Finite Group Databasebeta

← #16950314

Subgroups — #16950314

Order 4352.

588 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 93 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#14531508C211yesmaximal, characteristic
2C2×#11762601C212yesmaximal
3#14528770C212yesmaximal, complemented
4#14526909C211yesmaximal, characteristic, complemented
5C17×#1393C211yesmaximal, characteristic, complemented
6C2×#10587626C2211yescharacteristic
7C2×D4×Dic17C2211yescharacteristic
8C2×#10587579C411yescharacteristic, complemented
9C2×#10587505C2212yes—
10#11762711C411yescharacteristic, complemented
11#11762686—22no—
12#11762685—22no—
13#11762674C411yescharacteristic, complemented
14#11762667—22no—
15#11762655C411yescharacteristic, complemented
16#11762601—24no—
17#11762575—22no—
18C2×C17×#99C2212yes—
19C17×#390C2211yescharacteristic
20C22×D4×C17C2311yescharacteristic
21C22×C22⋊D17—22no—
22C23×Dic17D412yes—
23C22×D68C2×C411yescharacteristic
24C22×C4×D17—21no—
25C2×C4⋊C4×C17C2×C411yescharacteristic
26C2×C22⋊C4×C17—22no—
27C2×C22⋊C4×C17—22no—
28C2×C42×C17C2×C411yescharacteristic
29C2×C22⋊Dic17—22no—
30C2×C22⋊Dic17D412yes—
31#10587626—22no—
32#10587626D412yes—
33#10587626Q812yes—
34D4×Dic17—42no—
35C2×#63227—21no—
36C2×#63227—22no—
37C2×C4⋊Dic17C2×C411yescharacteristic
38C2×#63225—42no—
39C2×C4×Dic17C2×C411yescharacteristic
40C2×C4×Dic17C2×C411yescharacteristic
41#10587579—42no—
42C17×#99—22no—
43C17×#99—22no—
44#10587505—22no—
45#10587505—22no—
46#10587502—21no—
47#10587502—21no—
48#10587502—22no—
49#10587502—42no—
50C24×C17C2×D412yes—
51C23×D17—21no—
52C2×D4×C17—41no—
53C2×D4×C17—41no—
54C2×D4×C17C2×D411yescharacteristic
55C2×D4×C17C2×Q811yescharacteristic
56C2×D4×C17#4112yes—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—42no—
60C22×C4×C17C22×C411yescharacteristic
61C2×C22⋊D17—24no—
62C2×C22⋊D17—24no—
63C2×C22⋊D17—82no—
64C2×C22⋊D17—82no—
65C22×Dic17—21no—
66C22×Dic17—22no—
67C22×Dic17—22no—
68C22×Dic17—42no—
69C22×Dic17—82no—
70C22×Dic17C22⋊C411yescharacteristic
71C22×Dic17C22⋊C412yes—
72C22×Dic17C22⋊C411yescharacteristic
73C2×D68—21no—
74C2×D68—41no—
75C2×D68—41no—
76C2×D68C4⋊C412yescomplemented
77C2×C4×D17—42no—
78C2×C4×D17—82no—
79C4⋊C4×C17—41no—
80C22⋊C4×C17—42no—
81C22⋊C4×C17—42no—
82C22⋊C4×C17—82no—
83C42×C17—41no—
84C22⋊Dic17—42no—
85C22⋊Dic17—42no—
86C22⋊Dic17—42no—
87#63227—42no—
88#63227—82no—
89C4⋊Dic17—41no—
90#63225—84no—
91C4×Dic17—21no—
92C4×Dic17—22no—
93C4×Dic17C4⋊C412yescomplemented
94#12799—171nomaximal, Sylow
95C23×C17—22no—
96C23×C17—22no—
97C23×C17—42no—
98C23×C17—42no—
99C23×C17C2×C22⋊C411yescharacteristic
100C23×C17#12112yes—
101C23×C17C22⋊Q812yes—
102C22×D17—22no—
103C22×D17—42no—
104C22×D17—42no—
105C22×D17—41no—
106C22×D17—41no—
107D4×C17—41no—
108D4×C17—41no—
109D4×C17—42no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—41no—
113C2×C4×C17—42no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—82no—
118C2×C4×C17—82no—
119C2×C4×C17—82no—
120C2×C4×C17C2×C4⋊C411yescharacteristic
121C2×C4×C17#11711yescharacteristic
122C22⋊D17—84no—
123C22⋊D17—84no—
124C2×Dic17—22no—
125C2×Dic17—21no—
126C2×Dic17—22no—
127C2×Dic17—41no—
128C2×Dic17—44no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—42no—
133C2×Dic17—82no—
134C2×Dic17—82no—
135C2×Dic17—82no—
136C2×Dic17—82no—
137C2×Dic17#9912yes—
138D68—42no—
139D68—42no—
140C4×D17—84no—
141#2328—171no—
142C2×#353—172no—
143#1799—172no—
144#1393—171no—
145#1393D1711yescharacteristic, complemented
146C22×C17—21no—
147C22×C17—21no—
148C22×C17—22no—
149C22×C17—22no—
150C22×C17—42no—
151C22×C17—42no—
152C22×C17—42no—
153C22×C17—42no—
154C22×C17—82no—
155C22×C17#38011yescharacteristic
156C22×C17C2×#9911yescharacteristic
157C22×C17#41011yescharacteristic
158C2×D17—44no—
159C2×D17—42no—
160C2×D17—82no—
161C2×D17—81no—
162C2×D17—81no—
163C4×C17—21no—
164C4×C17—21no—
165C4×C17—81no—
166C4×C17—81no—
167C4×C17—162no—
168Dic17—22no—
169Dic17—42no—
170Dic17—42no—
171Dic17—41no—
172Dic17—162no—
173C2×#127—171no—
174C2×C4×D4—171no—
175C2×C4×D4—171no—
176C2×#99—172no—
177C2×#99C2×D1712yes—
178#390—171no—
179#390C2×D1711yescharacteristic
180#386—342no—
181#386—342no—
182#382—171no—
183#380—342no—
184#376—171no—
185#353—344no—
186#342—342no—
187C2×C17—21no—
188C2×C17—21no—
189C2×C17—42no—
190C2×C17—42no—
191C2×C17#179911yescharacteristic
192C2×C17#212711yescharacteristic
193C2×C17#212811yescharacteristic
194D17—82no—
195C22×D4—171no—
196C22×D4—342no—
197C22×D4C22×D1711yescharacteristic
198C23×C4—172no—
199C23×C4—341no—
200#127—172no—
201#127—172no—
202#127—342no—
203C4×D4—682no—
204C4×D4—682no—
205C2×C4⋊C4—171no—
206C2×C4⋊C4—682no—
207C2×C4⋊C4C4×D1711yescharacteristic
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—172no—
211C2×C22⋊C4—341no—
212C2×C22⋊C4—342no—
213C2×C22⋊C4—342no—
214C2×C42—171no—
215C2×C42—171no—
216C2×C42C4×D1711yescharacteristic
217#99—22no—
218#99—22no—
219#99—342no—
220#99—342no—
221#95—341no—
222#95—341no—
223#95—342no—
224#95—682no—
225C17#1279911yescharacteristic, complemented, Sylow
226C24—341no—
227C24D4×D1712yes—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—172no—
231C2×D4—344no—
232C2×D4—344no—
233C2×D4—341no—
234C2×D4—681no—
235C2×D4—681no—
236C2×D4—1362no—
237C2×D4—1362no—
238C2×D4#6325212yes—
239C2×D4D4×D1711yescharacteristic
240C2×D4Q8×D1711yescharacteristic
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—42no—
244C22×C4—171no—
245C22×C4—171no—
246C22×C4—172no—
247C22×C4—342no—
248C22×C4—342no—
249C22×C4—341no—
250C22×C4—682no—
251C22×C4—682no—
252C22×C4—1362no—
253C22×C4—1362no—
254C22×C4C2×C4×D1711yescharacteristic
255C4⋊C4—41no—
256C4⋊C4—681no—
257C4⋊C4—1364no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—82no—
261C22⋊C4—682no—
262C22⋊C4—682no—
263C22⋊C4—682no—
264C22⋊C4—682no—
265C22⋊C4—1362no—
266C42—41no—
267C42—172no—
268C42—342no—
269C42—341no—
270C23—22no—
271C23—22no—
272C23—42no—
273C23—42no—
274C23—342no—
275C23—682no—
276C23—682no—
277C23—681no—
278C23—681no—
279C23C22⋊C4×D1711yescharacteristic
280C23#1058757012yes—
281C23#1058758212yes—
282D4—42no—
283D4—41no—
284D4—41no—
285D4—682no—
286D4—682no—
287D4—1364no—
288D4—1364no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—172no—
300C2×C4—341no—
301C2×C4—342no—
302C2×C4—342no—
303C2×C4—681no—
304C2×C4—684no—
305C2×C4—681no—
306C2×C4—681no—
307C2×C4—682no—
308C2×C4—681no—
309C2×C4—1364no—
310C2×C4—1362no—
311C2×C4—1362no—
312C2×C4—1362no—
313C2×C4—1362no—
314C2×C4#1058755811yescharacteristic
315C2×C4C4⋊C4×D1711yescharacteristic
316C22—21no—
317C22—21no—
318C22—22no—
319C22—22no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—82no—
325C22—684no—
326C22—682no—
327C22—1362no—
328C22—1361no—
329C22—1361no—
330C22#1176266811yescharacteristic
331C22#99×D1711yescharacteristic
332C22#1176278111yescharacteristic
333C4—21no—
334C4—21no—
335C4—81no—
336C4—81no—
337C4—162no—
338C4—2722no—
339C4—342no—
340C4—682no—
341C4—682no—
342C4—681no—
343C2—21no—
344C2—21no—
345C2—42no—
346C2—42no—
347C2—1362no—
348C2#1452877111yescharacteristic
349C2#1453045211yescharacteristic
350C2#1453045411yescharacteristic
subgroupG/H
centerC22#99×D17
commutatorC22×C4×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#1393C2
socleC22×C17C2×#99
prime radicalC2×#10587626C22
cosocleC22×D4×C17C23
hypercenter#1393D17
hypocenterC17#12799
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1393

Complements

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

NH ≅ G/Ntype
3. #14528770C2semidirect
4. #14526909C2semidirect
5. C17×#1393C2semidirect
8. C2×#10587579C4semidirect
10. #11762711C4semidirect
13. #11762674C4semidirect
15. #11762655C4semidirect
76. C2×D68C4⋊C4semidirect
93. C4×Dic17C4⋊C4semidirect
145. #1393D17semidirect
225. C17#12799semidirect

Maximal subgroups

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

subgroupindex
#145315082
C2×#117626012
#145287702
#145269092
C17×#13932
#1279917