—
164C4×C17—41no—
165C4×C17—41no—
166C4×C17#32712yes—
167C4×C17#32712yes—
168Dic17—82no—
169C22×M16—341no—
170C2×#127—171no—
171C2×#121—341no—
172C2×#117C2×D1711yescharacteristic
173#431—21no—
174#394—172no—
175#394—341no—
176#392—172no—
177#392—341no—
178#390—341no—
179#382—171no—
180#343—172no—
181#343—341no—
182#324C2×D1712yes—
183#323—21no—
184C2×C17—21no—
185C2×C17—21no—
186C2×C17—42no—
187C2×C17C2×#32711yescharacteristic
188C2×C17#212211yescharacteristic
189C2×C17#213411yescharacteristic
190D17—162no—
191C22×D4—172no—
192C22×D4—342no—
193C22×D4—341no—
194C22×D4—341no—
195C23×C4D6811yescharacteristic
196C2×M16—344no—
197C2×M16—342no—
198C2×M16—681no—
199C2×M16—682no—
200C22×C8—681no—
201#127—681no—
202#127—681no—
203#121—682no—
204#121—682no—
205#117—41no—
206#117—41no—
207C2×C4⋊C4—341no—
208C2×C4⋊C4D6811yescharacteristic
209C2×C4⋊C4C22×D1711yescharacteristic
210C2×C22⋊C4—342no—
211C2×C22⋊C4—342no—
212C2×C22⋊C4—342no—
213C2×C22⋊C4C4×D1711yescharacteristic
214C2×C22⋊C4C22⋊D1711yescharacteristic
215C2×C42C4×D1711yescharacteristic
216C2×C42C22⋊D1711yescharacteristic
217C4⋊C8—41no—
218C22⋊C8—22no—
219C4×C8—41no—
220#95—172no—
221#95—341no—
222C17#1169911yescharacteristic, complemented, Sylow
223C24—342no—
224C24#6322711yescharacteristic
225C2×D4—682no—
226C2×D4—682no—
227C2×D4—684no—
228C2×D4—681no—
229C2×D4—681no—
230C2×D4—682no—
231C2×D4—682no—
232C2×D4—1362no—
233C2×D4—1362no—
234C2×D4—1362no—
235C2×D4—1362no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—342no—
240C22×C4#6321912yes—
241C22×C4D13612yes—
242C22×C4#6322711yescharacteristic
243C22×C4#6322711yescharacteristic
244C22×C4#6322711yescharacteristic
245C22×C4C2×C4×D1711yescharacteristic
246C22×C4C2×D6811yescharacteristic
247C22×C4C2×C22⋊D1711yescharacteristic
248M16—344no—
249M16—684no—
250M16—682no—
251C2×C8—42no—
252C2×C8—682no—
253C2×C8—684no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—682no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—682no—
260C22⋊C4—682no—
261C22⋊C4—1362no—
262C22⋊C4—1362no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23—682no—
272C23—682no—
273C23—682no—
274C23—682no—
275C23—682no—
276C23—682no—
277C23—682no—
278C23#1058749212yes—
279C23C2×#6322711yescharacteristic
280D4—1364no—
281D4—1364no—
282D4—1368no—
283D4—1362no—
284D4—1362no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—682no—
300C2×C4—682no—
301C2×C4—682no—
302C2×C4#1058747711yescharacteristic
303C2×C4#1058747711yescharacteristic
304C2×C4#1058749212yes—
305C2×C4#1058749612yes—
306C2×C4C2×#6321911yescharacteristic
307C2×C4C2×D13611yescharacteristic
308C8—82no—
309C8—684no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22—1362no—
318C22—1362no—
319C22—1362no—
320C22—1362no—
321C22—1362no—
322C22—1362no—
323C22—1362no—
324C22C2×#1058747711yescharacteristic
325C22C2×#1058749211yescharacteristic
326C22C2×#1058749611yescharacteristic
327C4—21no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—1362no—
334C4#1176254412yes—
335C4#1176254412yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—2722no—
340C2C2×#1176254411yescharacteristic
341C2#1453043311yescharacteristic
342C2#1453047511yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1458C2
socleC22×C17C2×#99
prime radical#14534534C2
cosocleC2×C4⋊C4×C17C23
hypercenter#1458D17
hypocenterC17#11699
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. #14528523C2semidirect
3. #14526788C2semidirect
4. C17×#1458C2semidirect
141. #1458D17semidirect
222. C17#11699semidirect

Maximal subgroups

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

subgroupindex
#145345342
#145285232
#145267882
C17×#14582
#1169917